arXiv subjects

Giacomo Scornavacca

Publications and source records attributed to Giacomo Scornavacca.

6 recordsLinked to original sources

arXiv2020

Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks

Consider the following process on a network: Each agent initially holds either opinion blue or red; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. This process is known as the 2-Choices dynamics and is arguably the most basic non-trivial opinion dynamics modeling voting behavior on social networks. Despite its apparent simplicity, 2-Choices has been analytically characterized only on restricted network classes---under assumptions on the initial configuration that establish it as a fast majority consensus protocol. In this work, we aim at contributing to the understanding of the 2-Choices dynamics by considering its behavior on a class of networks with core-periphery structure, a well-known topological assumption in social networks. In a nutshell, assume that a densely-connected subset of agents, the core, holds a different opinion from the rest of the network, the periphery. Then, depending on the strength of the cut between the core and the periphery, a phase-transition phenomenon occurs: Either the core's opinion rapidly spreads among the rest of the network, or a metastability phase takes place, in which both opinions coexist in the network for superpolynomial time. The interest of our result is twofold. On the one hand, by looking at the 2-Choices dynamics as a simplistic model of competition among opinions in social networks, our theorem sheds light on the influence of the core on the rest of the network, as a function of the core's connectivity toward the latter. On the other hand, we provide one of the first analytical results which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network. Finally, we validate our theoretical predictions with extensive experiments on real networks.

cs.SI↗

arXiv2020

Cutting Bamboo Down to Size

This paper studies the problem of programming a robotic panda gardener to keep a bamboo garden from obstructing the view of the lake by your house. The garden consists of $n$ bamboo stalks with known daily growth rates and the gardener can cut at most one bamboo per day. As a computer scientist, you found out that this problem has already been formalized in [Gąsieniec et al., SOFSEM'17] as the Bamboo Garden Trimming (BGT) problem, where the goal is that of computing a perpetual schedule (i.e., the sequence of bamboos to cut) for the robotic gardener to follow in order to minimize the makespan, i.e., the maximum height ever reached by a bamboo. Two natural strategies are Reduce-Max and Reduce-Fastest(x). Reduce-Max trims the tallest bamboo of the day, while Reduce-Fastest(x) trims the fastest growing bamboo among the ones that are taller than $x$. It is known that Reduce-Max and Reduce-Fastest(x) achieve a makespan of $O(\log n)$ and $4$ for the best choice of $x=2$, respectively. We prove the first constant upper bound of $9$ for Reduce-Max and improve the one for Reduce-Fastest(x) to $\frac{3+\sqrt{5}}{2} < 2.62$ for $x=1+\frac{1}{\sqrt{5}}$. Another critical aspect stems from the fact that your robotic gardener has a limited amount of processing power and memory. It is then important for the algorithm to be able to quickly determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a Trimming Oracle data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by Reduce-Max and Reduce-Fastest(x).

cs.DS↗

arXiv2018

Distributed Community Detection via Metastability of the 2-Choices Dynamics

We investigate the behavior of a simple majority dynamics on networks of agents whose interaction topology exhibits a community structure. By leveraging recent advancements in the analysis of dynamics, we prove that, when the states of the nodes are randomly initialized, the system rapidly and stably converges to a configuration in which the communities maintain internal consensus on different states. This is the first analytical result on the behavior of dynamics for non-consensus problems on non-complete topologies, based on the first symmetry-breaking analysis in such setting. Our result has several implications in different contexts in which dynamics are adopted for computational and biological modeling purposes. In the context of Label Propagation Algorithms, a class of widely used heuristics for community detection, it represents the first theoretical result on the behavior of a distributed label propagation algorithm with quasi-linear message complexity. In the context of evolutionary biology, dynamics such as the Moran process have been used to model the spread of mutations in genetic populations [Lieberman, Hauert, and Nowak 2005]; our result shows that, when the probability of adoption of a given mutation by a node of the evolutionary graph depends super-linearly on the frequency of the mutation in the neighborhood of the node and the underlying evolutionary graph exhibits a community structure, there is a non-negligible probability for species differentiation to occur.

cs.SI↗

arXiv2018

Consensus Needs Broadcast in Noiseless Models but can be Exponentially Easier in the Presence of Noise

Consensus and Broadcast are two fundamental problems in distributed computing, whose solutions have several applications. Intuitively, Consensus should be no harder than Broadcast, and this can be rigorously established in several models. Can Consensus be easier than Broadcast? In models that allow noiseless communication, we prove a reduction of (a suitable variant of) Broadcast to binary Consensus, that preserves the communication model and all complexity parameters such as randomness, number of rounds, communication per round, etc., while there is a loss in the success probability of the protocol. Using this reduction, we get, among other applications, the first logarithmic lower bound on the number of rounds needed to achieve Consensus in the uniform GOSSIP model on the complete graph. The lower bound is tight and, in this model, Consensus and Broadcast are equivalent. We then turn to distributed models with noisy communication channels that have been studied in the context of some bio-inspired systems. In such models, only one noisy bit is exchanged when a communication channel is established between two nodes, and so one cannot easily simulate a noiseless protocol by using error-correcting codes. An $Ω(ε^{-2} n)$ lower bound on the number of rounds needed for Broadcast is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] in one such model (noisy uniform PULL, where $ε$ is a parameter that measures the amount of noise). In such model, we prove a new $Θ(ε^{-2} n \log n)$ bound for Broadcast and a $Θ(ε^{-2} \log n)$ bound for binary Consensus, thus establishing an exponential gap between the number of rounds necessary for Consensus versus Broadcast.

cs.DC↗

arXiv2018

A Tight Analysis of the Parallel Undecided-State Dynamics with Two Colors

The \emph{Undecided-State Dynamics} is a well-known protocol for distributed consensus. We analyze it in the parallel \pull\ communication model on the complete graph for the \emph{binary} case (every node can either support one of \emph{two} possible colors, or be in the undecided state). An interesting open question is whether this dynamics \emph{always} (i.e., starting from an arbitrary initial configuration) reaches consensus \emph{quickly} (i.e., within a polylogarithmic number of rounds) in a complete graph with $n$ nodes. Previous work in this setting only considers initial color configurations with no undecided nodes and a large \emph{bias} (i.e., $Θ(n)$) towards the majority color. In this paper we present an \textit{unconditional} analysis of the Undecided-State Dynamics that answers to the above question in the affirmative. We prove that, starting from \textit{any} initial configuration, the process reaches a monochromatic configuration within $O(\log n)$ rounds, with high probability. This bound turns out to be tight. Our analysis also shows that, if the initial configuration has bias $Ω(\sqrt{n\log n})$, then the dynamics converges toward the initial majority color, with high probability.

cs.DS↗

arXiv2017

Rational Fair Consensus in the GOSSIP Model

The \emph{rational fair consensus problem} can be informally defined as follows. Consider a network of $n$ (selfish) \emph{rational agents}, each of them initially supporting a \emph{color} chosen from a finite set $ Σ$. The goal is to design a protocol that leads the network to a stable monochromatic configuration (i.e. a consensus) such that the probability that the winning color is $c$ is equal to the fraction of the agents that initially support $c$, for any $c \in Σ$. Furthermore, this fairness property must be guaranteed (with high probability) even in presence of any fixed \emph{coalition} of rational agents that may deviate from the protocol in order to increase the winning probability of their supported colors. A protocol having this property, in presence of coalitions of size at most $t$, is said to be a \emph{whp\,-$t$-strong equilibrium}. We investigate, for the first time, the rational fair consensus problem in the GOSSIP communication model where, at every round, every agent can actively contact at most one neighbor via a \emph{push$/$pull} operation. We provide a randomized GOSSIP protocol that, starting from any initial color configuration of the complete graph, achieves rational fair consensus within $O(\log n)$ rounds using messages of $O(\log^2n)$ size, w.h.p. More in details, we prove that our protocol is a whp\,-$t$-strong equilibrium for any $t = o(n/\log n)$ and, moreover, it tolerates worst-case permanent faults provided that the number of non-faulty agents is $Ω(n)$. As far as we know, our protocol is the first solution which avoids any all-to-all communication, thus resulting in $o(n^2)$ message complexity.

cs.GT↗