Absorbing States of Binary Trust Gossip Are Counted by Plane Partitions

Abstract

We study an opinion dynamics model in which $n$ agents hold directed trust or distrust opinions about one another, represented as a matrix $M \in \{0,1\}^{n \times n}$ in which 1 represents trust and 0 represents distrust. A gossip event $(a, z, y)$ causes agent $z$ to adopt agent $a$'s opinion of $y$, provided that $z$ trusts $a$. We characterize the absorbing states of this process, i.e. the states in which no further opinion change can take place: we find that they are the states in which agents are partitioned into isolated factions, each faction containing a subset of core members who share mutual trust, while the remaining peripheral members trust all core members but receive no trust in return. This structure establishes a bijection between absorbing states on $[n]$ and pairs consisting of a set partition $π$ of $[n]$ together with a choice of non-empty subset of each faction of $π$. The number of such absorbing states is therefore given by OEIS A143405, with exponential generating function $\exp(\exp(x) \cdot (\exp(x) - 1))$. In addition, up to isomorphism, the count equals the number of plane partitions of $n$, given by OEIS A000219, recovering MacMahon's classical product formula $\prod_{k \geq 1} 1/(1 - x^k)^k$. Exhaustive computation for $n \leq 7$ confirms both counts.