Homophily, the tendency of humans to attract each other when sharing similar features, traits, or opinions, has been identified as one of the main driving forces behind the formation of structured societies. Here we ask to what extent homophily can explain the formation of social groups, particularly their size distribution. We propose a spin-glass-inspired framework of self-assembly, where opinions are represented as multidimensional spins that dynamically self-assemble into groups; individuals within a group tend to share similar opinions (intragroup homophily), and opinions between individuals belonging to different groups tend to be different (intergroup heterophily). We compute the associated nontrivial phase diagram by solving a self-consistency equation for “magnetization” (combined average opinion). Below a critical temperature, there exist two stable phases: one ordered with nonzero magnetization and large clusters, the other disordered with zero magnetization and no clusters. The system exhibits a first-order transition to the disordered phase. We analytically derive the group-size distribution that successfully matches empirical group-size distributions from online communities.Received 15 July 2022Revised 18 October 2022Accepted 30 November 2022DOI:https://doi.org/10.1103/PhysRevLett.130.057401© 2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasClassical statistical mechanicsNetwork formation & growthNetwork phase transitionsPhase diagramsPhase transitionsSelf-assemblySocial systemsPhysical SystemsNetworks & random structuresSocial dynamicsSocial networksSpin glassesTechniquesData analysisIsing modelMarkovian processesMean field theoryMetropolis algorithmSimulated annealingStatistical PhysicsInterdisciplinary PhysicsNetworks 

Homophily-Based Social Group Formation in a Spin Glass Self-Assembly Framework

Networks with fat-tailed degree distributions are omnipresent across many scientific disciplines. Such systems are characterized by so-called hubs, specific nodes with high numbers of connections to other nodes. By this property, they are expected to be key to the collective network behavior, e.g., in Ising models on such complex topologies. This applies in particular to the transition into a globally ordered network state, which thereby proceeds in a hierarchical fashion, and with a nontrivial local structure. Standard mean-field theory of Ising models on scale-free networks underrates the presence of the hubs, while nevertheless providing remarkably reliable estimates for the onset of global order. Here we expose that a spurious self-feedback effect, inherent to mean-field theory, underlies this apparent paradox. More specifically, we demonstrate that higher order interaction effects precisely cancel the self-feedback on the hubs, and we expose the importance of hubs for the distinct onset of local versus global order in the network. Due to the generic nature of our arguments, we expect the mechanism that we uncover for the archetypal case of Ising networks of the Barab\'asi-Albert type to be also relevant for other systems with a strongly hierarchical underlying network structure.

Global hierarchy vs local structure: Spurious self-feedback in scale-free networks

Networks with fat-tailed degree distributions are omnipresent across many scientific disciplines. Such systems are characterized by so-called hubs, specific nodes with high numbers of connections to other nodes. By this property, they are expected to be key to the collective network behavior, e.g., in Ising models on such complex topologies. This applies in particular to the transition into a globally ordered network state, which thereby proceeds in a hierarchical fashion, and with a non-trivial local structure. Standard mean-field theory of Ising models on scale-free networks underrates the presence of the hubs, while nevertheless providing remarkably reliable estimates for the onset of global order. Here, we expose that a spurious self-feedback effect, inherent to mean-field theory, underlies this apparent paradox. More specifically, we demonstrate that higher order interaction effects precisely cancel the self-feedback on the hubs, and we expose the importance of hubs for the distinct onset of local versus global order in the network. Due to the generic nature of our arguments, we expect the mechanism that we uncover for the archetypal case of Ising networks of the Barabasi-Albert type to be also relevant for other systems with a strongly hierarchical underlying network structure.

Claudia Merger

Papers

Global hierarchy vs local structure: Spurious self-feedback in scale-free networks

Global hierarchy vs. local structure: spurious self-feedback in scale-free networks.