arXiv subjects

Michele Salvi

Publications and source records attributed to Michele Salvi.

At least 19 recordsLinked to original sources

arXiv2026

Random spanning trees in random environment

We introduce a new spanning tree model called the random spanning tree in random environment (RSTRE), which interpolates between the uniform spanning tree and the minimum spanning tree as the inverse temperature (disorder strength) $β$ varies. On the complete graph with $n$ vertices and i.i.d.\ uniform disorder variables on the edges, we identify: (1) a low disorder regime with $β\leq C n/\log n$, where the diameter of the random spanning tree is typically of order $n^{1/2}$, the same as for the uniform spanning tree; (2) a high disorder regime with $β\geq n^{4/3} \log n$, where the diameter is typically of order $n^{1/3}$, the same as for the minimum spanning tree. We conjecture that for $β=n^α$ with $α\in (1, 4/3)$, the diameter is of order $n^{γ+o(1)}$ for some $γ=γ(α)$ strictly between $1/2$ and $1/3$.

math.PR↗

arXiv2026

Fractal and Regular Geometry of Deep Neural Networks

We study the geometric properties of random neural networks by investigating the boundary volumes of their excursion sets for different activation functions, as the depth increases. More specifically, we show that, for activations which are not very regular (e.g., the Heaviside step function), the boundary volumes exhibit fractal behavior, with their Hausdorff dimension monotonically increasing with the depth. On the other hand, for activations which are more regular (e.g., ReLU, logistic and $\tanh$), as the depth increases, the expected boundary volumes can either converge to zero, remain constant or diverge exponentially, depending on a single spectral parameter which can be easily computed. Our theoretical results are confirmed in some numerical experiments based on Monte Carlo simulations.

math.PR↗

arXiv2025

Scaling limit of the complex mobility matrix for the random conductance model on $\mathbb{T}^d_N$

We consider a continuous-time random walk on the $d$-dimensional torus $\mathbb{T}^d_{N}=\mathbb{Z}^d/N \mathbb{Z}^d$, possibly with long-range, but finite, jumps. The law of the jumps is regulated by a random environment $ξ$ yielding a stationary and ergodic field of random conductances. The complex mobility matrix $σ_N^ξ(ω)$ measures the linear response of the random walk to a $\cos(ωt)$-type oscillating external field. By investigating the homogenization properties of the medium, and assuming in addition that the conductances have finite second moment, we show that, for almost every realization of the environment $ξ$, the complex mobility matrix $σ_N^ξ(ω)$ converges as $N\to+\infty$ to a deterministic limiting matrix $σ(ω)$ and provide different characterizations of $σ(ω)$.

math.PR↗

arXiv2025

Payment-failure times for random Lightning paths

We study a random process over graphs inspired by the way payments are executed in the Lightning Network, the main layer-two solution on top of Bitcoin. We first prove almost tight upper and lower bounds on the time it takes for a payment failure to occur, as a function of the number of nodes and the edge capacities, when the underlying graph is complete. Then, we show how such a random process is related to the edge-betweenness centrality measure and we prove upper and lower bounds for arbitrary graphs as a function of edge-betweenness and capacity. Finally, we validate our theoretical results by running extensive simulations over some classes of graphs, including snapshots of the real Lightning Network.

cs.NI↗

arXiv2025

Threshold-Driven Streaming Graph: Expansion and Rumor Spreading

A randomized distributed algorithm called RAES was introduced in [Becchetti et al., SODA 2020] to extract a bounded-degree expander from a dense $n$-vertex expander graph $G = (V, E)$. The algorithm relies on a simple threshold-based procedure. A key assumption in [Becchetti et al., SODA 2020] is that the input graph $G$ is static - i.e., both its vertex set $V$ and edge set $E$ remain unchanged throughout the process - while the analysis of RAES in dynamic models is left as a major open question. In this work, we investigate the behavior of RAES under a dynamic graph model induced by a streaming node-churn process (also known as the sliding window model), where, at each discrete round, a new node joins the graph and the oldest node departs. This process yields a bounded-degree dynamic graph $\mathcal{G} =\{ G_t = (V_t, E_t) : t \in \mathbb{N}\}$ that captures essential characteristics of peer-to-peer networks -- specifically, node churn and threshold on the number of connections each node can manage. We prove that every snapshot $G_t$ in the dynamic graph sequence has good expansion properties with high probability. Furthermore, we leverage this property to establish a logarithmic upper bound on the completion time of the well-known PUSH and PULL rumor spreading protocols over the dynamic graph $\mathcal{G}$.

cs.DC↗

arXiv2025

Spectral complexity of deep neural networks

It is well-known that randomly initialized, push-forward, fully-connected neural networks weakly converge to isotropic Gaussian processes, in the limit where the width of all layers goes to infinity. In this paper, we propose to use the angular power spectrum of the limiting field to characterize the complexity of the network architecture. In particular, we define sequences of random variables associated with the angular power spectrum, and provide a full characterization of the network complexity in terms of the asymptotic distribution of these sequences as the depth diverges. On this basis, we classify neural networks as low-disorder, sparse, or high-disorder; we show how this classification highlights a number of distinct features for standard activation functions, and in particular, sparsity properties of ReLU networks. Our theoretical results are also validated by numerical simulations.

stat.ML↗

arXiv2025

The contact process on Scale-Free Percolation

We consider the contact process on scale-free percolation, a spatial random graph model where the degree distribution of the vertices follows a power law with exponent $β$. We study the extinction time $τ_{G_n}$ of the contact process on the graph restricted to a d-dimensional box of volume n, starting from full occupancy. In the regime $β\in (2, 3)$, where the degrees have finite mean but infinite variance and the graph exhibits the ultra-small world behaviour, we adapt the techniques of [Linker et al., 2021] to show that $τ_{G_n}$ is exponential in n. Our main contribution, though, deals with the case $β\geq 3$, where the degrees have finite variance and the graph is small-world. We prove that also in this case $τ_{G_n}$ grows exponentially, at least up to a logarithmic correction reflecting the sparser graph structure. The proof requires the generalization of a result from [Mountford et al., 2016] and combines a multi-scale analysis of the graph, the study of the chemical distance between vertices and percolation arguments.

math.PR↗

arXiv2025

The spectrum of dense kernel-based random graphs

Kernel-based random graphs (KBRGs) are a broad class of random graph models that account for inhomogeneity among vertices. We consider KBRGs on a discrete $d-$dimensional torus $\mathbf{V}_N$ of size $N^d$. Conditionally on an i.i.d.~sequence of {Pareto} weights $(W_i)_{i\in \mathbf{V}_N}$ with tail exponent $τ-1>0$, we connect any two points $i$ and $j$ on the torus with probability $$p_{ij}= \frac{κ_σ(W_i,W_j)}{\|i-j\|^α} \wedge 1$$ for some parameter $α>0$ and $κ_σ(u,v)= (u\vee v)(u \wedge v)^σ$ for some $σ\in(0,τ-1)$. We focus on the adjacency operator of this random graph and study its empirical spectral distribution. For $α 2$, we show that a non-trivial limiting distribution exists as $N\to\infty$ and that the corresponding measure $μ_{σ,τ}$ is absolutely continuous with respect to the Lebesgue measure. $μ_{σ,τ}$ is given by an operator-valued semicircle law, whose Stieltjes transform is characterised by a fixed point equation in an appropriate Banach space. We analyse the moments of $μ_{σ,τ}$ and prove that the second moment is finite even when the weights have infinite variance. In the case $σ=1$, corresponding to the so-called scale-free percolation random graph, we can explicitly describe the limiting measure and study its tail.

math.PR↗

arXiv2024

Diameter of uniform spanning trees on random weighted graphs

For any edge weight distribution, we consider the uniform spanning tree (UST) on finite graphs with i.i.d. random edge weights. We show that, for bounded degree expander graphs and finite boxes of ${\mathbb Z}^d$, the diameter of the UST is of order $n^{1/2+o(1)}$ with high probability, where $n$ is the number of vertices.

math.PR↗

arXiv2024

A Systematization of the Wagner Framework: Graph Theory Conjectures and Reinforcement Learning

In 2021, Adam Zsolt Wagner proposed an approach to disprove conjectures in graph theory using Reinforcement Learning (RL). Wagner's idea can be framed as follows: consider a conjecture, such as a certain quantity f(G) < 0 for every graph G; one can then play a single-player graph-building game, where at each turn the player decides whether to add an edge or not. The game ends when all edges have been considered, resulting in a certain graph G_T, and f(G_T) is the final score of the game; RL is then used to maximize this score. This brilliant idea is as simple as innovative, and it lends itself to systematic generalization. Several different single-player graph-building games can be employed, along with various RL algorithms. Moreover, RL maximizes the cumulative reward, allowing for step-by-step rewards instead of a single final score, provided the final cumulative reward represents the quantity of interest f(G_T). In this paper, we discuss these and various other choices that can be significant in Wagner's framework. As a contribution to this systematization, we present four distinct single-player graph-building games. Each game employs both a step-by-step reward system and a single final score. We also propose a principled approach to select the most suitable neural network architecture for any given conjecture, and introduce a new dataset of graphs labeled with their Laplacian spectra. Furthermore, we provide a counterexample for a conjecture regarding the sum of the matching number and the spectral radius, which is simpler than the example provided in Wagner's original paper. The games have been implemented as environments in the Gymnasium framework, and along with the dataset, are available as open-source supplementary materials.

cs.LG↗

arXiv2023

A Quantitative Functional Central Limit Theorem for Shallow Neural Networks

We prove a Quantitative Functional Central Limit Theorem for one-hidden-layer neural networks with generic activation function. The rates of convergence that we establish depend heavily on the smoothness of the activation function, and they range from logarithmic in non-differentiable cases such as the Relu to $\sqrt{n}$ for very regular activations. Our main tools are functional versions of the Stein-Malliavin approach; in particular, we exploit heavily a quantitative functional central limit theorem which has been recently established by Bourguin and Campese (2020).

math.PR↗

arXiv2023

Branching process and homogeneization for epidemics on spatial random graphs

Consider a graph where the sites are distributed in space according to a Poisson point process on $\mathbb R^n$. We study a population evolving on this network, with individuals jumping between sites with a rate which decreases exponentially in the distance. Individuals give also birth (infection) and die (recovery) at constant rate on each site. First, we construct the process, showing that it is well-posed even when starting from non-bounded initial conditions. Secondly, we prove hydrodynamic limits in a diffusive scaling. The limiting process follows a deterministic reaction diffusion equation. We use stochastic homogenization to characterize its diffusion coefficient as the solution of a variational principle. The proof involves in particular the extension of a classic Kipnis-Varadhan estimate to cope with the non-reversiblity of the process, due to births and deaths. This work is motivated by the approximation of epidemics on large networks and the results are extended to more complex graphs including percolation of edges.

math.PR↗

arXiv2021

Scale-free percolation mixing time

Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index $τ-1>0$. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent $α>0$ in their distance. The resulting graph is called scale-free percolation. The goal of this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in $α$ and $τ$. In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each phase, the most interesting of which is a bootstrap procedure to reduce the model from a phase where the degrees have bounded averages to a setting with unbounded averages.

math.PR↗

arXiv2019

Scaling Limit of Sub-ballistic 1D Random Walk among Biased Conductances: a Story of Wells and Walls

We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at $+\infty$ or at $0$. We prove that the scaling limit of the process is the inverse of an $α$-stable subordinator, which indicates an aging phenomenon, expressed in terms of the generalized arcsine law. In analogy with the case of an i.i.d. random environment studied in details in [Enriquez, Sabot, Zindy, Bull. Soc. Math. 2009; Enriquez, Sabot, Tournier, Zindy, Ann. Appl. Probab. 2013], some `traps' are responsible for the slowdown of the random walk. However, the phenomenology is somehow different (and richer) here. In particular, three types of traps may occur, depending on the fine properties of the tails of the conductances: (i) a very large conductance (a well in the potential); (ii) a very small conductance (a wall in the potential); (iii) the combination of a large conductance followed shortly after by a small conductance (a well-and-wall in the potential).

math.PR↗

arXiv2019

Scale-free percolation in continuum space: quenched degree and clustering coefficient

Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuum space version of scale-free percolation introduced in [DW18]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in $\mathbb R^d$. Each vertex is equipped with a random weight and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the the annealed clustering coefficient of one point, which is a strictly positive quantity.

math.PR↗

arXiv2018

Regularity of biased 1D random walks in random environment

We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity $λ\in\mathbb{R}$. For ergodic shift-invariant environments, we show that the limiting velocity $v(λ)$ is always increasing and that it is everywhere analytic except at most in two points $λ_-$ and $λ_+$. When $λ_-$ and $λ_+$ are distinct, $v(λ)$ might fail to be continuous. We refine the assumptions in \cite{Z} for having a recentered CLT with diffusivity $σ^2(λ)$ and give explicit conditions for $σ^2(λ)$ to be analytic. For the random conductance model we show that, in contrast with the deterministic case, $σ^2(λ)$ is not monotone on the positive (resp.~negative) half-line and that it is not differentiable at $λ=0$. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of \cite{LD16}.

math.PR↗

arXiv2017

Scaling of sub-ballistic 1D Random Walks among biased Random Conductances

We consider two models of one-dimensional random walks among biased i.i.d. random conductances: the first is the classical exponential tilt of the conductances, while the second comes from the effect of adding an external field to a random walk on a point process (the bias depending on the distance between points). We study the case when the walk is transient to the right but sub-ballistic, and identify the correct scaling of the random walk: we find $α\in[0,1]$ such that $\log X_n / \log n \to α$. Interestingly, $α$ does not depend on the intensity of the bias in the first case, but it does in the second case.

math.PR↗

arXiv2016

The velocity of 1D Mott variable range hopping with external field

Mott variable range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps. We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias.

math.PR↗