Existence, Cost and Robustness of Spatial Small-World Networks

Small-world networks embedded in Euclidean space represent useful cartoon models for a number of real systems such as electronic circuits, communication systems, the large-scale brain architecture and others. Since the small-world behavior relies on the presence of long-range connections that are likely to have a cost which is a growing function of the length, we explore whether it is possible to choose suitable probability distributions for the shortcut lengths so as to preserve the small-world feature and, at the same time, to minimize the network cost. The flow distribution for such networks, and their robustness, are also investigated.

[1]  M. Newman,et al.  Scaling and percolation in the small-world network model. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  M. Newman,et al.  Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Massimo Marchiori,et al.  Economic small-world behavior in weighted networks , 2003 .

[4]  V. Latora,et al.  Structural properties of planar graphs of urban street patterns. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Jon M. Kleinberg,et al.  Navigation in a small world , 2000, Nature.

[6]  U. Brandes.  A faster algorithm for betweenness centrality , 2001 .

[7]  Dmitri B. Chklovskii,et al.  Wiring Optimization in Cortical Circuits , 2002, Neuron.

[8]  L. Amaral,et al.  Small-World Networks: Evidence for a Crossover Picture , 1999, cond-mat/9903108.

[9]  Parongama Sen,et al.  Phase transitions in a network with a range-dependent connection probability. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Payman Zarkesh-Ha,et al.  Prediction of net-length distribution for global interconnects in a heterogeneous system-on-a-chip , 2000, IEEE Trans. Very Large Scale Integr. Syst..

[11]  Terrence J Sejnowski,et al.  Communication in Neuronal Networks , 2003, Science.

[12]  Vitaly A Klyachko,et al.  Connectivity optimization and the positioning of cortical areas , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[13]  L. Amaral,et al.  Erratum: Small-World Networks: Evidence for a Crossover Picture [Phys. Rev. Lett. 82, 3180 (1999)] , 1999, cond-mat/9906247.

[14]  M. Biskup.  On the scaling of the chemical distance in long-range percolation models , 2003, math/0304418.

[15]  N Mathias,et al.  Small worlds: how and why. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Jespersen,et al.  Small-world networks: links with long-tailed distributions , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[18]  M. A. de Menezes,et al.  Shortest paths on systems with power-law distributed long-range connections. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[20]  Hawoong Jeong,et al.  Modeling the Internet's large-scale topology , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[21]  M. Newman,et al.  Renormalization Group Analysis of the Small-World Network Model , 1999, cond-mat/9903357.

[22]  R. Pastor-Satorras,et al.  Structure of cycles and local ordering in complex networks , 2004 .

[23]  S. N. Dorogovtsev,et al.  Evolution of networks , 2001, cond-mat/0106144.

[24]  T. Petermann,et al.  Physical realizability of small-world networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..