Bistability through triadic closure

We propose and analyse a class of evolving network models suitable for describing a dynamic topological structure. Applications include telecommunication, on-line social behaviour and information processing in neuroscience. We model the evolving network as a discrete time Markov chain, and study a very general framework where, conditioned on the current state, edges appear or disappear independently at the next timestep. We show how to exploit symmetries in the microscopic, localized rules in order to obtain conjugate classes of random graphs that simplify analysis and calibration of a model. Further, we develop a mean field theory for describing network evolution. For a simple but realistic scenario incorporating the triadic closure effect that has been empirically observed by social scientists (friends of friends tend to become friends), the mean field theory predicts bistable dynamics, and computational results confirm this prediction. We also discuss the calibration issue for a set of real cell phone data, and find support for a stratified model, where individuals are assigned to one of two distinct groups having different within-group and across-group dynamics

Mining Triadic Closure Patterns in Social Networks

ABSTRACT A closed triad is a group of three people who are connected with each other. It is the most basic unit for studying group phenomena in social networks. In this paper, we study how closed triads are formed in dynamic networks. More specifically, given three persons, what are the fundamental factors that trigger the formation of triadic closure? There are various factors that may influence the formation of a relationship between persons. Can we design a unified model to predict the formation of triadic closure? Employing a large microblogging network as the source in our study, we formally define the problem and conduct a systematic investigation. The study uncovers how user demographics and network topology influence the process of triadic closure. We also present a probabilistic graphical model to predict whether three persons will form a closed triad in dynamic networks. The experimental results on the microblogging data demonstrate the efficiency of our proposed model for the prediction of triadic closure formation

Block matrix formulations for evolving networks

Many types of pairwise interaction take the form of a fixed set of nodes with edges that appear and disappear over time. In the case of discrete-time evolution, the resulting evolving network may be represented by a time-ordered sequence of adjacency matrices. We consider here the issue of representing the system as a single, higher dimensional block matrix, built from the individual time-slices. We focus on the task of computing network centrality measures. From a modeling perspective, we show that there is a suitable block formulation that allows us to recover dynamic centrality measures respecting time's arrow. From a computational perspective, we show that the new block formulation leads to the design of more effective numerical algorithms.Comment: 18 pages, 2 figure

Do Friends-of-Friends Become Friends?

Social scientists have hypothesised that new social contacts arise preferentially between those who currently share neighbours: friends-of-friends have an increased chance of becoming friends. Such a triadic closure eect was quantied through an evolving network model in [Bistability through triadic closure, P. Grindrod, D. J. Higham and M. C. Parsons, Internet Mathematics, to appear]. Here we show how this mathematical model can be used in order to develop a statistical test for the presence of triadic closure in a large scale evolving network. This new tool has the potential to help our understanding of online social interaction and also to predict future network behaviour