arXiv subjects

Luca Becchetti

Publications and source records attributed to Luca Becchetti.

At least 19 recordsLinked to original sources

arXiv2026

A Tour of Locality Sensitive Filtering on the Sphere

The Approximate Near Neighbor (ANN) problem is a cornerstone of high-dimensional data analysis. While Locality Sensitive Hashing (LSH) has been the classical paradigm, recent work has investigated Locality Sensitive Filtering (LSF), which can afford greater expressivity by allowing asymmetric regions to independently control queries and data updates. In its full generality, however, this framework can obscure the essential algorithmic ideas beneath substantial technical complexity. In this work, we bridge classical LSH and modern LSF by providing a self-contained, streamlined treatment of angular distance on the unit sphere under symmetric Gaussian filters. Although specialized, this canonical setting -- monotonically equivalent to the widely used cosine similarity -- captures all key aspects of the problem, delivering a "guided tour" of the core mechanisms of locality-sensitive filtering and allowing us to expose all fundamental algorithmic and probabilistic ingredients, while retaining transparent notation and pedagogical clarity. More precisely, we develop a treatment of Spherical-LSF that makes explicit the connection between the probabilistic behavior of Gaussian filters and the performance of the resulting data structure. Using elementary calculus, we derive sharp probability bounds and obtain a transparent proof of asymptotic optimality among schemes whose filter distribution is fixed independently of the input dataset. We further exploit the specific structure of Gaussian filters to obtain an efficient implementation while avoiding much of the technical machinery required by the general LSF framework. Overall, our results offer a self-contained and accessible entry point to locality-sensitive filtering, while providing a unified view of its main probabilistic and algorithmic principles.

cs.DS↗

arXiv2025

Approximate $2$-hop neighborhoods on incremental graphs: An efficient lazy approach

In this work, we propose, analyze and empirically validate a lazy-update approach to maintain accurate approximations of the $2$-hop neighborhoods of dynamic graphs resulting from sequences of edge insertions. We first show that under random input sequences, our algorithm exhibits an optimal trade-off between accuracy and insertion cost: it only performs $O(\frac{1}{\varepsilon})$ (amortized) updates per edge insertion, while the estimated size of any vertex's $2$-hop neighborhood is at most a factor $\varepsilon$ away from its true value in most cases, regardless of the underlying graph topology and for any $\varepsilon > 0$. As a further theoretical contribution, we explore adversarial scenarios that can force our approach into a worst-case behavior at any given time $t$ of interest. We show that while worst-case input sequences do exist, a necessary condition for them to occur is that the girth of the graph released up to time $t$ be at most $4$. Finally, we conduct extensive experiments on a collection of real, incremental social networks of different sizes, which typically have low girth. Empirical results are consistent with and typically better than our theoretical analysis anticipates. This further supports the robustness of our theoretical findings: forcing our algorithm into a worst-case behavior not only requires topologies characterized by a low girth, but also carefully crafted input sequences that are unlikely to occur in practice. Combined with standard sketching techniques, our lazy approach proves an effective and efficient tool to support key neighborhood queries on large, incremental graphs, including neighborhood size, Jaccard similarity between neighborhoods and, in general, functions of the union and/or intersection of $2$-hop neighborhoods.

cs.DS↗

arXiv2023

The Minority Dynamics and the Power of Synchronicity

We study the minority-opinion dynamics over a fully-connected network of $n$ nodes with binary opinions. Upon activation, a node receives a sample of opinions from a limited number of neighbors chosen uniformly at random. Each activated node then adopts the opinion that is least common within the received sample. Unlike all other known consensus dynamics, we prove that this elementary protocol behaves in dramatically different ways, depending on whether activations occur sequentially or in parallel. Specifically, we show that its expected consensus time is exponential in $n$ under asynchronous models, such as asynchronous GOSSIP. On the other hand, despite its chaotic nature, we show that it converges within $O(\log^2 n)$ rounds with high probability under synchronous models, such as synchronous GOSSIP. Finally, our results shed light on the bit-dissemination problem, that was previously introduced to model the spread of information in biological scenarios. Specifically, our analysis implies that the minority-opinion dynamics is the first stateless solution to this problem, in the parallel passive-communication setting, achieving convergence within a polylogarithmic number of rounds. This, together with a known lower bound for sequential stateless dynamics, implies a parallel-vs-sequential gap for this problem that is nearly quadratic in the number $n$ of nodes. This is in contrast to all known results for problems in this area, which exhibit a linear gap between the parallel and the sequential setting.

cs.DC↗

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 Role of Memory in Robust Opinion Dynamics

We investigate opinion dynamics in a fully-connected system, consisting of $n$ identical and anonymous agents, where one of the opinions (which is called correct) represents a piece of information to disseminate. In more detail, one source agent initially holds the correct opinion and remains with this opinion throughout the execution. The goal for non-source agents is to quickly agree on this correct opinion, and do that robustly, i.e., from any initial configuration. The system evolves in rounds. In each round, one agent chosen uniformly at random is activated: unless it is the source, the agent pulls the opinions of $\ell$ random agents and then updates its opinion according to some rule. We consider a restricted setting, in which agents have no memory and they only revise their opinions on the basis of those of the agents they currently sample. As restricted as it is, this setting encompasses very popular opinion dynamics, such as the voter model and best-of-$k$ majority rules. Qualitatively speaking, we show that lack of memory prevents efficient convergence. Specifically, we prove that no dynamics can achieve correct convergence in an expected number of steps that is sub-quadratic in $n$, even under a strong version of the model in which activated agents have complete access to the current configuration of the entire system, i.e., the case $\ell=n$. Conversely, we prove that the simple voter model (in which $\ell=1$) correctly solves the problem, while almost matching the aforementioned lower bound. These results suggest that, in contrast to symmetric consensus problems (that do not involve a notion of correct opinion), fast convergence on the correct opinion using stochastic opinion dynamics may indeed require the use of memory. This insight may reflect on natural information dissemination processes that rely on a few knowledgeable individuals.

cs.DC↗

arXiv2022

Biological Random Walks: multi-omics integration for disease gene prioritization

Motivation: Over the past decade, network-based approaches have proven useful in identifying disease modules within the human interactome, often providing insights into key mechanisms and guiding the quest for therapeutic targets. This is all the more important, since experimental investigation of potential gene candidates is an expensive task, thus not always a feasible option. On the other hand, many sources of biological information exist beyond the interactome and an important research direction is the design of effective techniques for their integration. Results: In this work, we introduce the Biological Random Walks (BRW) approach for disease gene prioritization in the human interactome. The proposed framework leverages multiple biological sources within an integrated framework. We perform an extensive, comparative study of BRW's performance against well-established baselines. Availability and implementation: All code is publicly available and can be downloaded at \url{https://github.com/LeoM93/BiologicalRandomWalks}. We used publicly available datasets, details on their retrieval and preprocessing are provided in the supplementary material.

q-bio.QM↗

arXiv2022

Network Based Approach to Gene Prioritization at Genome-Wide Association Study Loci

Motivation: Genome-wide association studies (GWAS) have successfully identified thousands of genetic risk loci for complex traits and diseases. Most of these GWAS loci lie in regulatory regions of the genome and the gene through which each GWAS risk locus exerts its effects is not always clear. Many computational methods utilizing biological data sources have been proposed to identify putative casual genes at GWAS loci; however, these methods can be improved upon. Results: We present the Relations-Maximization Method, a dense module searching method to identify putative causal genes at GWAS loci through the generation of candidate sub-networks derived by integrating association signals from GWAS data into the gene co-regulation network. We employ our method in a chronic obstructive pulmonary disease GWAS. We perform an extensive, comparative study of Relations-Maximization Method's performance against well-established baselines.

q-bio.QM↗

arXiv2022

On the Multidimensional Random Subset Sum Problem

In the Random Subset Sum Problem, given $n$ i.i.d. random variables $X_1, ..., X_n$, we wish to approximate any point $z \in [-1,1]$ as the sum of a suitable subset $X_{i_1(z)}, ..., X_{i_s(z)}$ of them, up to error $\varepsilon$. Despite its simple statement, this problem is of fundamental interest to both theoretical computer science and statistical mechanics. More recently, it gained renewed attention for its implications in the theory of Artificial Neural Networks. An obvious multidimensional generalisation of the problem is to consider $n$ i.i.d. $d$-dimensional random vectors, with the objective of approximating every point $\mathbf{z} \in [-1,1]^d$. In 1998, G. S. Lueker showed that, in the one-dimensional setting, $n=\mathcal{O}(\log \frac 1\varepsilon)$ samples guarantee the approximation property with high probability.In this work, we prove that, in $d$ dimensions, $n = \mathcal{O}(d^3\log \frac 1\varepsilon \cdot (\log \frac 1\varepsilon + \log d))$ samples suffice for the approximation property to hold with high probability. As an application highlighting the potential interest of this result, we prove that a recently proposed neural network model exhibits universality: with high probability, the model can approximate any neural network within a polynomial overhead in the number of parameters.

cs.NE↗

arXiv2022

Bond Percolation in Small-World Graphs with Power-Law Distribution

\emph{Full-bond percolation} with parameter $p$ is the process in which, given a graph, for every edge independently, we delete the edge with probability $1-p$. Bond percolation is motivated by problems in mathematical physics and it is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Full-bond percolation is also equivalent to the \emph{Reed-Frost process}, a network version of \emph{SIR} epidemic spreading, in which the graph represents contacts among people and $p$ corresponds to the probability that a contact between an infected person and a susceptible one causes a transmission of the infection. We consider \emph{one-dimensional power-law small-world graphs} with parameter $\alpha$ obtained as the union of a cycle with additional long-range random edges: each pair of nodes $(u,v)$ at distance $L$ on the cycle is connected by a long-range edge $(u,v)$, with probability proportional to $1/L^\alpha$. Our analysis determines three phases for the percolation subgraph $G_p$ of the small-world graph, depending on the value of $\alpha$. 1) If $\alpha < 1$, there is a $p<1$ such that, with high probability, there are $\Omega(n)$ nodes that are reachable in $G_p$ from one another in $O(\log n)$ hops; 2) If $1 < \alpha < 2$, there is a $p<1$ such that, with high probability, there are $\Omega(n)$ nodes that are reachable in $G_p$ from one another in $\log^{O(1)}(n)$ hops; 3) If $\alpha > 2$, for every $p<1$, with high probability all connected components of $G_p$ have size $O(\log n)$. The setting of full-bond percolation in finite graphs studied in this paper, which is the one that corresponds to the network SIR model of epidemic spreading, had not been analyzed before.

math.PR↗

arXiv2021

Network and Sequence-Based Prediction of Protein-Protein Interactions

Background:Typically, proteins perform key biological functions by interacting with each other. As a consequence, predicting which protein pairs interact is a fundamental problem. Experimental methods are slow, expensive, and may be error prone.Many computational methods have been proposed to identify candidate interacting pairs. When accurate, they can serve as an inexpensive, preliminary filtering stage, to be followed by downstream experimental validation. Among such methods, sequence-based ones are very promising.Results:We present, a new algorithm that leverages both topological and biological information to predict protein-protein interactions. We comprehensively compare our Framework with state-of-the-art approaches on reliable PPIs datasets, showing that they have competitive or higher accuracy on biologically validated test sets.Conclusion:We shown that topological plus sequence-based computational methods can effectively predict the entire human interactome compared with methods that leverage only one source of biological information.

q-bio.BM↗

arXiv2021

Percolation and Epidemic Processes in One-Dimensional Small-World Networks

We obtain tight thresholds for bond percolation on one-dimensional small-world graphs, and apply such results to obtain tight thresholds for the \emph{Independent Cascade} process and the \emph{Reed-Frost} process in such graphs. These are the first fully rigorous results establishing a phase transition for bond percolation and SIR epidemic processes in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative epidemiological phenomena emerge from our analysis, including the epidemic spread through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.

math.PR↗

arXiv2020

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 $\alpha$, and with probability $1 - \alpha$ 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

Expansion and Flooding in Dynamic Random Networks with Node Churn

We study expansion and information diffusion in dynamic networks, that is in networks in which nodes and edges are continuously created and destroyed. We consider information diffusion by {\em flooding}, the process by which, once a node is informed, it broadcasts its information to all its neighbors. We study models in which the network is {\em sparse}, meaning that it has $\mathcal{O}(n)$ edges, where $n$ is the number of nodes, and in which edges are created randomly, rather than according to a carefully designed distributed algorithm. In our models, when a node is "born", it connects to $d=\mathcal{O}(1)$ random other nodes. An edge remains alive as long as both its endpoints do. If no further edge creation takes place, we show that, although the network will have $\Omega_d(n)$ isolated nodes, it is possible, with large constant probability, to inform a $1-exp(-\Omega(d))$ fraction of nodes in $\mathcal{O}(\log n)$ time. Furthermore, the graph exhibits, at any given time, a "large-set expansion" property. We also consider models with {\em edge regeneration}, in which if an edge $(v,w)$ chosen by $v$ at birth goes down because of the death of $w$, the edge is replaced by a fresh random edge $(v,z)$. In models with edge regeneration, we prove that the network is, with high probability, a vertex expander at any given time, and flooding takes $\mathcal{O}(\log n)$ time. The above results hold both for a simple but artificial streaming model of node churn, in which at each time step one node is born and the oldest node dies, and in a more realistic continuous-time model in which the time between births is Poisson and the lifetime of each node follows an exponential distribution.

cs.DC↗

arXiv2020

Biological Random Walks: integrating heterogeneous data in disease gene prioritization

This work proposes a unified framework to leverage biological information in network propagation-based gene prioritization algorithms. Preliminary results on breast cancer data show significant improvements over state-of-the-art baselines, such as the prioritization of genes that are not identified as potential candidates by interactome-based algorithms, but that appear to be involved in/or potentially related to breast cancer, according to a functional analysis based on recent literature.

q-bio.MN↗

arXiv2019

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

Principal Fairness: Removing Bias via Projections

Reducing hidden bias in the data and ensuring fairness in algorithmic data analysis has recently received significant attention. We complement several recent papers in this line of research by introducing a general method to reduce bias in the data through random projections in a "fair" subspace. We apply this method to densest subgraph problem. For densest subgraph, our approach based on fair projections allows to recover both theoretically and empirically an almost optimal, fair, dense subgraph hidden in the input data. We also show that, under the small set expansion hypothesis, approximating this problem beyond a factor of 2 is NP-hard and we show a polynomial time algorithm with a matching approximation bound.

cs.DS↗

arXiv2018

Finding a Bounded-Degree Expander Inside a Dense One

It follows from the Marcus-Spielman-Srivastava proof of the Kadison-Singer conjecture that if $G=(V,E)$ is a $\Delta$-regular dense expander then there is an edge-induced subgraph $H=(V,E_H)$ of $G$ of constant maximum degree which is also an expander. As with other consequences of the MSS theorem, it is not clear how one would explicitly construct such a subgraph. We show that such a subgraph (although with quantitatively weaker expansion and near-regularity properties than those predicted by MSS) can be constructed with high probability in linear time, via a simple algorithm. Our algorithm allows a distributed implementation that runs in $\mathcal O(\log n)$ rounds and does $\mathcal O(n)$ total work with high probability. The analysis of the algorithm is complicated by the complex dependencies that arise between edges and between choices made in different rounds. We sidestep these difficulties by following the combinatorial approach of counting the number of possible random choices of the algorithm which lead to failure. We do so by a compression argument showing that such random choices can be encoded with a non-trivial compression. Our algorithm bears some similarity to the way agents construct a communication graph in a peer-to-peer network, and, in the bipartite case, to the way agents select servers in blockchain protocols.

cs.DC↗

arXiv2018

Pooling or Sampling: Collective Dynamics for Electrical Flow Estimation

The computation of electrical flows is a crucial primitive for many recently proposed optimization algorithms on weighted networks. While typically implemented as a centralized subroutine, the ability to perform this task in a fully decentralized way is implicit in a number of biological systems. Thus, a natural question is whether this task can provably be accomplished in an efficient way by a network of agents executing a simple protocol. We provide a positive answer, proposing two distributed approaches to electrical flow computation on a weighted network: a deterministic process mimicking Jacobi's iterative method for solving linear systems, and a randomized token diffusion process, based on revisiting a classical random walk process on a graph with an absorbing node. We show that both processes converge to a solution of Kirchhoff's node potential equations, derive bounds on their convergence rates in terms of the weights of the network, and analyze their time and message complexity.

cs.DC↗