Multilevel structural evaluation of signed directed social networks based on balance theory

Abstract

Balance theory explains how network structural configurations relate to tension in social systems, which are commonly modeled as static undirected signed graphs. We expand this modeling approach by incorporating directionality of edges and considering three levels of analysis for balance assessment: triads, subgroups, and the whole network. For triad-level balance, we develop a new measure by utilizing semicycles that satisfy the condition of transitivity. For subgroup-level balance, we propose measures of cohesiveness (intra-group solidarity) and divisiveness (inter-group antagonism) to capture balance within and among subgroups. For network-level balance, we re-purpose the normalized line index to incorporate directionality and assess balance based on the proportion of edges whose position suits balance. Through comprehensive computational analyses, we quantify, analyze, and compare patterns of social structure in triads, subgroups, and the whole network across a range of social settings. We then apply our multilevel framework to examine balance in temporal and multilayer networks to demonstrates the generalizability of our approach. In most cases, we find relatively high balance across the three levels; providing another confirmation of balance theory. We also deliver empirical evidence for the argument that balance at different levels is not the same social phenomenon measured at different scales, but represents different properties (triadic balance, internal cohesion and external division of subgroups, and overall network polarization), and should therefore be evaluated independently from one another. We propose a comprehensive yet parsimonious approach to address this need.

Directed Network Embedding with Virtual Negative Edges

“…We also note that, among six variants, DIVINE(RWR) and DIVINE(ATP) showed the lowest AUC in most cases. Further investigation reveals that, when RWR and ATP are used, both (1) the degrees of negativity between a node 𝑣 𝑖 and each node non-reachable from 𝑣 𝑖 and (2) those between 𝑣 𝑖 and each zero in-degree node are predicted to be high. As a result, when equipped with RWR or ATP, DIVINE selects VNEs randomly among them, failing to choose VNEs effectively for accurate embedding.…”

“…In Step 2, we select VNEs from each node 𝑣 𝑖 based on each 𝑥 𝑖 𝑗 , i.e., 𝑣 𝑖 's degree of negativity for each of other nodes, 𝑣 𝑗 . In Step 3, after determining the number of VNEs from each node, based on the structural balance [1,4,9] of signed directed networks, we add VNEs to G and model it as a signed directed network S = (V, E + , E − ), where V = {𝑣 1 , 𝑣 2 , • • • , 𝑣 𝑛 } denotes the set of 𝑛 nodes and E + and E − denote the sets of directed positive edges and directed VNEs, respectively. We let 𝑒 + 𝑖 𝑗 ∈ E + and 𝑒 − 𝑖 𝑗 ∈ E − be a directed positive edge and a VNE from 𝑣 𝑖 (i.e., source) to 𝑣 𝑗 (i.e., target), respectively.…”

“…For each existent edge 𝑒 𝑖 𝑗 , which we directly observe, we assign the highest value of 1 to the corresponding entry of W (i.e., if 𝑎 𝑖 𝑗 = 1, then 𝑤 𝑖 𝑗 = 1). For all non-existent edges, we assign an equal weight less than 1 to the corresponding entries of W. 1 Then, we factorize A into the product of two low-rank matrices P and Q by performing weighted alternating least squares using W as weights. Specifically, to obtain P and Q, which are latent factors representing the properties of nodes as sources and targets, respectively, we aim to minimize the following objective function:…”

“…Moreover, among a large number of potential VNEs, only a small fraction of them clearly represent negative relationships. Below, we further discuss the effect of the parameter 𝜃 on structural balance [1,4,9], an important property of signed networks.…”

“…Next, for each node 𝑖, we find a proper number of nodes with respect to which 𝑖 has the highest degree of negativity; then, we add VNEs from 𝑖 to such nodes to the input network (Sections 3.3 and 3.4). We here deal with the issue of determining the number of VNEs from each node based on structural balance [1,4,9], which is a well-known property of signed networks.…”