Network Creation Games Research Articles

The diameter of sum basic equilibria games

We study the sum basic network creation game introduced in 2010 by Alon, Demaine, Hajiaghai and Leighton. In this game, an undirected and unweighted graph G is said to be a sum basic equilibrium if and only if, for every edge uv and any vertex v′ in G, swapping edge uv with edge uv′ does not decrease the total sum of the distances from u to all the other vertices. This concept lies at the heart of the network creation games, where the central problem is to understand the structure of the resulting equilibrium graphs, and in particular, how well they globally minimize the diameter. In this sense, in 2013 Alon et al. showed an upper bound of 2O(log⁡n) on the diameter of sum basic equilibria, and they also proved that if a sum basic equilibrium graph is a tree, then it has diameter at most 2. In this paper, we prove that the upper bound of 2 also holds for bipartite graphs and even for some non-bipartite classes like block graphs and cactus graphs.

On the PoA Conjecture: Trees versus Biconnected Components

In the classical model of network creation games introduced by Fabrikant et al. [On a network creation game, in Proceedings of the Twenty-Second Annual Symposium on Principles of Distributed Computing (PODC‘03), 2003, pp. 347–351], players correspond to the nodes of a network buying links of price and to the other players with the goal of being well-connected to the resulting network. Still as an open problem, the constant PoA conjecture states that the Price of Anarchy (PoA) is constant for any . When tackling this problem distinct behaviors must be taken into the account depending on whether has either large or low value. It is known that for every ne is a tree and for with the diameter of networks that are in equilibrium when restricting to deviations that consist only in buying links (buying equilibria) is at most a constant. These results imply that the PoA is constant for the disjoint union of the two ranges, and thus the constant PoA conjecture seems to be true for most of all the possible values . In this paper we study the PoA for the remaining range of and we show the following: (i) For the PoA is constant by proving that the size of any biconnected component of an equilibrium graph is constant. (ii) For we have that , where is the maximum diameter of an equilibrium graph for the same range of . Therefore if the constant PoA conjecture was false, it would suffice to construct equilibria of nonconstant diameter. Towards this direction we find nontrivial buying equilibria of nonconstant diameter when and , exploring new intimate relationships between distance-uniform graphs and buying equilibria.

Social Distancing Network Creation

During a pandemic people have to find a trade-off between meeting others and staying safely at home. While meeting others is pleasant, it also increases the risk of infection. We consider this dilemma by introducing a game-theoretic network creation model in which selfish agents can form bilateral connections. They benefit from network neighbors, but at the same time, they want to maximize their distance to all other agents. This models the inherent conflict that social distancing rules impose on the behavior of selfish agents in a social network. Besides addressing this familiar issue, our model can be seen as the inverse to the well-studied Network Creation Game by Fabrikant et al. (in: PODC 2003, pp 347–351, 2003. https://doi.org/10.1145/872035.872088), where agents aim at being as central as possible in the created network. We look at two variants of network creation governed by social distancing. Firstly, a variant without connection restrictions, where we characterize optimal and equilibrium networks, and derive asymptotically tight bounds on the Price of Anarchy and Price of Stability. The second variant allows connection restrictions. As our main result, we prove that Swap-Maximal Routing-Cost Spanning Trees, an efficiently computable weaker variant of Maximum Routing-Cost Spanning Trees, actually resemble equilibria for a significant range of the parameter space. Moreover, we give almost tight bounds on the Price of Anarchy and Price of Stability. These results imply that under social distancing the agents’ selfishness has a strong impact on the quality of the equilibria.

Selfish Creation of Social Networks

Understanding real-world networks is a core research endeavor within the last two decades. Network Creation Games are a promising approach for this from a game-theoretic perspective. In these games, selfish agents corresponding to nodes in a network strategically decide which links to form to optimize their centrality. Many versions have been introduced and analyzed, but none of them fits to modeling the evolution of social networks. In real-world social networks connections are often established by recommendations from common acquaintances or by a chain of such recommendations. Thus establishing and maintaining a contact with a friend of a friend is easier than connecting to complete strangers. This explains the high clustering, i.e., the abundance of triangles, in real-world social networks. We propose and analyze a network creation model inspired by real-world social networks. In our model edges are formed via bilateral consent of both endpoints and the cost for establishing and maintaining an edge is proportional to the distance of the endpoints before establishing the connection. We provide results for generic cost functions which essentially only must be convex functions in the distance of the endpoints without the respective edge. For this broad class of cost functions we provide many structural properties of equilibrium networks and prove (almost) tight bounds on the diameter, the Price of Anarchy and the Price of Stability. Moreover, as a proof-of-concept we show via experiments that the created equilibrium networks of our model indeed closely mimic real-world social networks. We observe degree distributions that seem to follow a power-law, high clustering, and low diameters. This can be seen as a promising first step towards game-theoretic network creation models that predict networks featuring all core real-world properties.

Network Creation Games with Traceroute-Based Strategies

Network creation games have been extensively used as mathematical models to capture the key aspects of the decentralized process that leads to the formation of interconnected communication networks by selfish agents. In these games, each user of the network is identified by a node and selects which link to activate by strategically balancing his/her building cost with his/her usage cost (which is a function of the distances towards the other player in the network to be built). In these games, a widespread assumption is that players have a common and complete information about the evolving network topology. This is only realistic for small-scale networks as, when the network size grows, it quickly becomes impractical for the single users to gather such a global and fine-grained knowledge of the network in which they are embedded. In this work, we weaken this assumption, by only allowing players to have a partial view of the network. To this aim, we borrow three popular traceroute-based knowledge models used in network discovery: (i) distance vector, (ii) shortest-path tree view, and (iii) layered view. We settle many of the classical game theoretic questions in all of the above models. More precisely, we introduce a suitable (and unifying) equilibrium concept which we then use to study the convergence of improving and best response dynamics, the computational complexity of computing a best response, and to provide matching upper and lower bounds to the price of anarchy.

On the Tree Conjecture for the Network Creation Game

Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al. (2003) is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of α per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all α and that for α ≥ n all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edge-price α and employ it to improve on the best known bound for the latter conjecture. In particular we show that for α > 4n − 13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of α. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.

On Strong Equilibria and Improvement Dynamics in Network Creation Games

We study strong equilibria in network creation games. These form a classical and well-studied class of games where a set of players form a network by buying edges to their neighbors at a cost of a fixed parameter $\alpha$. The cost of a player is defined to be the cost of the bought edges plus the sum of distances to all the players in the resulting graph. We identify and characterize various structural properties of strong equilibria, which lead to a characterization of the set of strong equilibria for all $\alpha$ in the range $(0,2)$. For $\alpha > 2$, Andelman et al. (2009) prove that a star graph in which every leaf buys one edge to the center node is a strong equilibrium, and conjecture that in fact any star is a strong equilibrium. We resolve this conjecture in the affirmative. Additionally, we show that when $\alpha$ is large enough ($\geq 2n$) there exist non-star trees that are strong equilibria. For the strong price of anarchy, we provide precise expressions when $\alpha$ is in the range $(0,2)$, and we prove a lower bound of $3/2$ when $\alpha \geq 2$. Lastly, we aim to characterize under which conditions (coalitional) improvement dynamics may converge to a strong equilibrium. To this end, we study the (coalitional) finite improvement property and (coalitional) weak acyclicity property. We prove various conditions under which these properties do and do not hold. Some of these results also hold for the class of pure Nash equilibria.

On network formation games with heterogeneous players and basic network creation games

We consider two variants of the network formation game that aims to study the creation of large-scale networks and to capture the impact of selfish behavior, on behalf of the network administrators, on the overall network structure and performance. In particular, we study basic network creation games, where each player aims to minimize her distance to the remaining players, and we present an improved lower bound on the graph diameter of equilibria of this game. We also consider network formation games with a large number of heterogeneous players and monetary transfers, and prove tight bounds on the price of anarchy under realistic assumptions about the cost function. Finally, we argue about the setting where these heterogeneous players must be connected with additional edge-disjoint paths to mitigate the impact of failures.

Celebrity games

We introduce Celebrity games, a new model of network creation games. In this model players have weights (W being the sum of all the player's weights) and there is a critical distance β as well as a link cost α. The cost incurred by a player depends on the cost of establishing links to other players and on the sum of the weights of those players that remain farther than the critical distance. Intuitively, the aim of any player is to be relatively close (at a distance less than β) from the rest of players, mainly of those having high weights. The main features of celebrity games are that: computing the best response of a player is NP-hard if β>1 and polynomial time solvable otherwise; they always have a pure Nash equilibrium; the family of celebrity games having a connected Nash equilibrium is characterized (the so called star celebrity games) and bounds on the diameter of the resulting equilibrium graphs are given; a special case of star celebrity games shares its set of Nash equilibrium profiles with the MaxBD games with uniform bounded distance β introduced in Bilò et al. [6]. Moreover, we analyze the Price of Anarchy (PoA) and of Stability (PoS) of celebrity games and give several bounds. These are that: for non-star celebrity games PoA=PoS=max{1,W/α}; for star celebrity games PoS=1 and PoA=O(min{n/β,Wα}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when β=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for β=2.

Locality-Based Network Creation Games

Network creation games have been extensively studied, both by economists and computer scientists, due to their versatility in modeling individual-based community formation processes. These processes, in turn, are the theoretical counterpart of several economics, social, and computational applications on the Internet. In their several variants, these games model the tension of a player between the player’s two antagonistic goals: to be as close as possible to the other players and to activate a cheapest possible set of links. However, the generally adopted assumption is that players have a common and complete information about the ongoing network, which is quite unrealistic in practice. In this article, we consider a more compelling scenario in which players have only limited information about the network in whicy they are embedded. More precisely, we explore the game-theoretic and computational implications of assuming that players have a complete knowledge of the network structure only up to a given radius k , which is one of the most qualified local-knowledge models used in distributed computing. In this respect, we define a suitable equilibrium concept, and we provide a comprehensive set of upper and lower bounds to the price of anarchy for the entire range of values of k and for the two classic variants of the game, namely, those in which a player’s cost—besides the activation cost of the owned links—depends on the maximum/sum of all distances to the other nodes in the network, respectively. These bounds are assessed through an extensive set of experiments.

The Impact of Communication Patterns on Distributed Self-Adjusting Binary Search Tree

This paper introduces the problem of communication pattern adaption for a distributed self-adjusting binary search tree. We propose a simple local algorithm that is closely related to the over thirty-year-old idea of splay trees and evaluate its adaption performance in the distributed scenario if different communication patterns are provided. To do so, the process of self-adjustment is modeled similarly to a basic network creation game in which the nodes want to communicate with only a certain subset of all nodes. We show that, in general, the game (i.e., the process of local adjustments) does not converge, and that convergence is related to certain structures of the communication interests, which we call conflicts. We classify conflicts and show that for two communication scenarios in which convergence is guaranteed, the self-adjusting tree performs well. Furthermore, we investigate the different classes of conflicts separately and show that, for a certain class of conflicts, the performance of the tree network is asymptotically as good as the performance for converging instances. However, for the other conflict classes, a distributed self-adjusting binary search tree adapts poorly.

Tree Nash Equilibria in the Network Creation Game

In the network creation game with n vertices, every vertex (a player) buys a set of adjacent edges, each at a fixed amount α > 0. It has been conjectured that for α ≥ n, every Nash equilibrium is a tree, and has been confirmed for every α ≥ 273·n. We improve upon this bound and show that this is true for every α ≥ 65·n. To show this, we provide new and improved results on the local structure of Nash equilibria. Technically, we show that if there is a cycle in a Nash equilibrium, then α < 65·n. Proving this, we only consider relatively simple strategy changes of the players involved in the cycle. We further show that this simple approach cannot be used to show the desired upper bound α < n (for which a cycle may exist), but conjecture that a slightly worse bound α < 1.3 ·n can be achieved with this approach. Towards this conjecture, we show that if a Nash equilibrium has a cycle of length at most 10, then indeed α < 1.3·n. We further provide experimental evidence suggesting that when the girth of a Nash equilibrium is increasing, the upper bound on α obtained by the simple strategy changes is not increasing. To the end, we investigate the approach for a coalitional variant of Nash equilibrium, where coalitions of two players cannot collectively improve, and show that if α ≥ 41·n, then every such Nash equilibrium is a tree.

A Bounded Budget Network Creation Game

We introduce a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In this model, each link has a unit price, and each agent tries to minimize its cost, which is either its eccentricity or its total distance to other players in the underlying (undirected) graph of the created network. Two versions of the game are studied: In the MAX version, the cost incurred to a vertex is the maximum distance between the vertex and other vertices, and, in the SUM version, the cost incurred to a vertex is the sum of distances between the vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. We take the social cost of the created network to be its diameter, and next we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. When the sum of players’ budgets is n − 1, the equilibrium graphs are always trees, and we prove that their maximum diameter is Θ( n ) and Θ(log n ) in MAX and SUM versions, respectively. When each vertex has a unit budget (i.e., can establish a link to just one vertex), the diameter of any equilibrium graph in either version is Θ(1). We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Ω(√log n ). This interesting (and perhaps counterintuitive) result shows that increasing the budgets may increase the diameter of equilibrium graphs and hence deteriorate the network structure. Then we prove that every equilibrium graph in the SUM version has diameter 2 O (√log n ) . Finally, we show that if the budget of each player is at least k , then every equilibrium graph in the SUM version is k -connected or has a diameter smaller than 4.

On the structure of equilibria in basic network formation

We study network connection games where the nodes of a network perform edge swaps in order to improve their communication costs. For the model proposed by [2], in which the selfish cost of a node is the sum of all shortest path distances to the other nodes, we use the probabilistic method to provide a new, structural property of equilibrium graphs. We show how to use this property in order to prove upper bounds on the diameter of equilibrium graphs in terms of the size of the largest k-vicinity (defined as the set of vertices within distance k from a vertex), for any k≥1 and in terms of the number of edges, thus settling positively a conjecture of [2] in the cases of graphs of large k-vicinity size (including graphs of large maximum degree) and of graphs which are dense enough.Next, we present a new swap-based network creation game, in which selfish costs depend on the immediate neighborhood of each node; in particular, the profit of a node is defined as the sum of the degrees of its neighbors. We prove that, in contrast to the previous model, this network creation game admits an exact potential, and also that any equilibrium graph contains an n-vertex star as a spanning subgraph. The existence of the potential function is exploited in order to show that an equilibrium can be reached in expected polynomial time even in the case where nodes can only acquire limited knowledge concerning non-neighboring nodes.

The max-distance network creation game on general host graphs

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 her 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 very simple host graph topologies. More precisely, we first exhibit a lower bound of Ω(n/(1+α)) for any α=o(n). Notice that this implies a counter-intuitive lower bound of Ω(n) even for α=0, i.e., when edges can be activated for free. Then, we show that when the host graph is restricted to be either k-regular (for any constant k≥3), or a 2-dimensional 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 that the PoA is at most 2. 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.

On Nash Equilibria for a Network Creation Game

We study a basic network creation game proposed by Fabrikant et al. [2003]. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players. Fabrikant et al. conjectured that there exists a constant A such that, for any α &gt; A , all nontransient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this article, we disprove the tree conjecture. More precisely, we show that for any positive integer n 0 , there exists a graph built by n ≥ n 0 players which contains cycles and forms a nontransient Nash equilibrium, for any α with 1 &lt; α ≤ √ n /2. Our construction makes use of some interesting results on finite affine planes. On the other hand, we show that, for α ≥ 12 n ⌈log n ⌉, every Nash equilibrium forms a tree. Without relying on the tree conjecture, Fabrikant et al. proved an upper bound on the price of anarchy of O (√ α ), where α ∈ [2, n 2 . We improve this bound. Specifically, we derive a constant upper bound for α ∈ O (√ n ) and for α ≥ 12 n ⌈log n ⌉. For the intermediate values, we derive an improved bound of O (1 + (min{ α 2 / n , n 2 / α }) 1/3 ). Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing.

The Price of Anarchy in Network Creation Games Is (Mostly) Constant

We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1,2,…,n}, each identified with a vertex of a graph (network), where the strategy of player i, i=1,…,n, is to build some edges adjacent to i. The cost of building an edge is α>0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SumGame variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MaxGame variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α>273⋅n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that for α>129 all equilibria in MaxGame are trees and the price of anarchy is constant. For SumGame this answers (almost completely) one of the fundamental open problems in the field—is price of anarchy of the network creation game constant for all values of α?—in an affirmative way, up to a tiny range of α.

Basic Network Creation Games

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 local-average-distance version, we prove an upper bound of $2^{O(\sqrt{\lg n})}$, a lower bound of 3, and a tight bound of exactly 2 for trees, and give evidence of a general polylogarithmic upper bound. For the local-diameter version, we prove a lower bound of $\Omega(\sqrt{n})$ and a tight upper bound of 3 for trees. The same bounds apply, up to constant factors, to the price of anarchy. Our network creation games are closely related to the previously studied unilateral network creation game. The main difference is that our model has no parameter $\alpha$ for the link creation cost, so our results effectively apply for all values of $\alpha$ without additional effort; furthermore, equilibrium can be checked in polynomial time in our model, unlike in previous models. Our perspective enables simpler proofs that get at the heart of network creation games.

The price of anarchy in network creation games

We study Nash equilibria in the setting of network creation games introduced recently by Fabrikant, Luthra, Maneva, Papadimitriou, and Shenker. In this game we have a set of selfish node players, each creating some incident links, and the goal is to minimize α times the cost of the created links plus sum of the distances to all other players. Fabrikant et al. proved an upper bound O (√α) on the price of anarchy: the relative cost of the lack of coordination. Albers, Eilts, Even-Dar, Mansour, and Roditty show that the price of anarchy is constant for α = O (√ n ) and for α ≥ 12 n ⌈ lg n ⌉, and that the price of anarchy is 15(1+(min{α 2 / n , n 2 /α}) 1/3 ) for any α. The latter bound shows the first sublinear worst-case bound, O ( n 1/3 ), for all α. But no better bound is known for α between ω(√ n ) and o ( n lg n ). Yet α ≈ n is perhaps the most interesting range, for it corresponds to considering the average distance (instead of the sum of distances) to other nodes to be roughly on par with link creation (effectively dividing α by n ). In this article, we prove the first o ( n ε ) upper bound for general α, namely 2 O (√ lg n ) . We also prove a constant upper bound for α = O ( n 1-ε ) for any fixed ε &gt; 0, substantially reducing the range of α for which constant bounds have not been obtained. Along the way, we also improve the constant upper bound by Albers et al. (with the lead constant of 15 ) to 6 for α &lt; ( n /2) 1/2 and to 4 for α &lt; ( n /2) 1/3 . Next we consider the bilateral network variant of Corbo and Parkes, in which links can be created only with the consent of both endpoints and the link price is shared equally by the two. Corbo and Parkes show an upper bound of O (√α) and a lower bound of Ω(lgα) for α ≤ n . In this article, we show that in fact the upper bound O (√α) is tight for α ≤ n , by proving a matching lower bound of Ω(√α). For α &gt; n , we prove that the price of anarchy is Θ( n /√ α). Finally we introduce a variant of both network creation games, in which each player desires to minimize α times the cost of its created links plus the maximum distance (instead of the sum of distances) to the other players. This variant of the problem is naturally motivated by considering the worst case instead of the average case. Interestingly, for the original (unilateral) game, we show that the price of anarchy is at most 2 for α ≥ n , O (min {4 √lg n , ( n /α) 1/3 }) for 2√ lg n ≤ α ≤ n , and O ( n 2/α ) for α &lt; 2√ lg n . For the bilateral game, we prove matching upper and lower bounds of Θ( n /α + 1) for α ≤ n , and an upper bound of 2 for α &gt; n .

Constant Price of Anarchy in Network-Creation Games via Public-Service Advertising

Network creation games have been studied in many different settings recently. These games are motivated by social networks in which selfish agents want to construct a connection graph among themselves. Each node wants to minimize its average or maximum distance to the others, without paying much to construct the network. Many generalizations have been considered, including non-uniform interests between nodes, general graphs of allowable edges, bounded budget agents, etc. In all of these settings, there is no known constant bound on the price of anarchy. In fact, in many cases, the price of anarchy can be very large, namely, a constant power of the number of agents. This means that we have no control on the behavior of network when agents act selfishly. On the other hand, the price of stability in all these models is constant, which means that there is chance that agents act selfishly and we end up with a reasonable social cost.