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[El 38")! ......CN . :u . ...V 35301)) . .. g\)lwpomcmpcw $0 eczocmxumn zcwcowpzpo>m .H mcsmwd coyms$wwo :ovumsgomcH m>_pomcmch AT we mumm esp we Acowch \ /E COFonum w coppmsgo E : mwm>_mcm xcozpmz S OP o:Lm;u mmwmmmz 1t mymxpmc< xgoz mwvm 2mmmz )fl///z \\\\\« meFoFuom mczpochm Fmeoom :onmorczseou —morpmem;pmz xGOFOFoom cowuaovm w :owmeLomcw mo mappwvos some flewumwcpscmpmu \\\Mq afl/ a owpmm500pmv mmmooca msomepcoo covpaovm a cowpmsco$cw xcomcp ;% m: SFmvos Ilv m:WFwnoe some saw: mCOmemano /\ ./ \ §p msmspmz amorowswuwam I2 This is a very simplified overall evolutionary scheme and not a strict chronological ordering. The purpose of Figure l and the following discussion is to give the reader a general background overview to the problem addressed by this research. As mentioned above, many social science areas have been inter- ested in the interactive information diffusion process. However, major theoretical contributions by social scientists have been largely based on developments in the fields of epidemiology and mathematics. Because contagious diseases have had devastating effects on the human population, biologists have long been interested in the Spread of such diseases. To understand the process and to predict the rate of spread, mathematical epidemiologists have devel— Oped highly sophisticated models (deterministic and stochastic). This area of research is exemplified by the work of Bailey (I957, l975). Because Of the theoretical modeling developments in epidemiol- ogy and the similar overall behaviors (i.e., S—shaped cumulative curve), numerous attempts have been made to apply epidemiological models to interactive information and innovation diffusions (e.g., Bartholomew, l973; Brown, I968; Cane, I966; Coleman, I964; Daley & Kendall, I964; Dietz, I967; Goddman, l966; Goffman & Newill, I964; Lave & March, I975; Monin, Benayoun & Sert, I976; RapOport, l956). The work by Dodd (l953, I955) triggered many comparisons of interactive information diffusion with contagious disease. Dodd, a sociologist, was the first to suggest that the accumulated curve of interactive information diffusion is logistic (S-shaped). However, equating the components of interactive information diffusion (e.g. , I3 knowers, nonknowers, number of tellings, etc.) to the components of the contagious disease process (e.g., infected, susceptibles, infec- tious period, etc.) has not resulted in any theoretical advances in interactive information diffusion. The major contribution of epi- demiology to interactive information diffusion has been to serve as a theoretical role model for social scientists, in particular, mathematical sociologists (e.g., Coleman, I964; Hamblin, Jacobsen, & Miller, l973; S¢rensen, I978). Parallel research to epidemiology and mathematical sociology was conducted in the new emerging area of human communication. The major focus by communication scientists has been largely on the broadcast process of information diffusion (i.e., mass media research). The contribution by mass media research to interactive information diffusion has had a minor focus on message characteristics which may influence the rate of spread (e.g., Budd, Maclean, & Barnes, l966; Deutschman & Danielson, l960; Funkhouser, I970; Greenberg, Briton, & Farr, I965; Haroldsen & Harvey, I979). However, no study systemati- cally examines message characteristics and resulting changes in diffusion rates for either the broadcast or interactive processes. National news is primarily transmitted by the mass media. How— ever, mass media researchers have commented on the frequent co-occurrence of interactive diffusion with the broadcast process as far back as Lazarsfeld and Katz (Katz & Larzarsfeld, I955; Lazarsfeld, Berelson, & Gaudet, I948). To explain these observations mass media 'researchers focused on the two—step and subsequent multi—step flow I4 hypotheses of information proposed by Lazarsfeld et al. (I948) and Menzel and Katz (I955), respectively. This line of research high- lighted the role of opinion leadership in bridging the broadcast and interactive information diffusion processes. However, only a few studies have examined the social structure involved in interactive diffusion (Dodd, l958/59; Garabedian & Dodd, I962; Larson & Hill, I954, I958; Lionberger & Hassinger, I954). Except for the classical study by Coleman, Katz, & Menzel (I957) and a few subsequent others (Becker, I970; Czepiel, I974; Rogers, I979; Rogers & Kincaid, l98l), innovation diffusion has also suffered from a lack of research focus on the social structure. Almost 20 years ago, Katz, Levin, and Hamilton (I963) criticized innovation diffusion research for a lack of focus on ways in which different kinds of structural arrangements in a group influence the diffusion of a given item. For further progress in the field, Katz et al. stated that: interpersonal channels of communication must be viewed as elements of social structure; (6) that work is urgently needed on the comparative study of the same item diffus- ing in different social structures . . . (p. 252). ‘The same criticism and recommendation could have been made of inter- active information diffusion; it is still valid today for both processes. The absence of studies on communication structure has largely been due to a lack of appropriate analytic methodology. However, Iarxgely based on the work of Harary, Norman, and Cartwright (I965), Iaotfli communication scientists and sociologists have deveIOped what I5 has come to be known as network analysis. The principles of network analysis are largely drawn from graph theory, a branch of mathematics. Network analysis provides a way of examining the communication structure within which information diffuses via the interactive process. With recent developments in computer programs for analyzing networks (e.g., Richards, I975), network analysis allows examination of communication structural (i.e., network) variables which may influence interactive information diffusion. In conclusion, the one generally accepted finding about inter- active information diffusion is that, over time, the accumulated number of knowers produces the logistic or S—shaped curve. The logistic curve represents the overall behavior of the process. The two major components of the process are: (a) the communication structure (network) in which a (b) message diffuses. Only limited research has been conducted on message/information variables and network variables which may influence the speed at which a message diffuses in a network. As yet, there has been virtually no attempt to integrate past research findings into a theory which would explain and predict the growth rate of the logistic curve for interactive information diffusion. The following sections of this chapter present a review of the literature on mathematical modeling, network structure, and message characteristics. The section on mathematical modeling reviews the history of the logistic curve and its associated equation along with a theoretical modeling strategy for relating component variables I6 (network and message) to the overall behavior. The sections on net- work structure and message characteristics review the literature on variables which influence the rate of interactive infonnation diffu- sion. Mathematical Modeling This section presents a brief discussion of deterministic versus stochastic models, followed by a discussion of the logistic equation. The last part of this section presents the modeling strategy adopted for relating network and message variables to the logistic equation. Deterministic vs. Stochastic Most mathematical work on information diffusion (interactive and broadcast) has been on stochastic models (e.g., Bartholomew, I973, I976; Funkhouser, I970; Funkhouser & McCombs, I972; Gray & von Broembsen, l974; Karmeshu & Pathria, I980; Rapoport, I953a, l953b, I979; Rapoport & Rebhun, I952; S¢rensen, I978; Taga & Ish, I959). Stochastic models focus on the probability of a given person being affected, i.e., the probability of individuals moving from one state to another (e.g., nonknower to knower or nonteller to teller). Deterministic models focus directly on change in the variable.values or on the number of people affected (Monin et al., I976; Sdrenson, I978L Deterministic models can include a random term which is intro— duced into the solution of the defining equation. In contrast, the defining equation in stochastic models uses changes in probability I7 levels as the dependent variable (Sdrensen, I978). While stochastic models may be more predictively accurate, deterministic models often provide good approximations, since the deterministic form can often be taken as the expected or mean value of the probability distribution (Coleman, I964, p. 527; Olinick, I978, p. ll). Thus, the determinis- tic model can reflect in a simpler form the same basic process as does the stochastic model (Coleman, I964, p. 527; Olinick, I978, p. 327). The major difficulty with stochastic models is that with increased complexity they quickly become intractable (Coleman, I964; Olinick, I978; Sprensen, I978). Therefore, deterministic models are preferred. The commonly used deterministic model of complete interactive diffusion of information is the logistic equation (e.g., Bartholomew, I976; Dodd, I953, I955; Feldman &Lie, l974; Goffman, l966; Goffman & Newill, I964; Landahl, I953; Landau & Rapoport, I953; Lave & March, I975; Monin et al., I976; S¢rensen, I978). It represents the S-shaped or logistic curve and is simpler than its stochastic counterpart. The logistic equation for interactive information diffusion is: k "=1+e—(a+bt) (1) where n is the cumulative proportion of knowers of the information, k is the total proportion of possible knowers (i.e., size of the social system), t is time, a is a constant scaling parameter to be estimated, and b is the parameter which summarizes the rate of growth, also to be estimated. I8 The logistic equation is a description of the overall behavior on the global level by condensing data into two parameters (a and b) (Feldman & Lie, I978). Since it is the basic model of interactive information diffusion, the logistic equation is discussed further in the next section. Logistic Equation The logistic curve, which is an S-shaped symmetrical, cumulative growth curve, has been found to be applicable to pOpulations with limited growth (von Bertalanffy, I950). The rate of growth is not constant as in unlimited exponential growth, but is dependent on the maximum possible size of the population, e.g., k in the logistic equation (I). Historically, the logistic equation was first developed in the l840's by a Belgium mathematician, Pierre-Francois Verhulst, to model population growth (Olinick, I978). Because the existing census data at that time were inadequate to form any effective test of his model, Verhulst's discovery of the logistic curve and its application to population growth was forgotten for almost 80 years. It was redis- covered in the l920's independently by two American biologists, Ray- mond Pearl and Lowell J. Reed (Olinick, I978). In addition to pOpulation or biological growth, the logistic curve has also been found applicable in describing the cumulative ggrowth of autocatalytic chemical reactions (Coleman, I964; Hamblin et:al., I973), the epidemic of communicable diseases (e.g., Bailey, l5957, I975; Bartholomew, l973; Dietz, I967; Monin, Benayoun & Sert, F'— __ I9 I976; Serfling, I952), the diffusion of innovations (e.g., Brown, I968; Feldman & Lie, l974; Griliches, I957; Hamblin et al., l973; Rogers with Shoemaker, l97l), and the interactive diffusion of information (e.g., Dodd, I953, I955; Bartholomew, I973, I976; Feld— man & Lie, l974; Funkhouser, I970; Gray & von Broembsen, I974; Landau & Rapoport, I953; Lave & March, I975; Rapoport, l979/80; Rogers with Shoemaker, l97l). The first theoretical work on and application of the logistic equation to the S-shaped curve for suc- cessful interactive information diffusion was done by Dodd (I953, I955). Complete interactive information diffusion results in logistic growth rather than exponential, decaying exponential, or any other growth curve because of: (a) the interactive or chain-reaction process in a (b) finite population (network) where someone who knows the information tells it to other people (e.g., Dodd, I953, I955; Feldman & Lie, l974; Lave & March, I975). The growth limit (k) is the size of the communication network. In the early stage, the inter- active process will be relatively efficient because there is a high probability of talking to someone who does not have the information. Thus, the growth rate of the proportion of knowers increases and the lower half of the logistic curve is exponential growth. Toward the end of the process, the growth rate of the cumulative proportion of knowers is very slow. By this time most of the contacts will be with people who already have received the information rather than with the few remaining people who have not yet received it (Lave & March, l975). 20 The upper half of the logistic curve is asymptotic growth resulting from a decreasing rate of the proportion of knowers. Two other mathematical functions, the Gompertz and the normal ogive, have also been used to describe the S-shaped curve of success- ful diffusion (both information and innovation). The Gompertz equa— tion has been largely rejected because it describes an asymmetrical S-shaped curve, yet the accumulated S-shaped curves of innovation and interactive information diffusion data are generally quite symmetrical (Hart, I945; Hamblin et al., I973).1 Both the logistic and normal ogive curves are symmetrical. Working with diffusion of innovation data, Pemberton (I936) was the first to suggest that the normal ogive, the integrated normal frequency distribution, was the 2 However, the equation that best described the S-shaped curve. logistic equation is based on a chain-reaction process in a finite population and is, therefore, the preferred equation for mathemati- cally describing successful innovation diffusion and successful interactive information diffusion (Feldman & Lie, I974). The logis- tic equation is also preferred over the normal ogive because it can be easily transformed to a linear (in parameters) equation for a least-squares estimation of the two parameters: In ( k'jn)=a+bt, (2) or, in linear regression form: * n = a + bt, (3) where n* = ln(n/k - n).3 2I The logistic curve is illustrated in Figure 2 where k repre- sents the total possible population of knowers, i.e., the maximum possible limit of growth for interactive information diffusion. Across different diffusion studies, high b parameters represent faster overall growth rates. Mathematical Modeling Strategy The equation chosen to model the overall behavior of interactive information diffusion is the single deterministic logistic equation (I) with one exogenous variable, i.e., time. The logistic equation is adequate for describing the overall process's behavior for any one particular interactive diffusion study. However, it does not provide any explanation as to why the overall growth rate of the accumulated knowers (i.e., the b parameter) differs across messages and across communication networks. The focus of this research is to deveIOp a theory which will explain why the overall diffusion time as repre- sented by the b parameter is greater or lesser for one interactive information diffusion study than another. S¢rensen (I978) presents two main strategies for modifying sim- ple models to improve their theoretical adequacy. The first strategy is to model the variation in parameters of simple models. This is done by writing parameters as functions of independent variables (i.e., deterministic modeling) or by modeling the distribution of parameters (i.e., stochastic modeling). The second strategy is to modify the dependent variable. Modifying the dependent variable can be dealt with by introducing time—dependent variables as independent Cumulative Proportion of Knowers (n) Figure 2. 22 point of / inflection\<\>x I exponential 1 growth I’ I. / V Time (t) Logistic growth curve where n is the cumulative proportion of knowers, k is the limit or total possible proportion of knowers. Curve 1 represents a more rapid spread of informa- tion than does curve 2, i.e., b1 b2. 23 variables or by combining several dependent variables, e.g., simul- taneous equations model. The first strategy focuses on variables to explain and predict the variation in parameters of the basic model. For interactive information diffusion, the parameter of interest is the b in equa- tion (I). The b parameter summarizes the accumulated growth rate (or speed) of diffusion for a specific message in a specific communi- cation network. The questions of interest in this research are: why does a message diffuse faster (or slower) in one network than another and why will one message diffuse faster than another in the same network? Theoretically, these questions ask what variables are needed to explain the variation of the logistic b parameter. Thus, the logistic b parameter is the dependent variable of interest.4 Writing the logistic b parameter as a function of independent variables has been applied by Hamblin et al. (I973) to explain the different rates of adoption with a wide variety of innovations. Hamblin et al. model the logistic b parameter as a function of the level of reinforcement for adoption. Thus, they consider the logistic b a variable to be explained. In addition, Chadha and Chitgopekar (l97l) also use this strategy to explain changes in the potential market (k in the logistic equation) for the adoption of residence telephones. For interactive information diffusion, there does not appear to have been any attempts at applying Sorensen's first strategy to the logistic b parameter. Thus, similar to the work by Hamblin et al., 24 a set of independent variables will be sought to explain and predict the variation of the logistic b parameter resulting from different messages diffusing in different networks. The following two sections of this chapter review the literature on network structure and mes- sage characteristics, respectively. From this view, variables are selected or developed to explain the variation of the logistic b parameter. The model (or theory) relating the selected variables is presented in Chapter III along with the measurement models for each selected variable. Structural Variables As described above, interactive information diffusion refers to the spread of a message (information) in a social system of inter- acting members. The social system is defined as a communication net- work where communication is the basis of interaction. The location of the point where the message enters the network has also been suggested as important to the overall diffusion rate (Bavelas, I948; Becker, I970; Czepiel, l975). This section reviews the literature for possible network and message entrance location variables which might most affect the overall rate of interactive diffusion, i.e., the logistic b parameter. The discussion of structural variables is divided into two parts: Network structure and network location of message entrance. Network Structure Communication networks are defined "as the set of stable person- to—person relationships through which information flows . 25 (Monge, Edwards, & Kirste, I978, p. 3l2). The network structure is the actual configuration of the communication relationships. Indi- viduals in the network are referred to as nodes and the communication relationship between two people (nodes) is called a link (Richards, I976). The strength of the link is based on frequency of communica— tion interaction, duration of interaction, and/or intensity of inter- action (Richards, l976). Thus, a network is a configuration of a set of nodes interconnected by communication links. The network structure influences the speed at which the message moves from person to person because it imposes constraints on the possible paths the message can traverse (e.g., Bavelas, I948). How- ever, only a small number of studies actually compare different net— works (social systems) and the flow of information (e.g., Dodd, I953, I955; l958/59; Larson & Hill, I954, I958; Lionberger & Hassinger, I954). Dodd found that as city size and population density increased, the rate and extent of diffusion of airborne leaflets increased. Larson and Hill found that the rate of message spread is greater in more stable networks (i.e., communities) than in less stable ones. In comparing sources of farm information, Lionberger and Hassinger found that neighborhood residents most frequently named friends and neighbors as their most important sources while non—neighborhood residents most often named the mass media. Most networks are compared for adoption rates or innovativeness levels (Allen, I970; Becker, I970; Coleman et al., I957; Czepiel, I975; Guimaraes, I972; Liu & Duff, I972; Rogers, I979; Rogers & 26 Kincaid, l98l; Shoemaker, l97l). Innovativeness has been found to increase with increases in system openness (Allen, I970) and communi- cation integration (Allen, I970; Giumaraes, I972; Shoemaker, l97l). Increased adoption rates have been found to be positively associated with centrality (Becker, I970; Czepiel, I975), integration (Coleman et al., I957), heterophilous relationships (Liu & Duff, I972; Rogers & Kincaid, l98l), weak ties (Liu & Duff; Rogers & Kincaid), and system openness (Rogers & Kincaid). In addition, linkage distance (i.e., reachability), has been found to be negatively related to the adoption rate (Rogers & Kincaid). Because there are only a few network studies on interactive information diffusion, network studies on the adoption process are also included in this section. Many ways have been developed to characterize various aspects of a network. They can be classified as: (a) link of dyadic proper- ties (e.g., symmetry, strength, and reciprocity), (b) network roles (e.g., isolates, group members, bridge, and liaison), (c) network variables at the individual or node level (e.g., centrality, choice status, and reachability), (d) network variables at the group or clique level (e.g., connectedness, clique size, openness and domi- nance), and (e) network variables at the network level (e.g., density, reachability, Openness, and anchorage). The variables of interest are those which provide description of structure at the network level rather than at the individual or dyadic (link) level. Since all network level variables are defined in terms of links, the link or dyadic properties are subsumed in the network level variables. 27 Network variables at the node level along with network anchorage are discussed in the section on network location of message entrance. In the diffusion literature, several network related variables are suggested as influencing the rate of diffusion. A list of the variables found in the literature is provided in Table l. Almost half focus on the node or dyadic levels rather than on the larger network level. The node and dyadic variables include: centrality, heterophilous relations, interaction frequency, valence of dyadic relationship, and weak ties. Centrality can be both a node and network level variable. Except for centrality, these variables appear to be very similar on the operational level. Interaction frequency and valence are all measures of link strength (Richards, I976), where weak ties (low link strength) fall at the lower end of the link strength continuum (Granovetter, l973; Liu & Duff, l972; Rogers, l979). While heterophilous relations often refer to the degree of dissimilarity of values, education, and social status (Rogers & Bhowmik, l97l; Rogers with Shoemaker, l97l; Rogers & Kincaid, l98l), they can be operationalized as the amount of interaction frequency (Rogers, I979; Rogers & Kincaid, l98l). For example, two people with a high degree of dissimilarity in attitudes and values tend to communicate with one another less frequently than do two peOple with a low degree of dissimilarity. Thus, heterophilous relations fall at the lower end of the link strength continuum of interaction frequency the same as weak ties (Granovetter, l973; Liu & Duff, l972; Rogers, l979). While not identical, heterophilous 28 TABLE 1. Network Factors Influencing the Rate of Interactive Information Diffusion (from the Literature) Factors Researchers l. Centrality Bavelas (I948); Becker (I970);Czepiel (I975). 2. Clique size Dodd&McCurtain (I965); Garabedian & . Dodd (I962). 3. Connectedness Bott (I955); Rogers (I979); Rogers & Kincaid (I98l). 4. Density Czepiel (I975); Erbe (I962); Granovetter (I973). 5. Heterophilous Barnett (I975); Davis (I963); Erbe (I962); relations Liu & Duff (I972); Rogers & Bhowmik (l97l); Rogers with Shoemaker (l97l); Rogers & Kincaid (l98l). 6. Interaction Erbe (I962). frequency 7. Integration Coleman, Katz, & Menzel (I957); Erbe (I962); Guimaraes (I972); Lin & Burt (I976); Rogers (I979); Shoemaker (l97l). 8. Valence of dyadic Davis (I963); Fathi (I973). relationship 9. Weak ties Davis (I963); Friedkin (I980); Granovetter (I973); Liu & Duff (l972)- Rogers 1979 ; Rogers & Kincaid (I98 ). l0. System openness Allen (I970);Rogers (I979); Rogers & Kincaid (l98l) ll. Reachability Rogers & Kincaid (l98l). (Connectedness II) 29 relations and weak ties appear to be closely related in some causal way. 'For the purpose of describing communication networks based on interaction frequency, heterophilous relations and weak ties are considered to be functionally the same. Weak ties are also related to density and network (or clique) openness and will be discussed below. Centrality has been operationalized in several different ways. In tracing the history of the concept of centrality, Freeman (l978/ 79) found that there was little agreement on the definition and measurement of centrality. The only consensus Freeman found is that centrality is considered an important structural attribute of net- works. Based on his literature review, Freeman categorizes the vari- ous centrality definitions into those based on: (a) direct communi- cation activity or degree measures (e.g., number of direct links), (b) closeness (or distance)-based measures, and (c) betweenness or communication control-based measures. In the diffusion literature, centrality is used as either a communication activity (or degree) or a closeness (or distance)-based measure. Centrality is used as a direct communication activity-based measure by Becker (I970) and Czepiel (l975). Alternatively, Bavelas (I948), who originated the term centrality (Freeman, l978/79), defines it based on distance. Although Bavelas, Becker, and Czepiel use point centrality (i.e., node centrality), the variable can be used on the network level as discussed below (Freeman). Point centrality will be further discussed in the section on network location of message entrance. 30 When centrality is used as an activity-based measure at the network level, it is operationally the same as density (Czepiel, l975; Erbe, I962; Granovetter, I973). Density was first operationally defined by Prihar in I956 as the ratio of the actual number of links to the total number possible in a network (Barnes, I969). Barnes was the first to refer to this measure as density. Thus, density is a direct communication activity-based measure of centrality at the network level. As discussed by Barnes (I969), connectedness also has a variety of different definitions. However, based on the categories set forth by Freeman (l978/79), connectedness is used as a direct activity—based measure by Bott (I955), Rogers (l979), and Rogers and Kincaid (l98l). As an activity-based measure on the network level, connectedness is the same variable as density. However, connectedness is also used as a closeness-based measure by Rogers and Kincaid, which they refer to as connectedness II. Rogers and Kincaid's connectedness II meas- ure is the same variable as reachability, a distance-based measure. The same situation also occurs for integration, a sociological term. Integration is used as an activity-based measure by Coleman et al. (I957), Erbe (I962), Guimaraes (I972), Lin and Burt (l979), and Rogers (l979). As an activity-based measure, integration is the same variable as density. Alternatively, Shoemaker (l97l) uses inte- gration as a distance-based measure. In summary, there are two major types of centrality variables used in the diffusion literature. They are based on either direct 3I communication activity or distance. For purposes of distinguishing between the two and for consistency with most of the communication network literature, the direct activity-based variable, hereafter, will be referred to as density. The distance-based variable will be referred to as reachability. A completely connected network has no path constraints since every node directly connects every other node. An increase in density means there is less constraint on the paths which a message can take; therefore, a message should diffuse faster in a network with higher density. At the group level only clique size and connectedness have been suggested as affecting the rate of interactive diffusion. Based on Monte Carlo simulations, Dodd and McCurtain (I965) and Garabedian and Dodd (l962) found that as clique size increased, the rate of information diffusion increased within a network. For the simula- tions, this is explained by the accompanying increase in the number of connections (links) the larger clique has with the rest of the network. Thus, in the simulations, increases in clique size also resulted in an increase in network density. The strength of links connecting a clique to the rest of the network (or other cliques in the network) is often weaker (i.e., weak ties) than those of intraclique links (Friedkin, I980; Granovetter, l973; Liu & Duff, l972; Rogers, I979; Rogers & Kincaid, l98l). As mentioned above, weak ties indicate heterOphilous rela- tions and facilitate the flow of new information from one clique to 32 another (Granovetter, I973; Liu & Duff, l972; Rogers, l979). Alterna- tively, stronger ties within a clique facilitate intraclique diffusion. Thus, the greater the number of weak ties between cliques within a network, the more cliques overlap (Rogers, I979) and the greater the density of the network as a whole. Therefore, the number of weak ties within a network is subsumed in the summary variable density. The number of weak ties has also been suggested as a measure of the degree of openness of the clique or network (Rogers, I979; Rogers & Kincaid, l98l). Openness is defined as the ”degree to which a group has linkages with its environment“ (Farace, Monge, & Russell, l977, p. 202). When the degree of openness is applied to a clique, it is related to the density of the network. However, degree of openness as applied to a network does not appear to be a useful variable to account for the rate of diffusion in the network. Its usefulness seems to be more relevant to diffusion across networks. Reachability, which has been briefly discussed above, can be regarded as a distance—based centrality measure, i.e., an overall measure of the shortest length or distance of a network. Conceptually, reachability is defined as the "extent to which other system members can be connected with a minimum of intermediaries . . .“ (Farace et al., p. 202), i.e., the geodesic (Barnes, I969; Freeman, I977). Reachability takes into account both direct and indirect pathways; whereas, density takes into account only direct paths. In addition, if the links are asymmetric, the reachability score can reflect the paths that must be followed when the "one-way streets" of asymmetric 33 relations are present (Farace et al.). Thus, a message diffusing in a network with low reachability should spread faster than in a net- work with high reachability. Based on the literature and the above discussion, it appears that a network's density and reachability are two very useful net- work variables and subsume most of the variables at the group/clique, dyad, and individual levels. In a larger context, density (mass divided by volume) and reachability (distance) are important vari- ables (characteristic properties of objects) in theories of physical and biological systems (e.g., fluid dynamics, kinetic theory of gases, and the diffusion of molecules) (Giancoli, I980). Therefore, to explain and predict the rate of interactive diffusion of information, the amount of network density and the length of network reachability will be the two variables used to describe the network structure. Network Location of Message Entrance The orientation point or reference for a network is referred to as its anchorage (Farace et al.; Mitchell, I969). For information diffusion, the node where the message enters the network will be referred to as the anchor. Because it is theoretically possible for information to be introduced into a network via any node having external contact, the network location of the anchor should affect the rate of diffusion in the network (Bavelas, I948; Becker, I970; Czepiel, l975). For example, if an anchor with eight links is com— pared with an anchor with only two links, the accumulated number 34 of knowers will grow faster if the node with the eight links is the message point of entrance. It has been suggested that the most likely anchor will be a node with a weak, as opposed to a strong, tie to the environment (e.g., Friedkin, I980; Granovetter, I973; Liu & Duff; Rogers, I979; Rogers & Kincaid). While weak ties may be the links which facilitate information diffusion across networks (or cliques), the anchor's location in the network will be important for the network diffusion rate. To maximize the initial transmission of a message in a network, the optimum anchor location would be the most central node (i.e., the node with the most direct links to other network members). In both the information and innovation diffusion literature, the most central position is related to opinion leadership which has been found to be influential in innovation diffusion (Becker, I970; Czepiel, l975; Rogers, I979; Rogers & Kincaid; Rogers with Shoemaker). However, the most central position based on the number of direct links is only one of many possible locations in a network. Used in this way, centrality is a communication activity variable describing one node's activity with all other network nodes (Freeman, l978/79). The location of the anchor based on communication activity will be referred to as the amount of anchor centrality. Not only does it seem important to know the anchor's direct communication activity, but also the minimum communication distance necessary for the message to reach the other network members. 35 Bavelas (I948) suggests that the minimum amount of time for complete information spread throughout a network is achieved if the spread starts with the most central node. Bavelas's measure of length of centrality is based on distance. Thus, the average shortest distance of the anchor to all other nodes would provide additional information on the location of the anchor. This distance-based variable of location for the anchor will be referred to as the length of anchor reachability. Using both the amount of anchor centrality and the length of anchor reachability should provide sufficient location information about the network entrance point of the message. If one selectes the node where a message will be introduced, the optimum anchor for the most rapid spread would be the node with the highest amount of anchor centrality and the IOWest length of anchor reachability. Message Characteristics Certain characteristics of the message or information have been suggested as influencing the rate of diffusion. Before proceeding to a discussion of characteristics, the entity being diffused is first discussed. In the information diffusion literature, the terms “message“ and "information” are not defined nor distinguished from one another. Often they are used interchangeably, resulting in the same operation— alizations (e.g., Deutschman & Danielson, l960; Funkhouser & McCombs, l972; Liu & Duff, l972; Schneider & Fett, I978; Spitzer, l964-65). The lack of definitions for message and information in the diffusion 36 literature is justifiable on the basis that a commonly used word in "everyday” communication does not need to be defined. If a word is used in a more restricted or technical sense than its common usage, then the word warrants definition. For example, Shannon and Weaver (l949) used the word “information" in a more restricted and technical sense than its everyday meaning. Therefore, they provided a definition for their meaning to distinguish information from its ”everyday" meaning/use. The context in which message and informa— tion are used in the diffusion literature indicates that their every- day meaning is implied and that they are not used in any restricted or technical sense. A list of message characteristics which have been suggested as influencing the rate of information diffusion is provided in Table 2. The variables are divided into two categories for purpose of dis- cussion. Upon close examination, the first eight characteristics in Table 2 are functionally equivalent. In fact, quite often the terms are used interchangeably. For example, news value is used inter— changeably with age of a message (Landau & Rapoport, I953) and importance (e.g., Budd, MacLean, & Barnes, l966; Deutschman & Danielson, l960; Hill & Bonjean, I964). Utility is used interchange— ably with importance (Greenberg, l964b) and relevance (Schneider & Fett, I978). The only empirical investigation on message characteristic relationships was conducted on interest and utility; they were highly correlated (Greenberg, Brinton, & Farr, I955). Even comparative 37 TABLE 2. Message Characteristics Influencing the Rate of Interactive Information Diffusion (from the Literature) Characteristics Researchers l. Age of message Landau & Rapoport (I953). 2. Comparative novelty Greenberg, Brinton, & Farr (I965). 3. Complexity (relative) Greenberg, Brinton, & Farr (l965); of information Lave & March (l975). 4. Importance Budd, MacLean, & Barnes (I966); Deutschman & Danielson (I960); Greenberg (l964b); Greenberg, Brinton, 8 Farr (l965); Haroldsen & Harvey (I979); Lave & March (l975). 5. Interest value Bartholomew (I976); Funkhouser (I970); Funkhouser & McCombs (I972); Gray von Broembsen (I974); Greenberg, Brinton & Farr (l965); Hyman & Sheatsley (I947); Landahl (I953); Lave & March (l975). 6. News value Budd, MacLean, & Barnes (I966); Deutschman & Danielson (I960); Faithi (I973); Hill & Bonjean (I964); Landau & Rapoport (I953). 7. Relevance Schneider & Fett (I978). 8. Shocking value Haroldsen & Harvey (l979). 9' Tabooness 0f message Rogers (I979); Rogers & Kincaid (l98l). topic 10. Utility Greenberg, Brinton, & Farr (l965); Schneider & Fett (I978). 38 novelty and shocking value appear to be similar, based on degree of "unexpectedness." For the full set of message variables, conceptual definitions are not offered by the researchers so that one message characteristic cannot be distinguished from other closely related ones. The only measurements made are for interest (Funkhouser, I970; Funkhouser & McCombs, l972; Hyman & Sheatsley, I947) and shocking value (Haroldsen & Harvey, I970). The first eight message characteristics seem to indicate a common underlying variable. A clue to this may be in the frequent slang expressions: "1 have a hot tip for you" or ”I have some hot news for you." Slang is based on immediate experience and offers us an immediate index to changing perceptions (McLuhan, I964). Thus, the expression, "I have some hot news for you” indicates the speaker's immediate experience with the information. The two expressions seem to indicate highly charged affect (e.g., excitement and enthusiasm) in the speaker regarding the information about to be transmitted to another person. Just as increases in temperature cause particles to more faster, we would expect a person to want to quickly pass on a ”hot" message to many of his/her contacts. The perception of a message as ”hot” can be called message temperature. The higher the temperature of the message, the faster it should spread. The first eight message characteristics in Table 2 which appear to be functionally equivalent can be reformulated as indicators of the underlying message tempera- ture variable. For example, a new message would be expected to have 39 a higher message temperature than an older one. Novelty, importance, relevance, utility, and news value of the information would also be expected to contribute to the message's temperature. In addition, increasing the interest and shocking value of a message should also increase the message's temperature. Therefore, the first eight variables in Table 2 are viewed as observed variables (indicators) of the unobserved variable called message temperature. This approach provides a framework for organizing functionally similar variables. The complexity of information has been suggested as influencing the rate of spread (Greenberg et al., I965; Lave & March, l975). However, no empirical studies were found in the diffusion literature supporting this. Yet, it seems reasonable that complex information will not diffuse as fast as simple infonnation because of the greater difficulty in transmitting (i.e., "telling”) the complex message. Related to complexity of information is the amount of time required to "tell" the information. If a more complex message takes longer to tell than a simple message, we would expect fewer tellings per person. Both complexity and amount of time appear to be indicating a resis- tance factor in passing on a message. In a similar light, the tabooness of the message topic also seems to contribute to the resistance in telling the information (Rogers, I979; Rogers & Kincaid). Generalizing from both characteristics, there appears to be a general factor of resistance to telling a message. ‘ Support of a resistance to transmit factor is the MUM effect (keeping mum about gndesirable messages to the recipient) studied 40 by Rosen and Tesser (Rosen & Tesser, I970; Tesser & Rosen, l975). They define the MUM effect as the "reluctance to communicate informa- tion that one could assume to be noxious for a potential audience” (Rosen & Tesser, I970, p. 260). Thus, the tabooness of a message can be considered noxious for a potential receiver. Such a per- ceived message tends to be communicated less frequently and less quickly than non-noxious messages (Tesser & Rosen, l975). An unobserved variable which could capture a general reluctance or hesitance to transmit will be referred to as the message transmission resistance. It should be mentioned that the topic of the message influences the configuration of the network (Farace et al., I977; Monge et al., I978; Bavelas, I948). Thus, for comparing the rates of spread for different messages in the same network, the topic of the message should be held constant. In conclusion, no study was found which examines the influences of message characteristics on the rate of interactive information diffusion in a network. In addition, very little research defines (conceptually and operationally) the message characteristic of inter- est. In fact, in almost all of the literature reviewed, message characteristics are ad hoc discussions. However, two major dimensions of information can be identified from the literature. The two resulting variables considered important to the rate of interactive information diffusion are message temperature and message trans- Tnission resistance. 41 Summary The first section of this chapter presented a theoretical back- ground for interactive information diffusion. The research history of interactive information diffusion is closely tied to the research ‘history of innovation diffusion. Epidemiology theory has provided a role model for theoretical development of interactive information diffusion. Alternatively, graph theory and related computer programs have contributed a methodology (i.e., network analysis) for studying the communication structure through which information spreads. Mass media research was the only area found to discuss message character- istics. However, there does not appear to be a theory which relates communication network variables and message variables to the varia- tions in the overall rate of interactive information diffusion. The section on mathematical modeling presented the determinis- tic logistic equation and its history. The logistic equation has been used by numerous researchers to represent the overall S-shaped accumulated growth behavior of interactive information diffusion. In the last part of the section, a strategy for modifying the logis- tic equation to include independent variables was presented. The third and fourth sections of this chapter reviewed litera- ture on network structure and message characteristics. From the discussion of network structure, two variables were selected for an overall description of a network: I. Amount of network density, and 2. Length of network reachability. 42 The discussion on networks led to a discussion on the network location of the message's entry point (anchor). The following two variables were selected to describe the anchor point of the message: 3. Amount of anchor centrality, and 4. Length of anchor reachability. In the discussion on messages, two distinct variables were developed for the message (or information) being diffused: 5. Amount of message temperature, and 6. Amount of message transmission resistance. These six variables were selected to explain and predict the logistic equation b parameter for interactive information diffusion. The next chapter presents a theory of the rate of interactive information diffusion composed of the above six exogenous variables. FOOTNOTES--CHAPTER II 1The Gompertz equation is: where n is the cumulative number of knowers or adopters and t is time (Feldman & Lie, I964). 2The normal ogive equation is: t 2 2 ”t = aria— fi ”(u-t") ”0 )du —(X) where nt is the cumulative number of knowers at time t, o is the standard deviation of n, u is the population mean of n, e is the base of natural logarithms (Feldman & Lie, I974). The rate equa- tion for the normal ogive is the cumulative density function for the normal distribution. 3Transforming the logistic equation into simple linear regres- sions form (Brown, I968): k n = _ l + e (a + bt) II = I k = n + ne'(a + bt) k-n _ (a + bt) ——-- e 43 44 n _ 1Oge(k _ n) - a + bt n* = a + bt n . . . where n* = loge (E—3—fi). The last equation 15 linear in parameters and can be used for OLS estimation of the parameters (a and b). 4The growth rate, dn/dt, of the logistic equation is the velocity (speed) of the information spread per time unit. Modeling dn/dt as a function of exogenous variables focuses on explaining the growth rate at each time interval for a single diffusion process. The research focus is on explaining the variations in the overall or summary growth rate resulting from many different diffusions, rather than on accounting for each time interval rate in a single diffusion. Thus, the research focus is on the logistic b parameter rather than on the first derivative (dn/dt) of the logistic equation. 5Message distortion will not be addressed in this paper. CHAPTER III THEORY This chapter is divided into two sections: theoretical model and measurement models. The strategy for developing the theoretical model is to model the logistic b parameter as a function of exogenous variables (discussed in Chapter II). The purpose of the theory is to explain and predict variations in the logistic b parameter as a result of different messages diffusing in different networks. There- fore, the logistic b is written as a function of the six exogenous variables selected from the literature review. The theoretical model section provides: (a) conceptual definitions of the variables, (b) the bivariate relationship of each exogenous variable with the logis- tic b parameter, (c) the multi-exogenous equation representing the theory, and (d) the constraints of the theoretical model. The section on measurement models provides the operational definition and bound- ary condition for each observed variable. Theoretical Model Conceptual Definitions The logistic b is a summary measure of the growth rate of the paroportion of knowers over time. It is an estimated parameter in the logistic equation (I) and is the dependent variable to be explained in this study. 45 46 Network links. The first four variables below (network density, network reachability, anchor centrality, and anchor reachability) are based on communication network links. A network link represents a certain minimum level of regularly occurring direct communication activity between two network members (nodes). Amount of network density is defined as the average amount of direct communication activity in a network.1 This can be interpreted as the average degree to which network members are interlinked, i.e., the extent of interconnectedness (e.g., Farace, Monge, & Russell, 1977). Maximum network density would indicate that the network members are completely interconnected with maximum communication activity. Length of network reachability is defined as the average minimum distance between all pairs of network members. The average geodesic (i.e., minimum distance) can be interpreted as the minimum diameter of the network (Bavelas, 1948; Harary, Norman, & Cartwright, 1965). Reachability takes into account all direct and indirect pathways the message can follow. If network density is at its maximum, then network reachability is at a minimum. However, the relationship between the two variables is not linear.2 Amount of anchor centrality is defined as the anchor person's average amount of out-going direct communication activity with all other network members. The anchor is the first network node to receive a message entering the network. Because the message radiates out from the anchor, anchor centrality is based on only out-going linkages. Anchor centrality reflects one node's average out-going 47 communication activity; whereas, network density reflects the average amount of direct communication activity in the network as a whole. Lepgth of anchor reachability is defined as the average minimum distance from the anchor to all other network members. Amount of message temperature is defined as the average level of arousal from the message among the members of the network. Amount of message transmission resistance is defined as the average level of reluctance or hesitancy about passing on the message among the members of the network. Bivariate Relationships The bivariate relationships of the logistic b parameter with each of the above six exogenOus variables (along with boundary condi- tions for the relationship) are given next: As network density (g1) increases, the logistic b (n) increases. If network density is zero, there will be no interactive diffusion. The overall growth rate of the accumulated proportion of knowers should be positively related to the average amount of direct commun- ication activity in the network. That is, the greater the amount of direct communication activity, the greater the growth rate of knowers in the network. If there is little direct communication activity occurring, messages should take longer to spread through the network. By definition, if there is no direct communication activity among a set of people, there is no network, and therefore, interactive infor- mation diffusion is impossible. 48 As network reachability (£2) increases, the logistic b (n) decreases (i.e., a slower rate of spread). As network reach- ability approaches infinity, the logistic b approaches zero, i.e., no diffusion. The overall growth rate of accumulated knowers should be nega- tively related to the communication distance the message has to travel. Holding the speed of the message constant, the longer the average distance that the message must travel, the more time it will take for the message to spread through the network (i.e., speed is a measure of distance divided by time). As anchor centrality (53) increases, the logistic b (n) increases. If anchor centrality is zero (i.e., an isolate is the anchor), there will be no interactive diffusion. Anchor centrality is based on direct out-going communication activity of the first network message sender (i.e., anchor). There— fore, the more direct out-going communication an anchor has, the faster should be the initial growth rate of accumulated knowers, which should be reflected in a higher overall growth rate for the network. If an anchor has no direct out-going communication activity with any other network member (i.e., if s/he is an isolate), then anchor centrality will be zero and the message will not spread beyond the anchor person. As anchor reachability (6,) increases, the logistic b (n) decreases. If anchor reachability is infinite (i.e., an isolate is the anchor), there will be no diffusion. Holding the speed of the message constant, the greater the average distance from the anchor to all other nodes, the greater the diffusion time required. Thus, the overall growth rate of the accumulated knowers will be slower. If the average communication 49 distance from the anchor to all other network members is infinite, then there will be no interactive diffusion. As message temperature (as) increases, the logistic b (n) increases. If there is absolutely no message temperature, there will be no diffusion and, therefore, no b parameter to estimate. Also, if the message temperature is extremely high, the message could diffuse almost instantaneously which would most likely not produce the logistic curve. The more excited, enthusiastic, or interested network members are about the information, the more people they are likely to pass it on to and the sooner they are likely to pass it on. Holding the network and anchor variables constant, increases in message tempera- ture should result in a faster overall growth rate of accumulated knowers. If there is absolutely no excitement, enthusiasm, or inter- est in the information, people will have little inclination to pass on the information. However, if the message temperature is extremely high, the anchor of a small network will soon pass on the message to all other network members. This would likely produce some other curve than the logistic growth. As message transmission resistance (as) increases, the logistic b (n) decreases. Message transmission resistance probably has a critical point, beyond which no diffusion will occur. The more people are reluctant to pass on the information, the fewer people they will tell and the longer the time delay between receiving and sending the information. Thus, the overall growth rate of accumu- lated knowers should be much slower. However, there is probably some upper level of resistance beyond which people will simply not tell one another. For example, an anchor person who is very sensitive 50 to group norms may consider the message taboo and not pass it on to others. This appears to have been the case for family planning methods in many Korean villages prior to the national information campaign in the l960's (Rogers & Kincaid, l98l). Even today in the U.S., abortion information does not circulate in certain networks. Thus, any amount of message transmission resistance will have the effect of slowing the rate of message spread and, therefore, slowing the growth rate of accumulated knowers. Multi-exogenous Equation Combining these six bivariate relationships produces the follow- ing multi-exogenous proposition for the logistic b parameter: The logistic b parameter (n) will increase as network density (g1) increases, network reachability (52) decreases, anchor centrality (E3) increases, anchor reachability (Eu) decreases, message temperature (as) increases, and message transmission resistance (as) decreases. The multi-exogenous equation for the logistic b parameter is: O = YO€1Y152-Y2€3Y3€A-YQESYSge-YS C ’ (4) or, alternatively: _ £1Y1E3Y3€5Y5 (5) n _ YO ngzguYungs , where Yo is a scaling constant, Yi are parameters to be estimated, and C is the error of prediction. 5I The scaling constant, Yo, could represent some external input (e.g., input by the mass media) where more than one network member receives the message from the environment. Constraints Based on the boundary conditions of the exogenous variables (e.g., greater than zero) and the bivariate relationships, the multi-exogenous equation representing the above pr0position is written as multiplicative. That is, if any one of the positively related exogenous variables is zero or any one of the negatively related exogenous variables approaches infinity, there will be no interactive diffusion. Thus, the multiplicative form is required to estimate the equation's parameters. In equation (5), as the denominator increases, n will decrease. Alternatively, as the numerator increases, n will increase. When exogenous variables are multiplicative, the error term is often also multiplicative rather than additive. Multiplicative error terms are frequently found to be heteroscedastic and nonnormally dis- tributed (e.g., Danes, I978; Laroche, I977; Welch, I978). Transform- ing both sides of equation (4) by taking logarithms will create an equation which is linear in parameters with an additive logarithmic error term, which is assumed to be normally distributed (i.e., Log-normal). Linearizing equation (4) by taking natural logarithms (ln) results in: 52 Inn = Invo + vllnil - YzIn€2 + Y3In€3 - YAIDEA + YsINEs ' YOIOCO + InC- (6) Substituting equation (4) into equation (I) yields: k 1 + e'La + (Y051Y1gz-Y2Eayagu-Yugsysie-Y6C)t] The difference between equations (4) and (7) is that equation (7) allows the six exogneous variables to vary during any one diffusion study (i.e., one message in one network). Equation (4) constrains the exogenous variables to be constant (i.e., stationary) during any one diffusion study but alloWs them to vary across studies (i.e., across messages and/or across networks). Thus, equation (7) could be a nonrecursive model for a single message diffused in one network, while equation (4) is a recursive model for multiple messages and/or multiple networks. It may be the case that ultimately the diffusion of a single message in one network may lead to subsequent adoption of some new idea or technology, which, in turn, may lead to altering the network structure. However, the purpose behind developing the theory of the rate of interactive information diffusion is to account for the variations in the overall rates from the spread of different Inessages in different networks. For this purpose, equation (4) best represents the theory. 53 Measurement Models The measurement model for each of the seven theoretical (unob- served)variables in equation (4) is presented below. The measure- ment model consists of both the operational definition and boundary condition for each observed variable. For all seven theoretical variables, the measurement model is a single indicator. Logistic b parameter is estimated from the linearized logistic equation (3): n*=a+8t (3) where n* = ln(n/K - n) and t is time. Strength of network links must be operationally defined before the measurement models for the two network and two anchor variables can be presented. For interactive information diffusion, link strength is operationally defined as the number of hours in a typical week that one person directly communicates with another person. The link strength scale is, therefore, bounded at zero and I68 hours (i.e., 24 hours times seven days). A link exists if the strength is greater than zero. This definition allows relatively weak links (i.e., weak ties) to exist along with strong links for tracing the paths of a message in a network. The measurement models for network density, network reachability, anchor centrality, and anchor reachability are based on a value directed graph (vigraph) (Peay, I980). A vigraph allows directed links with the link strength values retained. Two vigraph networks with the same 54 configuration and network size can usually be distinguished from one another based on the link strength values. On the other hand, two ordinary graphs or two directed graphs with the same configuration and network size cannot be distinguished from one another. Basing the network measures on vigraphs allows a greater possibility of distinguishing between two networks with the same size and the same number of links. Thus, vigraphs are used for both network and anchor variables. Amount of network density is operationally defined as: , (8) where S '(adj) is the strength of the link between two adjacent nodes 13 n k 0 I O . . ’ C O , . . . O h i and J (I e a direct link) iEl jE1SIJ(adJ) is t e sum of rows and columns in the adjacency matrix (i.e., sum of all directional link strengths in the network), n is the number of nodes, and n(n - I) is the number of off-diagonal elements in the adjacency matrix. Equation (8) is an extension of the density measurement model used for ordinaty graphs (e.g., Czepiel, l975; Friedkin, l98l; Grano- vetter, I976; Richards, I976) and directed graphs (e.g., Edwards & Monge, I977; Guimareas, l972; Rogers & Kincaid, l98l) where the strength of actual links is summed rather than the number of links. The density measure can be interpreted as the average direct communi- cation activity of the average direct link strength between any two n k adjacent nodes in a network. Dividing Z z S. i=l j=l ‘3Iadjl by n(n - 1) 55 results in an average score, allowing for the comparability of net- works of differing sizes (Freeman, I978/79). Based on link strength scale of zero to I68, amount of network density can range between zero and I68. A maximum density score of I68 means that all the nodes in a network have maximum direct communication activity (i.e., I68 hours per week) with all other nodes. While a maximum network density score of I68 is pragmatically impossible, it is theoretically possible. Length of network reachability is defined as the average geodesic (shortest distance) from node i to node j. For ordinary or directed graphs, the geodesic is the minimum number of links or steps from node i to node j (Harary, Norman, & Cartwright, I965; Peay, I980). The length of network reachability for a directed graph (digraph) is (e.g., Edwards & Monge, I977): n k E Z L1.(dist) #13:] J ’ if i-' L= ‘J, O: (9) n(n - I) where Lij(dist) is the minimum number of links from node i to node j in the distance matrix. For a metric distance between two adjacent nodes in a communi- cation network, Farace and Mabee (I980) suggest the inverse of the link strength (I/Sij)' The assumption is that the stronger the link .strength, the shorter the communication distance. Alternatively, the weaker the link strength, the greater the communication distance. 56 Taking the inverse of the link strength converts strong link strengths into short distances and weak link strengths into long distances. The link strengths (Sij) are retained in the adjacency matrix for the computation of the amount of network density. Thus, extending the operational definition from digraphs to vigraphs, the length of network reachability is defined as: n k E ZI/S.. . ._ ._ ij(dist) 1“ 3“ , if i=j. i/x = o. (10) n(n - I) where I/Sij(dist) is the distance (inverse of link strength) from node i to node j in the distance matrix, and n(n—l) is the number of finite off-diagonal elements in the distance matrix. On close examination, there is a problem with equation (l0) and equation (9) for directed graphs. Since digraphs and vigraphs do not assume symmetrical links, it is possible to have a directed path from node i to node j, but no path from node j back to node i. Thus, node j is reachable from node i, but node i is not reachable from node j. When node i is not reachable from node j, the distance is infinite (Barnes, I969; Harary et al., I965). An infinite distance for a cell element in the distance matrix results in an incomplete matrix of finite numbers. Since zero on the main diagonal represents the distance from every node to itself (Harary et al.), zero would be inappropriate to represent infinite reachability. Using equation (9) and (IQ), i.e., dividing by n(n-l), implies a full matrix of finite numbers. 57 The number n(n-l) for a full matrix represents the number of off-diagonal finite elements for computing an average (mean) reach- ability score for a network. Therefore, n(n-l) can be generalized to Zf(D), which is the sum of the number of off-diagnoal finite elements in the distance matrix.2 Replacing n(n-l) with Zf(D), equation (ID) for a vigraph becomes: n k ,2] jE11/Sij(dist) ’ ui=j.ws=O (n) Zf(D) Using Zf(D) as the denominator restricts the interpretation of the resulting score to the average geodesic of the nodes that are reachable. Based on link strengths of one to I68 for existing links, the distance between twO adjacent nodes can range from .0060 (l/l68) to one (l/I). The length of network reachability can range from greater than zero to infinity. A zero or infinite score would imply that no interactive diffusion of a message occurred. Although, as the amount of network density increases, the length of network reachabil- ity decreases, the relationship between the two variables is non- linear. The inverse of link strength for computing distance is a monotonic nonlinear transformation. In addition, the sum of the inverse of each link is not the same as the inverse of the link strength (i.e., Zl/S # l/ZS). Therefore, network reachability is a different structural measure than network density and the reachability 58 score cannot be directly derived from the density score, i.e., they are not multicollinear. Amount of anchor centrality is operationally defined as: k X .. . j=1SlJ(adJ), (n-l) n h 2 .. . w ere j=1SIJ(adJ) the anchor (node i) to all other nodes in the adjacency matrix and is the sum of the out-going link strengths from (n-l) is the number of other network members. Like the amount of network density, the amount of anchor cen- trality score can range from zero to I68. Thus, the amount of anchor centrality score can be interpreted as the anchor's average number of hours per week of out-going direct communication with other network members. Length of anchor reachability is operationally defined as: k .E11/51j(dist) J : if i=j, l/S = 0, (13) (n - I) k u o a where jEII/Sij(dist) IS the sum of the out-gOing distances (i.e., inverse link strength) from the anchor (node i) to all other network members. Because a message will spread out from the anchor, all other network nodes that ultimately receive the message will be reachable 59 from the anchor. The length of anchor reachability score can range from greater than zero to infinity. Both a zero or infinite score would imply that an isolate (i.e., a node with no communication links to other network members) is the anchor and, therefore, no inter- active message diffusion has occurred. (A zero score represents the distance to oneself while a score of infinity represents the unreachability to other nodes.) Amount of message temperature is operationally defined as the average amount of excitement, enthusiasm, or interest a message gen- erates in the network. The measurement scale and general question to be asked of network members are: If 0 (zero) is NOT AT ALL, and IOO is AVERAGE: How EXCITED, ENTHUSIASTIC, or INTERESTED are you in the message news? The above is a ratio scale bounded at zero with IOO given as a modulus (reference point). To create the average score for the people in the network, individual members' scores are summed and divided by the number of network members (N).4 Thus, amount of message temperature scores for networks can range from zero to infinity. A score of less than IOO would indicate a network's excitement, enthusiasm, or interest in the message is less than average. Amount of message transmission resistance is operationally based on the same measurement scale used for amount of message temperature above with the following question: How HESTITANT (for any reason) are you to pass the information on to someone you know might be interested in the news? 60 To create the average score for the network, individual members' scores are summed and divided by the number of network members (N). Amount of message transmission resistance scores for networks can also range from zero to infinity. Single indicators for the two message variables were chosen based on the study's design and the sample which are presented in the next chapter. Summary This chapter has presented a theory (model) of the rate of inter- active information diffusion. As part of the theory, both the theoretical and measurement models were presented. The theoretical model consists of a single multi-exogenous equation with the logistic b parameter as the dependent variable. The theory specifies that the functional form of exogenous variable relationships is multiplicative, or alternatively, the linearized form is the log-log equation. The measurement model for each variable is a single indicator. The measurement models for amount of network density, length of net- work reachability, amount of anchor centrality, and length of anchor reachability are all based on the analysis of value directed graphs (vigraphs). Analysis of vigraphs allows the greater precision in distinguishing between networks and between anchors than vigraphs. However, to allow for a possible incomplete distance matrix (i.e., nodes of infinite reachability), the often used formula for network reachability was modified by replacing n(n-l) with the more general Zf(D) in the denominator. 6I The next chapter will present a study conducted to test the theory presented in this chapter. The research is a non-experimental field study of the diffusion of a single message in multiple net- works. FOOTNOTES--CHAPTER III 1For brevity, all exogenous variables are referred to by their key identifying name, rather than their full name after they are conceptually defined, e.g., amount of network density is subsequently referred to only as network density. 2This will be explained in the section on measurement models. 3Two other possible alternatives are to replace the infinite element with: (a) the median reachability between nodes i and j or (b) some finite upper limit value (e.g., the maximum reachability between nodes i and j) (personal conversation with Edward L. Fink, January I982). The difficulty with these two alternatives is that the reachability score will be biased upwardly. For a distance matrix with more than 25% infinite elements, the upward bias could be quite substantial. However, until an empirical comparison is made between these two alternatives and Zf(D), no one alternative is clearly superior. 4Empirically, the mean may not be the best aggregation if the scores do not approximate a normal distribution. However, using the mean for message temperature and message transmission resistance is consistent with the measures of network density, network reachability, anchor centrality, and anchor reachability which are also mean measures . 62 CHAPTER IV METHODS To test the theory Of interactive information diffusion, i.e., equation (6), a non—experimental field study was conducted. The same message was diffused in multiple networks. For the first test of the theory, it was decided to hold constant the amount of message temperature and the amount of message transmission resistance while allowing the other four exogenous variables to vary. Holding the two message variables constant allows for better parameter estimates (i.e., smaller standard errors) for the other four exogenous variables with the same sample size. Allowing all six exogenous variables to vary would require a very large sample of networks and would be a very complex study to conduct and analyze. Thus, the study's major focus is to test the functional form of the theory and estimate the parameters of the amount of network density, length of network reach— ability, amount of anchor centrality, and length of anchor reach- ability. Groups/Subjects To test equation (6), groups of dormitory residents from the Michigan State University campus were contacted. The criteria for selecting groups were: (a) maximum group size of IOO, (b) anticipated 63 64 cooperation with the researcher, (c) sufficient probable variability of the two network and the two anchor variables for parameter esti- mation, and (d) topic interest commonality among the groups so that the same message could be reasonably used for all of the groups. The dormitory living groups on the MSU campus appeared to meet these four criteria. Each dormitory group was composed of a resident assistant (RA) and approximately 50 residents (hereafter referred to as an RA-group). RA-group members live on the same wing and floor of a dormitory. The dormitory system at MSU is one of the largest in the country. Students are housed in several different dorm structures. The floor structure of the dorms are of three basic types: (a) straight hall- ways, (b) single jointed hallways (45° and 90° joints) where only half of the hallway can be seen at a time, and (c) curved hallways where only one-fourth to one-third of the hallway can be seen at a time. In addition, while most dorm floors are divided into suites with a bath between two rooms, several other dorms have a community bath for each floor/wing. Because the different hallway structures and bath arrangements could reasonably affect the communication patterns on the floors, it was felt that RA-groups from different dorms would have a high probability of exhibiting variable network structures (i.e., network density and network reachability). All but one dorm on MSU's campus houses undergraduate students. Sonm topic interest commonality was expected for undergraduate dorm residents as a whole. It was felt that dorm residents would 65 cooperate in the study if: (a) the message to be diffused was of sufficient interest to dorm residents, (b) the procedures were made as simple and as effortless to carry out as possible, and (c) partici- pation time in the study was kept to no more than IO minutes. There- fore, because RA-groups appeared to meet all of the above criteria, they were selected as the sample units of analysis in the study. For all groups, the RA was chosen to be the first person to receive the message (anchor node). This was done for two reasons. To conduct a ”survey” on each dorm floor, the permission and coopera- tion of the RA was needed. Second, RAs are frequently the initial sources of information which circulates in the dorms and, therefore, it was felt that the dissemination of the message would more likely follow the usual pattern of flow. Permission to conduct a survey in the dorms was obtained by contacting the Director of University Housing Programs. The Director was informed of the nature of the study and the topic of the message to be diffused. An offer was made by the researcher to conduct a training workshop on communication networking for participating dorm directors and RAs. Because of the offer, the Director of University Housing Programs deferred the decision to his six Area Dorm Directors. One Area Director gave permission to contact her respective five dorm directors (again, this was because of the training workshop offer). All five dorm directors gave their permission to contact RAs. During the meeting with each dorm director, the researcher left copies of a two—page handout for all RAs in the dorm. The 66 handout contained a very brief description of the study with a list of advantages to participating in the study (see Appendix A). A week after meeting with the five dorm directors, RAs were contacted by telephone and asked if they would be willing to partici- pate. RAs were informed about the message topic. It was emphasized that participation by an RA did not obligate other floor residents to participate. An RA's participation allowed other residents the opportunity to receive the information and instructions. Residents could decide for themselves whether to participate by completing a short questionnaire and passing the message on to other floor resi- dents. If the RA agreed to participate, a meeting time was set for the researcher to deliver the materials to the RA. Out of 48 RAs contacted, five refused to participate. A summary description of the 43 RA groups from the five dorms is presented in Table 3. Ten groups with three or fewer participants in the study were dropped from further analysis. Message The criteria for selecting the single message were: (a) the message should be highly interesting to the majority of undergraduate dorm residents, and (b) there would be little resistance or reluctance by undergraduate dorm residents to pass the message on to others. A message which meets these two criteria should maximize undergraduate dorm residents' willingness to participate in the study. The message selected was a list and schedule of the winter term RHA (Resident Hall Association) movies. During the week and on 67 .azogm mzp cw mucmawovpcma Lacy mo Ezswcwe m we: mm; mcmgp w? mmmemcm new mFQEmm mzp soc» emaaocu mm; azogmuczo mcmx< mpcmucoammg Aumaaosuv Amwagocnv Aumaaocvv Aumaaocnv mace go a cow: masocmi are very low. This is not surprising given the diffusion of a single Itiessage and the criterion of t0pic interest commonality for the sample sselection. Among the exogenous variables, there appears to be a high, Eiltmough not perfect, degree of multicollinearity between network reach- éa.bility and anchor reachability (r = .9l6). The high correlation t:>etween these variables is a result of using Zf(D) rather than n(n— 1) 23.5 the denominator in computing network reachability. For four of 1:;he vigraphs, Zf(D) is the same value as (n-—l) which is the denomina- ‘t:¢or for computing anchor reachability, i.e., four vigraphs resulted ‘i r1 network reachability scores identical to their respective anchor Y‘teachability scores. Many of the other vigraphs have network reach- Eltaility scores very close to their anchor reachability scores because 13Iie message diffused very little beyond the first-order zone. 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Nmm.mmfl m-.~m~ om~.~mfi oo.H.oH.- assumgwaewp wmmmmmz £5: vfime. mfiwm. ofiofi. emo.nw~ omm.omH mom.~mfi om.H.o~.- zuwpwamguuwm Locuc< 85: wwwe. mwmm. ammo. oeo.~wfi NH~.om~ m¢m.~w~ ma..oH.- xuw—cgucwu Loguc< 5.: mqu. NMHN. Hmmo. ¢om.om~ wHH.owH omm.mmH mo..oH.- xu_p_ne;ummz xLoZumz $8.: omom. mmmm. vaH. mwm.NmH mvm.~wfi voe.me mv.H.oH.- xuwmcoo xsozuwz an: evfim. eemm. memo. vmm.mw~ on.NmH mow.nm~ mm.H.o~.- mmcoz a a o A a XME a :F :F ncp =_ Adv :F cp cg ”We H ”we 4 w m :owmmmgmom so» a Low mm so» u so» 4 Low 4 Ezewuao Ezswuqo mgu seem um>oEwm mm—nm_gu> .11-): 11111 mxsozumz mm we wNTm wpqsmm a new: mzowmmmemmx :o_umecoem:msh xou: was 99 The theoretically specified log-log (ln,ln) transformation falls within the 95% C.I. (L ln,ln) = l86.984) as did the log transfor- max( mation of the dependent variable only (L (ln,1) = l87.3l6, i.e., max regression of the logarithmic transformation of the logistic b with the untransformed set of six exogenous variables. While the (ln, ln) transformation is theoretically relevant, the (ln,1) transforma- tion is empirically relevant based on the positively skewed dis- tribution of the logistic b and the outlier identified in the bivariate scatterplots. Because (ln,1) transformation with all six exogenous variables in the equation resulted in a higher Lmax value than the Lmax(ln,ln) and both transformations are within the 95% C.I., Lmax(ln,1) results are included in Table 8 for comparison. Since R2, R2, and R are not valid criteria for the selection or test of an appropriate functional form (Anscombe, l974; Bauer, l98l), they are presented for only the theoretical (ln,ln) transformation as additional information on the removal of exogenous variables from the equation.6 The results of the separate removal of each exogenous variable are presented in Table 8. Both the L (ln,1) and L (ln,ln) for max max each equation are within the respective 95% C.I., i.e., between L (1,0) and (L (i5) - 2.996). max max Rather than using the change in R2 to assess the effects of Eicdcjing or deleting an exogenous variable from an equation, the change 7 ’1 Lmax(A,p) is used. As demonstrated by Bauer (l98l), Lmax(A,u) is 2 £3 TTiore stable and, therefore, a more appropriate criterion than R to 100 use in selecting transformation parameters for functional forms. Since a functional form consists of both the transformation para- meters and the addition or deletion of variables from an equation, the change in Lmax(A,p) values are used to evaluate the effects of removing variables from the equation. Removing one exogenous variable at a time, the removal of net- work density results in the highest Lmax(ln,ln) (i.e., l87.363), while the removal of message transmission resistance results in the highest LmaX(A,0) (188.491) and highest Lmax(ln,1) (187.911). The single removal of the other exogenous variables had varying minimal 6.13), L positive and negative effects on Lma X(ln,1) and X ma Lmax(ln,ln). Since a single message was diffused in the multiple networks, the effect of the joint removal of message temperature and message transmission resistance is examined. The Lmax for (A,fi), (ln,1), and (ln,ln) increased compared to the respective Lmax for all six variables included (i.e., none removed). However, the differences between Lmax (A,0) for the removal of message temperature and message transmission resistance and L (A,u) for the equation with all six max exogenous variables included is less than 2.996. Therefore, the two inessage variables can be considered constants (i.e., set to 1) and £2)‘ility is jointly removed with the two message variables since lOl anchor reachability and network reachability are highly correlated (r = .9l) and the removal of anchor reachability resulted in a (A,u), Lmax(ln,1) and Lmax(ln,ln) for the joint removal of the three variables are higher Lmax than network reachability. The Lmax higher than respective Lmax values for the inclusion of all six exogenous variables. However, the joint removal results in a lower Lmax than the single removal of network density. Among the single removal of variables, the removal of network density results in the highest Lmax‘ Thus, the joint removal of network density, message temperature, and message transmission resis- tance was conducted. The joint removal of these three variables results in the highest obtained Lmax(A,p), Lmax(ln,1) and Lmax(ln,ln). AA The highest L (A,p) in Table 8 is 188.819 (X = -.10, u = 1.50) max for the joint removal of network density, message temperature, and message transmission resistance. In constructing the 95% C.I., the two degrees of freedom are based on the two transformation parameters (A,0) to be estimated rather than on the number of partial slopes (yi). Thus, the ex§(l - q), i.e., 2.996, remains the same for each 0.1. Therefore, each Lmax(A,u) can be evaluated against the 95% C.I. around (-.l0,l.50) which is referred to as the overall 95% 0.1. For the Lmax(-.l0,l.50), the approximate 95% C.I. is: LmaX(-10,l.50) - L l88.8l9 - Lmax(A,u) :_2.996 (23) 102 185.823 : Lmax(A,p) (24) which results in the Lmax interval of l85.823 to l88.8l9. All of the Lmax values in Table 8 fall within this overall 95% C.I. Hence, all three sets of transformations on the various com- binations of exogenous variables are empirically appropriate. How- ever, since the functional form specified by the theory is (ln,ln), it is the preferred transformation. For (ln,ln), the various com— binations of exogenous variables all are within the overall 95% C.I. However, the equation with the removal of network density, message temperature, and message resistance has the highest Lmax(ln,ln). This equation includes network reachability (£2), anchor centrality (g3), and anchor reachability (£4). The theoretical form of the equation is written: €23 n = Yo] _?2~;4~ Ca (25) £2 54 where Y0] = (Y0515556) and Y0 is the intercept in equation (5). Linearizing equation (25) by taking natural logarithms (ln) results in: lnn = lny01 — yelng2 + y3lng3 - y4lng4 + lng. (26) 103 The partial slopes and their S.E.s are presented for both equations (6), i.e., the equation with all six exogenous variables, and (26) in a later table. Looking down the column labeled ”optimum A,fi” in Table 8, it can be seen that the removal of different exogenous variables effects 0 for the set of exogenous variables and not A for the logistic b (endogenous variable). The A for all combinations of exogenous vari- ables is -.lO while 0 ranges from .05 to l.75. Table 9 presents the descriptive statistics for the logarithmic (ln) variables. Comparing the skewness and kurtosis statistics in Table 9 with Table 6, transforming the logistic b by ln removes most of the nonnormality in the untransformed variable. Except for anchor centrality and message temperature, the ln transformation over-corrects the positive skewness in the untransformed exogenous variables. How- ever, as mentioned previously, the nonnormality of the exogenous variables is not relevant to the appropriate functional form for linear analysis. The descriptive statistics for the residuals (at the bottom of the table) are discussed in a later section on residual analysis. Because a logistic b outlier was discovered in the untransformed bivariate scatterplots, the effect of removing the case with the out- l ier is examined.7 The descriptive statistics of the untransformed \rariable with the removal of the outlier case (i.e., N = 32) are FD resented in Table l0. Comparing Table l0 with Table 6 (i.e., lalrfitransformed variables with N = 33), the removal of the outlier TABLE 9. Descriptive Statistics of the Logarithmic (ln) Variables with a Sample Size of 33 Networks Variables A— 8.0. Skewness Kurtosis Logistic b -5.928 1.365 .425 .298 Network Density .575 .843 -.305 —.877 Network Reachability - 892 .525 -.542 —.514 Anchor Centrality 1.118 .789 .189 -.560 Anchor Reachability —.738 .527 —.216 -.448 Message Temperature 5.050 .301 .737 1.367 Message Transmission 3.201 2.094 -3.362 11.330 Resistance Residuals from 0.0 1.171 .681 1.105 Equation (6) Residuals from 0.0 1.216 .368 1.235 Equation (26) 105 TABLE 10. Descriptive Statistics of the Untransformed Variables with a Sample Size of 32 Networks Variables 7' 5.0. Skewness Kurtosis Logistic B .005 .008 2.786 7.387 Network Density 2.345 1.768 .973 .127 Network Reachability .472 .212 .386 -.141 Anchor Centrality 4.070 3.611 1.937 4.003 Anchor Reachability .551 .279 .918 .809 Message Temperature 162.694 57.345 2.102 6.484 Message Transmission 47.528 36.467 2.024 4.491 Resistance 106 decreases both the skewness (from 3.758 to 2.786) and the kurtosis (from l5.8l0 to 7.387) of the logistic b. The transformation analysis is repeated with a sample size of 32 networks (i.e., the outlier case removed) for equation (6) (all six exogenous variables, i.e., none removed) and for equation (26) (network density, message temperature, and message transmission resistance removed). The results are presented in Table ll. Remov- ing the outlier case raises (A,fi) for both equations. The Lmax(A,fi) for both equations with N = 32 increased from the L (A,fi) for the max same equations with N = 33. However, L ln,1) and L (ln,ln) with max( max N = 32 decreased. The Lmax values in Table 11 are within the overall 95% C.I. and, therefore, there is not significant improvement in removing the outlier case. The descriptive statistics of the logarithmic transformed vari- ables with N = 32 are presented in Table l2. Compared to N = 33 (Table 9), the ln transformation with N = 32 results in a more normal distribution of the logistic b (i e., from a skewness of .425 to .l70 and from a kurtosis of .298 to .025). As discussed in the previous section (see Table 5), eight of the ratios of the logistic b to its S.E. are less than two. The corresponding eight cases were removed and a parallel set of analyses to N = 32 was conducted for N = 25. For a sample size of 25, the descriptive statistics of the untransformed variables are presented in Table l3. 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Ill ....GOUOEOO 0:920 .20.. i . .80: ....e» “we... fie... ....o» H 5.03.3.0! :35! «u: . 050! >255 0:. 009.0. 3:0! £30352: "23.3.... 9.3023 w... .5323 33: szmeE mi: <1”. 1952.. 151 MICHIGAN STATE UNIVERSITY COLLEGE or COMMUNICATION ARTS AND SCIENCES EAST LANSING - MICHIGAN ~ 43324 DEPARTMENT or COMMUNICATION November, 1981 Dear RA: Thank you for expressing your willingness to participate in a study of communication behavior (information diffusion) on your floor. You are among the first people on campus to receive information on the RHA films scheduled for Winter term. Please note: This study is flQI_a test or evaluation of your communication behavior. I am only interested in the overall picture of the communication on your floor. Please follow the instructions on the white sheet (the message-questionnaire) that accompanies this letter. Give a copy to only those floor residents that you would normally tell informal information to. QQ_flQI_give a copy of the message to all of your floor residents. If you have any questions, please ask the person delivering this letter or call me. Cordially, Connie L. Bauer 441 Communication Arts Bldg. Dept. of Communication 355-6666 (office) 484-3962 (home) MSL' u on Affirmative Arnom Equal Opportumty lnstimhon APPENDIX B TABLES OF VARIABLE VALUES FOR EACH NETWORK 152 TABLE Bl. Logistic Parameters for each Network . A 2 Network T a b r F ID (S.E.) (S.E.) (r) significance 01 12 -.355 .000331 .873 .000 (.354) (.000040 (.934) 02 6 -1.260 .000566 .294 .266 (2.636) (.000438) (.542) O3 7 -1.622 .003466 .582‘ .046 (1.484) (.001312) (.763) 04 12 —.808 .002996 .815 .000 (.468) (.000451) (.903) 05 7 -3.643 .020881 .333 .175 (3.315) (.013216) (.577) 06 9 —2.924 .002853 .926 .000 (.535) (.000306) (.962) 07 12 —5.423 .003542 .659 .001 (1.492) (.000805) (.812) 08 5 —2.405 .001536 .970 .002 (.577) (.000157) (.985) 09 4 -.370 .001639 .921 .040 (1.002) (.000339) (.960) 10 5 —1.567 .001209 .982 .001 (.396) (.000094) (.991) 11 15 .049 .000377 .717 .000 (.418) (.000066) (.847) 12 8 -l.548 .010292 .543 .037 (1.353) (.003857) (.737) 153 TABLE Bl (cont'd) 154 > Network T a 6 r F ID (S.E.) (S.E.) (r) significance 13 8 -1.930 .004800 .816 .002 (.834) .000929) (.904) 14 8 -.011 .003791 .893 .000 (.476) .000537) (.945) 15 21 -1.652 .000195 .452. .001 (.675) .000049) (.673) 16 3 -.495 .005544 .962 .126 (1.122) .001109) (.981) 17 4 -2.157 .077288 .269 .481 (5.237) .090065) (.519) 18 5 -2.420 .000949 .475 .198 (3.037) .000577) (.689) 19 11 -.523 .000438 .900 .000 (.354) .000049) (.949) 20 8 -1.515 .001349 .557 .033 (1.307) .000491) (.746) 21 6 -.221_ .001303. .943. .001 (.479) .000160) (.971) 22 5 -1.408 .029596 .973 .002 (.474) .002830) (.987) 23 5 -1.331 .002646 .343 .300 (2.926) .002115) (.586) 24 7 -2.868 .003056 .874 .002 (.891) .000518) (.935) 155 TABLE B1 (cont'd) 2 A Network T a 6 r F ID (S.E.) (S.E.) (r) significance 25 8 -1.583 .007658 .984 .000 (.222) (.000398) (.992) 26 5 -.428 .005338 .969 .002 (.453) (.000553) (.984) 27 7 -3.525 .003057 .367 .149 (3.089) (.001795) (.606) 28 10 -3.903 .000711 .316 .091 (2.659) (.000370) (.562) 29 3 -1.543 .034598 .989 .067 (.676) (.002367) (.994) 30 7 -.185 .000934 .919 .001 (.478) (.000124) (.958) 31 4 -1.384 .002499 .276 .474 (4.877) (.002861) (.526) 32 7 -.790 .002544 .578 .047 (1.226) (.000973) (.760) 33 5 -.430 .004732 .968 .002 (.460) (.000498) .984) I‘ Note. T is the number of different time points in estimating the logistic parameters. TABLE 82. VariabTe Values for Each Network Logistic Network Network Anchor Anchor Message Message Network N 6 Density ReachabiTity Centrality Reachability Temperature Transmission ID (S.D.) (S.D.) (S.D.) (S.D.) (S.D.) (5.0.) 01 13 .000331 1.1603 .2617 5.0833 .3385 199.67 61.25 (5.8060) (.3290) (5.4349) (.3490) (252.37) (66.20) 02 6 .000566 2.6667 1.0214 1.6000 1.2314 128.80 42.00 (12.7829 (.7273) (3.0496) (.7853) (62.18) (88.43) 03 7 .003466 1.9048 .4920 4.5000 .5971 179.80 20.00 (8.0541) (.4405) (6.0249) (.4547) (83.31) (44.72) 04 15 .002996 1.2381 .3229 1.5000 .5262 111.73 33.40 (6.3155) (.3976) (3.1317) (.4512) (50.61) (58.72) 05 10 .020881 4.3444 .1866 16.5556 .2080 156.90 25.10 (17.4409) (.4549) (31.4528) (.1899) (94.59) 42.43) 06 10 .002853 6.9667 .2908 1.5556 .3295 195.50 40.00 (20.0955) (.3947) (4.6667) (.4311) (116.11) (51.64) 07 13 .003542 2.4615 .7328 1.1667 1.2905 176.92 30.77 (11.1282) (.6180) (2.8551) (.7132) (63.30) (43.49) 08 5 .001536 1.4000 .5387 7.0000 .5387 384.50 12.50 (3.9921) (.5327) (6.9761) (.5327) (403.02) (25.00) 09 8 .001639 .6429 .3934 1.7143 .4536 276.86 28.57 (2.4601) (.2142) (1.7043) (.1944) (316.78) (39.34) 10 5 .001209 5.8000 .7036 1.0000 .8772 183.33 00.00 5.0023) (.4452) (.8165) (.2515) (76.38) (00.00) 11 20 .000377 2.0000 .4617 2.8421 .3247 176.90 34.00 (10.3786) (2.0421) (3.2191) (.1713) (201.52) (45.93) 12 8 .010292 3.2500 .3587 12.1429 .3716 209.36 32.14 (15.1493) (.3093) (26.0284) (.3638) (163.63) (47.25) 13 11 .004800 2.0273 .4553 1.4000 .5853 156.67 38.89 (13.4890) (.3882) (1.5055) (.3794) (99.34) (54.65) 14 11 .003791 .6091 .6096 3.3000 .6209 164.67 33.33 (3.0473) (.4766) (5.4579) (.4681) (97.17) (35.36) TABLE 82 (cont'd) 157. Logistic Network Network Anchor Anchor Message Message Network N 6 Density ReachabiTity Centrality ReachabiTity Temperature Transmission ID (S.D.) (S.D.) (S.D.) (S.D.) (S.D.) (S.D.) 15 23 .000195 1.3317 .3933 1.0000 .5049 234.30 25.87 (11.1155) (1.5767) (2.4137) (.4803) (246.10) (41.63) 16 6 .005544 4.9333 .1289 10.8000 .1614 126.67 16.67 (15.9869) (.1544) (10.0349) (.1916) (25.82) (25.82) 17 6 .077288 4.4667 .1726 6.6000 .2945 191.67 00.00 (8.7247) (.2408) (10.6911) (.3996) (49.16) (00.00) 18 5 .000949 4.7500 .6711 1.2500 .8361 116.67 50.00 (20.0785) (.4695) (1.2583) (.3352) (28.67) (50.00) 19 14 .000438 .4231 .7023 3.2308 .7692 160.50 32.25 (2.0578) (.5025) (5.0192) (.5209) (81.96) (70.91) 20 10 .001349 1.7000 .4917 8.1111 .5954 176.00 125.80 (7.0654) (.5019) (10.6706) (.4983) (136.97) (307.42) 21 6 .001303 .6667 .7238 2.0000 .8486 88.20 50.00 (2.1867) (.4684) (2.8284) (.3969) (54.09) (50.00) 22 5 .029596 .6000 .4875 3.0000 .4875 103.75 50.00 (1.4290) (.3660) (1.8257) (.3660) (41.51) (40.83) 23 6 .002646 1.6667 .6901 3.0000 .8224 137.50 50.83 (6.6661) (.4916) (5.0498) (.4133) (54.20) (76.84) 24 7 .003056 3.1667 .1508 5.6667 .2090 115.00 52.86 (9.6219) (.1213) (8.0416) (.1062) (90.78) (62.37) 25 10 .007658 1.0222 .4111 3.0000 .4537 148.33 22.22 (6.4459) (.3313) (3.5000) (.3573) (71.76) (44.10) 26 5 .005338 .9000 .2524 4.5000 .2524 125.00 175.00 (1.9974) (.0963) (1.9149) (.0963) (28.87) (95.74) 27 7 .003057 4.5238 .2571 3.3333 .3389 168.57 57.14 (18.2560) (.1880) (1.2111) (.1319) (79.04) (83.81) 28 10 .000711 .3667 .5750 3.6667 .5750 125.00 42.50 (1.5024) (.4187) (3.3912) (.4187) (33.33) (55.34) 29 4 .034598 3.0833 .5393 .6667 .7095 175.00 100.00 (10.0676) (.5335) (1.5000) (.5031) (35.36) (00.00) 158 TABLE 82 (cont'd) Logistic Network Network Anchor Anchor Message Message Network N 6 Density ReachabiTity Centrality Reachability Temperature Transmission 10 (5.0.) (S.D.) (S.D.) (S.D.) (S.D.) (S.D.) 3O 7 .000934 .3750 .7620 2.2857 .8083 112.86 64.29 (1.3289) (.6104) (2.6904) (.6574) (54.69) (47.56) 31 4 .002499 1.9167 .5458 7.3333 .3944 137.50 25.00 (4.2738) (.5245) (6.4291) (.5245) (47.87) (50 00) 32 8 .002544 5.0000 .1950 4.2857 .1838 143.75 123.50 (20.3720) (.3098) (4.5356) (.0857) (41.73) (349.31) 33 5 .004732 2.1500 .2875 1.7500 .4032 110.00 25.00 (5.6221) (.3571) (2.8723) (.3988) (31.62) (50.00) Note. Some networks have missing va1ues for message temperature and message resistance. N is the size of the network. transmission APPENDIX C SOCIOGRAMS 159 01 03 a? 952 «5‘2? 90 ‘11 lq " 7' 07 14 08 (,0 Figure C1. Sociogram of network 01. 160 161 Figure C2. Sociogram of network 02. 01 a: 3 °9 0‘ Figure C3. Sociogram of network 03. 162 05' 01 3'3 3 #0 06 03 1 5' ‘3 01 5b a? o 1 04} 3° 1? 7 08 15‘ 7 09 to 10 Sb 1.1. 10 12. Figure C4. Sociogram of network 04. 163 Figure CS. Sociogram of network 05. Figure C6. Sociogram of network 06. 164 Figure C7. Sociogram of network 07. 02 01 Figure C8. Sociogram of network 08. 165 Figure C9. Sociogram of network 09. Figure 610. Sociogram of network 10. 167 Figure C12. Sociogram of network 12. 08 em 01 no 07 3 03 6 07 ll 09 9. 01 3 05 1 06 1 C13. Sociogram of network 13. 168 01 O3 25' IO 09 [2 05' 0‘7 1 .11 09 Figure C14. Sociogram of network 14. 169 I! I? IA I7 05 I60 .20 1 12> 2’ (11 98 1; 1 13 1 19 Figure 615. Sociogram of network 15. 170 01 )0 29’ 06 Figure C16. Sociogram of network 16. Figure C17. Sociogram of network 17. 171 02. 01 3 03 Figure 618. Sociogram of network 18. 07 /o 03. 08 1 . 8 QB 1 3 09 05 15 1 05 )0 01 7/ 15' 06 1 IO 5' 11 .1 11 13 Ii Figure 619. Sociogram of network 19. 172 oz W 03 30 07 1 01 ___——EL>£xr O7 10 08 Figure C20. Sociogram of network 20. Figure C21. Sociogram of network 21. 173 01 1 03 01 r 04 of Figure C22. Sociogram of network 22. Figure C23. Sociogram of network 23. 174 Figure 624. Sociogram of network 24. 03. 01 Figure C25. Sociogram of network 25. 175 01 3 03 .5 01 3 0., 7 05 Figure 626. Sociogram of network 26. 01 a) 03 0‘1 3 01 1 os’ 65’ 35 07 Figure 627. Sociogram of network 27. 176 0.1. 0'1 01 as 06 01 07 oz 0? IO Figure 628. Sociogram of network 28. 02 01 Figure 29. Sociogram of network 29. 177 Figure C30. Sociogram of network 30. 01 ll 01 IO 03 1 0’7I Figure C31. Sociogram of network 31. 178 Figure C32. Sociogram of network 32. Figure C33. Sociogram of network 33. APPENDIX D SCATTERPLOTS OF UNTRANSFORMED VARIABLES 179 .AmmFQmw;m> umELommchpczv xpwmcmn xgozpmc .m> a ovpmwmo— asp yo poraemppmum coco omoh C fiQ nun-1H— Q Hunt-nu ¢~~1~~ Q u-u—I—ou-u o Q 6 HO—it—di—i 0 nut-own- .h—u—n—n- O ocom ooom C 4 _ _ fi fl 2 a L HHHHMHF‘HMHHh-‘h—IHD-Obdfid omom xpwmcmo xgozpmz owe: oooe oNon ocom c¢ou on. 0 =0 0 o a. c c c c 5 fl 5 e c c _ o «0 c o o . e 03.: H r ~~~~~~~HHH~~~H~ Mum—Hmumhmn—o—tu—u—n—ohu— o¢.~ ‘DIID.----.-D'-.-'-"-'-'.----‘l---.----0-"-."l..-.0-.-..-“'0'-.----.--".|"'§-'-"0'-'.""‘--"‘ ooo~ owo@ meow mod .Ho o 1l r O 0 unwa Q m—Iv-no-d Q Hv—t-d— Quad-du- 9 u—a—uu—o—no u-‘Hp—u—l O mun—H 6 ant-amt— Q ~~u—.—-¢ o—n—«o—c—n Q g as. mczmwa ~o. No. mo. .3. Mo. «0. we. to. Odo 4 31181001 180 181 vaLowmcmgpcsv HHHHHnmcome xLozpm: .m> a qumHmoH 6;“ Ho poHamepmum nnoH mH.H sHHHHnmeummm 3203662 mooH Ho. 55. 0:. .HmmHanLm> on. Hm. ho. 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QH. .No mesmwa 4 31181501 182 vmsgommcmgpczv HHHngucwo Losucm .m> a owpmwmoH esp we poHagmuymom DoomH omooH oeooH omoNH Humegpcmu Losoc< amoOH coco omoh o¢om .HmmHanLm> econ oooH o o 0 3‘9..th 6 t—u—o—It-n Q D-IHHI—I § HMO-i... Q n—no—u—u-a ‘HHHM. Hung—a Q uncut-a o u—u—t—o—q o cam—.... o OHohH anomH amonH o c H c H H C HHMHuuMHHHHHl—u—Hu H H HHHH—‘hflHHHHO-‘U—QHHH o—nu—ni—‘t—nwu—n—u—uo—u—ov—u—amh—auno—u—n HHHo—p-H—q—HHHHHHMHo—I osoHH owow nHom ‘0'--.'I'-.-’-l§'|".'l¢'-§--l'.'-".'-l-.-"'.'--'.-‘0-.-'--."-"--'-."l'00---.00'6‘0'0-‘9'-'."'|. once o co. cocoa C .0 CC. c a ace 0 C o O Fahd—b‘. H—nu—‘o—n Q pan-n»..— . u-iu-no—o—Q 5.40—0.1— Q taunt—1rd. Haunt-dud Q ~H—h-t Qu—ou—o—n— Q ~—--—--—- 0 om.a DH.~ om. Ho. no. Mo. 5o. me. no. 50. mo. to. OH. .mo wezmwa 4 91181601 183 uwsgommcmgpczv poHHnmcummg Legucm .m> n oHpmHmoH 65p 40 uoHamepmom ocoH @NoH NHoH moo HBHHwnmeummm Loeue< cm. oh. mm. 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H H H H H H H H H H H H H H H H o H H o H H H H H H H H H H H H H H H H o H H 5 H H H H H H H H H H c H H H 9 H H o .‘-'-"-'-.--'-.'-'-.--'-’----.-0'-.-'-'.-'-'.-'--.--'0.--‘-‘--'-.'I.'-.'-"§'Il-""".----.‘V'DQOI'IO O oc.ooH onoo~H cooomH ocoomH oo.oHH on.oo oo.o~ Cooom oo.on Oo.oH Ho. Mo. :0. Mo. we. no. we. to. 3. .80 mezmHH 4 31151601 APPENDIX E RESIDUAL SCATTERPLOTS 186 .o :oHHmzcm LOH poHQHmHHmum HmzuHmmm .Hm ocsmHH H ao.H1 e~.Ho o:.~n cH.no oo.nu em.:u ow.m1 oo.mc 00.01 on.~1 co 1 O 0-0-0.0---‘----‘----‘DOOO‘--.-.OO'DO‘---"----‘D'O'O'EOOOO'DD‘DOOD‘----‘V---‘0'-..---|"--|¢.-|0"-. é o I O b C 0+ Hedi-4H ¢ HHHH éHHHH¢HHHH¢HHHH o HHHH +HHHH¢HHHH6 HHHH ¢ HHHH ¢ on "n 99.31 o~.n1 o:.~1 oo.H1 WHHHHHHHHHHHHHHH HHHHHHHHH—l—‘HHHHHH H . H r o H . H o 1 H H A cool “a H H «L H N t 5 co H H L. H o H I cool D. H H . . r. H H s S v H s on. I n H o Ho.H a:.~ om.n HHHHHHHHHHHHHHH—‘H HHHHH HHF'OH HHHHHHHH no.3 0 OHHHH ‘HHHHOHHHHOHHH-O‘HHHH O MHHOHHHH‘HHHHOHWH ‘ HMO-4 6 mn.«1 mo.~c m~.~c m:.nn mH.:1 mo.:1 mm.mo m~.ol mo.¢1 mo.~a 187 188 O‘HHHHOHHW‘ HM ‘HHHH ‘WHHOHHO-‘i‘ HM 6 WHO-0 ‘HHHH‘ .4.-Om. 0d- mn.nc ouoa- me.~- aao~l ms.~n Odom: HHH’I‘HHHO—HH WHO-‘0‘.“ HWHHHHHwHHO-fl-IHHH m:.nn coon- m«.:- .om cowpmscw L0$ pOFQLmuumom szuwmmm > ( om.:u O 0 mn.¢- o~.mn I Q HHHHHHHHHHO-O-OHHH I HHHHHHHHHHHHHHMH HHHHHHHHHHHHHHHHH mm.ml ao.m- O .----.----‘-.--‘----.----O----.----O----. .-.-.----.----‘----.--.-."'-§-'.-.."I'.---'.'---.--". ---" 0 a m~.ol 00.0: D I. mwoon on.~n mw.~n .Nm mgsm?m coon: O‘HHHHOHHHHO HHo-oHo—aHu-u-I ‘WHHOHHO-O-I‘HWOWH‘HHHHOH—om‘ oa.4- o:.~o oo.«n I. I. slenpgsaa coon o:.~ om.n ao.: REFERENCES 189 REFERENCES Allen, R.K. A comparison of communication behaviors in innovative and non-innovative secondary schools. Unpublished Ph.D. thesis, Department of Communication, Michigan State University, l970. Anscombe, F.J. 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