A Geometric Approach to Robustness in Complex Networks

We explore the geometry of networks in terms of an n-dimensional Euclidean embedding represented by the Moore-Penrose pseudo-inverse of the graph Laplacian (L+). The reciprocal of squared distance from each node i to the origin in this n-dimensional space yields a structural centrality index (C*(i)) for the node, while the harmonic sum of individual node structural centrality indices, Pi 1/C * (i), i.e. the trace of L+, yields the well-known Kirchoff index (K), an overall structural descriptor for the network. In addition to its geometric interpretation, we provide alternative interpretation of the proposed structural centrality index (C*(i)) of each node in terms of forced detour costs and recurrences in random walks and electrical networks. Through empirical evaluation over example and real world networks, we demonstrate how structural centrality is better able to distinguish nodes in terms of their structural roles in the network and, along with Kirchoff index, is appropriately sensitive to perturbations/rewirings in the network.

[1]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[2]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[3]  Massimo Marchiori,et al.  Error and attacktolerance of complex network s , 2004 .

[4]  Leonard M. Freeman,et al.  A set of measures of centrality based upon betweenness , 1977 .

[5]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  M. Randic,et al.  Resistance distance , 1993 .

[7]  Prabhakar Raghavan,et al.  The electrical resistance of a graph captures its commute and cover times , 2005, computational complexity.

[8]  François Fouss,et al.  Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation , 2007, IEEE Transactions on Knowledge and Data Engineering.

[9]  Walter Willinger,et al.  A first-principles approach to understanding the internet's router-level topology , 2004, SIGCOMM 2004.

[10]  J. Kemeny,et al.  Denumerable Markov chains , 1969 .

[11]  A. Barabasi,et al.  Spectra of "real-world" graphs: beyond the semicircle law. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[13]  M. Randic.  Characterization of molecular branching , 1975 .

[14]  L. Freeman.  Centrality in social networks conceptual clarification , 1978 .

[15]  J. A. Rodríguez-Velázquez,et al.  Subgraph centrality in complex networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  F. Göbel,et al.  Random walks on graphs , 1974 .

[17]  P. Tetali.  Random walks and the effective resistance of networks , 1991 .

[18]  Béla Bollobás,et al.  Graphs of Extremal Weights , 1998, Ars Comb..