Evaluating balance on social networks from their simple cycles

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 R of negative simple cycles of length for each 20 which generalises the triangle index, and a ratio K which extends the relative signed clustering coefficient introduced by Kunegis. To do so, we use a Monte Carlo implementation of a novel exact formula for counting the simple cycles on any weighted directed graph. Our method is free from the double-counting problem affecting previous cycle-based approaches, does not require edge-reciprocity of the underlying network, provides a gray-scale measure of balance for each cycle length separately and is sufficiently tractable that it can be implemented on a standard desktop computer. We observe that social networks exhibit strong inter-edge correlations favouring balanced situations and we determine the corresponding correlation length ξ . For longer simple cycles, R undergoes a sharp transition to values expected from an uncorrelated model. This transition is absent from synthetic random networks, strongly suggesting that it carries a sociological meaning warranting further research.

Balance and frustration in signed networks

“…While this study provided an overview of the state-of-the-art of the numerical computations on signed graphs and the vast range of applications to which it can be applied, it is by no means an exhaustive survey on the applications of the frustration index. From a computational perspective, this and some other recent studies [40,39] call for more advanced computational models that put larger networks within the reach of exact analysis. As another future research direction, one may consider formulating edge-based measures of stability for directed signed networks based on theories involving directionality and signed ties [61,87].…”

A General Purpose Algorithm for Counting Simple Cycles and Simple Paths of Any Length

“…Instead, the algorithm is best used in conjunction with Monte Carlo methods. We demonstrate this procedure in a separate publication [19], where we use it in a sociological context to obtain the ratios of negative to positive simple cycles of length up to 20 on several real-world directed signed networks with up to 130,000+ vertices.…”

Discovering Interesting Cycles in Directed Graphs

“…Discovering cyclic pa erns in graphs has not received much a ention in the data-mining community. Giscard et al [10] evaluate the balance of a signed social network by nding simple cycles. Kumar and Calders [15] propose an algorithm for enumerating all simple cycles in a directed temporal network, by extending Johnson's algorithm [12] to a temporal se ing.…”