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11
First passage percolation on the ErdősRényi random graph
, 2010
"... In this paper we explore first passage percolation (FPP) on the ErdősRényi random graph Gn(pn), where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when npn → λ> 1, we find refined asymptotics both for the minimal weight of the path between un ..."
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Cited by 17 (4 self)
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In this paper we explore first passage percolation (FPP) on the ErdősRényi random graph Gn(pn), where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when npn → λ> 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 λ/(λ − 1) log n. Furthermore, we prove that the minimal weight centered by log n/(λ − 1) converges in distribution. We also investigate the dense regime, where npn → ∞. We find that although the base graph is a 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. Indeed, the hopcount Hn satisfies the universality property that whatever be the value of pn, Hn / log n → 1 in probability and, more precisely, (Hn−βn log n) / √ log n, where βn = λn/(λn− 1), has a limiting standard normal distribution. The constant βn can be replaced by 1 precisely when λn ≫ √ log n, a case that has appeared in the literature (under stronger conditions on λn) in [2, 12]. We also find bounds for the maximal weight and maximal hopcount between vertices in the graph. This paper continues the investigation of FPP initiated in [2] and [3]. Compared to the setting on the configuration model studied in [3], the proofs presented here are much simpler due to a direct relation between FPP on the ErdősRényi random graph and thinned continuoustime branching processes.
Diameters in preferential attachment models
, 2009
"... In this paper, we investigate the diameter in preferential attachment (PA) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PAmo ..."
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Cited by 16 (1 self)
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In this paper, we investigate the diameter in preferential attachment (PA) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PAmodels. There is a substantial amount of literature proving that, quite generally, PAgraphs possess powerlaw degree sequences with a powerlaw exponent τ> 2. We prove that the diameter of the PAmodel is bounded above by a constant times log t, where t is the size of the graph. When the powerlaw exponent τ exceeds 3, then we prove that log t is the right order for the diameter, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for τ> 3, distances are of the order log t. For τ ∈ (2, 3), we improve the upper bound to a constant times log log t, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order log log t. These bounds partially prove predictions by physicists that the typical distance in PAgraphs are similar to the ones in other scalefree random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order log log t when τ ∈ (2, 3), and of order log t when τ > 3.
Random graph asymptotics on highdimensional tori
 Comm. Math. Phys
"... Abstract We investigate the scaling of the largest critical percolation cluster on a large ddimensional torus, for nearestneighbor percolation in high dimensions, or when d > 6 for sufficient spreadout percolation. We use a relatively simple coupling argument to show that this largest critica ..."
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Cited by 8 (5 self)
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Abstract We investigate the scaling of the largest critical percolation cluster on a large ddimensional torus, for nearestneighbor percolation in high dimensions, or when d > 6 for sufficient spreadout percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V 2/3 and below by a small constant times V 2/3 (log V ) −4/3 , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation twopoint function on Z d under which the lower bound can be improved to small constant times V 2/3 , i.e. we prove random graph asymptotics for the largest critical cluster on the highdimensional torus. This establishes a conjecture by Our method is crucially based on the results in
Flooding in Weighted Random Graphs
"... In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen n ..."
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Cited by 7 (1 self)
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In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node, and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of n, when the size of the graph n tends to infinity. We also derive the exact value for the prefactors. These allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average. 1
Universality for distances in powerlaw random graphs
, 2008
"... We survey the recent work on phase transition and distances in various random graph models with general degree sequences. We focus on inhomogeneous random graphs, the configuration model and affine preferential attachment models, and pay special attention to the setting where these random graphs hav ..."
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Cited by 4 (0 self)
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We survey the recent work on phase transition and distances in various random graph models with general degree sequences. We focus on inhomogeneous random graphs, the configuration model and affine preferential attachment models, and pay special attention to the setting where these random graphs have a powerlaw degree sequence. This means that the proportion of vertices with degree k in large graphs is approximately proportional to k −τ, for some τ> 1. Since many real networks have been empirically shown to have powerlaw degree sequences, these random graphs can be seen as more realistic models for real complex networks. It is often suggested that the behavior of random graphs should have a large amount of universality, meaning, in this case, that random graphs with similar degree sequences share similar behavior. We survey the available results on graph distances in powerlaw random graphs that are consistent with this prediction.
LAW OF LARGE NUMBERS FOR THE SIR EPIDEMIC ON A RANDOM GRAPH WITH GIVEN DEGREES
, 2013
"... We study the susceptibleinfectiverecovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective ..."
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Cited by 3 (3 self)
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We study the susceptibleinfectiverecovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak may occur. We prove that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time. In contrast to earlier results for this model, our results only require basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex.
A phase transition for the diameter of the configuration model
 INTERNET MATH
, 2008
"... In this paper, we study the configuration model (CM) with i.i.d. degrees. We establish a phase transition for the diameter when the powerlaw exponent τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ> 2 and when vertices with degree 1 or 2 are present with positive probability, t ..."
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Cited by 3 (2 self)
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In this paper, we study the configuration model (CM) with i.i.d. degrees. We establish a phase transition for the diameter when the powerlaw exponent τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ> 2 and when vertices with degree 1 or 2 are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for τ ∈ (2, 3), the diameter of the graph is, with high probability, bounded from above by a constant times the log log of the size of the graph.
Sparse Random Graphs Methods, Structure, and Heuristics
, 2007
"... This dissertation is an algorithmic study of sparse random graphs which are parametrized by the distribution of vertex degrees. Our contributions include: a formula for the diameter of various sparse random graphs, including the ErdősRényi random graphs Gn,m and Gn,p and certain powerlaw graphs; a ..."
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Cited by 1 (1 self)
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This dissertation is an algorithmic study of sparse random graphs which are parametrized by the distribution of vertex degrees. Our contributions include: a formula for the diameter of various sparse random graphs, including the ErdősRényi random graphs Gn,m and Gn,p and certain powerlaw graphs; a heuristic for the korientability problem, which performs optimally for certain classes of random graphs, again including the ErdősRényi models Gn,m and Gn,p; an improved lower bound for the independence ratio of random 3regular graphs. In addition to these structural results, we also develop a technique for reasoning abstractly about random graphs by representing discrete structures topologically.