We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with a type 1 and type 2 infection, respectively, and the infection then spreads via nearest neighbors in the graph. The time it takes for the type 1 (resp. 2) infection to traverse an edge e is given by a random variable $X_1(e)$ (resp. $X_2(e)$) and, if the vertex at the other end of the edge is still uninfected, it then becomes type 1 (resp. 2) infected and immune to the other type. Assuming that the degrees follow a power-law distribution with exponent $\tau \in (1,2)$, we show that with high probability as the number of vertices tends to infinity, one of the infection types occupies all vertices except for the starting point of the other type. Moreover, both infections have a positive probability of winning regardless of the passage-time distribution. The result is also shown to hold for the erased configuration model, where self-loops are erased and multiple edges are merged, and when the degrees are conditioned to be smaller than $n^\alpha$ for some $\alpha\gt 0$.

In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.

We consider three different schemes for signal routeing on a tree. The vertices of the tree represent transceivers that can transmit and receive signals, and are equipped with independent and identically distributed weights representing the strength of the transceivers. The edges of the tree are also equipped with independent and identically distributed weights, representing the costs for passing the edges. For each one of our schemes, we derive sharp conditions on the distributions of the vertex weights and the edge weights that determine when the root can transmit a signal over arbitrarily large distances.

In this paper we investigate first passage percolation on an inhomogeneous random graph model introduced by Bollobás et al. (2007). Each vertex in the graph has a type from a type space, and edge probabilities are independent, but depend on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal-weight path, properly normalized, follows a central limit theorem. We handle the cases where the average number of neighbors λ̃n of a vertex tends to a finite λ̃ in full generality and consider λ̃ = ∞ under mild assumptions. This paper is a generalization of the paper of Bhamidi et al. (2011), where first passage percolation is explored on the Erdős-Rényi graphs.

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

We establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour and Reinert (2013) from bounded degree graphs to general sparse random graphs with degrees having finite second moments as n → ∞, with an appropriate X2log+X condition. We also study the epidemic trail between the source and typical vertices in the graph.

We pursue the study of a random coloring first passage percolation model introduced byFontes and Newman. We prove that the asymptotic shape of this first passage percolationmodel continuously depends on the law of the coloring. The proof uses several couplings,particularly with greedy lattice animals.

We consider the standard first passage percolation model in ℤd for d ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to n and whose height is h(n) for a certain height function h. We denote this maximal flow by τn (respectively φn). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than ν(v) + ε for some positive ε, where ν(v) is the almost sure limit of the rescaled variable τn when n goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable τn depends on the tail of the distribution of the capacities of the edges: it can decay exponentially fast with nd−1, or with nd−1min(n,h(n)), or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable φn decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that ν(v) is not in general the almost sure limit of the rescaled maximal flow φn, but it is the case at least when the height h(n) of the cylinder is negligible compared to n.

We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.

Equip the edges of the lattice ℤ2 with i.i.d. random capacities. A law oflarge numbers is known for the maximal flow crossing a rectangle in ℝ2 when theside lengths of the rectangle go to infinity. We prove that the lower large deviations areof surface order, and we prove the corresponding large deviation principle from below.This extends and improves previous large deviations results of Grimmett and Kesten [9] obtained for boxes of particular orientation.

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times ln between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of ln / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

We analyze a class of signal-to-interference-and-noise-ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. In this paper we study optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both ‘positive’ and ‘negative’ results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to ∞. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that, when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite.

We study first passage percolation (FPP) on the configuration model (CM) having power-law degrees with exponent τ ∈ [1, 2) and exponential edge weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal-weight path, both of which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via construction of infinite limiting objects describing the FPP problem in the densely connected core of the network. We consider two separate cases, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, and the erased CM, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly, the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the mean-field setting or the locally tree-like setting, which occurs as τ > 2, and for which the hopcount between typical vertices scales as log n. In our setting the hopcount is tight and has an explicit limiting distribution, showing that information can be transferred remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models; see Aldous and Bhamidi (2010), Bhamidi (2008), and Bhamidi, van der Hofstad and Hooghiemstra (2009).

A Euclidean first passage percolation model describing the competing growth between k different types of infection is considered. We focus on the long-time behavior of this multitype growth process and we derive multitype shape results related to its morphology.

A continuum growth model is introduced. The state at time t, St, is a subset of ℝd and consists of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their centre points. An outburst occurs somewhere in St after an exponentially distributed time with expected value |St|-1 and the location of the outburst is uniformly distributed over St. The main result is that, if the distribution of the radii of the outburst balls has bounded support, then St grows linearly and St/t has a nonrandom shape as t → ∞. Due to rotational invariance the asymptotic shape must be a Euclidean ball.

An interacting particle system modelling competing growth on the ℤ2 lattice is defined as follows. Each x ∈ ℤ2 is in one of the states {0,1,2}. 1's and 2's remain in their states for ever, while a 0 flips to a 1 (a 2) at a rate equal to the number of its neighbours which are in state 1 (2). This is a generalization of the well-known Richardson model. 1's and 2's may be thought of as two types of infection, and 0's as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event that both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.