arXiv subjects

Emilio Cruciani

Publications and source records attributed to Emilio Cruciani.

16 recordsLinked to original sources

arXiv2026

The Careless Coupon Collector's Problem

We initiate the study of the Careless Coupon Collector's Problem (CCCP), a novel variation of the classical coupon collector, that we envision as a model for information systems such as web crawlers, dynamic caches, and fault-resilient networks. In CCCP, a collector attempts to gather $n$ distinct coupon types by obtaining one coupon type uniformly at random in each discrete round, however the collector is \textit{careless}: at the end of each round, each collected coupon type is independently lost with probability $p$. We analyze the number of rounds required to complete the collection as a function of $n$ and $p$. In particular, we show that it transitions from $\Theta(n \ln n)$ when $p = o\big(\frac{\ln n}{n^2}\big)$ up to $\Theta\big((\frac{np}{1-p})^n\big)$ when $p=\omega\big(\frac{1}{n}\big)$ in multiple distinct phases. Interestingly, when $p=\frac{c}{n}$, the process remains in a metastable phase, where the fraction of collected coupon types is concentrated around $\frac{1}{1+c}$ with probability $1-o(1)$, for a time window of length $e^{\Theta(n)}$. Finally, we give an algorithm that computes the expected completion time of CCCP in $O(n^2)$ time.

cs.DM↗

arXiv2026

Incremental (k, z)-Clustering on Graphs

Given a weighted undirected graph, a number of clusters $k$, and an exponent $z$, the goal in the $(k, z)$-clustering problem on graphs is to select $k$ vertices as centers that minimize the sum of the distances raised to the power $z$ of each vertex to its closest center. In the dynamic setting, the graph is subject to adversarial edge updates, and the goal is to maintain explicitly an exact $(k, z)$-clustering solution in the induced shortest-path metric. While efficient dynamic $k$-center approximation algorithms on graphs exist [Cruciani et al. SODA 2024], to the best of our knowledge, no prior work provides similar results for the dynamic $(k,z)$-clustering problem. As the main result of this paper, we develop a randomized incremental $(k, z)$-clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of $\tilde O(k m^{1+o(1)}+ k^{1+\frac{1}{\lambda}} m)$, where $\lambda \geq 1$ is an arbitrary fixed constant. Our incremental algorithm consists of two stages. In the first stage, we maintain a constant-factor bicriteria approximate solution of size $\tilde{O}(k)$ with a total update time of $m^{1+o(1)}$ over all adversarial edge insertions. This first stage is an intricate adaptation of the bicriteria approximation algorithm by Mettu and Plaxton [Machine Learning 2004] to incremental graphs. One of our key technical results is that the radii in their algorithm can be assumed to be non-decreasing while the approximation ratio remains constant, a property that may be of independent interest. In the second stage, we maintain a constant-factor approximate $(k,z)$-clustering solution on a dynamic weighted instance induced by the bicriteria approximate solution. For this subproblem, we employ a dynamic spanner algorithm together with a static $(k,z)$-clustering algorithm.

cs.DS↗

arXiv2025

A Phase Transition for Opinion Dynamics with Competing Biases

We study a nonlinear dynamics of binary opinions in a population of agents connected by a directed network, influenced by two competing forces. On the one hand, agents are stubborn, i.e., have a tendency for one of the two opinions; on the other hand, there is a disruptive bias, $p\in[0,1]$, that drives the agents toward the other opinion. The disruptive bias models external factors, such as market innovations or social controllers, aiming to challenge the status quo, while agents' stubbornness reinforces the initial opinion making it harder for the external bias to drive the process toward change. Each agent updates its opinion according to a nonlinear function of the states of its neighbors and of the bias $p$. We consider the case of random directed graphs with prescribed in- and out-degree sequences and we prove that the dynamics exhibits a phase transition: when the disruptive bias $p$ is larger than a critical threshold $p_c$, the population converges in constant time to a consensus on the disruptive opinion. Conversely, when the bias $p$ is less than $p_c$, the system enters a metastable state in which only a fraction of agents $q_\star(p)<1$ will share the new opinion for a long time. We characterize $p_c$ and $q_\star(p)$ explicitly, showing that they only depend on few simple statistics of the degree sequences. Our analysis relies on a dual system of branching, coalescing, and dying particles, which we show exhibits equivalent behavior and allows a rigorous characterization of the system's dynamics. Our results characterize the interplay between the degree of the agents, their stubbornness, and the external bias, shedding light on the tipping points of opinion dynamics in networks.

cs.SI↗

arXiv2025

Opinion Dynamics: A Comprehensive Overview

Opinion dynamics, the evolution of individuals through social interactions, is an important area of research with applications ranging from politics to marketing. Due to its interdisciplinary relevance, studies of opinion dynamics remain fragmented across computer science, mathematics, the social sciences, and physics, and often lack shared frameworks. This survey bridges these gaps by reviewing well-known models of opinion dynamics within a unified framework and categorizing them into distinct classes based on their properties. Furthermore, the key findings on these models are covered in three parts: convergence properties, viral marketing, and user characteristics. We first analyze the final configuration (consensus vs polarized) and convergence time for each model. We then review the main algorithmic, complexity, and combinatorial results in the context of viral marketing. Finally, we explore how node characteristics, such as stubbornness, activeness, or neutrality, shape diffusion outcomes. By unifying terminology, methods, and challenges across disciplines, this paper aims to foster cross-disciplinary collaboration and accelerate progress in understanding and harnessing opinion dynamics.

cs.SI↗

arXiv2025

Choosing to Be Green: Advancing Green AI via Dynamic Model Selection

Artificial Intelligence is increasingly pervasive across domains, with ever more complex models delivering impressive predictive performance. This fast technological advancement however comes at a concerning environmental cost, with state-of-the-art models - particularly deep neural networks and large language models - requiring substantial computational resources and energy. In this work, we present the intuition of Green AI dynamic model selection, an approach based on dynamic model selection that aims at reducing the environmental footprint of AI by selecting the most sustainable model while minimizing potential accuracy loss. Specifically, our approach takes into account the inference task, the environmental sustainability of available models, and accuracy requirements to dynamically choose the most suitable model. Our approach presents two different methods, namely Green AI dynamic model cascading and Green AI dynamic model routing. We demonstrate the effectiveness of our approach via a proof of concept empirical example based on a real-world dataset. Our results show that Green AI dynamic model selection can achieve substantial energy savings (up to ~25%) while substantially retaining the accuracy of the most energy greedy solution (up to ~95%). As conclusion, our preliminary findings highlight the potential that hybrid, adaptive model selection strategies withhold to mitigate the energy demands of modern AI systems without significantly compromising accuracy requirements.

cs.CY↗

arXiv2025

Towards Constant Time Multi-Call Rumor Spreading on Small-Set Expanders

We study a multi-call variant of the classic PUSH&PULL rumor spreading process where nodes can contact $k$ of their neighbors instead of a single one during both PUSH and PULL operations. We show that rumor spreading can be made faster at the cost of an increased amount of communication between the nodes. As a motivating example, consider the process on a complete graph of $n$ nodes: while the standard PUSH&PULL protocol takes $\Theta(\log n)$ rounds, we prove that our $k$-PUSH&PULL variant completes in $\Theta(\log_{k} n)$ rounds, with high probability. We generalize this result in an expansion-sensitive way, as has been done for the classic PUSH&PULL protocol for different notions of expansion, e.g., conductance and vertex expansion. We consider small-set vertex expanders, graphs in which every sufficiently small subset of nodes has a large neighborhood, ensuring strong local connectivity. In particular, when the expansion parameter satisfies $\phi > 1$, these graphs have a diameter of $o(\log n)$, as opposed to other standard notions of expansion. Since the graph's diameter is a lower bound on the number of rounds required for rumor spreading, this makes small-set expanders particularly well-suited for fast information dissemination. We prove that $k$-PUSH&PULL takes $O(\log_{\phi} n \cdot \log_{k} n)$ rounds in these expanders, with high probability. We complement this with a simple lower bound of $\Omega(\log_{\phi} n+ \log_{k} n)$ rounds.

cs.DS↗

arXiv2023

Dynamic algorithms for k-center on graphs

In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to its closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on point sets in arbitrary metric spaces. In this paper, we give a deterministic decremental $(2+\epsilon)$-approximation algorithm and a randomized incremental $(4+\epsilon)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+\epsilon)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art fully dynamic $(1+\epsilon)$-approximation single-source shortest paths algorithm in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+\epsilon)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances.

cs.DS↗

arXiv2023

On a Voter Model with Context-Dependent Opinion Adoption

Opinion diffusion is a crucial phenomenon in social networks, often underlying the way in which a collective of agents develops a consensus on relevant decisions. The voter model is a well-known theoretical model to study opinion spreading in social networks and structured populations. Its simplest version assumes that an updating agent will adopt the opinion of a neighboring agent chosen at random. The model allows us to study, for example, the probability that a certain opinion will fixate into a consensus opinion, as well as the expected time it takes for a consensus opinion to emerge. Standard voter models are oblivious to the opinions held by the agents involved in the opinion adoption process. We propose and study a context-dependent opinion spreading process on an arbitrary social graph, in which the probability that an agent abandons opinion $a$ in favor of opinion $b$ depends on both $a$ and $b$. We discuss the relations of the model with existing voter models and then derive theoretical results for both the fixation probability and the expected consensus time for two opinions, for both the synchronous and the asynchronous update models.

cs.MA↗

arXiv2023

On the convergence of nonlinear averaging dynamics with three-body interactions on hypergraphs

Complex networked systems in fields such as physics, biology, and social sciences often involve interactions that extend beyond simple pairwise ones. Hypergraphs serve as powerful modeling tools for describing and analyzing the intricate behaviors of systems with multi-body interactions. Herein, we investigate a discrete-time nonlinear averaging dynamics with three-body interactions: an underlying hypergraph, comprising triples as hyperedges, delineates the structure of these interactions, while the vertices update their states through a weighted, state-dependent average of neighboring pairs' states. This dynamics captures reinforcing group effects, such as peer pressure, and exhibits higher-order dynamical effects resulting from a complex interplay between initial states, hypergraph topology, and nonlinearity of the update. Differently from linear averaging dynamics on graphs with two-body interactions, this model does not converge to the average of the initial states but rather induces a shift. By assuming random initial states and by making some regularity and density assumptions on the hypergraph, we prove that the dynamics converges to a multiplicatively-shifted average of the initial states, with high probability. We further characterize the shift as a function of two parameters describing the initial state and interaction strength, as well as the convergence time as a function of the hypergraph structure.

math.DS↗

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↗

arXiv2021

Biased Opinion Dynamics: When the Devil Is in the Details

We investigate opinion dynamics in multi-agent networks when a bias toward one of two possible opinions exists; for example, reflecting a status quo vs a superior alternative. Starting with all agents sharing an initial opinion representing the status quo, the system evolves in steps. In each step, one agent selected uniformly at random adopts the superior opinion with some probability $α$, and with probability $1 - α$ it follows an underlying update rule to revise its opinion on the basis of those held by its neighbors. We analyze convergence of the resulting process under two well-known update rules, namely majority and voter. The framework we propose exhibits a rich structure, with a non-obvious interplay between topology and underlying update rule. For example, for the voter rule we show that the speed of convergence bears no significant dependence on the underlying topology, whereas the picture changes completely under the majority rule, where network density negatively affects convergence. We believe that the model we propose is at the same time simple, rich, and modular, affording mathematical characterization of the interplay between bias, underlying opinion dynamics, and social structure in a unified setting.

cs.MA↗

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

Election Control through Social Influence with Unknown Preferences

The election control problem through social influence asks to find a set of nodes in a social network of voters to be the starters of a political campaign aiming at supporting a given target candidate. Voters reached by the campaign change their opinions on the candidates. The goal is to shape the diffusion of the campaign in such a way that the chances of victory of the target candidate are maximized. Previous work shows that the problem can be approximated within a constant factor in several models of information diffusion and voting systems, assuming that the controller, i.e., the external agent that starts the campaign, has full knowledge of the preferences of voters. However this information is not always available since some voters might not reveal it. Herein we relax this assumption by considering that each voter is associated with a probability distribution over the candidates. We propose two models in which, when an electoral campaign reaches a voter, this latter modifies its probability distribution according to the amount of influence it received from its neighbors in the network. We then study the election control problem through social influence on the new models: In the first model, under the Gap-ETH, election control cannot be approximated within a factor better than $1/n^{o(1)}$, where $n$ is the number of voters; in the second model, which is a slight relaxation of the first one, the problem admits a constant factor approximation algorithm.

cs.SI↗

arXiv2020

Step-by-Step Community Detection in Volume-Regular Graphs

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph). Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden $k$ partition and characterized by a milder form of volume-regularity. We show that the class of $k$-volume-regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits $k$ "stepwise" eigenvectors (i.e., vectors that are constant over each set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first $k$ eigenvalues and the gap between the $k$-th and the ($k$+1)-th eigenvalues is sufficiently large, the averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability.

cs.DC↗

arXiv2019

Exploiting Social Influence to Control Elections Based on Scoring Rules

Online social networks are used to diffuse opinions and ideas among users, enabling a faster communication and a wider audience. The way in which opinions are conditioned by social interactions is usually called social influence. Social influence is extensively used during political campaigns to advertise and support candidates. Herein we consider the problem of exploiting social influence in a network of voters in order to change their opinion about a target candidate with the aim of increasing his chance to win/lose the election in a wide range of voting systems. We introduce the Linear Threshold Ranking, a natural and powerful extension of the well-established Linear Threshold Model, which describes the change of opinions taking into account the amount of exercised influence. We are able to maximize the score of a target candidate up to a factor of $1-1/e$ by showing submodularity. We exploit such property to provide a $\frac{1}{3}(1-1/e)$-approximation algorithm for the constructive election control problem. Similarly, we get a $\frac{1}{2}(1-1/e)$-approximation ratio in the destructive scenario. The algorithm can be used in arbitrary scoring rule voting systems, including plurality rule and borda count. Finally, we perform an experimental study on real-world networks, measuring Probability of Victory (PoV) and Margin of Victory (MoV) of the target candidate, to validate the model and to test the capability of the algorithm.

cs.SI↗

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↗