Role of Graph Theory in Computational Neuroscience

Neuroscience is the study of the brain and its impact on behavior and cognitive functions. Computational neuroscience is the subfield that deals with the study of the ability of the brain to think and compute. It also analyzes various electrical and chemical signals that take place in the brain to represent and process the information. In this chapter, a special focus will be given on the processing of signals by the brain to solve the problems. In the second section of the chapter, the role of graph theory is discussed to analyze the pattern of neurons. Graph-based analysis reveals meaningful information about the topological architecture of human brain networks. The graph-based analysis also discloses the networks in which most nodes are not neighbors of each other but can be reached from every other node by a small number of steps. In the end, it is concluded that by using the various operations of graph theory, the vertex centrality, betweenness, etc. can be computed to identify the dominant neurons for solving different types of computational problems.

A Distributed Bagging Ensemble Methodology for Community Prediction in Social Networks

Presently, due to the extended availability of gigantic information networks and the beneficial application of graph analysis in various scientific fields, the necessity for efficient and highly scalable community detection algorithms has never been more essential. Despite the significant amount of published research, the existing methods—such as the Girvan–Newman, random-walk edge betweenness, vertex centrality, InfoMap, spectral clustering, etc.—have virtually been proven incapable of handling real-life social graphs due to the intrinsic computational restrictions that lead to mediocre performance and poor scalability. The purpose of this article is to introduce a novel, distributed community detection methodology which in accordance with the community prediction concept, leverages the reduced complexity and the decreased variance of the bagging ensemble methods, to unveil the subjacent community hierarchy. The proposed approach has been thoroughly tested, meticulously compared against different classic community detection algorithms, and practically proven exceptionally scalable, eminently efficient, and promisingly accurate in unfolding the underlying community structure.

Graph Energies of Egocentric Networks and Their Correlation with Vertex Centrality Measures

Graph energy is the energy of the matrix representation of the graph, where the energy of a matrix is the sum of singular values of the matrix. Depending on the definition of a matrix, one can contemplate graph energy, Randić energy, Laplacian energy, distance energy, and many others. Although theoretical properties of various graph energies have been investigated in the past in the areas of mathematics, chemistry, physics, or graph theory, these explorations have been limited to relatively small graphs representing chemical compounds or theoretical graph classes with strictly defined properties. In this paper we investigate the usefulness of the concept of graph energy in the context of large, complex networks. We show that when graph energies are applied to local egocentric networks, the values of these energies correlate strongly with vertex centrality measures. In particular, for some generative network models graph energies tend to correlate strongly with the betweenness and the eigencentrality of vertices. As the exact computation of these centrality measures is expensive and requires global processing of a network, our research opens the possibility of devising efficient algorithms for the estimation of these centrality measures based only on local information.

CS_TOTR: A new vertex centrality method for directed signed networks based on status theory

Measuring the importance (or centrality) of vertices in a network is a significant topic in complex network analysis, which has significant applications in diverse domains, for example, disease control, spread of rumors, viral marketing and so on. Existing studies mainly focus on social networks with only positive (or friendship) relations, while signed networks with also negative (or enemy) relations are seldom studied. Various signed networks commonly exist in real world, e.g. a network indicating friendship/enmity, love/hate or trust/mistrust relationships. In this paper, we propose a new centrality method named CS_TOTR to give a ranking of vertices in directed signed networks. To design this new method, we use the “status theory” for signed networks, and also adopt the vertex ranking algorithm for a tournament and the topological sorting algorithm for a general directed graph. We apply this new centrality method on the famous Sampson Monastery dataset and obtain a convincing result which shows its validity.