Network discovery by generalized random walks

We investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large graphs. We derive an exact formula valid for arbitrary graphs and arbitrary walks with stationary transition probabilities (STP), for the average number of discovered edges as a function of time. We show that for STP walks site and edge exploration obey the same scaling ~nλ as a function of time n. Therefore, edge exploration on graphs with many loops is always lagging compared to site exploration, the revealed graph being sparse until almost all nodes have been discovered. We then introduce the edge explorer model (EEM), which presents a novel class of adaptive walks, that perform faithful network discovery even on dense networks.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[2]  Petter Holme,et al.  Congestion and Centrality in Traffic Flow on Complex Networks , 2003, Adv. Complex Syst..

[3]  D. Cassi,et al.  Random walks on graphs: ideas, techniques and results , 2005 .

[4]  Heiko Rieger,et al.  Random walks on complex networks. , 2004, Physical review letters.

[5]  László Lovász,et al.  Random Walks on Graphs: A Survey , 1993 .

[6]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  E Almaas,et al.  Scaling properties of random walks on small-world networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  A. Barabasi,et al.  Hierarchical Organization of Modularity in Metabolic Networks , 2002, Science.

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

[10]  R. Durrett.  Random walks and random environments. Volume 1: Random walks , 1996 .

[11]  . Markov Chains and Random Walks on Graphs , .

[12]  A. Ramezanpour.  Intermittent exploration in a scale-free network , 2007 .

[13]  P. M. Lee,et al.  Random Walks and Random Environments: Volume 1: Random Walks , 1995 .

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

[15]  Hierarchical Organization of Modularity in Metabolic Networks Supporting Online Material , 2002 .

[16]  L. Asz.  Random Walks on Graphs: a Survey , 2022 .

[17]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Cristopher Moore,et al.  On the bias of traceroute sampling: Or, power-law degree distributions in regular graphs , 2005, JACM.

[19]  K. Kaski,et al.  Scaling of random spreading in small world networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[21]  Kai Lai Chung,et al.  Markov Chains with Stationary Transition Probabilities , 1961 .

[22]  David J. Aldous,et al.  Lower bounds for covering times for reversible Markov chains and random walks on graphs , 1989 .

[23]  P. Blanchard,et al.  Epidemic spreading in a variety of scale free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Lada A. Adamic,et al.  Search in Power-Law Networks , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  S. Havlin,et al.  Scaling theory of transport in complex biological networks , 2007, Proceedings of the National Academy of Sciences.

[26]  J. A. d'Auriac,et al.  Random walk on fractals: numerical studies in two dimensions , 1983 .

[27]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[28]  S. Vempala.  Geometric Random Walks: a Survey , 2007 .

[29]  E. Montroll.  Random walks on lattices , 1969 .

[30]  Boleslaw K. Szymanski,et al.  Efficient and inefficient ant coverage methods , 2001, Annals of Mathematics and Artificial Intelligence.

[31]  George Weiss,et al.  Random walks and random environments, volume 1: Random walks , 1996 .

[32]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[33]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[34]  Uriel Feige,et al.  Short random walks on graphs , 1993, SIAM J. Discret. Math..