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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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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
Diameter and Rumour Spreading in RealWorld Network Models
, 2015
"... The socalled ‘smallworld phenomenon’, observed in many realworld networks, is that there is a short path between any two nodes of a network, whose length is much smaller that the network’s size, typically growing as a logarithmic function. Several mathematical models have been defined for socia ..."
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The socalled ‘smallworld phenomenon’, observed in many realworld networks, is that there is a short path between any two nodes of a network, whose length is much smaller that the network’s size, typically growing as a logarithmic function. Several mathematical models have been defined for social networks, the WWW, etc., and this phenomenon translates to proving that such models have a small diameter. In the first part of this thesis, we rigorously analyze the diameters of several random graph classes that are introduced specifically to model complex networks, verifying whether this phenomenon occurs in them. In Chapter 3 we develop a versatile technique for proving upper bounds for diameters of evolving random graph models, which is based on defining a coupling between these models and variants of random recursive trees. Using this technique we prove, for the first time,
Not Always Sparse: Flooding Time in Partially Connected Mobile Ad Hoc Networks
, 2013
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Submitted to the Annals of Applied Probability VIRAL PROCESSES BY RANDOM WALKS ON RANDOM REGULAR GRAPHS
"... We study the SIR epidemic model with infections carried by k particles making independent random walks on a random regular graph. Here we assume k ≤ n, where n is the number of vertices in the random graph and is some sufficiently small constant. We give an edgeweighted graph reduction of the dyna ..."
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We study the SIR epidemic model with infections carried by k particles making independent random walks on a random regular graph. Here we assume k ≤ n, where n is the number of vertices in the random graph and is some sufficiently small constant. We give an edgeweighted graph reduction of the dynamics of the process that allows us to apply standard results of ErdősRényi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, O(ln k) particles are infected. In the supercritical regime, for a constant β ∈ (0, 1) determined by the parameters of the model, βk get infected with probability β, and O(ln k) get infected with probability (1−β). Finally, there is a regime in which all k particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of