First-passage properties of the Erdos–Renyi random graph

Probability in the Engineering and Informational Sciences, 2001

We study first passage percolation on the random graph G p (N ) with exponentially distributed weights on the links. For the special case of the complete graph, we show that this problem can be described in terms of a continuous time Markov chain and recursive trees. The Markov chain X(t) describes the number of nodes that can be reached from the initial node in time t. The recursive trees, which are uniform trees of N nodes, describe the structure of the cluster once it contains all the nodes of the complete graph. We compute the distribution of the number hops of the shortest path between two arbitrary nodes.

Combinatorics, Probability & Computing, 2011

log n, a case that has appeared in the literature (under stronger conditions on λn) in [4, 13]. We also find lower bounds for the maximum, over all pairs of vertices, of the optimal weight and hopcount. This paper continues the investigation of FPP initiated in [4] and [5]. Compared to the setting on the configuration model studied in [5], the proofs presented here are much simpler due to a direct relation between FPP on the Erdos–Rényi random graph and thinned continuous-time branching processes.

Combinatorics Probability and Computing

In this paper we explore first passage percolation (FPP) on the Erd\H{o}s-R\'enyi random graph $G_n(p_n)$, where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when $np_n\to \lambda>1,$ we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to $\lambda/(\lambda-1)\log{n}$. Furthermore, we prove that the minimal weight centered by $\log{n}/(\lambda-1)$ converges in distribution. We also investigate the dense regime, where $np_n \to \infty$. We find that although the base graph is a {\it ultra small} (meaning that graph distances between uniformly chosen vertices are $o(\log{n})$), attaching random edge weights changes the geometry of the network completely. Indee...

The Annals of Applied Probability, 2010

We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount.

2012

Abstract: We consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform X^ 2\ log {X}-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path or hopcount.

We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in accordance with results obtained for infinite random graphs using the emergence of a giant connected component as marking the percolation transition. Our approach is general and can be applied to all graph models for which an algebraic formulation of the adjacency matrix is available.

We study the optimal distance opt in random networks in the presence of disorder implemented by assigning random weights to the links. The optimal distance between two nodes is the length of the path for which the sum of weights along the path ("cost") is a minimum. We study the case of strong disorder for which the distribution of weights is so broad that its sum along any path is dominated by the largest link weight in the path. We find that in Erdős-Rényi (ER) random graphs, opt scales as N 1/3 , where N is the number of nodes in the graph. Thus, opt increases dramatically compared to the known small world result for the minimum distance min , which scales as log N . We also find the functional form for the probability distribution P ( opt ) of optimal paths. In addition we show how the problem of strong disorder on a random graph can be mapped onto a percolation problem on a Cayley tree and using this mapping, obtain the probability distribution of the maximal weight on the optimal path.

We present a scaling Ansatz for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for (i) the e ect of the ÿnite size of the system, and (ii) the dependence of this distribution on the site occupancy probability p. We present evidence supporting the scaling Ansatz for the case of two-dimensional percolation.

2008

We study various properties of least cost paths under independent and identically distributed (iid) disorder for the complete graph and dense Erdős–Rényi random graphs in the connected phase, with iid exponential and uniform weights on edges.

Journal of Statistical Physics, 2000

We present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for (i) the effect of the finite size of the system, and (ii) the dependence of this distribution on the site occupancy probability p. We test the hypothesis for the case of two-dimensional percolation.