Message passing and cyclicity transition

Abstract

Message passing, also known as belief propagation, is a versatile framework for analyzing models defined on networks. Its most prototypical application is percolation; yet, the interpretation of the message passing formulation of percolation remains elusive. We show that the message passing solutions commonly associated with the probability of belonging to the giant component actually identify reachability from cycles. This interpretation applies to bond and site percolation on arbitrary undirected or directed networks. Our findings emphasize the distinction between transition in cyclicity and the emergence of the giant component.

Figures (4)

• Figure 1: Directed graph $G$, corresponding dual graph $M_G$, and $M_G$ after root removal. The nodes in $G$ and $M_G$ are denoted by squares and circles, respectively. The color of each node represents its message or node marginal value: one (dark purple), zero (yellow), and non-convergence (white).
• Figure 2: Node-wise comparison between node marginal $y_i$ obtained by solving Eqs. \ref{['eq:mp_bond']} and \ref{['eq:marginal_bond']} and empirical probabilities $\hat{p}^\mathrm{A}_i$, $\hat{p}^\mathrm{U}_i$, $\hat{p}^\mathrm{M}_i$, and $\hat{p}^\mathrm{L}_i$ of belonging to an acyclic, unicyclic, multicyclic, and the largest component in bond percolation. Empirical probabilities are computed from 5000 independent realizations. The networks are (a) an Erdős--Rényi graph with $n = 1000$ nodes and edge probability $p = 0.006$, and (b) a random geometric graph with $n = 1000$ nodes and connection radius $\ell = \sqrt{0.006 / \pi}$.
• Figure 3: Mean node marginal ($\bar{y}$) and the mean fractions of nodes in acyclic components ($S^\mathrm{A}$), in multicyclic components ($S^\mathrm{M}$), and in the largest component ($S^\mathrm{L}$), as a function of edge retention probability $q$ in bond percolation. The fractions are averaged over 500 realizations. The networks are the same as in Fig. \ref{['fig:nodewise']}.
• Figure 4: The deviation of the marginal values obtained by the message passing algorithm from the empirical probabilities for bond and site percolation in real-world networks. For each network and each value of $q$ between 0 and 1 in increments of 0.01, the empirical probabilities for each node are estimated from $10^4$ realizations. The differences in probability are first averaged over all nodes for each $q$, and then aggregated across $q$ either by taking the mean (a) or the maximum (b). The dashed line denotes identity.