Structurally balanced growing network as randomized Pólya urn process

TL;DR

The paper addresses how polarization emerges in structurally balanced, growing signed networks by modeling newcomer attachments with an input bias $p$ as a randomized Pólya urn process. Through rate equations, master equations, and fluctuation analyses, it shows that mean faction sizes obey $m(t)=\tfrac{t}{2}+C t^{2p-1}$ and that faction-size differences scale as $\Delta m(t)\sim t^{2p-1}$, with a bimodality threshold $p^{ch}$ that depends on initial conditions. In symmetric initial conditions, bimodality appears for $p>p^{ch}\approx 0.836$, while for asymmetric starts, bimodality is absent and fluctuations mask asymmetry until very large $p$. A separate bimodality criterion indicates a finite-time threshold approaching $p=3/4$, but fluctuations push the practical onset to higher values, governed by a crossover near $p\approx 0.75$ and a long-run regime near $p=1$. The findings provide a growth-based mechanism for polarization and link structural balance with Polya-type reinforcement, with implications for understanding polarization in expanding social systems.

Abstract

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 Pólya 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.

Figures (6)

• Figure 1: A schematic representation of the growing process can be presented both as a network-based framework and an equivalent urn process. The growth dynamics of the signed network adheres to structural balance and resembles a generalized Pólya urn process. Here, we start with a single node (white ball). At each time $t$, the generated complete graph organizes into two mutually hostile factions. The solid blue lines correspond to friendly links, and the dashed red lines correspond to hostile links.
• Figure 2: Evolution of sizes of two factions $m_\pm(t)$ for different attachment biases $p$. Expected faction sizes given by the rate equation (RE, solid lines) closely match the respective means of the agent-based model (ABM, dashed lines) simulation. (a) For a small value of $p=0.3$, the mean sizes of the factions are equal. (b) For a large value $p=0.9$, the means $\left< m_\pm(t) \right>$ diverge. Thin lines show individual trajectories of 100 agent-based simulations of the two faction sizes. The initial condition was a single starting node belonging to $m_+$. The inset in each panel shows the corresponding normalized mean over time, which converges over a long time.
• Figure 3: In time, the normalized group sizes approach equality. Convergence rates depend on the bias $p$. The system starts at $t_0=1$ with a single node belonging to group $m_+$. Exact results derived from the master equation (ME) (\ref{['main:eq:meq']}) are shown as solid lines, while approximate values from the Pólya process formula (\ref{['main:eq:-m-pm-polya']}) and the rate equation (RE) (\ref{['main:eq:general_meanm']}) (dashed) are shown as dotted and dashed lines, respectively.
• Figure 4: Evolution of the three faction size distributions for three values of the attachment bias parameter $p$. Each panel shows the exact distributions derived from the master equation, starting from a single-node system. Two distributions (with dotted outlines) in each panel correspond to an asymmetric initial condition, where the starting node belongs to the faction $m_+$, while the other group $m_-$ is empty. The blue and orange fill are used for the resulting distributions for the initially smaller faction $P(m_-,t\mid m_-^{t_0=1}=0)$ and the larger one $P(m_+,t\mid m_+^{t_0=1}=1)$, respectively. Solid outline with green fill represents the distribution under a symmetric initial condition, where the first node is equally likely to join any faction. It also reflects the case where no specific group is observed, and one asks for the probability that any faction is of size $m$. That is why it is presented as $P(m_-,t) + P(m_+,t)$. (a) At $p=0.7$, all distributions are unimodal and approximately Gaussian. (b) At $p=0.8$, although the asymmetric distributions diverge, the combined result remains unimodal, and no bimodal behavior is observed. (c) At $p=0.9$, the symmetric distribution becomes distinctly bimodal, with the two peaks gradually diverging as new nodes are added.
• Figure 5: Observing bimodality. The second derivative of the distribution evaluated at $m = t/2$ serves as an indicator of bimodality; a positive value signifies the presence of a bimodal distribution. For symmetric initial conditions, bimodality persists in the limit $t \rightarrow \infty$ when the attachment bias $p$ exceeds the characteristic value $p^{ch}\approx 0.836$ indicated by a red cross.