Evaluating balance on social networks from their simple cycles
P.-L. Giscard, P. Rochet, R. C. Wilson

TL;DR
This paper introduces novel, scalable measures of social network balance based on simple cycles, using a Monte Carlo method to analyze real and synthetic networks, revealing sociologically meaningful transition phenomena.
Contribution
It presents a new exact formula and Monte Carlo implementation for counting simple cycles, enabling detailed balance analysis without double-counting or reciprocity constraints.
Findings
Networks show strong inter-edge correlations favoring balance.
A sharp transition in cycle balance measures occurs at a certain cycle length.
The transition is absent in random networks, indicating sociological significance.
Abstract
Signed networks have long been used to represent social relations of amity (+) and enmity (-) between individuals. Group of individuals who are cyclically connected are said to be balanced if the number of negative edges in the cycle is even and unbalanced otherwise. In its earliest and most natural formulation, the balance of a social network was thus defined from its simple cycles, cycles which do not visit any vertex more than once. Because of the inherent difficulty associated with finding such cycles on very large networks, social balance has since then been studied via other means. In this article we present the balance as measured from the simple cycles and primitive orbits of social networks. We specifically provide two measures of balance: the proportion of negative simple cycles of length for each which generalises the triangle index, and a ratio…
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