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Extreme value theory, PoissonDirichlet distributions, and first passage percolation on random networks
 Adv. in Appl. Probab
"... Abstract We study first passage percolation on the configuration model (CM) having powerlaw degrees with exponent τ ∈ [1, 2). To this end we equip the edges with exponential weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the ..."
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Cited by 14 (4 self)
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Abstract We study first passage percolation on the configuration model (CM) having powerlaw degrees with exponent τ ∈ [1, 2). To this end we equip the edges with exponential 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, which can be computed in terms of the PoissonDirichlet distribution. We explicitly describe these limits via the construction of an infinite limiting object describing the FPP problem in the densely connected core of the network. We consider two separate cases, namely, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, as well as 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 meanfield setting or the locally treelike 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 one can transfer information 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
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
Random Shortest Paths: NonEuclidean Instances for Metric Optimization Problems
"... Abstract. Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtai ..."
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Cited by 1 (1 self)
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Abstract. Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The length of an edge is then the length of a shortest path (with respect to the weights drawn) that connects its two endpoints. We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the kcenter problem, as well as the runningtime of the 2opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worstcase bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons. 1
DEGREE DISTRIBUTION OF SHORTEST PATH TREES AND BIAS OF NETWORK SAMPLING ALGORITHMS
, 2014
"... In this article, we explicitly derive the limiting distribution of the degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the powerlaw exponent of the degree distribution of this tree and compare it to the degree dist ..."
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In this article, we explicitly derive the limiting distribution of the degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the powerlaw exponent of the degree distribution of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic meanfield model of distance) as well as on the configuration model with edgeweights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree powerlaw exponent. We also consider random rregular graphs for large r, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean field model of distance. We use our results to explain an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [7, 8, 6] of analyzing the effect of attaching random edge lengths on the geometry of random network models.
The winner takes it all
, 2013
"... We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate λ1 (λ2) times the number of edges conne ..."
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We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate λ1 (λ2) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a powerlaw with exponent τ ∈ (2, 3), then, as the number of vertices tends to infinity, one of the infection types will almost surely occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of λ1 and λ2. The picture is similar with multiple starting points for the infections.