arXiv subjects

Angelica Pachon

Publications and source records attributed to Angelica Pachon.

9 recordsLinked to original sources

arXiv2025

Clustering in a preferential attachment network with triangles

We study a generalization of the affine preferential attachment model where triangles are randomly added to the graph. We show that the model exhibits an asymptotically power-law degree distribution with adjustable parameter $γ\in (1,\infty)$, and positive clustering. However, the clustering behaviour depends on how it is measured. With high probability, the average local clustering coefficient remains positive, independently of $γ$, whereas the expectation of the global clustering coefficient does not vanish only when $γ>3$.

math.PR↗

arXiv2025

Phase transition in preferential attachment-detachment through embedding

We study a random graph model with preferential edge attachment and detachment through the embedding into a generalized Yule model. We show that the in-degree distribution of a vertex chosen uniformly at random follows a power law in the supercritical regime but has an exponential decay in the subcritical. We provide the corresponding asymptotics. In the critical regime we observe an intermediate decay. The regimes are clearly defined in terms of parameter ranges.

math.PR↗

arXiv2023

Upper bounds for the largest components in critical inhomogeneous random graphs

We consider the Norros-Reittu random graph $NR_n(\textbf{w})$, where edges are present independently but edge probabilities are moderated by vertex weights, and use probabilistic arguments based on martingales to analyse the component sizes in this model when considered at criticality. In particular, we obtain stronger upper bounds (with respect to those available in the literature) for the probability of observing unusually large maximal clusters, and simplify the arguments needed to derive polynomial upper bounds for the probability of observing unusually small largest components.

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

On discrete-time semi-Markov processes

In the last years, many authors studied a class of continuous time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop the discrete-time version of such a theory. We show that a class of discrete-time semi-Markov chains can be seen as time-changed Markov chains and we obtain governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.

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

Phase transition for the Maki-Thompson rumour model on a small-world network

We consider the Maki-Thompson model for the stochastic propagation of a rumour within a population. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model. This structure is realized starting from a $k$-regular ring and by inserting, in the average, $c$ additional links in such a way that $k$ and $c$ are tuneable parameter for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter $c$. A quantitative estimate for the critical value of $c$ is obtained via extensive numerical simulations.

math.PR↗

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

Reconstruction of a multidimensional scenery with a branching random walk

In this paper we consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to [22] where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of a branching random walk, but millions of colors are necessary.

math.PR↗