arXiv subjects

Akshay Sakanaveeti

Publications and source records attributed to Akshay Sakanaveeti.

5 recordsLinked to original sources

arXiv2026

Network evolution with Macroscopic Delays: asymptotics and condensation

Preferential attachment models typically assume that each arriving vertex observes the current network before choosing its connection. Motivated by distributed systems and social networks, we study network delay, where this decision uses only a time-delayed snapshot. We focus on macroscopic delays, for which the delay is proportional to the current network size and hence removes a non-vanishing fraction of the available information. We identify the local weak limit as a continuous-time branching process whose reproduction point process has memory of its entire past. Since this non-Markovian description is difficult to analyze directly, we construct a dual branching process in which edges reproduce, recovering enough independence for quantitative analysis. This yields a detailed understanding of how the delay affects features such as the tail behavior of the asymptotic degree distribution, together with necessary and sufficient conditions for condensation-the phenomenon in which a positive fraction of the degree mass escapes to infinity. We conclude by studying the impact of the delay distribution on macroscopic functionals such as the root degree.

math.PR↗

arXiv2026

Subcritical percolation and network archaeology on random recursive tree substrate networks

We study a network-archaeology problem for a dynamic graph whose latent substrate is a random recursive tree and whose observed topology is enriched by an independent homogeneous Erdős-Rényi shortcut layer. From a single unlabeled snapshot, the goal is to construct a confidence set of deterministic size for the first vertex. Since shortcut edges create cycles, the usual tree-based arguments using Jordan centrality do not apply directly. Our method uses auxiliary subcritical bond percolation to expose a tree-like renormalized structure: retained recursive-tree clusters form heavy-tailed blobs, retained shortcuts connect these blobs through a subcritical rank-one random graph, and large components are leading backbone blobs decorated by subcritical shortcut pieces. Applying Jordan centrality inside the largest auxiliary percolation components then gives a deterministic-size root confidence set for the cyclic observed network.

math.PR↗

arXiv2025

Network evolution with mesoscopic delay

Owing to the influence of real-world networks both in science and society, numerous mathematical models have been developed to understand the structure and evolution of these systems, particularly in a temporal context. Recent advancements in fields like distributed cyber-security and social networks have spurred the creation of probabilistic models of evolution, where individuals make decisions based on only partial information about the network's current state. This paper seeks to explore models incorporating network delay, where new participants receive information from a time-lagged snapshot of the system. In the context of mesoscopic network delays, we develop probabilistic tools built on stochastic approximation to understand asymptotics of both local functionals, such as local neighborhoods and degree distributions, as well as global properties, such as the evolution of the degree of the network's initial founder. A companion paper explores the regime of macroscopic delays in the evolution of the network.

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

Functional Central limit theorems for microscopic and macroscopic functionals of inhomogeneous random graphs

We study inhomogeneous random graphs with a finite type space. For a natural generalization of the model as a dynamic network-valued process, the paper establishes the following results: (a) Functional central limit theorems for the infinite vector of microscopic type-densities and characterizations of the limits as infinite-dimensional conditionally Gaussian processes in a certain Banach space. (b) Functional (joint) central limit theorems for macroscopic observables of the giant component in the supercritical regime including size, surplus and number of vertices of various types in the giant component. As a corollary this provides central limit theorems for the size of the largest connected component, its surplus, and its type vector, for percolation on dense graphs obtained from a finite type Graphon. (c) Central limit theorem for the weight of the minimum spanning tree with random i.i.d. Exponential edge weights on dense graph sequences driven by an underlying finite type graphon.

math.PR↗