arXiv subjects

Rajat Subhra Hazra

Publications and source records attributed to Rajat Subhra Hazra.

At least 37 records · Page 2Linked to original sources

arXiv2024

Inhomogeneous random graphs with infinite-mean fitness variables

We consider an inhomogeneous Erdős-Rényi random graph ensemble with exponentially decaying random disconnection probabilities determined by an i.i.d. field of variables with heavy tails and infinite mean associated to the vertices of the graph. This model was recently investigated in the physics literature in Garuccio et al. (2020) as a scale-invariant random graph within the context of network renormalization. From a mathematical perspective, the model fits in the class of scale-free inhomogeneous random graphs whose asymptotic geometrical features have been recently attracting interest. While for this type of graphs several results are known when the underlying vertex variables have finite mean and variance, here instead we consider the case of one-sided stable variables with necessarily infinite mean. To simplify our analysis, we assume that the variables are sampled from a Pareto distribution with parameter $α\in(0,1)$. We start by characterizing the asymptotic distributions of the typical degrees and some related observables. In particular, we show that the degree of a vertex converges in distribution, after proper scaling, to a mixed Poisson law. We then show that correlations among degrees of different vertices are asymptotically non-vanishing, but at the same time a form of asymptotic tail independence is found when looking at the behavior of the joint Laplace transform around zero. Moreover, we present some findings concerning the asymptotic density of wedges and triangles and show a cross-over for the existence of dust (i.e. disconnected vertices).

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arXiv2024

Largest eigenvalue of positive mean Gaussian matrices

This short note studies the fluctuations of the largest eigenvalue of symmetric random matrices with correlated Gaussian entries having positive mean. Under the assumption that the covariance kernel is absolutely summable, it is proved that the largest eigenvalue, after centering, converges in distribution to normal with an explicitly defined mean and variance. This result generalizes known findings for Wigner matrices with independent entries.

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arXiv2024

Functional Central Limit Theorem for the principal eigenvalue of dynamic Erdős-Rényi random graphs

In this paper we consider a dynamic version of the Erdős-Rényi random graph, in which edges independently appear and disappear in time, with the on- and off times being exponentially distributed. The focus lies on the evolution of the principle eigenvalue of the adjacency matrix in the regime that the number of vertices grows large. The main result is a functional central limit theorem, which displays that the principal eigenvalue essentially inherits the characteristics of the dynamics of the individual edges.

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arXiv2024

Meeting, coalescence and consensus time on random directed graphs

We consider Markovian dynamics on a typical realization of the so-called Directed Configuration Model (DCM), that is, a random directed graph with prescribed in- and out-degrees. In this random geometry, we study the meeting time of two random walks on a typical realization of the graph starting at stationarity, the coalescence time for a system of coalescent random walks, and the consensus time of the voter model. Indeed, it is known that the latter three quantities are related to each other when the underlying sequence of graphs satisfies certain mean field conditions. Such conditions can be summarized by requiring a fast mixing time of the random walk and some anti-concentration of its stationary distribution: properties that a typical random directed graph is known to have under natural assumptions on the degree sequence. In this paper we show that, for a typical large graph from the DCM ensemble, the distribution of the meeting time is well-approximated by an exponential random variable and we provide the first-order approximation of its expectation, showing that the latter is linear in the size of the graph, and the preconstant depends on some easy statistics of the degree sequence. As a byproduct, we are able to analyze the effect of the degree sequence in changing the meeting, coalescence and consensus time. Our approach follows the classical idea of converting meeting into hitting times of a proper collapsed chain, which we control by the so-called First Visit Time Lemma.

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arXiv2024

Discordant edges for the voter model on regular random graphs

We consider the two-opinion voter model on a regular random graph with n vertices and degree $d \geq 3$. It is known that consensus is reached on time scale n and that on this time scale the volume of the set of vertices with one opinion evolves as a Fisher-Wright diffusion. We are interested in the evolution of the number of discordant edges (i.e., edges linking vertices with different opinions), which can be thought as the perimeter of the set of vertices with one opinion, and is the key observable capturing how consensus is reached. We show that if initially the two opinions are drawn independently from a Bernoulli distribution with parameter $u \in (0, 1)$, then on time scale 1 the fraction of discordant edges decreases and stabilises to a value that depends on d and u, and is related to the meeting time of two random walks on an infinite tree of degree d starting from two neighbouring vertices. Moreover, we show that on time scale n the fraction of discordant edges moves away from the constant plateau and converges to zero in an exponential fashion. Our proofs exploit the classical dual system of coalescing random walks and use ideas from Cooper et al. (2010) built on the so-called First Visit Time Lemma. We further introduce a novel technique to derive concentration properties from weak-dependence of coalescing random walks on moderate time scales.

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arXiv2024

Eigenvalues outside the bulk of inhomogeneous Erdős-Rënyi random graphs

The article considers an inhomogeneous Erdős-Rënyi random graph on $\{1,\ldots, N\}$, where an edge is placed between vertices $i$ and $j$ with probability $\varepsilon_N f(i/N,j/N)$, for $i\le j$, the choice being made independent for each pair. The function $f$ is assumed to be non-negative definite, symmetric, bounded and of finite rank $k$. We study the edge of the spectrum of the adjacency matrix of such an inhomogeneous Erdős-Rényi random graph under the assumption that $N\varepsilon_N\to \infty$ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by $\sqrt{N\varepsilon_N}$, is compactly supported, the $k$-th largest eigenvalue goes to infinity. It turns out that the largest eigenvalue after appropriate scaling and centering converge to a Gaussian law, if the largest eigenvalue of $f$ has multiplicity $1$. If $f$ has $k$ distinct non-zero eigenvalues, then the joint distribution of the $k$ largest eigenvalues converge jointly to a multivariate Gaussian law. The first order behaviour of the eigenvectors is derived as a by-product of the above results. The results complement the homogeneous case derived by Erdős et al.(2013).

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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 $δ$. We identify the critical percolation threshold as $$π_c=\fracδ{2\big(m(m+δ)+\sqrt{m(m-1)(m+δ)(m+1+δ)}\big)}$$ for $δ$ positive and $π_c=0$ for non-positive values of $δ$. Therefore the giant component is robust for $δ\in(-m,0]$, while it is not for $δ>0$. 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 $1/π_c$ as the spectral radius of the mean offspring operator of the Pólya point tree, the local limit of preferential attachment models. Lastly, we prove that the critical percolation threshold for the Pólya point tree is the inverse of the spectral radius of the mean offspring operator. For positive $δ$, 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 $δ\leq 0$ and any $π>0$ instead, we prove that the percolated Pólya point tree dominates a supercritical branching process, proving that the critical percolation threshold equals $0$.

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arXiv2023

Largest Eigenvalue of the Configuration Model and Breaking of Ensemble Equivalence

We analyse the largest eigenvalue of the adjacency matrix of the configuration model with large degrees, where the latter are treated as hard constraints. In particular, we compute the expectation of the largest eigenvalue for degrees that diverge as the number of vertices $n$ tends to infinity, uniformly on a scale between $1$ and $\sqrt{n}$, and show that a weak law of large numbers holds. We compare with what was derived in our earlier paper "Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs" for the Chung-Lu model, which in the regime considered represents the corresponding configuration model with soft constraints, and show that the expectation is shifted down by $1$ asymptotically. This shift is a signature of breaking of ensemble equivalence between the hard and soft (also known as micro-canonical and canonical) versions of the configuration model. The latter result generalizes a previous finding in "A spectral signature of breaking of ensemble equivalence for constrained random graphs" obtained in the case when all degrees are equal.

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arXiv2023

Limiting Spectra of inhomogeneous random graphs

We consider sparse inhomogeneous Erdős-Rényi random graph ensembles where edges are connected independently with probability $p_{ij}$. We assume that $p_{ij}= \varepsilon_N f(w_i, w_j)$ where $(w_i)_{i\ge 1}$ is a sequence of deterministic weights, $f$ is a bounded function and $N\varepsilon_N\to λ\in (0,\infty)$. We characterise the limiting moments in terms of graph homomorphisms and also classify the contributing partitions. We present an analytic way to determine the Stieltjes transform of the limiting measure. The convergence of the empirical distribution function follows from the theory of local weak convergence in many examples but we do not rely on this theory and exploit combinatorial and analytic techniques to derive some interesting properties of the limit. We extend the methods of Khorunzhy et al. (2004) and show that a fixed point equation determines the limiting measure. The limiting measure crucially depends on $λ$ and it is known that in the homogeneous case, if $λ\to\infty$, the measure converges weakly to the semicircular law (Jung and Lee (2018)). We extend this result of interpolating between the sparse and dense regimes to the inhomogeneous setting and show that as $λ\to \infty$, the measure converges weakly to a measure which is known as the operator-valued semicircular law.

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arXiv2022

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.

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arXiv2022

Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs

A Chung-Lu random graph is an inhomogeneous Erdős-Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung-Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph.

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arXiv2022

Branching random walk with infinite progeny mean: a tale of two tails

We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the $n$-th generation when the tail of the displacement behaves like $\exp(-K(x))$, where either $K$ is a regularly varying function of index $r> 0$, or $K$ has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when $r> 1$.

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arXiv2021

Spectral properties of the Laplacian of a generalized Wigner matrix

In this article we consider the spectrum of a Laplacian matrix, also known as the Markov matrix, under the independence assumption. We assume that the entries have a variance profile. Motivated by recent works on generalized Wigner matrices we assume that variance profile gives rise to a sequence of graphons. Under the assumption that these graphons converge, we show that the limiting spectral distribution of the scaled Laplacian converge. We give an expression for the moments of the limiting measure in terms of graph homomorphisms. In some special cases we identify the limit explicitly. We also study the spectral norm and derive the order of the maximum eigenvalue. We show that our results cover Laplacian of various random graphs include inhomogeneous Erdős- R\' enyi random graph, sparse W-random graphs, stochastic block matrices and constrained random graphs.

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arXiv2020

Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs

We consider an inhomogeneous Erdős-Rényi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $λ_N$ be the maximal eigenvalue of the adjacency matrix of $G_N$. It is known that $λ_N/N$ satisfies a large deviation principle as $N \to \infty$. The associated rate function $ψ_r$ is given by a variational formula that involves the rate function $I_r$ of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of $ψ_r$, specially when the reference graphon is of rank 1.

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arXiv2020

The scaling limit of the $(\nabla+Δ)$-model

In this article we study the scaling limit of the interface model on $\mathbb{Z}^d$ where the Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension the scaling limit is given by the Gaussian free field. We discuss the appropriate spaces in which the convergence takes place. While in infinite volume the proof is based on Fourier analytic methods, in finite volume we rely on some discrete PDE techniques involving finite-difference approximation of elliptic boundary value problems.

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arXiv2020

Scaling limit of semiflexible polymers: a phase transition

We consider a semiflexible polymer in $\mathbb Z^d$ which is a random interface model with a mixed gradient and Laplacian interaction. The strength of the two operators is governed by two parameters called lateral tension and bending rigidity, which might depend on the size of the graph. In this article we show a phase transition in the scaling limit according to the strength of these parameters: we prove that the scaling limit is, respectively, the Gaussian free field, a "mixed" random distribution and the continuum membrane model in three different regimes.

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arXiv2019

Spectra of Adjacency and Laplacian Matrices of Inhomogeneous Erdős-Rényi Random Graphs

Inhomogeneous Erdős-Rényi random graphs $\mathbb G_N$ on $N$ vertices in the non-dense regime are considered in this paper. The edge between the pair of vertices $\{i,j\}$ is retained with probability $\varepsilon_N\,f(\frac{i}{N},\frac{j}{N})$, $1 \leq i \neq j \leq N$, independently of other edges, where $f\colon\,[0,1] \times [0,1] \to [0,\infty)$ is a continuous function such that $f(x,y)=f(y,x)$ for all $x,y \in [0,1]$. We study the empirical distribution of both the adjacency matrix $A_N$ and the Laplacian matrix $Δ_N$ associated with $\mathbb G_N$ in the limit as $N \to \infty$ when $\lim_{N\to\infty} \varepsilon_N = 0$ and $\lim_{N\to\infty} N\varepsilon_N = \infty$. In particular, it is shown that the empirical spectral distributions of $A_N$ and $Δ_N$, after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where $f(x,y) = r(x)r(y)$ with $r\colon\,[0,1] \to [0,\infty)$ a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, applications of the results to constrained random graphs, Chung-Lu random graphs and social networks are shown.

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arXiv2019

The scaling limit of the membrane model

On the integer lattice we consider the discrete membrane model, a random interface in which the field has Laplacian interaction. We prove that, under appropriate rescaling, the discrete membrane model converges to the continuum membrane model in $d\ge 2$. Namely, it is shown that the scaling limit in $d=2,\,3$ is a Hölder continuous random field, while in $d\ge 4$ the membrane model converges to a random distribution. As a by-product of the proof in $d=2,\,3$, we obtain the scaling limit of the maximum. This work complements the analogous results of Caravenna and Deuschel (2009) in $d=1$.

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