arXiv subjects

Matteo Quattropani

Publications and source records attributed to Matteo Quattropani.

29 records · Page 2Linked to original sources

arXiv2023

Phase Transition of the k-Majority Dynamics in Biased Communication Models

Consider a graph where each of the $n$ nodes is either in state $\mathcal{R}$ or $\mathcal{B}$. Herein, we analyze the \emph{synchronous $k$-Majority dynamics}, where in each discrete-time round nodes simultaneously sample $k$ neighbors uniformly at random with replacement and adopt the majority state among those of the nodes in the sample (breaking ties uniformly at random). Differently from previous work, we study the robustness of the $k$-Majority in \emph{maintaining a $\mathcal{R}$ majority}, when the dynamics is subject to two forms of \emph{bias} toward state $\mathcal{B}$. The bias models an external agent that attempts to subvert the initial majority by altering the communication between nodes, with a probability of success $p$ in each round: in the first form of bias, the agent tries to alter the communication links by transmitting state $\mathcal{B}$; in the second form of bias, the agent tries to corrupt nodes directly by making them update to $\mathcal{B}$. Our main result shows a \emph{sharp phase transition} in both forms of bias. By considering initial configurations in which every node has probability $q \in (\frac{1}{2},1]$ of being in state $\mathcal{R}$, we prove that for every $k\geq3$ there exists a critical value $p_{k,q}^*$ such that, with high probability, the external agent is able to subvert the initial majority either in $n^{ω(1)}$ rounds, if $p p_{k,q}^*$. When $k<3$, instead, no phase transition phenomenon is observed and the disruption happens in $O(1)$ rounds for $p>0$.

cs.DC↗

arXiv2022

Cutoff for the Averaging process on the hypercube and complete bipartite graphs

We consider the averaging process on a graph, that is the evolution of a mass distribution undergoing repeated averages along the edges of the graph at the arrival times of independent Poisson processes. We establish cutoff phenomena for both the $L^1$ and $L^2$ distance from stationarity when the graph is a discrete hypercube and when the graph is complete bipartite. Some general facts about the averaging process on arbitrary graphs are also discussed.

math.PR↗

arXiv2022

Mixing of the Averaging process and its discrete dual on finite-dimensional geometries

We analyze the $L^1$-mixing of a generalization of the Averaging process introduced by Aldous. The process takes place on a growing sequence of graphs which we assume to be finite-dimensional, in the sense that the random walk on those geometries satisfies a family of Nash inequalities. As a byproduct of our analysis, we provide a complete picture of the total variation mixing of a discrete dual of the Averaging process, which we call Binomial Splitting process. A single particle of this process is essentially the random walk on the underlying graph. When several particles evolve together, they interact by synchronizing their jumps when placed on neighboring sites. We show that, given $k$ the number of particles and $n$ the (growing) size of the underlying graph, the system exhibits cutoff in total variation if $k\to\infty$ and $k=O(n^2)$. Finally, we exploit the duality between the two processes to show that the Binomial Splitting satisfies a version of Aldous' spectral gap identity, namely, the relaxation time of the process is independent of the number of particles.

math.PR↗

arXiv2021

Rankings in directed configuration models with heavy tailed in-degrees

We consider the extremal values of the stationary distribution of sparse directed random graphs with given degree sequences and their relation to the extremal values of the in-degree sequence. The graphs are generated by the directed configuration model. Under the assumption of bounded $(2+η)$-moments on the in-degrees and of bounded out-degrees, we obtain tight comparisons between the maximum value of the stationary distribution and the maximum in-degree. Under the further assumption that the order statistics of the in-degrees have a power-law behavior, we show that the extremal values of the stationary distribution also have a power-law behavior with the same index. In the same setting, we prove that these results extend to the PageRank scores of the random digraph, thus confirming a version of the so-called power-law hypothesis. Along the way, we establish several facts about the model, including the mixing time cutoff and the characterization of the typical values of the stationary distribution, which were previously obtained under the assumption of bounded in-degrees.

math.PR↗

arXiv2021

Efficiency of equilibria in games with random payoffs

We consider normal-form games with $n$ players and two strategies for each player, where the payoffs are i.i.d. random variables with some distribution $F$ and we consider issues related to the pure equilibria in the game as the number of players diverges. It is well-known that, if the distribution $F$ has no atoms, the random number of pure equilibria is asymptotically Poisson$(1)$. In the presence of atoms, it diverges. For each strategy profile, we consider the (random) average payoff of the players, called Average Social Utility (ASU). In particular, we examine the asymptotic behavior of the optimum ASU and the one associated to the best and worst pure Nash equilibria and we show that, although these quantities are random, they converge, as $n\to\infty$ to some deterministic quantities.

math.PR↗

arXiv2021

Mixing time of PageRank surfers on sparse random digraphs

We consider the generalised PageRank walk on a digraph $G$, with refresh probability $α$ and resampling distribution $λ$. We analyse convergence to stationarity when $G$ is a large sparse random digraph with given degree sequences, in the limit of vanishing $α$. We identify three scenarios: when $α$ is much smaller than the inverse of the mixing time of $G$ the relaxation to equilibrium is dominated by the simple random walk and displays a cutoff behaviour; when $α$ is much larger than the inverse of the mixing time of $G$ on the contrary one has pure exponential decay with rate $α$; when $α$ is comparable to the inverse of the mixing time of $G$ there is a mixed behaviour interpolating between cutoff and exponential decay. This trichotomy is shown to hold uniformly in the starting point and uniformly in the resampling distribution $λ$.

math.PR↗

arXiv2021

A probabilistic proof of Cooper and Frieze's "First Visit Time Lemma"

In this short note we present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first formulation in [21], and then used and refined in a list of papers by Cooper, Frieze and coauthors. We work in the original setting, considering a growing sequence of irreducible Markov chains on $n$ states. We assume that the chain is rapidly mixing and with a stationary measure having no entry which is too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state $x$ -- for the chain started at stationarity -- up to a small multiplicative correction. While the proof of the FVTL presented by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob's transform of the chain on the complement of the state $x$.

math.PR↗

arXiv2020

The Buck-Passing Game

We consider a game in which players are the vertices of a directed graph. Initially, Nature chooses one player according to some fixed distribution and gives her a buck, which represents the request to perform a chore. After completing the task, the player passes the buck to one of her out-neighbors in the graph. The procedure is repeated indefinitely and each player's cost is the asymptotic expected frequency of times that she receives the buck. We consider a deterministic and a stochastic version of the game depending on how players select the neighbor to pass the buck. In both cases we prove the existence of pure equilibria that do not depend on the initial distribution; this is achieved by showing the existence of a generalized ordinal potential. We then use the price of anarchy and price of stability to measure fairness of these equilibria. We also study a buck-holding variant of the game in which players want to maximize the frequency of times they hold the buck, which includes the PageRank game as a special case.

cs.GT↗

arXiv2020

Mixing time trichotomy in regenerating dynamic digraphs

We study convergence to stationarity for random walks on dynamic random digraphs with given degree sequences. The digraphs undergo full regeneration at independent geometrically distributed random time intervals with parameter $α$. Relaxation to stationarity is the result of a competition between regeneration and mixing on the static digraph. When the number of vertices $n$ tends to infinity and the parameter $α$ tends to zero, we find three scenarios according to whether $α\log n$ converges to zero, infinity or to some finite positive value: when the limit is zero, relaxation to stationarity occurs in two separate stages, the first due to mixing on the static digraph, and the second due to regeneration; when the limit is infinite, there is not enough time for the static digraph to mix and the relaxation to stationarity is dictated by the regeneration only; finally, when the limit is a finite positive value we find a mixed behaviour interpolating between the two extremes. A crucial ingredient of our analysis is the control of suitable approximations for the unknown stationary distribution.

math.PR↗

arXiv2020

Loop-erased partitioning of a graph: mean-field analysis

We consider a random partition of the vertex set of an arbitrary graph that can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter $q>0$, that we see as a tuning parameter.The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. We explore the emerging macroscopic structure by analyzing 2-point correlations. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block of the random partition. This interaction potential can be seen as an affinity measure for ``densely connected nodes'' and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structures. For the latter geometry we show a phase-transition for ``community detectability'' as a function of the tuning parameter and the edge weights.

math.PR↗

arXiv2019

Stationary distribution and cover time of sparse directed configuration models

We consider sparse digraphs generated by the configuration model with given in-degree and out-degree sequences. We establish that with high probability the cover time is linear up to a poly-logarithmic correction. For a large class of degree sequences we determine the exponent $γ\ge 1$ of the logarithm and show that the cover time grows as $n\log^γ(n)$, where $n$ is the number of vertices. The results are obtained by analysing the extremal values of the stationary distribution of the digraph. In particular, we show that the stationary distribution $π$ is uniform up to a poly-logarithmic factor, and that for a large class of degree sequences the minimal values of $π$ have the form $\frac1n\log ^{1-γ}(n)$, while the maximal values of $π$ behave as $\frac1n\log ^{1-κ}(n)$ for some other exponent $κ\in[0,1]$. In passing, we prove tight bounds on the diameter of the digraphs and show that the latter coincides with the typical distance between two vertices.

math.PR↗