Measuring partial balance in signed networks

Abstract

Is the enemy of an enemy necessarily a friend? If not, to what extent does this tend to hold? Such questions were formulated in terms of signed (social) networks and necessary and sufficient conditions for a network to be "balanced" were obtained around 1960. Since then the idea that signed networks tend over time to become more balanced has been widely used in several application areas. However, investigation of this hypothesis has been complicated by the lack of a standard measure of partial balance, since complete balance is almost never achieved in practice. We formalize the concept of a measure of partial balance, discuss various measures, compare the measures on synthetic datasets, and investigate their axiomatic properties. The synthetic data involves Erdős-Rényi and specially structured random graphs. We show that some measures behave better than others in terms of axioms and ability to differentiate between graphs. We also use well-known datasets from the sociology and biology literature, such as Read's New Guinean tribes, gene regulatory networks related to two organisms, and a network involving senate bill co-sponsorship. Our results show that substantially different levels of partial balance is observed under cycle-based, eigenvalue-based, and frustration-based measures. We make some recommendations for measures to be used in future work.

Balance and frustration in signed networks

“…Based on large directed signed networks such as Epinions, Slashdot, and Wikipedia, the binary predictions made by balance theory are incorrect almost half of the time [61]. This observation supports the inefficacy of balance theory for structural analysis of directed signed graphs [8].…”

“…For comparing the level of frustration among networks of different size and order, we use the normalised frustration index, F(G) = 1 − 2L(G)/m. This standard measure of partial balance is suggested in a comparative study of structural balance measures because it satisfies key axiomatic properties [8]. Values of F(G) are within the range of [0, 1] and greater values of F(G) represent closeness to a state of structural balance.…”

“…We focus on the frustration index of signed networks, a standard measure of balance mostly estimated or approximated for decades due to the inherent combinatorial complexity. We continue a line of research begun in [8] (which compared various measures of partial balance and suggested that the frustration index should be more widely used), [7] (which explained how integer linear optimisation models can be used to compute the frustration index) and [6] (which substantially improves the efficiency of such computations using algorithmic refinements and powerful mathematical programming solvers).…”

“…Satisfying essential axiomatic properties as a measure of partial balance [8], the frustration index is a key to frequently stated problems in many different fields of research [54,55,21,23]. In biological networks, optimal decomposition of a network into monotone subsystems is made possible by calculating the frustration index of the underlying signed graph [54].…”

Visualizing Structural Balance in Signed Networks

“…We develop Structural-balance-viz by using D3.js with a Java back-end. The visualization is made available by a web interface that allows the selection of the input dataset and of µ (i.e., the network measure that defines the angular coefficient of the x-axis) 3 . The current implementation can consider only the size of the sets of nodes as µ, but the code is easily extendable to consider other characteristics.…”