How Do Networks Become Navigable

Networks created and maintained by social processes, such as the human friendship network and the World Wide Web, appear to exhibit the property of navigability: namely, not only do short paths exist between any pair of nodes, but such paths can easily be found using only local information. It has been shown that for networks with an underlying metric, algorithms using only local information perform extremely well if there is a power-law distribution of link lengths. However, it is not clear why or how real networks might develop this distribution. In this paper we define a decentralized ``rewiring'' process, inspired by surfers on the Web, in which each surfer attempts to travel from their home page to a random destination, and updates the outgoing link from their home page if this journey takes too long. We show that this process does indeed cause the link length distribution to converge to a power law, achieving a routing time of O(log^2 n) on networks of size n. We also study finite-size effects on the optimal exponent, and show that it converges polylogarithmically slowly as the lattice size goes to infinity.

[1]  Jon M. Kleinberg,et al.  Small-World Phenomena and the Dynamics of Information , 2001, NIPS.

[2]  Ramesh Govindan,et al.  Using the small-world model to improve Freenet performance , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[3]  A. Vázquez.  Growing network with local rules: preferential attachment, clustering hierarchy, and degree correlations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  D. Watts,et al.  An Experimental Study of Search in Global Social Networks , 2003, Science.

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

[6]  S. S. Manna,et al.  Scale free networks from a self-organizing Hamiltonian dynamics , 2003 .

[7]  Sharon L. Milgram,et al.  The Small World Problem , 1967 .

[8]  S. Redner,et al.  Organization of growing random networks. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  A Ruschhaupt,et al.  Small world graphs by iterated local edge formation. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  M E J Newman,et al.  Identity and Search in Social Networks , 2002, Science.

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

[12]  Jon M. Kleinberg,et al.  The small-world phenomenon: an algorithmic perspective , 2000, STOC '00.

[13]  Ph. Blanchard,et al.  Small World Graphs by the iterated"My Friends are Your Friends'' Principle , 2003 .

[14]  S. S. Manna,et al.  LETTER TO THE EDITOR: Scale-free network on Euclidean space optimized by rewiring of links , 2003, cond-mat/0302224.

[15]  Ian Clarke,et al.  Freenet: A Distributed Anonymous Information Storage and Retrieval System , 2000, Workshop on Design Issues in Anonymity and Unobservability.

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