Fixed points of graph peeling

Degree peeling is used to study complex networks. It corresponds to a decomposition of the graph into vertex groups of increasing minimum degree. However, the peeling value of a vertex is non-local in this context since it relies on the connections the vertex has to groups above it. We explore a different way to decompose a network into edge layers such that the local peeling value of the vertices on each layer does not depend on their non-local connections with the other layers. This corresponds to the decomposition of a graph into subgraphs that are invariant with respect to degree peeling, i.e. they are fixed points. We introduce in this context a method to partition the edges of a graph into fixed points of degree peeling, called the iterative-edge-core decomposition. Information from this decomposition is used to formulate a notion of vertex diversity based on Shannon's entropy. We illustrate the usefulness of this decomposition in social network analysis. Our method can be used for community detection and graph visualization.

[1]  Dimitrios M. Thilikos,et al.  Evaluating Cooperation in Communities with the k-Core Structure , 2011, 2011 International Conference on Advances in Social Networks Analysis and Mining.

[2]  Vladimir Batagelj,et al.  An O(m) Algorithm for Cores Decomposition of Networks , 2003, ArXiv.

[3]  Yuval Shavitt,et al.  A model of Internet topology using k-shell decomposition , 2007, Proceedings of the National Academy of Sciences.

[4]  Charles F. Hockett,et al.  A mathematical theory of communication , 1948, MOCO.

[5]  B. Bollobás.  The evolution of random graphs , 1984 .

[6]  Stephen B. Seidman,et al.  Network structure and minimum degree , 1983 .

[7]  Guy Melançon,et al.  Continental integration in multilevel approach of world air transportation (2000-2004) , 2008 .

[8]  Alessandro Vespignani,et al.  Large scale networks fingerprinting and visualization using the k-core decomposition , 2005, NIPS.

[9]  P. Erdos,et al.  On chromatic number of graphs and set-systems , 1966 .

[10]  Guy Melançon,et al.  The Tulip 3 Framework: A Scalable Software Library for Information Visualization Applications Based on Relational Data , 2012 .

[11]  David Eppstein,et al.  Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time , 2010, Exact Complexity of NP-hard Problems.

[12]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[13]  Ulrik Brandes,et al.  Drawing the AS Graph in 2.5 Dimensions , 2004, GD.

[14]  D. R. Lick,et al.  k-Degenerate Graphs , 1970, Canadian Journal of Mathematics.

[15]  Joel H. Spencer,et al.  Sudden Emergence of a Giantk-Core in a Random Graph , 1996, J. Comb. Theory, Ser. B.

[16]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[17]  Michael T. Goodrich,et al.  External-Memory Network Analysis Algorithms for Naturally Sparse Graphs , 2011, ESA.

[18]  Francesco De Pellegrini,et al.  General , 1895, The Social History of Alcohol Review.

[19]  Pablo M. Gleiser,et al.  Community Structure in Jazz , 2003, Adv. Complex Syst..

[20]  G. Szekeres,et al.  An inequality for the chromatic number of a graph , 1968 .

[21]  Donald E. Knuth,et al.  The Stanford GraphBase - a platform for combinatorial computing , 1993 .

[22]  B. Arnold.  Majorization and the Lorenz Order: A Brief Introduction , 1987 .

[23]  P. Erdos,et al.  On chromatic number of graphs and set-systems , 1966 .

[24]  Tim Roughgarden,et al.  Preventing Unraveling in Social Networks: The Anchored k-Core Problem , 2012, SIAM J. Discret. Math..

[25]  Andrew Parker,et al.  The Hidden Power of Social Networks: Understanding How Work Really Gets Done in Organizations , 2004 .

[26]  Sune Lehmann,et al.  Link communities reveal multiscale complexity in networks , 2009, Nature.

[27]  Dorothea Wagner,et al.  Enumerating and Generating Labeled k-degenerate Graphs , 2010, ANALCO.

[28]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[29]  Lada A. Adamic,et al.  The political blogosphere and the 2004 U.S. election: divided they blog , 2005, LinkKDD '05.