arXiv subjects

Laura Sacerdote

Publications and source records attributed to Laura Sacerdote.

At least 19 recordsLinked to original sources

arXiv2025

Animal social networks as intersections graphs of random walks

We study here the social network generated by the asynchronous visits, to a fixed set of sites, of mobile agents modelled as independent random walks on the plane lattice. The social network is constructed by assuming that a group of agents are associated if they have visited the same set of sites within a finite time interval. This construction is an instance of a random intersection graph, and has been used in the literature to study association networks in a number of animal species. We characterize the mathematical structure of these networks, which we view as one-mode projections of suitable bipartite graphs or, equivalently, as 2-sections of the corresponding hypergraphs. We determine analytically the probability distribution of the random bipartite graphs and hypergraphs associated to this construction, and suggest that association networks generated by the use of common resources are better described by hypergraphs rather than simple projected graphs, that miss important information regarding the actual associations among the agents.

physics.soc-ph↗

arXiv2024

Supplementary Materials to Graph Convolutional Branch and Bound

This article explores the integration of deep learning models into combinatorial optimization pipelines, specifically targeting NP-hard problems. Traditional exact algorithms for such problems often rely on heuristic criteria to guide the exploration of feasible solutions. In this work, we propose using neural networks to learn informative heuristics, most notably, an optimality score that estimates a solution's proximity to the optimum. This score is used to evaluate nodes within a branch-and-bound framework, enabling a more efficient traversal of the solution space. Focusing on the Traveling Salesman Problem, we introduce Concorde, a state-of-the-art solver, and present a hybrid approach called Graph Convolutional Branch and Bound, which augments it with a graph convolutional neural network trained with a novel unsupervised training strategy that facilitates generalization to graphs of varying sizes without requiring labeled data. Empirical results demonstrate the effectiveness of the proposed method, showing a significant reduction in the number of explored branch-and-bound nodes and overall computational time. Some of the results concerning the use of the 1-tree relaxation are in the supplementary materials.

cs.LG↗

arXiv2023

Jacobi processes with jumps as neuronal models: a first passage time analysis

To overcome some limits of classical neuronal models, we propose a Markovian generalization of the classical model based on Jacobi processes by introducing downwards jumps to describe the activity of a single neuron. The statistical analysis of inter-spike intervals is performed by studying the first-passage times of the proposed Markovian Jacobi process with jumps through a constant boundary. In particular, we characterize its Laplace transform which is expressed in terms of some generalization of hypergeometric functions that we introduce, and deduce a closed-form expression for its expectation. Our approach, which is original in the context of first passage time problems, relies on intertwining relations between the semigroups of the classical Jacobi process and its generalization, which have been recently established in [11]. A numerical investigation of the firing rate of the considered neuron is performed for some choices of the involved parameters and of the jumps distributions.

math.PR↗

arXiv2023

Some exact results on Lindley process with Laplace jumps

We consider a Lindley process with Laplace distributed space increments. We obtain closed form recursive expressions for the density function of the position of the process and for its first exit time distribution from the domain $[0,h]$. We illustrate the results in terms of the parameters of the process. The work is completed by an open source version of the software.

math.PR↗

arXiv2022

Input-output consistency in integrate and fire interconnected neurons

Interspike intervals describe the output of neurons. Signal transmission in a neuronal network implies that the output of some neurons becomes the input of others. The output should reproduce the main features of the input to avoid a distortion when it becomes the input of other neurons, that is input and output should exhibit some sort of consistency. In this paper, we consider the question: how should we mathematically characterize the input in order to get a consistent output? Here we interpret the consistency by requiring the reproducibility of the input tail behaviour of the interspike intervals distributions in the output. Our answer refers to a system of interconnected neurons with stochastic perfect integrate and fire units. In particular, we show that the class of regularly-varying vectors is a possible choice to obtain such consistency. Some further necessary technical hypotheses are added.

math.PR↗

arXiv2020

On dynamic random graphs with degree homogenization via anti-preferential attachment probabilities

We analyze a dynamic random undirected graph in which newly added vertices are connected to those already present in the graph either using, with probability $p$, an anti-preferential attachment mechanism or, with probability $1-p$, a preferential attachment mechanism. We derive the asymptotic degree distribution in the general case and study the asymptotic behaviour of the expected degree process in the general and that of the degree process in the pure anti-preferential attachment case. Degree homogenization mainly affects convergence rates for the former case and also the limiting degree distribution in the latter.

math.PR↗

arXiv2020

On the continuous-time limit of the Barabási-Albert random graph

We prove that the Barabási-Albert model converges weakly to a set of generalized Yule models via an appropriate scaling. To pursue this aim we superimpose to its graph structure a suitable set of processes that we call the planted model and we introduce an ad-hoc sampling procedure. The use of the obtained limit process represents an alternative and advantageous way of looking at some of the asymptotic properties of the Barabási-Albert random graph.

math.PR↗

arXiv2020

A sharp bound on the expected local time of a continuous ${\cal L}_2$-bounded Martingale

For a continuous ${\cal L}_2$-bounded Martingale with no intervals of constancy, starting at $0$ and having final variance $σ^2$, the expected local time at $x \in \cal{R}$ is at most $\sqrt{σ^2+x^2}-|x|$. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $(x-\sqrt{σ^2+x^2},x+\sqrt{σ^2+x^2})$. Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals, have been established by Dubins and Schwarz (1988), Dubins, Gilat and Meilijson (2009) and by the authors (2017).

math.PR↗

arXiv2019

A study of dependency features of spike trains through copulas

Simultaneous recordings from many neurons hide important information and the connections characterizing the network remain generally undiscovered despite the progresses of statistical and machine learning techniques. Discerning the presence of direct links between neuron from data is still a not completely solved problem. To enlarge the number of tools for detecting the underlying network structure, we propose here the use of copulas, pursuing on a research direction we started in [1]. Here, we adapt their use to distinguish different types of connections on a very simple network. Our proposal consists in choosing suitable random intervals in pairs of spike trains determining the shapes of their copulas. We show that this approach allows to detect different types of dependencies. We illustrate the features of the proposed method on synthetic data from suitably connected networks of two or three formal neurons directly connected or influenced by the surrounding network. We show how a smart choice of pairs of random times together with the use of empirical copulas allows to discern between direct and un-direct interactions.

stat.AP↗

arXiv2018

Generalized Nonlinear Yule Models

With the aim of considering models with persistent memory we propose a fractional nonlinear modification of the classical Yule model often studied in the context of macrovolution. Here the model is analyzed and interpreted in the framework of the development of networks such as the World Wide Web. Nonlinearity is introduced by replacing the linear birth process governing the growth of the in-links of each specific webpage with a fractional nonlinear birth process with completely general birth rates. Among the main results we derive the explicit distribution of the number of in-links of a webpage chosen uniformly at random recognizing the contribution to the asymptotics and the finite time correction. The mean value of the latter distribution is also calculated explicitly in the most general case. Furthermore, in order to show the usefulness of our results, we particularize them in the case of specific birth rates giving rise to a saturating behaviour, a property that is often observed in nature. The further specialization to the non-fractional case allows us to extend the Yule model accounting for a nonlinear growth.

math.PR↗

arXiv2017

On a class of Time-fractional Continuous-state Branching Processes

We propose a class of non-Markov population models with continuous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton--Watson processes. The class includes as specific cases the classical continuous-state branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The generalized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general model.

math.PR↗

arXiv2017

Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale free behavior of the degree distribution. We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1-p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion (αn) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold. The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.

math.PR↗

arXiv2017

First passage times of two-correlated processes: analytical results for the Wiener process and a numerical method for diffusion processes

Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show that these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times. The convergence of the method is theoretically proved for bivariate diffusion processes. We derive explicit expressions for these and other quantities of interest in the case of a bivariate Wiener process, correcting previous misprints appearing in the literature. Finally we illustrate the application of the method through a set of examples.

math.PR↗

arXiv2016

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model

Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters. These parameters are readily associated with known physiological mechanisms. The other model is descriptive, Gamma renewal process, and its parameters only reflect the observed experimental data or assumed theoretical properties. Both of these commonly used models are related here. We show under which conditions the Gamma model is an output from the diffusion Ornstein-Uhlenbeck model. In some cases we can see that the Gamma distribution is unrealistic to be achieved for the employed parameters of the Ornstein-Uhlenbeck process.

q-bio.NC↗

arXiv2016

Random Graphs Associated to some Discrete and Continuous Time Preferential Attachment Models

We give a common description of Simon, Barabási--Albert, II-PA and Price growth models, by introducing suitable random graph processes with preferential attachment mechanisms. Through the II-PA model, we prove the conditions for which the asymptotic degree distribution of the Barabási--Albert model coincides with the asymptotic in-degree distribution of the Simon model. Furthermore, we show that when the number of vertices in the Simon model (with parameter $α$) goes to infinity, a portion of them behave as a Yule model with parameters $(λ,β) = (1-α,1)$, and through this relation we explain why asymptotic properties of a random vertex in Simon model, coincide with the asymptotic properties of a random genus in Yule model. As a by-product of our analysis, we prove the explicit expression of the in-degree distribution for the II-PA model, given without proof in \cite{Newman2005}. References to traditional and recent applications of the these models are also discussed.

math.PR↗

arXiv2015

A copula-based method to build diffusion models with prescribed marginal and serial dependence

This paper investigates the probabilistic properties that determine the existence of space-time transformations between diffusion processes. We prove that two diffusions are related by a monotone space-time transformation if and only if they share the same serial dependence. The serial dependence of a diffusion process is studied by means of its copula density and the effect of monotone and non-monotone space-time transformations on the copula density is discussed. This provides us a methodology to build diffusion models by freely combining prescribed marginal behaviors and temporal dependence structures. Explicit expressions of copula densities are provided for tractable models. A possible application in neuroscience is sketched as a proof of concept.

math.PR↗

arXiv2014

The role of detachment of in-links in scale-free networks

Real-world networks may exhibit detachment phenomenon determined by the cancelling of previously existing connections. We discuss a tractable extension of Yule model to account for this feature. Analytical results are derived and discussed both asymptotically and for a finite number of links. Comparison with the original model is performed in the supercritical case. The first-order asymptotic tail behavior of the two models is similar but differences arise in the second-order term. We explicitly refer to World Wide Web modeling and we show the agreement of the proposed model on very recent data. However, other possible network applications are also mentioned.

math.PR↗

arXiv2014

On Firing Rate Estimation for Dependent Interspike Intervals

If interspike intervals are dependent the instantaneous firing rate does not catch important features of spike trains. In this case the conditional instantaneous rate plays the role of the instantaneous firing rate for the case of samples of independent interspike intervals. If the conditional distribution of the interspikes intervals is unknown, it becomes difficult to evaluate the conditional firing rate. We propose a non parametric estimator for the conditional instantaneous firing rate for Markov, stationary and ergodic ISIs. An algorithm to check the reliability of the proposed estimator is introduced and its consistency properties are proved. The method is applied to data obtained from a stochastic two compartment model and to in vitro experimental data.

math.ST↗