arXiv subjects

Neeladri Maitra

Publications and source records attributed to Neeladri Maitra.

13 recordsLinked to original sources

arXiv2026

Estimating Community Boundaries in Geometric Random Graphs

The unit square $I=[0,1]^2$ is divided into two rectangles by the vertical line $x=p$, where $p\in(0,1)$. Consider $N$ independent uniformly distributed points on $I$, which we interpret as a population of individuals, with the line $x=p$ representing a community boundary that separates the population into two communities. Whether a pair of individuals share a connection depends on their locations in $I$ and on whether they belong to the same community. Specifically, two individuals in the same community are connected if they are within distance $R_N$ of each other, while two individuals in different communities are connected if they are within distance $R_N'$ of each other, giving rise to a geometric variant of the so-called \emph{stochastic block model}. A statistician observes the adjacency matrix of the resulting graph together with the geometric locations of the individuals and is tasked with estimating the boundary location $p$. Depending on how $R_N$ and $R_N'$ scale as $N\to\infty$, we establish necessary and sufficient conditions for consistent estimation of $p$. Whenever consistent estimation is possible, we devise an estimator that converges to $p$ as $N\to\infty$ and provide explicit bounds on its estimation error.

math.ST↗

arXiv2026

Long-range competition on the torus

We study competition between two growth models with long-range correlations on the torus $\mathbb T_n^d$ of size $n$ in dimension $d$. We append the edge set of the torus $\mathbb T_n^d$ by including all non-nearest-neighbour edges, and from two source vertices $v^\ominus$ and $v^\oplus$ in $\mathbb T_n^d$ two infection processes $\ominus$ and $\oplus$ start spreading to other vertices. Each susceptible vertex can be infected by at most one infection type and when infected stays infected forever (i.e.\ competing SI models). A vertex $v$ infected with type $\square\in\{\ominus,\oplus\}$ infects a susceptible vertex $u$ at rate $λ_\square \|u-v\|^{-α_\square}$, where $λ_{\ominus}=λ_\ominus(n),λ_\oplus=λ_\oplus(n)>0$ and $α_\ominus=α_\ominus(n),α_\oplus=α_\oplus(n)\in[0,d)$ are allowed to depend on $n$. We study \emph{coexistence}, the event that both infections reach an asymptotically positive proportion of the graph as $n$ tends to infinity, and identify precisely when coexistence occurs. In the case of absence of coexistence, we outline several phase transitions in the size of the infection that reaches a negligible proportion of the vertices, which depends on the ratio of the sum of infection rates across all vertices of type $\ominus$ and $\oplus$. The work extends known results for the case $α_\ominus(n)=α_\oplus(n)\equiv 0$ and $λ_\ominus(n)\equiv 1, λ_\oplus(n)\equiv λ>0$, and includes general and novel results that cannot be observed when the model parameters are fixed and independent of $n$. The main technical contribution is a coupling of the competition process with branching random walks, where we are able to use the coupling even when the coupling error between the competition process and the branching random walks is of the same order of magnitude as the size of the coupled processes.

math.PR↗

arXiv2026

Finding Adam in noisy trees

We consider the problem of finding the root vertex of a random uniform attachment tree, when the union of the unlabeled tree and an Erdős-Rényi random graph $\mathbb{G}(n,p)$ is observed. We prove that, as long as $p=o(\log n /n)$, for any $\varepsilon>0$, one can construct a confidence set of vertices of size $K(\varepsilon)$ that depends only on $\varepsilon$ and not on $n$, such that it contains the root with probability at least $1-\varepsilon$. This affirms a conjecture of Crane and Xu (2021). Our approach ranks vertices by their Jordan centrality in the largest component of the subgraph spanned by high-degree vertices. We show that the same approach works in other noise models as well.

math.PR↗

arXiv2026

Multipartite random graphs with given degrees: local limit, revisiting the giant, distances

We consider multipartite random graphs with given degree sequences, within and across different partitions. Under general assumptions, we prove the local limit of this graph is a multi-type branching process, establish that a giant component exists only when the local limit survives, and deduce that the typical distance is of logarithmic order in probability in the supercritical regime. Our analysis removes two major assumptions from Gamarnik and Misra (2015), where the giant component problem for this model was first considered. In particular, we do not assume irreducibility of the local limit, and provide a general framework to extract giant components even when the limiting branching process is reducible, which we hope to be useful in other contexts. We also provide a new simpler survival criterion of multi-type branching processes, which we hope to be useful when direct calculation of the spectral radius of the offspring matrix may prove to be difficult.

math.PR↗

arXiv2026

Algorithmic Phase Transition for Large Independent Sets in Dense Hypergraphs

We study the algorithmic tractability of finding large independent sets in dense random hypergraphs. In the sparse regime, much of the natural algorithms can be formulated within either the local or the low-degree polynomial (LDP) framework, and a rich literature has subsequently identified nearly sharp algorithmic thresholds within these classes by exploiting their stability. In the dense setting, however, the algorithmic paradigms are fundamentally different: they are online and thus need not be stable. Perhaps more crucially, even for the classical Erdős-Rényi random graph $G(n,p)$, LDPs are conjectured to fail in the 'easy' regime accessible to online algorithms, thereby challenging their viability for dense models. Our focus is on two models: (i) finding large independent sets in dense $r$-uniform Erdős-Rényi hypergraphs, and (ii) the more challenging problem of finding large $γ$-balanced independent sets in dense $r$-uniform $r$-partite hypergraphs, where the $i$-th coordinate of $γ\in\mathbb{Q}^r$ specifies the proportion of vertices from $V_i$ in the independent set. For both models, we pinpoint the size of the largest independent set and design online algorithms that achieve a multiplicative approximation factor of $r^{1/(r-1)}$ in the uniform and $(\max_i γ_i)^{-1/(r-1)}$ in the $r$-partite model. Furthermore, we establish matching algorithmic lower bounds, showing that these computational gaps are sharp: no online algorithms can breach these gaps.

cs.DS↗

arXiv2025

On exponentially height-penalized random trees

Given $n \in \mathbb{N}$ and $μ\in \mathbb{R}$, a $\textit{$μ$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-μh(t)}$, where $t$ ranges over fixed plane trees with $n$ vertices, and $h(t)$ is the height of $t$. Fix a sequence $(μ_n)_{n \ge 1}$ of real numbers, and for $n \ge 1$ let $\mathbf{T}_n$ be a $μ$-height-biased tree of size $n$. Durhuus and Ünel (2023) described the asymptotic behaviour of $h(\mathbf{T}_n)$ when $μ_n \equiv μ\in \mathbb{R}$ is fixed. In this work, we extend their results to arbitrary sequences of positive parameters depending on $n$. Most notably, we show that such a tree behaves like a height-biased Continuum Random Tree (CRT) when $μ_n$ is of order $1/\sqrt{n}$; that its height is asymptotically $(2π^2n/μ_n)^{1/3}$ when $μ_n$ is of larger order than $1/\sqrt{n}$ and of smaller order than $n$; and that its height converges to a fixed constant when $μ_n$ is of order at least $n$, with some random jumps under specific conditions on $μ_n$. We additionally prove various results on second order behaviours, and large deviation principles for the height, for different regimes of $μ_n$. Finally, we describe new statistics of these trees, covering their widths, their root degrees, and the local structure around their roots.

math.PR↗

arXiv2025

Evolution of recursive trees with limited memory

Motivated by questions in social networks, distributed computing and probabilistic combinatorics, the last few years have seen increasing interest in network evolution models where new vertices entering the system need to make decisions based on a partial snapshot of the current state of the network. This paper considers a specific variant of the classical random recursive tree dynamics, where a vertex at time $n+1$ has information only on those vertices that have arrived in the interval $[j(n), n]$ for a sequence $j(n) \uparrow \infty$, and connects to vertices uniformly at random amongst this set. We consider two different regimes on the density information, termed macroscopic and mesoscopic regimes, which respectively correspond to $j(n)=θn$ for some $θ\in (0,1)$, and $j(n)=n-n^β$ for some $β\in (0,1)$. Our main interest is in studying asymptotics of various local and global functionals of the network. We show that in the macroscopic regime, the local limit is expressed in terms of an associated continuous time branching process that depends on the parameter $θ$, while it is a $\mathrm{Poisson}(1)$-branching process in the mesoscopic regime for any $β\in (0,1)$. Furthermore, the height of the macroscopic tree is logarithmic, which we prove exploiting a connection with scaled-attachment random recursive trees (SARRTs) as studied by Devroye, Fawzi and Fraiman (RSA 2011), while it is polynomial in the mesoscopic regime; our argument in this latter case relies on a differential equation approach to track the ancestor indices of late-coming vertices, together with a multiscale analysis. Further, we develop an exploration algorithm to simultaneously reveal the ancestral path of youngest vertices. Using this algorithm, we show that in the mesoscopic regime, the global structure experiences a phase transition at $β=1/2$.

math.PR↗

arXiv2025

On random bipartite graphs evolving by degrees

In this paper, we study a bipartite analogue of the `random graphs evolving by degrees' process. We are given a bipartitioned set of vertices $V$ into two disjoint parts ${L}$ and ${R}$ and possibly unequal positive constants $α$ and $β$. The graph evolves starting from $B_0$, the empty graph (with only isolated vertices). Given $B_t$, a non-adjacent vertex pair $u \in {L}, v \in {R}$ is sampled with probability proportional to $(d_u(t)+α)(d_v(t)+β)$, and the edge $\{u,v\}$ is included to $B_t$ to form $B_{t+1}$, where $d_u(t)$ is the degree of $u$ in $B_t$. For this model, we establish the threshold for the appearance of a giant component, the connectivity threshold for the associated multigraph variant, and provide a superlinear lower bound on the connectivity threshold for the simple graph case. For the proof of the giant component result, our methods involve setting up an exact coupling of the multigraph case with a bipartite configuration model and using existing results on the giant of bipartite configuration models. This is an adaptation of the technique of Janson and Warnke (Ann. Appl. Probab. 2021) where they treat the unipartite case similarly. For the connectivity results, we first set up a formula for the exact probability of the occurrence of certain connectivity events in the multigraph process, which is interesting in its own right. To then derive the connectedness threshold, we analyze a particular case of it à la Pittel (Adv. Math. 2010). For the superlinear connectivity lower bound in the simple graph case, we establish and use a tail bound on the number of isolated vertices in the multigraph process, together with a change of measure statement to go from the multi to the simple graph process.

math.PR↗

arXiv2025

Sharp Online Hardness for Large Balanced Independent Sets

We study the algorithmic problem of finding large $γ$-balanced independent sets in dense random bipartite graphs; an independent set is $γ$-balanced if a $γ$ proportion of its vertices lie on one side of the bipartition. In the sparse regime, Perkins and Wang established tight bounds within the low-degree polynomial (LDP) framework, showing a factor-$1/(1-γ)$ statistical-computational gap via the Overlap Gap Property (OGP) framework tailored for stable algorithms. However, these techniques do not appear to extend to the dense setting. For the related large independent set problem in dense random graph, the best known algorithm is an online greedy procedure that is inherently unstable, and LDP algorithms are conjectured to fail even in the "easy" regime where greedy succeeds. We show that the largest $γ$-balanced independent set in dense random bipartite graphs has size $α:=\frac{\log_b n}{γ(1-γ)}$ whp, where $n$ is the size of each bipartition, $p$ is the edge probability, and $b=1/(1-p)$. We design an online algorithm that achieves $(1-ε)(1-γ)α$ whp for any $ε>0$. We complement this with a sharp lower bound, showing that no online algorithm can achieve $(1+ε)(1-γ)α$ with nonnegligible probability. Our results suggest that the same factor-$1/(1-γ)$ gap is also present in the dense setting, supporting its conjectured universality. While the classical greedy procedure on $G(n,p)$ is straightforward, our algorithm is more intricate: it proceeds in two stages, incorporating a stopping time and suitable truncation to ensure that $γ$-balancedness-a global constraint-is met despite operating with limited information. Our lower bound utilizes the OGP framework; we build on a recent refinement of this framework for online models and extend it to the bipartite setting.

cs.DS↗

arXiv2025

Annealed Potts models on rank-1 inhomogeneous random graphs

In this paper, we study the annealed ferromagnetic $q$-state Potts model on sparse rank-1 random graphs, where vertices are equipped with a vertex weight, and the probability of an edge is proportional to the product of the vertex weights. In an annealed system, we take the average on both numerator and denominator of the ratio defining the Boltzmann-Gibbs measure of the Potts model. We show that the thermodynamic limit of the pressure per particle exists for rather general vertex weights. In the infinite-variance weight case, we show that the critical temperature equals infinity. For finite-variance weights, we show that, under a rather general condition, the phase transition is {\em first order} for all $q\geq 3$. However, we cannot generally show that the discontinuity of the order parameter is {\em unique}. We prove this uniqueness under a reasonable condition that holds for various distributions, including uniform, gamma, log-normal, Rayleigh and Pareto distributions. Further, we show that the first-order phase transition {\em persists} even for some small positive external field. In the rather relevant case of Pareto distributions with power-law exponent $τ$, remarkably, the phase transition is first order when $τ\geq 4$, but not necessarily when the weights have an infinite third-moment, i.e., when $τ\in(3,4)$. More precisely, the phase transition is second order for $τ\in (3,τ(q)]$, while it is first order when $τ>τ(q)$, where we give an explicit equation that $τ(q)$ solves.

math.PR↗

arXiv2024

Condensation in scale-free geometric graphs with excess edges

We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. Our result covers the models of scale-free percolation, the Boolean model with heavy-tailed radius distribution, and the age-dependent random connection model. In all these cases the mechanism behind the large deviation is based on a condensation effect. Loosely speaking, the mechanism randomly selects a finite number of vertices and increases their power, so that they connect to a macroscopic number of vertices in the graph, while the other vertices retain a degree close to their expectation and thus make no more than the expected contribution to the large deviation event. We verify this intuition by means of limit theorems for the empirical distributions of degrees and edge-lengths under the conditioning. We observe that at large finite volumes, the edge-length distribution splits into a bulk and travelling wave part of asymptotically positive proportions.

math.PR↗

arXiv2023

Scaling of the clustering function in spatial inhomogeneous random graphs

We consider an infinite spatial inhomogeneous random graph model with an integrable connection kernel that interpolates nicely between existing spatial random graph models. Key examples are versions of the weight-dependent random connection model, the infinite geometric inhomogeneous random graph, and the age-based random connection model. These infinite models arise as the local limit of the corresponding finite models, see \cite{LWC_SIRGs_2020}. For these models we identify the scaling of the \emph{local clustering} as a function of the degree of the root in different regimes in a unified way. We show that the scaling exhibits phase transitions as the interpolation parameter moves across different regimes. In addition to the scaling we also identify the leading constants of the clustering function. This allows us to draw conclusions on the geometry of a \emph{typical} triangle contributing to the clustering in the different regimes.

math.PR↗

arXiv2022

Local limits of spatial inhomogeneous random graphs

Consider a set of $n$ vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$, and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations, and the vertex weights. Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on $\mathbb{R}^d$ with vertex locations given by a homogeneous Poisson point process, having weights which are i.i.d. copies of limiting vertex weights. Our setup covers many sparse geometric random graph models from the literature, including Geometric Inhomogeneous Random Graphs (GIRGs), Hyperbolic Random Graphs, Continuum Scale-Free Percolation and Weight-dependent Random Connection Models. We prove that the limiting degree distribution is mixed Poisson, and the typical degree sequence is uniformly integrable, and obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a by-product of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

math.PR↗