On bipartite sum basic equilibria

A connected and undirected graph G of size n≥1 is said to be a sum basic equilibrium iff for every edge uv from G and any node v′ from G, when performing the swap of the edge uv for the edge uv′ the sum of the distances from u to all the other nodes is not strictly reduced. This concept comes from the so called Network Creation Games, a wide subject inside Algorithmic Game Theory that tries to better understand how Internet-like networks behave. It has been shown that the diameter of sum basic equilibria is 2O(logn√) in general and at most 2 for trees. In this paper we see that the upper bound of 2 not only holds for trees but for bipartite graphs, too. Specifically, we show that the only bipartite sum basic equilibrium networks are the complete bipartite graphs Kr,s with r,s≥1 .This work has been partially supported by funds from the Spanish Ministry for Economy and Competitiveness (MINECO) and the European Union (FEDER funds) under grant GRAMM (TIN2017-86727-C2-1-R) and from the Catalan Agency for Management of University and Research Grants (AGAUR, Generalitat de Catalunya) under project ALBCOM 2017-SGR-786.Peer ReviewedPostprint (author's final draft

Distance-uniform graphs with large diameter

An ϵ-distance-uniform graph is one with a critical distance d such that from every vertex, all but at most an ϵ-fraction of the remaining vertices are at distance exactly d. Motivated by the theory of network creation games, Alon, Demaine, Hajiaghayi, and Leighton made the follow- ing conjecture of independent interest: that every ϵ-distance-uniform graph (and, in fact, a broader class of ϵ-distance-almost-uniform graphs) has critical distance at most logarithmic in the number of vertices n. We disprove this conjecture and characterize the asymptotics of this extremal prob- lem. Speci-cally, for 1/n ≤ ϵ ≤ 1 /log n , we construct ϵ-distance-uniform graphs with critical distance 2ω(log n/log ϵ-1). We also prove an upper bound on the critical distance of the form 2O(log n/log ϵ-1) for all ϵ and n. Our lower bound construction introduces a novel method inspired by the Tower of Hanoi puzzle and may itself be of independent interest.Peer ReviewedPostprint (author's final draft

Network creation games: structure vs anarchy

We study Nash equilibria and the price of anarchy in the classical model of Network Creation Games introduced by Fabrikant et al. In this model every agent (node) buys links at a prefixed price a > 0 in order to get connected to the network formed by all the n agents. In this setting, the reformulated tree conjecture states that for a > n, every Nash equilibrium network is a tree. Since it was shown that the price of anarchy for trees is constant, if the tree conjecture were true, then the price of anarchy would be constant for a > n. Moreover, Demaine et al. conjectured that the price of anarchy for this model is constant. Up to now the last conjecture has been proven in (i) the lower range, for a = O(n1-o¿) with o¿ = 1 and (ii) in the upper range, for a > 65n. In ¿log n contrast, the best upper bound known for the price of anarchy for the remaining range is 2O(vlog n). In this paper we give new insights into the structure of the Nash equilibria for different ranges of a and we enlarge the range for which the price of anarchy is constant. Regarding the upper range, we prove that every Nash equilibrium is a tree for a > 17n and that the price of anarchy is constant even for a > 9n. In the lower range, we show that any Nash equilibrium for a 4, induces an o¿-distance-almost- uniform graph.Postprint (published version

On the price of anarchy for high-price links

We study Nash equilibria and the price of anarchy in the classic model of Network Creation Games introduced by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker in 2003. This is a selfish network creation model where players correspond to nodes in a network and each of them can create links to the other n−1 players at a prefixed price α>0. The player’s goal is to minimise the sum of her cost buying edges and her cost for using the resulting network. One of the main conjectures for this model states that the price of anarchy, i.e. the relative cost of the lack of coordination, is constant for all α. This conjecture has been confirmed for α=O(n1−δ) with δ≥1/logn and for α>4n−13. The best known upper bound on the price of anarchy for the remaining range is 2O(logn√) . We give new insights into the structure of the Nash equilibria for α>n and we enlarge the range of the parameter α for which the price of anarchy is constant. Specifically, we prove that for any small ϵ>0, the price of anarchy is constant for α>n(1+ϵ) by showing that any biconnected component of any non-trivial Nash equilibrium, if it exists, has at most a constant number of nodes.This work has been partially supported by funds from the Spanish Ministry for Economy and Competitiveness (MINECO) and the European Union (FEDER funds) under grant GRAMM (TIN2017-86727-C2-1-R) and from the Catalan Agency for Management of University and Research Grants (AGAUR, Generalitat de Catalunya) under project ALBCOM 2017-SGR-786.Peer ReviewedPostprint (author's final draft

Nonexistence of almost Moore digraphs of degrees 4 and 5 with self-repeats

An almost Moore (d,k)-digraph is a regular digraph of degree d>1, diameter k>1 and order N(d,k)=d+d2+⋯+dk. So far, their existence has only been shown for k=2, whilst it is known that there are no such digraphs for k=3, 4 and for d=2, 3 when k≥3. Furthermore, under certain assumptions, the nonexistence for the remaining cases has also been shown. In this paper, we prove that (4,k) and (5,k)-almost Moore digraphs with self-repeats do not exist for k≥5.Nacho López: Supported in part by grants PID2020-115442RB-I00 and 2021 SGR-00434. Arnau Messegué: Supported in part by grants Margarita Sala and 2021SGR-00434. Josep M. Miret: Supported in part by grants PID2021-124613OB-I00 and 2021 SGR-00434.Peer ReviewedPostprint (published version

On the spectra of token graphs of cycles and other graphs

© 2023 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The k-token graph of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of equals the algebraic connectivity of G. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs for all r, and the multipartite complete graphs for all In the case of cycles, we present a new method that allows us to compute the whole spectrum of . This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of .Peer ReviewedPostprint (author's final draft

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 ß as well as a link cost a. 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 ß ) 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 ß>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 ß 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/a}; for star celebrity games PoS=1 and PoA=O(min{n/ß,Wa}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when ß=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for ß=2.Peer ReviewedPostprint (author's final draft

Stars and celebrities: A network creation game

CoRRCelebrity games, a new model of network creation games is introduced. The specific features of this model are that players have different celebrity weights and that a critical distance is taken into consideration. The aim of any player is to be close (at distance less than critical) to the others, mainly to those with high celebrity weights. The cost of each player depends on the cost of establishing direct links to other players and on the sum of the weights of those players at a distance greater than the critical distance. We show that celebrity games always have pure Nash equilibria and we characterize the family of subgames having connected Nash equilibria, the so called star celebrity games. Exact bounds for the PoA of non star celebrity games and a bound of O(n/ß+ß) for star celebrity games are provided. The upper bound on the PoA can be tightened when restricted to particular classes of Nash equilibria graphs. We show that the upper bound is O(n/ß) in the case of 2-edge-connected graphs and 2 in the case of trees.Preprin

Network creation games: structure vs anarchy

In an attempt to understand how Internet-like network and social networks behave, different models have been proposed and studied throughout history to capture their most essential aspects and properties. Network Creation Games are a class of strategic games widely studied in Algorithmic Game Theory that model these networks as the outcome of decentralised and uncoordinated interactions. In these games the different players model selfish agents that buy links towards the other agents trying to minimise an individual function. This cost is modelled as a function that usually decomposes into the creation cost (cost of buying links) and the usage cost (measuring the quality of the connection to the network). Due to the agents' selfish behaviour, stable configurations in which all players are happy with the current situation, the so-called Nash equilibria, do not have to coincide with any socially optimal configuration that could be established if a centralised authority could decide by all players. In this way, the price of anarchy is the measure that quantifies precisely the ratio between the most costly Nash equilibrium versus any optimal network from a social point of view. In this work, we study the price of anarchy and Nash equilibria in different scenarios and situations, in order to better understand how the selfish behaviour of agents in these networks affects their quality of the resulting networks. We propose this study from two different perspectives. In the first one, we study one of the most emblematic models of Network Creation Games called sum classical network creation game. This is a model that is based on two different parameters: n, the number of nodes, and a, a function of n that models the price per link. Throughout history it has been shown that the price of anarchy is constant for a =O(n^(1-6)) = 1/(log(n)), and for a > 4n-13. It has been conjectured that the price of anarchy is constant regardless of the parameter a. In this first part we show, first of all, that the price of anarchy is constant even when a > n(1+E) with =>0 any positive constant, thus enlarging the range of values a for which the price of anarchy is constant. Secondly, regarding the range a 4 any positive constant, we know that equilibria induce a class of graphs called distance-uniform. Then, we study the diameter of the distance-uniform graphs in an attempt to obtain information about the topology of equilibria for the range a4 any positive constant. Secondly, we study the diameter of the distance-uniform graphs in an attempt to obtain information about the topology of the equilibria for the range a 4 any positive constant. In the second perspective we propose and study two new models that we call celebrity games. These two models are based on the analysis of decentralized networks with heterogeneous players, that is, players with different degrees of relevance within the corresponding network, a feature that has not been studied in much detail in the literature. To capture this natural property, we introduce a weight for each player in the network. Furthermore, these models take into account a critical distance B, a threshold value. Each player aim to be not farther than ß from the other players and decides whether to buy links to other players depending on the price per link and their corresponding weights. Moreover, the larger is the weight of a player farther than B, larger is the corresponding penalty. Thus, in these new models players strive to have the minimum possible number of links and at the same time they want to minimise as much as possible the penalty for having players farther from B. They differ in how the penalty corresponding to the players further than ß is computed. For both models we obtain upper and lower bounds of the price of anarchy as well as the main topological properties and characteristics of their equilibria.En un intent per entendre com xarxes de naturalesa similar a les de l'Internet i les xarxes socials es comporten, al llarg de la història s'han proposat i estudiat diferents models que tracten de capturar-ne els aspectes i les propietats més essencials. Els jocs de formació de xarxes són una classe de jocs estratègics molt estudiats en teoria de jocs algorísmica que modelitzen aquestes xarxes com el resultat de la interacció descentralitzada dels agents que la integren. En aquests jocs els diferents jugadors modelitzen agents egoistes que compren enllaços cap als altres jugadors intentant minimitzar una funció individual. Aquest cost es modela com una funció que es descomposa en dues components: el cost de creació (cost relatiu a la compra dels enllaços del mateix jugador) i, en segon lloc, el cost d’utilització (mesura de la qualitat de connexió a la xarxa resultant). Degut al comportament egoista dels agents, les situacions estables que s'assoleixen, els anomenats equilibris de Nash, no tenen per què coincidir amb les configuracions òptimes des del punt de vista social que es podrien establir si existeix una entitat centralitzadora que decideix per tots els jugadors. Justament, el preu de l'anarquia és la mesura que quantifica la diferència que hi ha entre l'equilibri de Nash més costós versus l'òptim des del punt de vista social. En aquesta tesis, estudiarem aquests dos conceptes claus, el preu de l'anarquia i els equilibris de Nash, en escenaris i situacions diferents, per tal d'entendre millor com el comportament egoista dels agents d'aquestes xarxes n'afecta la seva qualitat. Proposem dues perspectives diferents a aquest estudi. En primer lloc, estudiem un dels models més emblemàtics dels jocs de formació de xarxes que anomenem sum classical network creation game. Aquest és un model de xarxes que es basa en dos paràmetres diferents: n, nombre de nodes, i α, una funció de n que modelitza el preu per enllaç. Al llarg de la història s'ha demostrat que el preu de l'anarquia és constant per a α= O (n1-δ ) i δ ≥1 log/n, així com per a α > 4n-13. A més s'ha conjecturat que el preu de l'anarquia és constant independentment del paràmetre α . Pel que fa al rang α 4 constant, sabem que els equilibris indueixen una classe de grafs que s'anomenen distance-uniform. En aquesta primera part es demostra, en primer lloc, que el preu de l'anarquia és constant inclús quan α> n(1 + ε) amb ε > 0 qualsevol constant positiva allargant, doncs, el rang de valors del paràmetre α pels quals el preu d'anarquia és constant. En segon lloc, s'estudia el diàmetre dels grafs distance-uniform en un intent d'obtenir informació sobre la topologia dels equilibris per al rang α 4 constant. El segon punt de vista que considerem consisteix en proposar i estudiar dos nous models de creació de xarxes que anomenem els celebrity games. Aquests dos models parteixen de l’anàlisi de les xarxes decentralitzades amb agents heterogenis, com és el cas d'agents que tenen diferents graus de rellevància dins de la xarxa corresponent, un tret fins ara poc estudiat en els models de la literatura. Per capturar aquesta característica natural s'introdueixen pesos, un per a cada agent de la xarxa. D'altra banda, una altra característica que es considera en la proposta d'aquests dos models nous és el concepte de distància critica que captura el llindar Β a partir del qual, nodes que estiguin més llunyans que el valor B del jugador en consideració penalitzen a tal jugador. Així, el que es persegueix en aquests dos nous models és tenir el mínim nombre d’enllaços possibles i al mateix temps, minimitzar el màxim possible la penalització dels jugadors més llunyans de Β d'acord amb els seus pesos. Els dos models que estudiem es diferencien en com es calcula l’afectació o penalització dels jugadors més llunyans de Β. Pels dos models obtenim fites superiors i inferiors del preu de l'anarquia així com les propietats i característiques topològiques principals dels equilibris.En un intento para entender como redes de naturaleza similar a las del Internet y las redes sociales se comportan, al largo de la historia se han propuesto y estudiado modelos que tratan de capturar los aspectos y las propiedades más esenciales. Los juegos de formación de redes son una clase de juegos estratégicos muy estudiados en la teoría de juegos algorítmica que modelizan estas redes como el resultado de la interacción descentralizada de los agentes que la integran. En estos juegos los distintos jugadores modelizan agentes egoístas que compran enlaces hacia los otros jugadores intentando minimizar una función individual. Este coste se modeliza como una función que se descompone en el coste de creación (coste relativo a la compra de los enlaces del mismo jugador) y, en segundo lugar, el coste de utilización (mesura de la cualidad de conexión a la red resultante). Debido al comportamiento egoísta de los agentes, las situaciones estables que se consiguen, los llamados equilibrios de Nash, no coinciden necesariamente con las configuraciones óptimas des del punto de vista social que se podrían establecer si existiera una entidad centralizadora que tomara una decisión por todos los jugadores. Justamente, el precio de la anarquía es la medida que cuantifica la diferencia que hay entre los equilibrios de Nash más costosos versus el óptimo des del punto de vista social. En esta tesis, estudiaremos estos dos conceptos claves, el precio de la anarquía y los equilibrios de Nash en escenarios y situaciones diferentes, con la intención de entender mejor como el comportamiento egoísta de los agentes de estas redes afecta su cualidad. Proponemos dos perspectivas distintas para este estudio. En primer lugar, estudiamos uno de los modelos más emblemáticos de los juegos de formación de redes que llamamos sum classical network creation game. Este es un modelo de redes que se basa en dos parámetros distintos: n, el número de nodos de la red, y _, una función de n que modeliza el precio por enlace. Al largo del tiempo se ha demostrado que el precio de la anarquía es constante para α= O (n1-δ ) y δ ≥1 log/n, como para α > 4n-13. Además se ha conjeturado que el precio de la anarquía es constante independientemente del parámetro α. Respecto al rango α 4 constante, sabemos que los equilibrios inducen una clase de grafos que se llama distancia-uniforme. En esta primera parte se demuestra, primero, que el precio de la anarquía es constante incluso cuando α> n(1 + ε) con ε > 0 con ε > 0 cualquier constante positiva engrandeciendo, pues, el rango de valores del parámetro _ por los cuales el precio de la anarquía es constante. En segundo lugar, se estudia el diámetro de los grafos distancia-uniforme en un intento de obtener información sobre la topología de los equilibrios para el rango α 4 constante. El segundo punto de vista que consideremos consiste en proponer y estudiar dos modelos de creación de redes nuevos que llamamos los celebrity games. Estos dos modelos parten del análisis de las redes descentralizadas con agentes heterogéneos, como puede ser el caso de agentes que tienen distintos grados de relevancia dentro de la red correspondiente, una característica hasta ahora poco estudiada en los modelos de la literatura. Para capturar esta característica natural se introducen pesos para cada agente de la red. Por otro lado, otra característica que se considera en la propuesta de estos dos modelos nuevos es el concepto de distancia crítica que captura el nivel _ a partir del cual, nodos que estén más lejanos que el valor _ del jugador en consideración penalizan a tal jugador. Así, lo que se persigue en estos dos nuevos modelos es tener el mínimo número de enlaces posibles y al mismo tiempo minimizar la máxima de las penalizaciones de los jugadores más lejanos de Β de acuerdo con sus pesos. Los dos modelos que estudiamos se diferencian en cómo se calcula la afectación o penalización de los jugadores más lejanos de Β. Para los dos modelos obtenemos cotas superiores e inferiores del precio de la anarquía, así como las propiedades y características topológicas principales de los equilibrios

Preu de l'anarquia en la creació de xarxes

This master thesis focuses its attention in the Sum Classic model introduced by Fabrikant et al. which is a simpli cation of the Internet and can be thought as a Strategic Game. In this model the distinct agents, who can be thought as nodes in a graph, wish to establish links of constant price to the others in order to be connected in the resulting network. Elements of interest are then studied, like equilibria, con gurations where every agent is not interested in deviating his current strategy, or the price of anarchy, a measure of how the e ciency of the system degrades due to sel sh behavior of its agents. After analyzing the work done so far in the area we have developed and proved original results which study the price of anarchy in di erent situations as well as the structure of equilibria. More precisely, given , the parameter corresponding to the cost of buying an edge, we give a new proof that the price of anarchy is constant for = O( p N) (being N the size of the network), we show that the price of anarchy is constant for the bounded degree graphs when = o(N) and we show that the structure of equilibria for < n=C for some constant C is a very restricted one. Also, we study the question of deciding if a given graph belonging to a certain family is an equilibrium and we show that the corresponding decisional problem is polynomial time solvable whenever the family considered are trees or bounded degree graphs