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# Graduate Student Seminar - 04/10/2009

**"Community Structure in Directed Networks"**

by **Sanjeev Chauhan**

Friday, April 10, 2009 -- 12:00 p.m.

Large Conference Room, 1207 Energy Research Facility

**Advisor: ** Professor Edward Ott and Asst. Professor Michelle Girvan

In many cases the spectrum of eigenvalues of the adjacency matrix of a network with community structure gives a clear indication of the number of communities in the network. In particular, for a network with N nodes and Nc communitieis there will typically be Nc eigenvalues that are significantly larger than the magnitudes of all the other (N - Nc) eigenvalues. We define an eigenvalue function of these Nc eigenvalues and investigate a network-function based definition for ending communities in complex networks. In particular, we consider networks whose function is enhanced by the ability to synchronize and/or by resilience to node failures. In the past, it was found that the largest eigenvalue of the network's adjacency matrix provides insight into both synchronization and percolation processes. Thus, for networks whose goal is to perform these functions, we propose a method that divides a given network into communities based on maximizing the largest eigenvalues of the adjacency matrices of the resulting ciommunities.