Meso-scale structures in signed networks

TL;DR

This work challenges the exclusive use of structural balance theory for interpreting meso-scale structure in signed networks by introducing a sign-agnostic, SBM-based framework that classifies inter-community relations via edge-density matrices. Analyzing 24 diverse networks, the authors find unbalanced meso-scale structures are common, with assortativity persisting across edge signs and core-periphery patterns prevalent in online social contexts. They also demonstrate that micro-scale triad balance and meso-scale balance are distinct, necessitating independent evaluation of both scales. The approach advances understanding of how complex signed networks organize and highlights the limitations of traditional balance-centric methods.

Abstract

Meso-scale structures in signed networks have been studied under the limiting assumption of the validity of social balance theory, which predicts positive connections within groups and negative connections between groups. Here, we propose and apply a methodology that overcomes this limitation and is able to find and characterize also the different possible unbalanced structures in signed networks. Applying our methodology to 24 empirical networks, from social-political, financial, and biological domains, we find that unbalanced meso-scale structures are prevalent in real-world networks, including cases with substantial balance at the micro-scale of triangles. In particular, we find that assortativity often prevails regardless of the interaction sign and that core-periphery structures are typical in online social networks. Our findings highlight the complexity of meso-scale relational structures, the importance of using computational methods that are a priori agnostic to specific patterns, and the importance of independently evaluating micro- and meso-scale predictions of social balance theory.

Figures (12)

• Figure 1: Structural balance theory from (a) micro-scale and (b) meso-scale perspectives. (a) shows fundamental triads or motifs considered as balanced and unbalanced by social balance theory. (b) Balanced (left) and anti-balanced (right) communities. The top panel shows simple signed networks, with positive links as solid lines and negative links as dashed lines. The two groups $r$ and $s$ are marked with circles. The bottom panels show the corresponding edge density matrices \ref{['eq.w']}.
• Figure 2: Classification of community types. (a) The three potential communities are represented by density matrices. The darker blocks indicate a higher density of edges. Assortative $A^{\pm}$ (disassortative $D^{\pm}$) cases have the two largest entries in the diagonal $w_{rr}^{\pm}$, $w_{ss}^{\pm}$ (antidiagonal $w_{rs}^{\pm}$,$w_{sr}^{\pm}$). The core–periphery type of relationship $C^{\pm}$ has one sparser community (periphery) linked strongly to the denser community (core), which is in turn more interconnected within itself. (b) The three possible community types for positive links and six for negative links, thus leading to a total of 18 possible combination types. Note, blocks are arranged such that the left column has the highest density of positive intra-block edges. This freedom is unavailable for the negative edges, so there are twice as many types.
• Figure 3: Classification of meso-scale structures in signed networks. '$A^{\pm}$', '$C^{\pm}$' and '$D^{\pm}$' stand for 'assortative', 'core–periphery' and 'disassortative' in positive ($+$) or negative ($-$) interaction, respectively, and are defined by Eq. \ref{['eq.adc']}. For example, the pair $(A^+, D^{'-})$ represents the positive edges are assortative and the negative edges are disassortative, and the mark $'$ labeled in $D^{'-}$ indicates that the negative diagonal entries $w_{rr}^- < w_{ss}^-$. From left to right, the 18 different possible configurations are divided according to their level of balance. A relationship is balanced when positive edges are assortative (dense within communities) and negative edges are disassortative (dense between communities). In contrast, a structure is anti-balanced when positive edges are disassortative and negative edges are assortative. These balanced (anti-balanced) structures are considered to have weak structural balance (anti-balance) since they allow the limited presence of other entries in the density matrix. The remaining unbalanced configurations are scored as follows: assortative is $+1$, core-periphery is $0$, and disassortative is $-1$. For negative edges, the scoring is: disassortative is $+1$, core-periphery is 0, and assortative is $-1$.
• Figure 4: Survey on 24 empirical networks. (a) Community types in empirical networks. Top: fraction of networks in which each community type is dominant (computed as the proportion of networks in which the type is the most prevalent). Bottom: fraction of networks in which each community type occurs (computed as the proportion of networks in which the type appears at least once). (b) Frustration index in empirical networks. The x-axis represents balance categories ranging from balanced, unbalanced, to anti-balanced configurations, while the y-axis shows the frustration index. Bars correspond to different methods, with the overall average shown in gray. The calculation of the pairwise frustration index refers to Methods \ref{['FI']}.
• Figure 5: Partition of Spanish High School network with (a) WSBM and (b) Louvain. Top: The blocks are ranked according to community size. The labels in each entry of the matrix correspond to the structure type and the robustness score (see Methods \ref{['Robustness']}). Bottom: The communities are grouped closely together, and the numbers denote the labels of each group. The colors of the nodes represent different age group memberships.