Edge flows in the complete random-lengths network

Consider the complete n-vertex graph whose edge-lengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the $n \to \infty$ limit we find explicitly the empirical distribution of these edge-flows, suitably normalized.

[1]  Johan Wästlund.  Random assignment and shortest path problems , 2006 .

[2]  J. Michael Steele,et al.  The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence , 2004 .

[3]  Béla Bollobás,et al.  Random Graphs , 1985 .

[4]  Johan Wästlund.  The travelling salesman problem in the stochastic mean field model , 2006 .

[5]  Frits Beukers,et al.  SPECIAL FUNCTIONS (Encyclopedia of Mathematics and its Applications 71) , 2001 .

[6]  D. Aldous.  The ζ(2) limit in the random assignment problem , 2000, Random Struct. Algorithms.

[7]  Shankar Bhamidia,et al.  First passage percolation on locally treelike networks , 2009 .

[8]  Shankar Bhamidi,et al.  First passage percolation on locally treelike networks. I. Dense random graphs , 2008 .

[9]  Piet Van Mieghem,et al.  The Flooding Time in Random Graphs , 2002 .

[10]  Remco van der Hofstad,et al.  Universality for the Distance in Finite Variance Random Graphs , 2006 .

[11]  Russell Lyons,et al.  Conceptual proofs of L log L criteria for mean behavior of branching processes , 1995 .

[12]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[13]  Béla Bollobás,et al.  On the Value of a Random Minimum Weight Steiner Tree , 2004, Comb..

[14]  R. Durrett.  Random Graph Dynamics: References , 2006 .

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

[16]  M. Mézard,et al.  On the solution of the random link matching problems , 1987 .

[17]  M. Mézard,et al.  The Cavity Method at Zero Temperature , 2002, cond-mat/0207121.

[18]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[19]  David Aldous,et al.  Asymptotic Fringe Distributions for General Families of Random Trees , 1991 .

[20]  Svante Linusson,et al.  A proof of Parisi’s conjecture on the random assignment problem , 2003, math/0303214.

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

[22]  Sergey N. Dorogovtsev,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW (Physics) , 2003 .

[23]  B. Prabhakar,et al.  Proofs of the Parisi and Coppersmith‐Sorkin random assignment conjectures , 2005 .

[24]  Alan M. Frieze,et al.  On the value of a random minimum spanning tree problem , 1985, Discret. Appl. Math..

[25]  Fritz Oberhettinger,et al.  Tables of Mellin Transforms , 1974 .

[26]  Svante Janson,et al.  One, Two and Three Times log n/n for Paths in a Complete Graph with Random Weights , 1999, Combinatorics, Probability and Computing.

[27]  David J. Aldous.  Percolation–like scaling exponents for minimal paths and trees in the stochastic mean field model , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  Allon G. Percus,et al.  Scaling and universality in continuous length combinatorial optimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[29]  R. Smythe,et al.  First-passage percolation on the square lattice. I , 1977, Advances in Applied Probability.

[30]  M. Mézard,et al.  A replica analysis of the travelling salesman problem , 1986 .