Methods and Applications for Detecting Structure in Complex Networks.

Abstract

The use of networks to represent systems of interacting components is now common in many fields including the biological, physical, and social sciences. Network models are widely applicable due to their relatively simple framework of vertices and edges. Network structure, patterns of connection between vertices, impacts both the functioning of networks and processes occurring on networks. However, many aspects of network structure are still poorly understood. This dissertation presents a set of network analysis methods and applications to real-world as well as simulated networks. The methods are divided into two main types: linear algebra formulations and probabilistic mixture model techniques.

Network models lend themselves to compact mathematical representation as matrices, making linear algebra techniques useful probes of network structure. We present methods for the detection of two distinct, but related, network structural forms. First, we derive a measure of vertex similarity based upon network structure. The method builds on existing ideas concerning calculation of vertex similarity, but generalizes and extends the scope to large networks. Second, we address the detection of communities or modules in a specific class of networks, directed networks. We propose a method for detecting community structure in directed networks, which is an extension of a community detection method previously only known for undirected networks.

Moving away from linear algebra formulations, we propose two methods for network structure detection based on probabilistic techniques. In the first method, we use the machinery of the expectation-maximization (EM) algorithm to probe patterns of connection among vertices in static networks. The technique allows for the detection of a broad range of types of structure in networks. The second method focuses on time evolving networks. We propose an application of the EM algorithm to evolving networks that can reveal significant structural divisions in a network over time.