Structurally balanced growing network as randomized P\'olya urn process

We investigate a process of growth of a signed network that strictly adheres to Heider structural balance rules, resulting in two opposing, growing factions. New agents make contact with a random existing agent and join one of the factions with the bias $p$ towards the group they made contact with. The evolution of the group sizes can be mapped to a randomized P\'olya urn model. Aside from $p=1$, the relative sizes of the two factions always tend towards $1/2$, but the behavior differs in the anti-bias regime ($p<1/2$) and the biased one ($p>1/2$). In the anti-bias regime, the expected faction sizes converge toward equality, regardless of initial differences, while in the biased regime, initial size difference persists over time. This difference is obscured by fluctuations, with the faction size distribution remaining unimodal even above $p>1/2$, up until a characteristic point $p^{ch}$, where it becomes bimodal, with initially larger and smaller factions featuring their own distinguishable peaks. We discuss several approaches to estimate this characteristic value. At $p=1$, differences between the relative sizes of factions can persist indefinitely, although still subject to fluctuations.