The average number of distinct sites visited by a random walker on random graphs

We study the linear large $n$ behavior of the average number of distinct sites $S(n)$ visited by a random walker after $n$ steps on a large random graph. An expression for the graph topology dependent prefactor $B$ in $S(n) = Bn$ is proposed. We use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message-passing equations to calculate its value. Numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs. Scaling with system size and average graph connectivity are also analysed.

[1]  Peter Sollich,et al.  Replica theory for learning curves for Gaussian processes on random graphs , 2012, 1202.5918.

[2]  N. Wormald,et al.  Models of the , 2010 .

[3]  Fan Chung Graham,et al.  The Spectra of Random Graphs with Given Expected Degrees , 2004, Internet Math..

[4]  J. Beeler.  DISTRIBUTION FUNCTIONS FOR THE NUMBER OF DISTINCT SITES VISITED IN A RANDOM WALK ON CUBIC LATTICES: RELATION TO DETECT ANNEALING , 1964 .

[5]  R. Monasson.  Diffusion, localization and dispersion relations on “small-world” lattices , 1999 .

[6]  J. Stoyanov.  A Guide to First‐passage Processes , 2003 .

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

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

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

[10]  M. Tamm,et al.  Number of common sites visited by N random walkers. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  S. Edwards,et al.  The eigenvalue spectrum of a large symmetric random matrix , 1976 .

[12]  G. Vineyard.  The Number of Distinct Sites Visited in a Random Walk on a Lattice , 1963 .

[13]  F Jasch,et al.  Target problem on small-world networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[15]  R. Kuehn.  Spectra of sparse random matrices , 2008, 0803.2886.

[16]  Adriano Barra,et al.  Extensive parallel processing on scale-free networks. , 2014, Physical review letters.

[17]  Paul Erdös,et al.  Some Problems on Random Walk in Space , 1951 .

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

[19]  F Jasch,et al.  Trapping of random walks on small-world networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[21]  Michael E. Fisher,et al.  Walks, walls, wetting, and melting , 1984 .

[22]  Stanley,et al.  Number of distinct sites visited by N random walkers. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[23]  Joel Friedman,et al.  On the second eigenvalue and random walks in randomd-regular graphs , 1991, Comb..

[24]  Muhammad Sahimi,et al.  Random walks on the Bethe lattice , 1982 .

[25]  Sidney Redner,et al.  A guide to first-passage processes , 2001 .

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

[27]  J. Klafter,et al.  Models for dynamically controlled relaxation , 1985 .

[28]  W. R. Buckland,et al.  Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. , 1952 .

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

[30]  János Komlós,et al.  The eigenvalues of random symmetric matrices , 1981, Comb..

[31]  Koujin Takeda,et al.  Cavity approach to the spectral density of sparse symmetric random matrices. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  S. Kak.  Information, physics, and computation , 1996 .

[33]  Paul Adrien Maurice Dirac,et al.  A new notation for quantum mechanics , 1939, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[35]  Vittorio Loreto,et al.  Collective dynamics of social annotation , 2009, Proceedings of the National Academy of Sciences.

[36]  O. Perron.  Zur Theorie der Matrices , 1907 .

[37]  Andrei Z. Broder,et al.  On the second eigenvalue of random regular graphs , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[38]  N. Wormald.  Models of random regular graphs , 2010 .

[39]  S. Majumdar,et al.  Exact distributions of the number of distinct and common sites visited by N independent random walkers. , 2013, Physical review letters.

[40]  Noga Alon,et al.  On the second eigenvalue of a graph , 1991, Discret. Math..

[41]  Extensive load in multitasking associative networks , 2014, 1404.3654.

[42]  Peter Sollich,et al.  Learning curves for Gaussian process regression on random graphs , 2013 .

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

[44]  Endre Szemerédi,et al.  On the second eigenvalue of random regular graphs , 1989, STOC '89.

[45]  Chi Ho Yeung,et al.  Networking—a statistical physics perspective , 2011, 1110.2931.