arXiv subjects

Rounak Ray

Publications and source records attributed to Rounak Ray.

7 recordsLinked to original sources

arXiv2026

Network evolution with self-reinforcement

We study a new class of preferential attachment trees with \emph{self-reinforcement}. At each time, each vertex is assigned a weight equal to the cumulative sum over past times of an affine function of its degree. A new vertex attaches itself via a single edge to an already present vertex with a probability proportional to the current weight of that vertex. This ``integrated popularity'' rule builds long memory directly into the attachment mechanism, thereby destroying the Markov and partial-exchangeability features that underlie the classical analysis of affine preferential attachment models. More broadly, the model connects to applied-probability work on long-memory self-interacting processes (such as the elephant random walk), emphasizing how non-Markovian reinforcement reshapes asymptotic behaviour. Despite this loss of structure, we identify an explicit exponent $\phi=\phi(\delta)$ governing both local and global growth: typical degrees at time $n$ scale as $n^{1/\phi}$, and the empirical degree distribution converges to a power-law with a tail exponent $\phi+1$. We further prove Benjamini--Schramm local convergence to an infinite random rooted tree characterized via an embedded continuous-time branching process. The limiting tree is a \texttt{sin}-tree, and is \emph{not} the P\'olya-type limiting tree arising in the non-reinforced setting. Our results provide a tractable probabilistic description of a natural ``memoryful'' network-growth mechanism, and quantify precisely how reinforcement renormalizes the classical preferential-attachment exponents.

math.PR↗

arXiv2025

Non-equilibrium coagulation processes and subcritical percolation on evolving networks

We investigate percolation on growing networks where the evolution of connected components resembles a non-equilibrium version of the multiplicative coalescent. The supercritical $\pi> \pi_c$ regime for a host of such models was conjectured in statistical physics, and then rigorously proven in mathematics, to exhibit behavior similar to the BKT infinite-order phase transition as $\pi\searrow \pi_c$. It has further been conjectured that the entire regime $\pi<\pi_c$ for such growing networks are ''critical'' with power-law cluster size distributions having a non-universal exponent for all values of $\pi \in (0, \pi_c)$. In this paper, we study percolation on the uniform attachment model, as a concrete template in order to develop general tools based on stochastic approximation, local convergence, branching random walks and tree-graph inequalities to prove the above conjectured phenomena. For each $\pi \in (0,\pi_c)$, we show there exists an explicit $\alpha(\pi) \in (0,\tfrac{1}{2}) $ such that the maximal component size, as well as the size of the component containing any fixed vertex, all re-scaled by $n^{\alpha(\pi)}$, converge almost surely to strictly positive random variables as the network size $n \to \infty$. These dynamics lead to novel phenomena, compared to classical 'static' models, including long-range dependence and fixation of the identity of the maximal component, within finite time, among a finite number of 'early' components. Moreover, in contrast with most static network models, we show that the susceptibility, that is, the expected size of the component of a uniformly chosen vertex, remains bounded as the network grows and $\pi$ approaches $\pi_c$ from below. The general tools developed in this paper will be used in follow-up work to understand percolation for general growing network evolution models.

math.PR↗

arXiv2025

Ising model on preferential attachment models

We study the Ising model on affine preferential attachment models with general parameters. We identify the thermodynamic limit of several quantities, arising in the large graph limit, such as pressure per particle, magnetisation, and internal energy for these models. Furthermore, for $m\geq 2$, we determine the inverse critical temperature for preferential attachment models as $\beta_c(m,\delta)=0$ when $\delta\in(-m,0]$, while, for $\delta>0$, $$\beta_c(m,\delta)= {\rm atanh}\left\{ \frac{\delta}{2\big( m(m+\delta)+\sqrt{m(m-1)(m+\delta)(m+\delta+1)} \big)} \right\}~.$$ Our proof for the thermodynamic limit of pressure per particle critically relies on the belief propagation theory for factor models on locally tree-like graphs, as developed by Dembo, Montanari, and Sun. It has been proved that preferential attachment models admit the P\'{o}lya point tree as their local limit under general conditions. We use the explicit characterisation of the P\'{o}lya point tree and belief propagation for factor models to obtain the explicit expression for the thermodynamic limit of the pressure per particle. Next, we use the convexity properties of the internal energy and magnetisation to determine their thermodynamic limits. To study the phase transition, we prove that the inverse critical temperature for a sequence of graphs and its local limit are equal. Finally, we show that $\beta_c(m,\delta)$ is the inverse critical temperature for the P\'{o}lya point tree with parameters $m$ and $\delta$, using results from Lyons who shows that the critical inverse temperature is closely related to the percolation critical threshold. This part of the proof heavily relies on the critical percolation threshold for P\'{o}lya point trees established earlier with Hazra.

math.PR↗

arXiv2024

Stochastic processes on preferential attachment models

In real life, networks are dynamic in nature; they grow over time and often exhibit power-law degree sequences. To model the evolving structure of the internet, Barab\'{a}si and Albert introduced a simple dynamic model with a power-law degree distribution. This model has since been generalised, leading to a broad class of affine preferential attachment models, where each new vertex connects to existing vertices with a probability proportional to the current degree of the vertex. While numerous studies have explored the global and local properties of these random graphs, their dynamic nature and the dependencies in edge-connection probabilities have posed significant analytical challenges. The first part of this thesis identifies the local limit of preferential attachment models in considerable generality. The second part focuses on stochastic processes on preferential attachment models, introducing an additional layer of randomness to the random graphs. Examples of such processes include bond and site percolation, random walks, the Ising and Potts models, and Gaussian processes on random graphs. In this thesis, we specifically examine percolation and the Ising model, exploring these processes using the local limit identified earlier.

math.PR↗

arXiv2023

Percolation on preferential attachment models

We study the percolation phase transition on preferential attachment models, in which vertices enter with $m$ edges and attach proportionally to their degree plus $\delta$. We identify the critical percolation threshold as $$\pi_c=\frac{\delta}{2\big(m(m+\delta)+\sqrt{m(m-1)(m+\delta)(m+1+\delta)}\big)}$$ for $\delta$ positive and $\pi_c=0$ for non-positive values of $\delta$. Therefore the giant component is robust for $\delta\in(-m,0]$, while it is not for $\delta>0$. Our proof for the critical percolation threshold consists of three main steps. First, we show that preferential attachment graphs are large-set expanders, enabling us to verify the conditions outlined by Alimohammadi, Borgs, and Saberi (2023). Within their conditions, the proportion of vertices in the largest connected component in a sequence converges to the survival probability of percolation on the local limit. In particular, the critical percolation threshold for both the graph and its local limit are identical. Second, we identify $1/\pi_c$ as the spectral radius of the mean offspring operator of the P\'olya point tree, the local limit of preferential attachment models. Lastly, we prove that the critical percolation threshold for the P\'olya point tree is the inverse of the spectral radius of the mean offspring operator. For positive $\delta$, we use sub-martingales to prove sub-criticality and apply spine decomposition theory to demonstrate super-criticality, completing the third step of the proof. For $\delta\leq 0$ and any $\pi>0$ instead, we prove that the percolated P\'olya point tree dominates a supercritical branching process, proving that the critical percolation threshold equals $0$.

math.PR↗

arXiv2022

Universality of the local limit of preferential attachment models

We study preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014). The degree distribution of this limiting random graph, which we call the random P\'{o}lya point tree, has a surprising size-biasing phenomenon. Many of the existing preferential attachment models can be viewed as special cases of our preferential attachment model with i.i.d. out-degrees. Additionally, our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees. Our proof of local convergence consists of two main steps: a P\'olya urn description of our graphs, and an explicit identification of the neighbourhoods in them. We provide a novel and explicit proof to establish a coupling between the preferential attachment model and the P\'{o}lya urn graph. Our result proves a density convergence result, for fixed ages of vertices in the local limit.

math.PR↗

arXiv2021

Maximum of the membrane model on regular trees

The discrete membrane model is a Gaussian random interface whose inverse covariance is given by the discrete biharmonic operator on a graph. In literature almost all works have considered the field as indexed over $\mathbb{Z}^d$, and this enabled one to study the model using methods from partial differential equations. In this article we would like to investigate the dependence of the membrane model on a different geometry, namely trees. The covariance is expressed via a random walk representation which was first determined by Vanderbei (1984). We exploit this representation on $m$-regular trees and show that the infinite volume limit on the infinite tree exists when $m\ge 3$. Further we determine the behavior of the maximum under the infinite and finite volume measures.

math.PR↗