"Many real world systems and relational data can be modeled as networks or graphs. With the availability of large amounts of network data, it is important to be able to reduce the network's dimensionality and extract useful information from it. A key approach to network data reduction is community detection. The objective of community detection is to summarize the network by a set of modules, where the similarity within the modules is maximized while the similarity between different modules is minimized. Early work in graph based community detection methods focused on static or single layer networks. This type of networks is usually considered as an oversimplification of many real world complex systems, such as social networks where there may be different types of relationships that evolve with time. Consequently, there is a need for a meaningful representation of such complex systems. Recently, multi-layer networks have been used to model complex systems where the objects may interact through different mechanisms. However, there is limited amount of work in community detection methods for dynamic and multi-layer networks. In this thesis, we focus on detecting and tracking the community structure in dynamic and multi-layer networks. Two particular applications of interest are considered including temporal social networks and dynamic functional connectivity networks (dFCNs) of the brain. In order to detect the community structure in dynamic single-layer and multi-layer networks, we have developed methods that capture the structure of these complex networks. In Chapter 2, a low-rank + sparse estimation based evolutionary spectral clustering approach is proposed to detect and track the community structure in temporal networks. The proposed method tries to decompose the network into low-rank and sparse parts and obtain smooth cluster assignments by minimizing the subspace distance between consecutive time points, simultaneously. Effectiveness of the proposed approach is evaluated on several synthetic and real social temporal networks and compared to the existing state-of-the-art algorithms. As the method developed in Chapter 2 is limited to dynamic single-layer networks and can only take limited amount of historic information into account, a tensor-based approach is developed in Chapter 3 to detect the community structure in dynamic single-layer and multi-layer networks. The proposed framework is used to track the change points as well as identify the community structure across time and multiple subjects of dFCNs constructed from resting state functional magnetic resonance imaging (rs-fMRI) data. The dFCNs are summarized into a set of FC states that are consistent over time and subjects. The detected community structures are evaluated using a consistency measure. In Chapter 4, an information-theoretic approach is introduced to aggregate the dynamic networks and identify the time points that are topologically similar to combine them into a tensor. The community structure of the reduced network is then detected using a tensor based approach similar to the one described in Chapter 3. In Chapter 5, a temporal block spectral clustering framework is introduced to detect and track the community structure of multi-layer temporal networks. A set of intra- and inter-adjacency matrices is constructed and combined to create a set of temporal supra-adjacency matrices. In particular, both the connections between nodes of the network within a time window, i.e. intra-layer adjacency, as well as the connections between nodes across different time windows, i.e. inter-layer adjacency are taken into account. The community structure is then detected by applying spectral clustering to these supra-adjacency matrices. The proposed approach is evaluated on dFCNs constructed from rs-fMRI across time and subjects revealing dynamic connectivity patterns between the resting state networks (RSNs)."--Pages ii-iii.
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