If two graphs are isomorphisms they will have exactly

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/12/2014 for the course CSCI 289 at UC Davis.

Ask a homework question - tutors are online