Label propagation on binomial random graphs
Marcos Kiwi, Lyuben Lichev, Dieter Mitsche, Paweł Prałat
TL;DR
This work analyzes a non-binary label propagation algorithm on the Erdős-Rényi random graph $\mathcal{G}(n,p)$, where vertices iteratively adopt the majority label in their neighborhood with specific tie-breaking rules. The authors develop a multi-stage edge-exposure framework and introduce the ALAP procedure to decouple dependencies, enabling rigorous analysis of convergence behavior. They prove that for $n^{5/8+\varepsilon}\le np\ll n$ the process a.a.s. ends with a single label after five rounds, and they precisely characterize the surviving label in broader regimes: it is typically $1$ when $n^{2/3}\ll np\ll n$, but not when $n^{5/8+\varepsilon}\le np\ll n^{2/3}$; at the critical scaling $np=\Theta(n^{2/3})$ the surviving label has a nontrivial distribution. A key technical tool is a monotonicity lemma for binomial tails (Lemma bis), together with couplings and Berry–Esseen/Slud-type bounds, which may have independent interest for related stochastic processes. The results illuminate how random initial labeling and dense graph topology interact to drive consensus (or a limited diversity) in majority-dynamics like LPA, with implications for understanding community detection under randomness.
Abstract
We study the behavior of a label propagation algorithm (LPA) on the Erdős-Rényi random graph $\mathcal{G}(n,p)$. Initially, given a network, each vertex starts with a random label in the interval $[0,1]$. Then, in each round of LPA, every vertex switches its label to the majority label in its neighborhood (including its own label). At the first round, ties are broken towards smaller labels, while at each of the next rounds, ties are broken uniformly at random. The algorithm terminates once all labels stay the same in two consecutive iterations. LPA is successfully used in practice for detecting communities in networks (corresponding to vertex sets with the same label after termination of the algorithm). Perhaps surprisingly, LPA's performance on dense random graphs is hard to analyze, and so far convergence to consensus was known only when $np\ge n^{3/4+\varepsilon}$, where LPA converges in three rounds. By defining an alternative label attribution procedure which converges to the label propagation algorithm after three rounds, a careful multi-stage exposure of the edges allows us to break the $n^{3/4+\varepsilon}$ barrier and show that, when $np \ge n^{5/8+\varepsilon}$, a.a.s.\ the algorithm terminates with a single label. Moreover, we show that, if $np\gg n^{2/3}$, a.a.s.\ this label is the smallest one, whereas if $n^{5/8+\varepsilon}\le np\ll n^{2/3}$, the surviving label is a.a.s.\ not the smallest one. En passant, we show a presumably new monotonicity lemma for Binomial random variables that might be of independent interest.