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Flooding in weighted sparse random graphs
, 2013
"... In this paper, we study the impact of edge weights on distances in sparse random graphs. We interpret these weights as delays and take them as independent and identically distributed exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all ..."
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Cited by 6 (2 self)
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In this paper, we study the impact of edge weights on distances in sparse random graphs. We interpret these weights as delays and take them as independent and identically distributed 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 standard 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 prefactor. These results 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.
Shortestweight paths in random regular graphs, arXiv preprint arXiv:1210.2657
, 2012
"... Abstract Consider a random regular graph with degree d and of size n. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortestweight paths between a fixed vertex and all the o ..."
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Abstract Consider a random regular graph with degree d and of size n. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortestweight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortestweight paths has about α log n edges where α is the unique solution of the equation α log
On the spectral distribution of large weighted random regular graphs
"... McKay proved the limiting spectral measures of the ensembles of dregular graphs with N vertices converge to Kesten’s measure as N →∞. Given a large dregular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiti ..."
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Cited by 1 (1 self)
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McKay proved the limiting spectral measures of the ensembles of dregular graphs with N vertices converge to Kesten’s measure as N →∞. Given a large dregular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiting spectral distribution obtained by averaging over the weighted graphs. We establish the existence of a unique ‘eigendistribution ’ (a weight distribution W such that the associated limiting spectral distribution is a rescaling ofW). Initial investigations suggested that the eigendistribution was the semicircle distribution, which by Wigner’s Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semicircular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d2)). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.
How to Optimally Allocate Your Budget of Attention in Social Networks
"... Abstract—We consider the performance of information propagation through social networks in a scenario where each user has a budget of attention, that is, a constraint on the frequency with which he pulls content from neighbors. In this context we ask the question “when users make selfish decisions o ..."
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Abstract—We consider the performance of information propagation through social networks in a scenario where each user has a budget of attention, that is, a constraint on the frequency with which he pulls content from neighbors. In this context we ask the question “when users make selfish decisions on how to allocate their limited access frequency among neighbors, does information propagate efficiently? ” For the metric of average propagation delay, we provide characterizations of the optimal social cost and the social cost under selfish user optimizations for various topologies of interest. Three situations may arise: wellconnected topologies where delay is small even under selfish optimization; treelike topologies where selfish optimization performs poorly while optimal social cost is low; and “stretched ” topologies where even optimal social cost is high. We propose a mechanism for incentivizing users to modify their selfish behaviour, and observe its efficiency in the family of treelike topologies mentioned above. I.
How to Optimally Allocate Your Budget of Attention in Social Networks
"... Abstract—We consider the performance of information propagation through social networks in a scenario where each user has a budget of attention, that is, a constraint on the frequency with which he pulls content from neighbors. In this context we ask the question “when users make selfish decisions o ..."
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Abstract—We consider the performance of information propagation through social networks in a scenario where each user has a budget of attention, that is, a constraint on the frequency with which he pulls content from neighbors. In this context we ask the question “when users make selfish decisions on how to allocate their limited access frequency among neighbors, does information propagate efficiently? ” For the metric of average propagation delay, we provide characterizations of the optimal social cost and the social cost under selfish user optimizations for various topologies of interest. Three situations may arise: wellconnected topologies where delay is small even under selfish optimization; treelike topologies where selfish optimization performs poorly while optimal social cost is low; and “stretched ” topologies where even optimal social cost is high. We propose a mechanism for incentivizing users to modify their selfish behaviour, and observe its efficiency in the family of treelike topologies mentioned above. I.
Budget of Attention in Social Networks
, 2013
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.