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85
Firstprice Path Auctions
, 2004
"... We study firstprice auction mechanisms for auctioning flow between given nodes in a graph.We assume edges are independent agents with fixed capacities and costs, and their objective is to maximize their profit. We characterize all strong fflNash equilibria of a firstprice auction for this problem ..."
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Cited by 29 (2 self)
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We study firstprice auction mechanisms for auctioning flow between given nodes in a graph.We assume edges are independent agents with fixed capacities and costs, and their objective is to maximize their profit. We characterize all strong fflNash equilibria of a firstprice auction for this problem, and show that the total payment is never significantly more than, and often less than, the well known dominant strategy VickreyClarkGroves (VCG) mechanism. We then present a randomized version of the firstprice auction, for which the equilibrium condition can be relaxed to fflNash equilibrium. We next consider a model in which the amount of demand is uncertain, but its probability distribution is known to the edges. For this model, we show that a simple ex ante firstprice auction may not have any fflNash equilibria. We then present a modified auction mechanism with 2parameter bids, and show that it has an
Social networks spread rumors in sublogarithmic time
 IN STOC
, 2011
"... With the prevalence of social networks, it has become increasingly important to understand their features and limitations. It has been observed that information spreads extremely fast in social networks. We study the performance of randomized rumor spreading protocols on graphs in the preferential a ..."
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Cited by 29 (6 self)
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With the prevalence of social networks, it has become increasingly important to understand their features and limitations. It has been observed that information spreads extremely fast in social networks. We study the performance of randomized rumor spreading protocols on graphs in the preferential attachment model. The wellknown random phone call model of Karp et al. (FOCS 2000) is a pushpull strategy where in each round, each vertex chooses a random neighbor and exchanges information with it. We prove the following. • The pushpull strategy delivers a message to all nodes within Θ(log n) rounds with high probability. The best known bound so far was O(log 2 n). • If we slightly modify the protocol so that contacts are chosen uniformly from all neighbors but the one contacted in the previous round, then this time reduces to Θ(log n / log log n), which is the diameter of the graph. This is the first time that a sublogarithmic broadcast time is proven for a natural setting. Also, this is the first time that avoiding doublecontacts reduces the runtime to a smaller order of magnitude.
Random dot product graph models for social network
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2007
"... Inspired by the recent interest in combining geometry with random graph models, we explore in this paper two generalizations of the random dot product graph model proposed by Kraetzl, Nickel and Scheinerman, and Tucker [1, 2]. In particular we consider the properties of clustering, diameter and deg ..."
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Cited by 26 (2 self)
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Inspired by the recent interest in combining geometry with random graph models, we explore in this paper two generalizations of the random dot product graph model proposed by Kraetzl, Nickel and Scheinerman, and Tucker [1, 2]. In particular we consider the properties of clustering, diameter and degree distribution with respect to these models. Additionally we explore the conductance of these models and show that in a geometric sense, the conductance is constant.
OPINION FLUCTUATIONS AND DISAGREEMENT IN SOCIAL NETWORKS
 SUBMITTED TO THE ANNALS OF APPLIED PROBABILITY
, 2010
"... We study a stochastic gossip model of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions and might represent ..."
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Cited by 26 (5 self)
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We study a stochastic gossip model of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions and might represent leaders, political parties or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, opinion dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society almost surely fail to converge, and the belief of each regular agent converges in law to a nondegenerate random variable. The model thus generates longrun disagreement and continuous opinion fluctuations. The structure of the social network and the location of stubborn agents within it shape opinion dynamics. When the society is “highly fluid”, meaning that the mixing time of the random walk on the graph describing the social network is small relative to (the inverse of) the relative size of the linkages to stubborn agents, the ergodic beliefs of most of the agents concentrate around a certain common value. We also show that under additional conditions, the ergodic beliefs distribution becomes “approximately chaotic”, meaning that the variance of the aggregate belief of the society vanishes in the large population limit while individual opinions still fluctuate significantly.
The cover time of the preferential attachment graph
 Journal of Combinatorial Theory Series B
, 2004
"... The preferential attachment graph Gm(n) is a random graph formed by adding a new vertex at each time step, with m edges which point to vertices selected at random with probability proportional to their degree. Thus at time n there are n vertices and mn edges. This process yields a graph which has be ..."
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Cited by 23 (14 self)
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The preferential attachment graph Gm(n) is a random graph formed by adding a new vertex at each time step, with m edges which point to vertices selected at random with probability proportional to their degree. Thus at time n there are n vertices and mn edges. This process yields a graph which has been proposed as a simple model of the world wide web [2]. In this paper we show that if m ≥ 2 then whp the cover time n log n. of a simple random walk on Gm(n) is asymptotic to 2m m−1 1
On the expected payment of mechanisms for task allocation
 In PODC
, 2004
"... We study a generic task allocation problem called shortest paths: Let G be a directed graph in which the edges are owned by self interested agents. Each edge has an associated cost that is privately known to its owner. Let s and t be two distinguished nodes in G. Given a distribution on the edge cos ..."
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Cited by 19 (1 self)
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We study a generic task allocation problem called shortest paths: Let G be a directed graph in which the edges are owned by self interested agents. Each edge has an associated cost that is privately known to its owner. Let s and t be two distinguished nodes in G. Given a distribution on the edge costs, the goal is to design a mechanism (protocol) which acquires a cheap st path. We first prove that the class of generalized VCG mechanisms has certain monotonicity properties. We exploit this observation to obtain, under an independence assumption, expected payments which are significantly better than the worst case bounds of [4, 7]. We then investigate whether these payments can be improved when there is a competition among paths. Surprisingly, we give evidence to the fact that typically such competition hardly helps incentive compatible mechanisms. In particular, we show this for the celebrated VCG mechanism. We then construct a novel general protocol combining the advantages of incentive compatible and nonincentive compatible mechanisms. Under reasonable assumptions on the agents we show that the overpayment of our mechanism is very small. Finally, we demonstrate that many task allocation problems can be reduced to shortest paths. 1
A randomsurfer webgraph model
 In ANALCO ’06: Proceedings of the 3rd Workshop on Analytic Algorithmics and Combinatorics
, 2006
"... In this paper we provide theoretical and experimental results on a randomsurfer model for construction of a random graph. In this model, a new node connects to the existing graph by choosing a start node uniformly at random and then performing a short random walk. We show that in certain formulatio ..."
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Cited by 19 (0 self)
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In this paper we provide theoretical and experimental results on a randomsurfer model for construction of a random graph. In this model, a new node connects to the existing graph by choosing a start node uniformly at random and then performing a short random walk. We show that in certain formulations, this results in the same distribution as the preferentialattachment randomgraph model, and in others we give a direct analysis of powerlaw distribution of degrees or “virtual degrees ” of the resulting graphs. We also present experimental results for a number of settings of parameters that we are not able to analyze mathematically. 1
Expanders via random spanning trees
 In SODA
, 2009
"... Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any boundeddegree nvertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to with ..."
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Cited by 18 (0 self)
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Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any boundeddegree nvertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(log n). For the random graph Gn,p, for p = Ω(log n/n), we give a randomized algorithm for constructing two spanning trees whose union is an expander. This is suggested by the case of the complete graph, where we prove that two random spanning trees give an expander. The construction of the splicer is elementary; each spanning tree can be produced independently using an algorithm by Aldous and Broder: A random walk in the graph with edges leading to previously unvisited vertices included in the tree. Splicers also turn out to have applications to graph cutsparsification where the goal is to approximate every cut using only a small subgraph of the original graph. For random graphs, splicers provide simple algorithms for sparsifiers of size O(n) that approximate every cut to within a factor of O(log n). 1