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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
• This is a necessary condition for isomorphism, but not sufﬁcient
– just because two graphs have same eigenvalues does not mean they are
– 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
• 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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- Winter '11
- Sort, Complex network, community structure, Laplacian matrix, Physical Review E