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18
Basic Network Creation Games
, 2010
"... We study a natural network creation game, in which each node locally tries to minimize its local diameter or its local average distance to other nodes, by swapping one incident edge at a time. The central question is what structure the resulting equilibrium graphs have, in particular, how well they ..."
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Cited by 22 (1 self)
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We study a natural network creation game, in which each node locally tries to minimize its local diameter or its local average distance to other nodes, by swapping one incident edge at a time. The central question is what structure the resulting equilibrium graphs have, in particular, how well they globally minimize diameter. For the localaveragedistance version, we prove an upper bound of 2 O( √ lg n), a lower bound of 3, a tight bound of exactly 2 for trees, and give evidence of a general polylogarithmic upper bound. For the localdiameter version, we prove a lower bound of Ω ( √ n), and a tight upper bound of 3 for trees. All of our upper bounds apply equally well to previously extensively studied network creation games, both in terms of the diameter metric described above and the previously studied price of anarchy (which are related by constant factors). In surprising contrast, our model has no parameter α for the link creation cost, so our results automatically apply for all values of α without additional effort; furthermore, equilibrium can be checked in polynomial time in our model, unlike previous models. Our perspective enables simpler and more general proofs that get at the heart of network creation games.
RTG: A Recursive Realistic Graph Generator using Random Typing
 DATA MINING AND KNOWLEDGE DISCOVERY
, 2009
"... We propose a new, recursive model to generate realistic graphs, evolving over time. Our model has the following properties: it is (a) flexible, capable of generating the cross product of weighted/unweighted, directed/undirected, uni/bipartite graphs; (b) realistic, giving graphs that obey eleven sta ..."
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Cited by 14 (4 self)
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We propose a new, recursive model to generate realistic graphs, evolving over time. Our model has the following properties: it is (a) flexible, capable of generating the cross product of weighted/unweighted, directed/undirected, uni/bipartite graphs; (b) realistic, giving graphs that obey eleven static and dynamic laws that real graphs follow (we formally prove that for several of the (power) laws and we estimate their exponents as a function of the model parameters); (c) parsimonious, requiring only four parameters. (d) fast, being linear on the number of edges; (e) simple, intuitively leading to the generation of macroscopic patterns. We empirically show that our model mimics two realworld graphs very well: Blognet (unipartite, undirected, unweighted) with 27K nodes and 125K edges; and CommitteetoCandidate campaign donations (bipartite, directed, weighted) with 23K nodes and 880K edges. We also show how to handle time so that edge/weight additions are bursty and selfsimilar. 1
THE PRICE OF ANARCHY IN COOPERATIVE NETWORK CREATION GAMES
, 2009
"... We analyze the structure of equilibria and the price of anarchy in the family of network creation games considered extensively in the past few years, which attempt to unify the network design and network routing problems by modeling both creation and usage costs. In general, the games are played o ..."
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We analyze the structure of equilibria and the price of anarchy in the family of network creation games considered extensively in the past few years, which attempt to unify the network design and network routing problems by modeling both creation and usage costs. In general, the games are played on a host graph, where each node is a selfish independent agent (player) and each edge has a fixed link creation cost α. Together the agents create a network (a subgraph of the host graph) while selfishly minimizing the link creation costs plus the sum of the distances to all other players (usage cost). In this paper, we pursue two important facets of the network creation game. First, we study extensively a natural version of the game, called the cooperative model, where nodes can collaborate and share the cost of creating any edge in the host graph. We prove the first nontrivial bounds in this model, establishing that the price of anarchy is polylogarithmic in n for all values of α in complete host graphs. This bound is the first result of this type for any version of the network creation game; most previous general upper bounds are polynomial in n. Interestingly, we also show that equilibrium graphs have polylogarithmic diameter for the most natural range of α (at most n polylg n). Second,
Reducibility among fractional stability problems
 in Proc. IEEE FOCS, 2009
"... Abstract — In a landmark paper [32], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [11], [16], [5], [6], [7], [8] has shown that finding Nash equilibria is complete for PPAD, a particular ..."
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Cited by 10 (1 self)
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Abstract — In a landmark paper [32], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [11], [16], [5], [6], [7], [8] has shown that finding Nash equilibria is complete for PPAD, a particularly notable subclass of TFNP. A major goal of this work is to expand the universe of known PPADcomplete problems. We resolve the computational complexity of a number of outstanding open problems with practical applications. Here is the list of problems we show to be PPADcomplete, along with the domains of practical significance: Fractional Stable
Bounded Budget Betweenness Centrality Game for Strategic Network Formations
"... Abstract. In this paper, we introduce the bounded budget betweenness centrality game, a strategic network formation game in which nodes build connections subject to a budget constraint in order to maximize their betweenness centrality, a metric introduced in the social network analysis to measure th ..."
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Cited by 7 (0 self)
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Abstract. In this paper, we introduce the bounded budget betweenness centrality game, a strategic network formation game in which nodes build connections subject to a budget constraint in order to maximize their betweenness centrality, a metric introduced in the social network analysis to measure the information flow through a node. To reflect real world scenarios where short paths are more important in information exchange, we generalize the betweenness definition to only consider shortest paths of length at most ℓ. We present both complexity and constructive existence results about Nash equilibria of the game. For the nonuniform version of the game where node budgets, link costs, and pairwise communication weights may vary, we show that Nash equilibria may not exist and it is NPhard to decide whether Nash equilibria exist in a game instance. For the uniform version of the game where link costs and pairwise communication weights are one and each node can build k links, we construct two families of Nash equilibria based on shift graphs, and study the properties of Nash equilibria. Moreover, we study the complexity of computing best responses and show that the task is polynomial for uniform 2B 3 C games and NPhard for other games.
Minimizing the diameter of a network using shortcut edges
 In Algorithm Theory  SWAT 2010
, 2010
"... Abstract. We study the problem of minimizing the diameter of a graph by adding k shortcut edges, for speeding up communication in an existing network design. We develop constantfactor approximation algorithms for different variations of this problem. We also show how to improve the approximation ra ..."
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Cited by 6 (0 self)
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Abstract. We study the problem of minimizing the diameter of a graph by adding k shortcut edges, for speeding up communication in an existing network design. We develop constantfactor approximation algorithms for different variations of this problem. We also show how to improve the approximation ratios using resource augmentation to allow more than k shortcut edges. We observe a close relation between the singlesource version of the problem, where we want to minimize the largest distance from a given source vertex, and the wellknown kmedian problem. First we show that our constantfactor approximation algorithms for the general case solve the singlesource problem within a constant factor. Then, using a linearprogramming formulation for the singlesource version, we find a (1 + ε)approximation using O(k log n) shortcut edges. To show the tightness of our result, we prove that any ( 3 2 − ε)approximation for the singlesource version must use Ω(k log n) shortcut edges assuming P = NP.
Contribution Games in Networks
, 2010
"... We consider network contribution games, where each agent in a network has a budget of effort that he can contribute to different collaborative projects or relationships. Depending on the contribution of the involved agents a relationship will flourish or drown, and to measure the success we use a re ..."
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Cited by 6 (4 self)
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We consider network contribution games, where each agent in a network has a budget of effort that he can contribute to different collaborative projects or relationships. Depending on the contribution of the involved agents a relationship will flourish or drown, and to measure the success we use a reward function for each relationship. Every agent is trying to maximize the reward from all relationships that it is involved in. We consider pairwise equilibriaofthisgame, andcharacterizetheexistence, computationalcomplexity, andquality of equilibrium based on the types of reward functions involved. When all reward functions are concave, we prove that the price of anarchy is at most 2. For convex functions the same only holds under some special but very natural conditions. Another special case extensively treated are minimum effort games, where the reward of a relationship depends only on the minimum effort of any of the participants. In these games, we can show existence of pairwise equilibrium and a price of anarchy of 2 for concave functions and special classes of games with convex functions. Finally, we show tight bounds for approximate equilibria and convergence of dynamics in these games.
Contribution games in social networks
 In Proc. 18th European Symposium on Algorithms (ESA
, 2010
"... We consider network contribution games, where each agent in a social network has a budget of effort that he can contribute to different collaborative projects or relationships. Depending on the contribution of the involved agents a relationship will flourish or drown, and to measure the success we u ..."
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Cited by 5 (1 self)
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We consider network contribution games, where each agent in a social network has a budget of effort that he can contribute to different collaborative projects or relationships. Depending on the contribution of the involved agents a relationship will flourish or drown, and to measure the success we use a reward function for each relationship. Every agent is trying to maximize the reward from all relationships that it is involved in. We consider pairwise equilibria of this game, and characterize the existence, computational complexity, and quality of equilibrium based on the types of reward functions involved. For example, when all reward functions are concave, we prove that the price of anarchy is at most 2. For convex functions the same only holds under some special but very natural conditions. A special focus of the paper are minimum effort games, where the reward of a relationship depends onlyonthe minimum effort ofanyofthe participants. Finally, we showtight bounds for approximate equilibria and convergence of dynamics in these games. 1
Hypergraphbased TaskBundle Scheduling Towards Efficiency and Fairness in Heterogeneous Distributed Systems
"... Abstract—This paper investigates scheduling loosely coupled taskbundles in highly heterogeneous distributed systems. Two allocation quality metrics are used in payperservice distributed applications: efficiency in terms of social welfare, and fairness in terms of envyfreeness. The first contribu ..."
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Abstract—This paper investigates scheduling loosely coupled taskbundles in highly heterogeneous distributed systems. Two allocation quality metrics are used in payperservice distributed applications: efficiency in terms of social welfare, and fairness in terms of envyfreeness. The first contribution of this work is that we build a unified hypergraph scheduling model under which efficiency and fairness are compatible with each other. Second, in the scenario of budgetunawareness, we formulate a strategic algorithm design for distributed negotiations among autonomous selfinterested computing peers and prove its convergence to complete local efficiency and envyfreeness. Third, we add budget limitation to the allocation problem and propose a class of hillclimbing heuristics in favor of different performance metrics. Finally we conduct extensive simulations to validate the performance of all the proposed algorithms. The results show that the decentralized hypergraph scheduling method is scalable, and yields desired allocation performance in various scenarios. Keywordstask scheduling; distributed systems; envyfree allocation; hypergraph I.
The maxdistance network creation game on general host graphs
 Internet and Network Economics, Lecture
"... Abstract. In this paper we study a generalization of the classic network creation game in the scenario in which the n players sit on a given arbitrary host graph, which constrains the set of edges a player can activate at a cost of α ≥ 0 each. This finds its motivations in the physical limitations ..."
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Abstract. In this paper we study a generalization of the classic network creation game in the scenario in which the n players sit on a given arbitrary host graph, which constrains the set of edges a player can activate at a cost of α ≥ 0 each. This finds its motivations in the physical limitations one can have in constructing links in practice, and it has been studied in the past only when the routing cost component of a player is given by the sum of distances to all the other nodes. Here, we focus on another popular routing cost, namely that which takes into account for each player its maximum distance to any other player. For this version of the game, we first analyze some of its computational and dynamic aspects, and then we address the problem of understanding the structure of associated pure Nash equilibria. In this respect, we show that the corresponding price of anarchy (PoA) is fairly bad, even for several basic classes of host graphs. More precisely, we first exhibit a lower bound of Ω( n/(1 + α)) for any α = o(n). Notice that this implies a counterintuitive lower bound of Ω( n) for very small values of α (i.e., edges can be activated almost for free). Then, we show that when the host graph is restricted to be either kregular (for any constant k ≥ 3), or a 2dimensional grid, the PoA is still Ω(1 + min{α, nα}), which is proven to be tight for α = Ω( n). On the positive side, if α ≥ n, we show the PoA is O(1). Finally, in the case in which the host graph is very sparse (i.e., E(H)  = n − 1 + k, with k = O(1)), we prove that the PoA is O(1), for any α.