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If two graphs are isomorphisms they will have exactly

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Unformatted text preview: 0 5 10 q q qq qq q q qq q 20 X q Aside: Some other important spectral properties • Eigenvalues of Adjacency matrix, Laplacian, rwalk matrix are invariant under node relabeling. • If two graphs are isomorphisms, they will have exactly the same eigenvalue spectrum. • This is a necessary condition for isomorphism, but not sufficient – just because two graphs have same eigenvalues does not mean they are isomorphic – but it is strong evidence for isomorphisms (sometimes just first few eigenvalues tested is considered evidence) • For more on graph isomorphisms, Fiedler vectors, graph embeddings, see http://cs-www.cs.yale.edu/homes/spielman/ From graph bisection to communities in networks • M. Girvan and M. E. J. Newman, Community structure in social and biological networks, Proceedings of the National Academy of Sciences, 99 (2002), 78217826. • The paper the launched “community structure” • Identify the edge with highest betweeness centrality, this edge partitions the graph into two pieces. • Rerun recursively on each sub-network. • Note, this is continued bi-section. • Very costly (need to calculate all shortest-paths between all pairs of nodes – recall definition of betweeness centrality). Relating this back to Kernighan-Lin and Fiedler • Newman, Girvan, Finding and evaluating community structure in networks, Physical Review E, 69, 026113 (2004). • Newman, Finding community structure in networks using the eigenvectors of matrices, Physical Review E, 74, 036104 (2006). • Note the next slides come from a presentation that Mark Newman gave at the Institute for Pure and Applied Math, Workshop on “Random and Dynamic Graphs and Networks”, May 2007. There are many other interesting presentations there: http://www.ipam.ucla.edu/programs/rsws3/ Spectral partitioning is related to min-cut/max-flow • Choose any pair of vertices (i, j ) the minimum set of edges whose deletion places the two vertices in disconnected components of the graph, carries the maximum flow between the two vertices. • Finding the optimal partitioning over all possible pairs of nodes is a difficult optimization problem that is in NP. • Maximizing modularity also is NP: U. Brandes, D. Delling, M. Gaertler, R. Goerke, M...
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