Large independent sets in recursive Markov random graphs

TL;DR

This work introduces G^{r}_{n,p}, a novel recursive Markov random graph model with decay parameter r∈(0,1], smoothly generalizing the Erdős–Rényi graph (recovered at r=1). The authors establish a phase-transition-like behavior: for any r<1, the independence number α(G^{r}_{n,p}) grows at least as Ω(n/log n) (in fact linked to the prime-counting function π(n)), while an explicit upper bound α(G^{r}_{n,p}) ≤ (c^{*}+ε) n is obtained via a first-moment analysis around a root c^{*} of a carefully crafted function φ_{r}. They prove sharp concentration results for the average degree and develop a detailed analysis of a dependent Bernoulli sequence arising from edge-generation probabilities, enabling both lower and upper bounds and nontrivial consequences for chromatic numbers and Hadwiger-type minors. Additionally, a greedy algorithm is analyzed, showing a near-optimal performance gap quantified by exponents in n, and the work discusses implications for line graphs and matchings. Overall, the paper advances understanding of how dependence in edge formation impacts independent sets and related graph parameters, with a rigorous probabilistic and combinatorial treatment in the nontrivial r<1 regime.

Abstract

Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical Erdős-Rényi-Gilbert random graph $G_{n,p}$ has been analysed and shown to have largest independent sets of size $Θ(\log{n})$ w.h.p. This classical model does not capture any dependency structure between edges that can appear in real-world networks. We initiate study in this direction by defining random graphs $G^{r}_{n,p}$ whose existence of edges is determined by a Markov process that is also governed by a decay parameter $r\in(0,1]$. We prove that w.h.p. $G^{r}_{n,p}$ has independent sets of size $(\frac{1-r}{2+ε}) \frac{n}{\log{n}}$ for arbitrary $ε> 0$, which implies an asymptotic lower bound of $Ω(π(n))$ where $π(n)$ is the prime-counting function. This is derived using bounds on the terms of a harmonic series, Turán bound on stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Since $G^{r}_{n,p}$ collapses to $G_{n,p}$ when there is no decay, it follows that having even the slightest bit of dependency (any $r < 1$) in the random graph construction leads to the presence of large independent sets and thus our random model has a phase transition at its boundary value of $r=1$. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most $1 + \frac{\log{n}}{(1-r)}$ w.h.p. when the lowest degree vertex is picked at each iteration, and also show that under any other permutation of vertices the algorithm outputs a set of size $Ω(n^{1/1+τ})$, where $τ=1/(1-r)$, and hence has a performance ratio of $O(n^{\frac{1}{2-r}})$.

Theorems & Definitions (55)

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