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Francesco Pasquale

Publications and source records attributed to Francesco Pasquale.

28 records · Page 2Linked to original sources
arXiv2015

Simple Dynamics for Plurality Consensus

We study a \emph{Plurality-Consensus} process in which each of $n$ anonymous agents of a communication network initially supports an opinion (a color chosen from a finite set $[k]$). Then, in every (synchronous) round, each agent can revise his color according to the opinions currently held by a random sample of his neighbors. It is assumed that the initial color configuration exhibits a sufficiently large \emph{bias} $s$ towards a fixed plurality color, that is, the number of nodes supporting the plurality color exceeds the number of nodes supporting any other color by $s$ additional nodes. The goal is having the process to converge to the \emph{stable} configuration in which all nodes support the initial plurality. We consider a basic model in which the network is a clique and the update rule (called here the \emph{3-majority dynamics}) of the process is the following: each agent looks at the colors of three random neighbors and then applies the majority rule (breaking ties uniformly). We prove that the process converges in time $\mathcal{O}( \min\{ k, (n/\log n)^{1/3} \} \, \log n )$ with high probability, provided that $s \geqslant c \sqrt{ \min\{ 2k, (n/\log n)^{1/3} \}\, n \log n}$. We then prove that our upper bound above is tight as long as $k \leqslant (n/\log n)^{1/4}$. This fact implies an exponential time-gap between the plurality-consensus process and the \emph{median} process studied by Doerr et al. in [ACM SPAA'11]. A natural question is whether looking at more (than three) random neighbors can significantly speed up the process. We provide a negative answer to this question: In particular, we show that samples of polylogarithmic size can speed up the process by a polylogarithmic factor only.

cs.DM↗
arXiv2013

Logit Dynamics with Concurrent Updates for Local-Interaction Games

Logit choice dynamics are a family of randomized best response dynamics based on the logit choice function [McFadden, 1974], used for modeling players with limited rationality and knowledge. In this paper we study the all-logit dynamics, where at each time step all players concurrently update their strategies according to the logit choice function. In the well studied one-logit dynamics [Blume, 1993] instead at each step only one randomly chosen player is allowed to update. We study properties of the all-logit dynamics in the context of local interaction games, a class of games that has been used to model complex social phenomena and physical systems. In a local interaction game, players are the vertices of a social graph whose edges are two-player potential games. Each player picks one strategy to be played for all the games she is involved in and the payoff of the player is the sum of the payoffs from each of the games. We prove that local interaction games characterize the class of games for which the all-logit dynamics is reversible. We then compare the stationary behavior of one-logit and all-logit dynamics. Specifically, we look at the expected value of a notable class of observables, that we call decomposable observables. We prove that the difference between the expected values of the observables at stationarity for the two dynamics depends only on the rationality level beta and on the distance of the social graph from a bipartite graph. In particular, if the social graph is bipartite then decomposable observables have the same expected value. Finally, we show that the mixing time of the all-logit dynamics has the same twofold behavior that has been highlighted in the case of the one-logit: for some games it exponentially depends on the rationality level beta, whereas for other games it can be upper bounded by a function independent from beta.

cs.GT↗
arXiv2013

Rumor Spreading in Random Evolving Graphs

Randomized gossip is one of the most popular way of disseminating information in large scale networks. This method is appreciated for its simplicity, robustness, and efficiency. In the "push" protocol, every informed node selects, at every time step (a.k.a. round), one of its neighboring node uniformly at random and forwards the information to this node. This protocol is known to complete information spreading in $O(\log n)$ time steps with high probability (w.h.p.) in several families of $n$-node "static" networks. The Push protocol has also been empirically shown to perform well in practice, and, specifically, to be robust against dynamic topological changes. In this paper, we aim at analyzing the Push protocol in "dynamic" networks. We consider the "edge-Markovian" evolving graph model which captures natural temporal dependencies between the structure of the network at time $t$, and the one at time $t+1$. Precisely, a non-edge appears with probability $p$, while an existing edge dies with probability $q$. In order to fit with real-world traces, we mostly concentrate our study on the case where $p=Ω(1/n)$ and $q$ is constant. We prove that, in this realistic scenario, the Push protocol does perform well, completing information spreading in $O(\log n)$ time steps w.h.p. Note that this performance holds even when the network is, w.h.p., disconnected at every time step (e.g., when $p << (\log n) / n$). Our result provides the first formal argument demonstrating the robustness of the Push protocol against network changes. We also address other ranges of parameters $p$ and $q$ (e.g., $p+q=1$ with arbitrary $p$ and $q$, and $p=1/n$ with arbitrary $q$). Although they do not precisely fit with the measures performed on real-world traces, they can be of independent interest for other settings. The results in these cases confirm the positive impact of dynamism.

cs.DM↗
arXiv2012

Convergence to Equilibrium of Logit Dynamics for Strategic Games

We present the first general bounds on the mixing time of the Markov chain associated to the logit dynamics for wide classes of strategic games. The logit dynamics with inverse noise beta describes the behavior of a complex system whose individual components act selfishly and keep responding according to some partial ("noisy") knowledge of the system, where the capacity of the agent to know the system and compute her best move is measured by the inverse of the parameter beta. In particular, we prove nearly tight bounds for potential games and games with dominant strategies. Our results show that, for potential games, the mixing time is upper and lower bounded by an exponential in the inverse of the noise and in the maximum potential difference. Instead, for games with dominant strategies, the mixing time cannot grow arbitrarily with the inverse of the noise. Finally, we refine our analysis for a subclass of potential games called graphical coordination games, a class of games that have been previously studied in Physics and, more recently, in Computer Science in the context of diffusion of new technologies. We give evidence that the mixing time of the logit dynamics for these games strongly depends on the structure of the underlying graph. We prove that the mixing time of the logit dynamics for these games can be upper bounded by a function that is exponential in the cutwidth of the underlying graph and in the inverse of noise. Moreover, we consider two specific and popular network topologies, the clique and the ring. For games played on a clique we prove an almost matching lower bound on the mixing time of the logit dynamics that is exponential in the inverse of the noise and in the maximum potential difference, while for games played on a ring we prove that the time of convergence of the logit dynamics to its stationary distribution is significantly shorter.

cs.GT↗
arXiv2012

Flooding Time in Opportunistic Networks under Power Law and Exponential Inter-Contact Times

Performance bounds for opportunistic networks have been derived in a number of recent papers for several key quantities, such as the expected delivery time of a unicast message, or the flooding time (a measure of how fast information spreads). However, to the best of our knowledge, none of the existing results is derived under a mobility model which is able to reproduce the power law+exponential tail dichotomy of the pairwise node inter-contact time distribution which has been observed in traces of several real opportunistic networks. The contributions of this paper are two-fold: first, we present a simple pairwise contact model -- called the Home-MEG model -- for opportunistic networks based on the observation made in previous work that pairs of nodes in the network tend to meet in very few, selected locations (home locations); this contact model is shown to be able to faithfully reproduce the power law+exponential tail dichotomy of inter-contact time. Second, we use the Home-MEG model to analyze flooding time in opportunistic networks, presenting asymptotic bounds on flooding time that assume different initial conditions for the existence of opportunistic links. Finally, our bounds provide some analytical evidences that the speed of information spreading in opportunistic networks can be much faster than that predicted by simple geometric mobility models.

cs.SI↗
arXiv2011

Mixing Time and Stationary Expected Social Welfare of Logit Dynamics

We study "logit dynamics" [Blume, Games and Economic Behavior, 1993] for strategic games. This dynamics works as follows: at every stage of the game a player is selected uniformly at random and she plays according to a "noisy" best-response where the noise level is tuned by a parameter $β$. Such a dynamics defines a family of ergodic Markov chains, indexed by $β$, over the set of strategy profiles. We believe that the stationary distribution of these Markov chains gives a meaningful description of the long-term behavior for systems whose agents are not completely rational. Our aim is twofold: On the one hand, we are interested in evaluating the performance of the game at equilibrium, i.e. the expected social welfare when the strategy profiles are random according to the stationary distribution. On the other hand, we want to estimate how long it takes, for a system starting at an arbitrary profile and running the logit dynamics, to get close to its stationary distribution; i.e., the "mixing time" of the chain. In this paper we study the stationary expected social welfare for the 3-player CK game, for 2-player coordination games, and for two simple $n$-player games. For all these games, we also give almost tight upper and lower bounds on the mixing time of logit dynamics. Our results show two different behaviors: in some games the mixing time depends exponentially on $β$, while for other games it can be upper bounded by a function independent of $β$.

cs.GT↗
arXiv2011

Information Spreading in Stationary Markovian Evolving Graphs

Markovian evolving graphs are dynamic-graph models where the links among a fixed set of nodes change during time according to an arbitrary Markovian rule. They are extremely general and they can well describe important dynamic-network scenarios. We study the speed of information spreading in the "stationary phase" by analyzing the completion time of the "flooding mechanism". We prove a general theorem that establishes an upper bound on flooding time in any stationary Markovian evolving graph in terms of its node-expansion properties. We apply our theorem in two natural and relevant cases of such dynamic graphs. "Geometric Markovian evolving graphs" where the Markovian behaviour is yielded by "n" mobile radio stations, with fixed transmission radius, that perform independent random walks over a square region of the plane. "Edge-Markovian evolving graphs" where the probability of existence of any edge at time "t" depends on the existence (or not) of the same edge at time "t-1". In both cases, the obtained upper bounds hold "with high probability" and they are nearly tight. In fact, they turn out to be tight for a large range of the values of the input parameters. As for geometric Markovian evolving graphs, our result represents the first analytical upper bound for flooding time on a class of concrete mobile networks.

cs.DM↗
arXiv2010

A note on uniform power connectivity in the SINR model

In this paper we study the connectivity problem for wireless networks under the Signal to Interference plus Noise Ratio (SINR) model. Given a set of radio transmitters distributed in some area, we seek to build a directed strongly connected communication graph, and compute an edge coloring of this graph such that the transmitter-receiver pairs in each color class can communicate simultaneously. Depending on the interference model, more or less colors, corresponding to the number of frequencies or time slots, are necessary. We consider the SINR model that compares the received power of a signal at a receiver to the sum of the strength of other signals plus ambient noise . The strength of a signal is assumed to fade polynomially with the distance from the sender, depending on the so-called path-loss exponent $α$. We show that, when all transmitters use the same power, the number of colors needed is constant in one-dimensional grids if $α>1$ as well as in two-dimensional grids if $α>2$. For smaller path-loss exponents and two-dimensional grids we prove upper and lower bounds in the order of $\mathcal{O}(\log n)$ and $Ω(\log n/\log\log n)$ for $α=2$ and $Θ(n^{2/α-1})$ for $α<2$ respectively. If nodes are distributed uniformly at random on the interval $[0,1]$, a \emph{regular} coloring of $\mathcal{O}(\log n)$ colors guarantees connectivity, while $Ω(\log \log n)$ colors are required for any coloring.

cs.DM↗
arXiv2009

MANETS: High mobility can make up for low transmission power

We consider a Mobile Ad-hoc NETworks (MANET) formed by "n" nodes that move independently at random over a finite square region of the plane. Nodes exchange data if they are at distance at most "r" within each other, where r>0 is the node transmission radius. The "flooding time" is the number of time steps required to broadcast a message from a source node to every node of the network. Flooding time is an important measure of the speed of information spreading in dynamic networks. We derive a nearly-tight upper bound on the flooding time which is a decreasing function of the maximal "velocity" of the nodes. It turns out that, when the node velocity is sufficiently high, even if the node transmission radius "r" is far below the "connectivity threshold", the flooding time does not asymptotically depend on "r". This implies that flooding can be very fast even though every "snapshot" (i.e. the static random geometric graph at any fixed time) of the MANET is fully disconnected. Data reach all nodes quickly despite these ones use very low transmission power. Our result is the first analytical evidence of the fact that high, random node mobility strongly speed-up information spreading and, at the same time, let nodes save energy.

cs.DM↗