Homophily Based on Few Attributes Can Impede Structural Balance

Piotr J. Górski, Klavdiya Bochenina, Janusz A. Hołyst, and Raissa M. D’Souza

Phys. Rev. Lett. 125, 078302 – Published 13 August 2020

Abstract

Homophily between agents and structural balance in connected triads of agents are complementary mechanisms thought to shape social groups leading to, for instance, consensus or polarization. To capture both processes in a unified manner, we propose a model of pair and triadic interactions. We consider $N$ fully connected agents, where each agent has $G$ underlying attributes, and the similarity between agents in attribute space (i.e., homophily) is used to determine the link weight between them. For structural balance we use a triad-updating rule where only one attribute of one agent is changed intentionally in each update, but this also leads to accidental changes in link weights and even link polarities. The link weight dynamics in the limit of large $G$ is described by a Fokker-Planck equation from which the conditions for a phase transition to a fully balanced state with all links positive can be obtained. This “paradise state” of global cooperation is, however, difficult to achieve requiring $G > O (N^{2})$ and $p > 0.5$ , where the parameter $p$ captures a willingness for consensus. Allowing edge weights to be a consequence of attributes naturally captures homophily and reveals that many real-world social systems would have a subcritical number of attributes necessary to achieve structural balance.

Received 24 January 2020
Revised 15 June 2020
Accepted 9 July 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.078302

Physics Subject Headings (PhySH)

Network evolution Network phase transitions Random walks Social systems Stochastic processes

Networks & random structures Social dynamics Stochastic dynamical systems Weighted networks

Diffusion & random walks Fokker–Planck equation

Interdisciplinary PhysicsNetworksNonlinear DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

Piotr J. Górski ¹, Klavdiya Bochenina², Janusz A. Hołyst ^1,2, and Raissa M. D’Souza ^3,4,*

¹Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL 00-662 Warsaw, Poland
²ITMO University, Kronverkskiy avenue 49, RU 197101 Saint Petersburg, Russia
³University of California, Davis, California 95616, USA
⁴Santa Fe Institute, Santa Fe, New Mexico 87501, USA

^*rmdsouza@ucdavis.edu

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Vol. 125, Iss. 7 — 14 August 2020

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Images

Figure 1
Coevolution of attributes, edge polarities and triadic relations. Nodes’ attributes (a) determine the relations between agents (b) and formation of balanced and unbalanced triads (c). Unbalanced triads drive the evolution of node attributes (a) and so on. In (b) positive and negative links are denoted with $+$ and $-$ , respectively. Panel (c) emphasizes the four types of possible triads (without presenting all links): two balanced ( $Δ_{0}$ and $Δ_{2}$ ) and two unbalanced ( $Δ_{1}$ and $Δ_{3}$ ) denoted with blue and red colors, respectively. If only $Δ_{0}$ and $Δ_{2}$ were present a group polarization would be observed, i.e., agents could be divided into two hostile groups (marked with ellipses) with all links within members of the given group positive and with cross-links negative. However, having some unbalanced triads such a division does not exist.
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Figure 2
Steady states of the ABLTD model for different relations between numbers of attributes $G$ and nodes $N$ . Shown is the density of positive links $ρ$ as a function of probability $p$ (desire for consensus). When $G > O (N^{2})$ the system displays a transition to the paradise state ( $ρ = 1$ ) provided $p$ is sufficiently large. Analytical solutions (dashed and solid lines) fit the numerical results (markers) well for $p < 0.5$ . Letting $G = O (N^{γ})$ we see the fit is more accurate when $γ$ is larger. The solid black and magenta curves show the solutions in the asymptotic limit $N \to \infty$ when $γ > 2$ and $γ < 2$ , respectively. The inset shows time evolution for two example systems. Both achieve a quasistationary phase, after which the smaller system ultimately reaches the paradise state. Main plot shows the time and ensemble average of the link density when the system is in the quasistationary state if such a state is observed.
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Figure 3
Phase diagram for the ABLTD model in the asymptotic limit emphasizes the abrupt change of phases in $γ = 2$ . Axes represent probability $p$ and exponent $γ$ , which describe agents’ willingness for consensus and relation between numbers of nodes $N$ and attributes $G$ [ $G = O (N^{γ})$ ], respectively. The brightness represents the expected density of positive links $ρ$ . When $γ > 2$ and $p \geq 0.5$ the only possible state is a paradise state ( $ρ = 1$ ). Otherwise the system reaches the quasistationary state. For $γ < 2$ for all $p$ the densities of positive and negative links are equal ( $ρ = 0.5$ ) as are the number of balanced and unbalanced triads. For $γ > 2$ such equality is obtained for $p = 1 / 3$ (marked by a dashed line).
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Figure 4
Observed transitions of fates for small systems of size (a) $N = 3$ and (b) $N = 11$ . Plots show the probability of reaching paradise $P_{P}$ in ABLTD as a function of incentive to achieve consensus $p$ . (a) In the case of a one-triad system there is no transition. The figure compares $P_{P}$ for the numerical results of two diverse numbers of attributes $G$ and the approximate $P_{P} (p)$ in the limit $G \to \infty$ . Numerical results for large $G$ agree with the analytical solution. The expected analytical results for the LTD model are also shown for comparison. (b) In the system with $N = 11$ the transition from never ( $P_{P} = 0$ ) to always ( $P_{P} = 1$ ) reaching paradise is observed and becomes sharper with increasing $G$ .
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