Dynamics in Affinity-Weighted Preferential Attachment Networks
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Dynamics in Affinity-Weighted Preferential Attachment Networks Diego Ruiz1
· Juan Campos2 · Jorge Finke2
Received: 9 December 2019 / Accepted: 15 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract During the formation process of stochastic networks, nodes tend to establish edges based on selective linkage mechanisms. In general, these mechanisms involve probability distributions that underlie the selection of target nodes. In social networks, edges are often associated to relationships that are homophilic with respect to individual traits. Such traits include, for example, gender and age, and are referred to as node types. Our work considers an affinityweighted preferential attachment model that characterizes the tendency of two group of nodes to connect to other nodes of the same type. We derive mathematical expressions for the dynamics and convergence of homophily measures at the node, group, and network level. Furthermore, we characterize the convergence of network modularity and show that the formation of community structures can be expressed as a function of network homophily. Keywords Community structure · Network homophily · Social networks
1 Introduction Understanding the formation of topological measures as the outcomes of selective linkage mechanisms has received increasing attention in recent years [1–10]. A number of network models aim to explain how the manner in which new edges are established ultimately shapes the evolution of measures of average degree and degree distributions [11–14]. In particular, the work in [14] introduces the notion of preferential attachment, a mechanism in which new
Communicated by Pierpaolo Vivo.
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Diego Ruiz [email protected] Juan Campos [email protected] Jorge Finke [email protected]
1
Universidad del Cauca–Universidad del Valle, Calle 5 # 4-70, Popayán, Cauca–Calle 13 # 100-00 Cali, Colombia
2
Pontificia Universidad Javeriana Cali, Calle 18 # 118-250 Cali, Colombia
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edges are established according to an underlying distribution that strongly favors nodes with a high degree. For linear preferential attachment, in particular, new edges are established in proportion to the degree of existing nodes. Preferential attachment models provide analytical frameworks that account for the formation of networks with a variety of single and extended power law degree distributions [1–7]. Although such models help us to understand conditions under which power laws in social networks emerge, they fall short in recreating the division of networks into communities [15–18]. Recent studies show that the topological measures of networks are often driven by specific node attributes [3–5,19–22]. To formalize the formation of communities from particular linkage mechanisms, different network models partition the set of nodes into groups whose members share a similar attribute (referred to as a node type) [5–8]. The work in [6] introduces the notion of group affinity, which captures the tendency of a group of nodes t
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