Community structure and modularity index determination of signed graphs: Pre- and post-ictal hippocampal depth recordings

In recent years, there is an increased interest in the role of static properties of and dynamics on signed graphs in physical, biological, and social networks. In particular, community structures of signed graphs and their modularity indices have drawn special attention, and there are numerous methods to identify them. Here we provide the technical details of implementing four important methods that in their nature are very different from one another; two spectral methods, one using the adjacency matrix and one using the Laplacian matrix, a heuristic graph theoretic method using a concept of edge centrality, and finally, a stochastic combinatorial optimization method. Because of the novel nature of the methods, we focus on an interesting problem of community structures in brain network of the CA1 region of the hippocampus of anesthetized rats to study the pre-ictal and post-ictal states. We use recent electroenecephalograph (EEG) depth recordings for a period of 117 minutes at a sampling rate of 1000Hz. Since the electrode used has 16-contact points, we construct non-overlapping but sliding 100,000 sample-windowed correlation matrices or functional connectivity (FC) matrices over the recorded time, yielding signed graphs. Our results from the clustering methods both at the pre-ictal and post-ictal periods of the depth recording reported here for the first time yield convincing answers to differing community structures.