arXiv subjects

Paola Bermolen

Publications and source records attributed to Paola Bermolen.

12 recordsLinked to original sources

arXiv2026

Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another

Hierarchical graphs embed in hyperbolic space with lower distortion than in Euclidean space owing to its negative curvature. However, their gradient-based learning is hampered at large radii, where the Poincaré ball and the Lorentz hyperboloid models fail numerically. Polar coordinates avoid this problem, but the hyperbolic metric scales the angular step by the hyperbolic sine of the radius, freezing angular motion. We observe that this factor is a choice, silently fixed by existing implementations: the Euclidean tangent parametrization, for instance, uses the radius itself. We show that other choices are not only possible but preferable. They are endpoints of a one-parameter family of optimization preconditioners with curvatures from $-1$ to $0$, while the embedding remains at curvature $-1$. We show that since the Euclidean preconditioner rearranges a layout but refines it poorly, while an intermediate one refines far better once a layout is in place, combining them in two stages reduces the loss on real-world trees by 46-74% over the best single curvature.

cs.LG↗

arXiv2026

Weighted Random Dot Product Graphs

Modeling of intricate relational patterns has become a cornerstone of contemporary statistical research and related data science fields. Networks, represented as graphs, offer a natural framework for this analysis. This paper extends the Random Dot Product Graph (RDPG) model to accommodate weighted graphs, markedly broadening the model's scope to scenarios where edges exhibit heterogeneous weight distributions. We propose a nonparametric weighted (W)RDPG model that assigns a sequence of latent positions to each node. Inner products of these nodal vectors specify the moments of their incident edge weights' distribution via moment-generating functions. In this way, and unlike prior art, the WRDPG can discriminate between weight distributions that share the same mean but differ in other higher-order moments. We derive statistical guarantees for an estimator of the nodal's latent positions adapted from the workhorse adjacency spectral embedding, establishing its consistency and asymptotic normality. We also contribute a generative framework that enables sampling of graphs that adhere to a (prescribed or data-fitted) WRDPG, facilitating, e.g., the analysis and testing of observed graph metrics using judicious reference distributions. The paper is organized to formalize the model's definition, the estimation (or nodal embedding) process and its guarantees, as well as the methodologies for generating weighted graphs, all complemented by illustrative and reproducible examples showcasing the WRDPG's effectiveness in various network analytic applications.

stat.ML↗

arXiv2026

Efficiency of Parallel and Restart Exploration Strategies in Model Free Stochastic Simulations

We analyze the efficiency of parallelization and restart mechanisms for stochastic simulations in model-free settings, where the underlying system dynamics are unknown. Such settings are common in Reinforcement Learning (RL) and rare event estimation, where standard variance-reduction techniques like importance sampling are inapplicable. Focusing on the challenge of reaching rare states under a finite computational budget, we model exploration via random walks and Lévy processes. Based on rigorous probability analysis, our work reveals a phase transition in the success probability as a function of the number of parallel simulations: an optimal number $N^*$ exists, balancing exploration diversity and time allocation per simulation. Beyond this threshold, performance degrades exponentially. Furthermore, we demonstrate that a restart strategy, which reallocates resources from stagnant trajectories to promising regions, can yield an exponential improvement in success probability. In the context of RL, these strategies can improve policy gradient methods by enabling more efficient state-space exploration, leading to more accurate policy gradient estimates.

math.PR↗

arXiv2026

Large Deviations and the Peano Phenomenon in Stochastic Differential Equations with Homogeneous Drift

We consider a diffusion equation in $\mathbb{R}^d$ with drift equal to the gradient of a homogeneous potential of degree $1+γ$, with $0<γ<1$, and local variance equal to $\varepsilon^2$ with $\varepsilon\to 0$. The associated deterministic system for $\varepsilon=0$ has a potential that is not a Lipschitz function at the origin. Therefore, an infinite number of solutions exist, known as the Peano phenomenon. In this work, we study first- and second-order large deviations for a noisy system, generalizing previous results for the specific potential $b(x)=x |x|^{γ-1}$. For the first-order large deviations, we recover the rate function from the well-known Freidlin-Wentzell work. For the second-order large deviation, we use a refinement of Carmona-Simon bounds for the eigenfunctions of a Schrödinger operator and prove that the exponential behavior of the process depends only on the ground state of such an operator. Moreover, a refined study of the ground state allows us to obtain the large deviation rate function explicitly and to deduce that the family of diffusions converges to the set of extreme solutions of the deterministic system.

math.PR↗

arXiv2023

Gradient-Based Spectral Embeddings of Random Dot Product Graphs

The Random Dot Product Graph (RDPG) is a generative model for relational data, where nodes are represented via latent vectors in low-dimensional Euclidean space. RDPGs crucially postulate that edge formation probabilities are given by the dot product of the corresponding latent positions. Accordingly, the embedding task of estimating these vectors from an observed graph is typically posed as a low-rank matrix factorization problem. The workhorse Adjacency Spectral Embedding (ASE) enjoys solid statistical properties, but it is formally solving a surrogate problem and can be computationally intensive. In this paper, we bring to bear recent advances in non-convex optimization and demonstrate their impact to RDPG inference. We advocate first-order gradient descent methods to better solve the embedding problem, and to organically accommodate broader network embedding applications of practical relevance. Notably, we argue that RDPG embeddings of directed graphs loose interpretability unless the factor matrices are constrained to have orthogonal columns. We thus develop a novel feasible optimization method in the resulting manifold. The effectiveness of the graph representation learning framework is demonstrated on reproducible experiments with both synthetic and real network data. Our open-source algorithm implementations are scalable, and unlike the ASE they are robust to missing edge data and can track slowly-varying latent positions from streaming graphs.

cs.LG↗

arXiv2022

Online Change Point Detection for Weighted and Directed Random Dot Product Graphs

Given a sequence of random (directed and weighted) graphs, we address the problem of online monitoring and detection of changes in the underlying data distribution. Our idea is to endow sequential change-point detection (CPD) techniques with a graph representation learning substrate based on the versatile Random Dot Product Graph (RDPG) model. We consider efficient, online updates of a judicious monitoring function, which quantifies the discrepancy between the streaming graph observations and the nominal RDPG. This reference distribution is inferred via spectral embeddings of the first few graphs in the sequence. We characterize the distribution of this running statistic to select thresholds that guarantee error-rate control, and under simplifying approximations we offer insights on the algorithm's detection resolution and delay. The end result is a lightweight online CPD algorithm, that is also explainable by virtue of the well-appreciated interpretability of RDPG embeddings. This is in stark contrast with most existing graph CPD approaches, which either rely on extensive computation, or they store and process the entire observed time series. An apparent limitation of the RDPG model is its suitability for undirected and unweighted graphs only, a gap we aim to close here to broaden the scope of the CPD framework. Unlike previous proposals, our non-parametric RDPG model for weighted graphs does not require a priori specification of the weights' distribution to perform inference and estimation. This network modeling contribution is of independent interest beyond CPD. We offer an open-source implementation of the novel online CPD algorithm for weighted and direct graphs, whose effectiveness and efficiency are demonstrated via (reproducible) synthetic and real network data experiments.

cs.LG↗

arXiv2021

Large deviations for the greedy exploration process on configuration models

We prove a large deviation principle for the greedy exploration of configuration models, building on a time-discretized version of the method proposed by Bermolen et al. and Brightwell et al. for jointly constructing a random graph from a given degree sequence and its exploration. The proof of this result follows the general strategy to study large deviations of processes proposed by Feng and Kurtz, based on the convergence of non-linear semigroups. We provide an intuitive interpretation of the LD cost function using Cramer's theorem for the average of random variables with appropriate distribution, depending on the degree distribution of explored nodes. The rate function can be expressed in a closed-form formula, and the large deviations trajectories can be obtained through explicit associated optimization problems. We then deduce large deviations results for the size of the independent set constructed by the algorithm. As a particular case, we analyze these results for d-regular graphs.

math.PR↗

arXiv2020

Sequential Algorithms and Independent Sets Discovering on Large Sparse Random Graphs

Computing the size of maximum independent sets is a NP-hard problem for fixed graphs. Characterizing and designing efficient algorithms to estimate this independence number for random graphs are notoriously difficult and still largely open issues. In a companion paper, we showed that a low complexity degree-greedy exploration is actually asymptotically optimal on a large class of sparse random graphs. Encouraged by this result, we present and study two variants of sequential exploration algorithms: static and dynamic degree-aware explorations. We derive hydrodynamic limits for both of them, which in turn allow us to compute the size of the resulting independent set. Whereas the former is simpler to compute, the latter may be used to arbitrarily approximate the degree-greedy algorithm. Both can be implemented in a distributed manner. The corresponding hydrodynamic limits constitute an efficient method to compute or bound the independence number for a large class of sparse random graphs. As an application, we then show how our method may be used to estimate the capacity of a large 802.11-based wireless network. We finally consider further indicators such as the fairness of the resulting configuration, and show how an unexpected trade-off between fairness and capacity can be achieved.

math.PR↗

arXiv2016

Scaling Limits and Generic Bounds for Exploration Processes

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become explored. Given an initial number of vertices $N$ growing to infinity, we study statistical properties of the proportion of explored nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the \emph{jamming constant}, through a diffusion approximation for the exploration process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e. random sequential adsorption. As opposed to homogeneous random graphs, these do not allow for a reduction in dimensionality. Instead we build on a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and obtain generic bounds: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, we give two trajectorial interpretations of our bounds by constructing two coupled processes that have the same fluid limits.

math.PR↗

arXiv2015

The Jamming Constant of Uniform Random Graphs

By constructing jointly a random graph and an associated exploration process, we define the dynamics of a "parking process" on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald.

math.PR↗

arXiv2015

Scaling limits for exploration algorithms

We consider an exploration algorithm where at each step, a random number of items become active while related items get explored. Given an initial number of items $N$ growing to infinity and building on a strong homogeneity assumption, we study using scaling limits of Markovian processes statistical properties of the proportion of active nodes in time. This is a companion paper that rigorously establishes the claims and heuristics presented in [5]. [5] Jaron Sanders, Matthieu Jonckheere, and Servaas Kokkelmans. Sub-Poissonian statistics of jamming limits in Rydberg gases. 2015. To appear.

math.PR↗

arXiv2014

Estimating the Spatial Reuse with Configuration Models

We propose a new methodology to estimate the spatial reuse of CSMA-like scheduling. Instead of focusing on spatial configurations of users, we model the interferences between users as a random graph. Using configuration models for random graphs, we show how the properties of the medium access mechanism are captured by some deterministic differential equations, when the size of the graph gets large. Performance indicators such as the probability of connection of a given node can then be efficiently computed from these equations. We also perform simulations to illustrate the results on different types of random graphs. Even on spatial structures, these estimates get very accurate as soon as the variance of the interference is not negligible.

cs.NI↗