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Andrea Clementi

Publications and source records attributed to Andrea Clementi.

31 records · Page 2Linked to original sources
arXiv2016

Find Your Place: Simple Distributed Algorithms for Community Detection

Given an underlying graph, we consider the following \emph{dynamics}: Initially, each node locally chooses a value in $\{-1,1\}$, uniformly at random and independently of other nodes. Then, in each consecutive round, every node updates its local value to the average of the values held by its neighbors, at the same time applying an elementary, local clustering rule that only depends on the current and the previous values held by the node. We prove that the process resulting from this dynamics produces a clustering that exactly or approximately (depending on the graph) reflects the underlying cut in logarithmic time, under various graph models that exhibit a sparse balanced cut, including the stochastic block model. We also prove that a natural extension of this dynamics performs community detection on a regularized version of the stochastic block model with multiple communities. Rather surprisingly, our results provide rigorous evidence for the ability of an extremely simple and natural dynamics to address a computational problem that is non-trivial even in a centralized setting.

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arXiv2016

Self-Stabilizing Repeated Balls-into-Bins

We study the following synchronous process that we call "repeated balls-into-bins". The process is started by assigning $n$ balls to $n$ bins in an arbitrary way. In every subsequent round, from each non-empty bin one ball is chosen according to some fixed strategy (random, FIFO, etc), and re-assigned to one of the $n$ bins uniformly at random. We define a configuration "legitimate" if its maximum load is $\mathcal{O}(\log n)$. We prove that, starting from any configuration, the process will converge to a legitimate configuration in linear time and then it will only take on legitimate configurations over a period of length bounded by any polynomial in $n$, with high probability (w.h.p.). This implies that the process is self-stabilizing and that every ball traverses all bins in $\mathcal{O}(n \log^2 n)$ rounds, w.h.p.

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arXiv2015

Stabilizing Consensus with Many Opinions

We consider the following distributed consensus problem: Each node in a complete communication network of size $n$ initially holds an \emph{opinion}, which is chosen arbitrarily from a finite set $Σ$. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be \emph{valid}, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of an adaptive, malicious adversary who can modify the opinions of a bounded number of nodes in every round. We consider the \emph{3-majority dynamics}: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let $k$ be the number of valid opinions. We show that, if $k \leqslant n^α$, where $α$ is a suitable positive constant, the 3-majority dynamics converges in time polynomial in $k$ and $\log n$ with high probability even in the presence of an adversary who can affect up to $o(\sqrt{n})$ nodes at each round. Previously, the convergence of the 3-majority protocol was known for $|Σ| = 2$ only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for $|Σ| > 2$.

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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.

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arXiv2013

Distributed Community Detection in Dynamic Graphs

Inspired by the increasing interest in self-organizing social opportunistic networks, we investigate the problem of distributed detection of unknown communities in dynamic random graphs. As a formal framework, we consider the dynamic version of the well-studied \emph{Planted Bisection Model} $\sdG(n,p,q)$ where the node set $[n]$ of the network is partitioned into two unknown communities and, at every time step, each possible edge $(u,v)$ is active with probability $p$ if both nodes belong to the same community, while it is active with probability $q$ (with $q< 0$), the protocol finds the right "planted" partition in $O(\log n)$ time even when the snapshots of the dynamic graph are sparse and disconnected (i.e. in the case $p=Θ(1/n)$).

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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.

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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.

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arXiv2012

Information Spreading in Dynamic Graphs

We present a general approach to study the flooding time (a measure of how fast information spreads) in dynamic graphs (graphs whose topology changes with time according to a random process). We consider arbitrary converging Markovian dynamic graph process, that is, processes in which the topology of the graph at time $t$ depends only on its topology at time $t-1$ and which have a unique stationary distribution. The most well studied models of dynamic graphs are all Markovian and converging. Under general conditions, we bound the flooding time in terms of the mixing time of the dynamic graph process. We recover, as special cases of our result, bounds on the flooding time for the \emph{random trip} model and the \emph{random path} models; previous analysis techniques provided bounds only in restricted settings for such models. Our result also provides the first bound for the \emph{random waypoint} model (which is tight for certain ranges of parameters) whose analysis had been an important open question.

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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.

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arXiv2011

Parsimonious Flooding in Geometric Random-Walks

We study the information spreading yielded by the \emph{(Parsimonious) $1$-Flooding Protocol} in geometric Mobile Ad-Hoc Networks. We consider $n$ agents on a convex plane region of diameter $D$ performing independent random walks with move radius $ρ$. At any time step, every active agent $v$ informs every non-informed agent which is within distance $R$ from $v$ ($R>0$ is the transmission radius). An agent is only active at the time step immediately after the one in which has been informed and, after that, she is removed. At the initial time step, a source agent is informed and we look at the \emph{completion time} of the protocol, i.e., the first time step (if any) in which all agents are informed. This random process is equivalent to the well-known \emph{Susceptible-Infective-Removed ($SIR$}) infection process in Mathematical Epidemiology. No analytical results are available for this random process over any explicit mobility model. The presence of removed agents makes this process much more complex than the (standard) flooding. We prove optimal bounds on the completion time depending on the parameters $n$, $D$, $R$, and $ρ$. The obtained bounds hold with high probability. We remark that our method of analysis provides a clear picture of the dynamic shape of the information spreading (or infection wave) over the time.

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arXiv2010

Fast Flooding over Manhattan

We consider a Mobile Ad-hoc NETwork (MANET) formed by n agents that move at speed V according to the Manhattan Random-Way Point model over a square region of side length L. The resulting stationary (agent) spatial probability distribution is far to be uniform: the average density over the "central zone" is asymptotically higher than that over the "suburb". Agents exchange data iff they are at distance at most R within each other. We study the flooding time of this MANET: the number of time steps required to broadcast a message from one source agent to all agents of the network in the stationary phase. We prove the first asymptotical upper bound on the flooding time. This bound holds with high probability, it is a decreasing function of R and V, and it is tight for a wide and relevant range of the network parameters (i.e. L, R and V). A consequence of our result is that flooding over the sparse and highly-disconnected suburb can be as fast as flooding over the dense and connected central zone. Rather surprisingly, this property holds even when R is exponentially below the connectivity threshold of the MANET and the speed V is very low.

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arXiv2010

Modelling Mobility: A Discrete Revolution

We introduce a new approach to model and analyze \emph{Mobility}. It is fully based on discrete mathematics and yields a class of mobility models, called the \emph{Markov Trace} Model. This model can be seen as the discrete version of the \emph{Random Trip} Model including all variants of the \emph{Random Way-Point} Model \cite{L06}. We derive fundamental properties and \emph{explicit} analytical formulas for the \emph{stationary distributions} yielded by the Markov Trace Model. Such results can be exploited to compute formulas and properties for concrete cases of the Markov Trace Model by just applying counting arguments. We apply the above general results to the discrete version of the \emph{Manhattan Random Way-Point} over a square of bounded size. We get formulas for the total stationary distribution and for two important \emph{conditional} ones: the agent spatial and destination distributions. Our method makes the analysis of complex mobile systems a feasible task. As a further evidence of this important fact, we first model a complex vehicular-mobile system over a set of crossing streets. Several concrete issues are implemented such as parking zones, traffic lights, and variable vehicle speeds. By using a \emph{modular} version of the Markov Trace Model, we get explicit formulas for the stationary distributions yielded by this vehicular-mobile model as well.

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