Degree correlations in signed social networks

Highlights

• Evidence shows that positive networks are assortative, whereas negative ones disassortative.

• We introduce simple models to uncover the relation between sign of links and degree correlations.

• If networks are balanced and split into unequal groups, negative subnetworks are disassortative.

• When unknown, the sign of links can be predicted from the type of degree correlations.

Abstract

We investigate degree correlations in two online social networks where users are connected through different types of links. We find that, while subnetworks in which links have a positive connotation, such as endorsement and trust, are characterized by assortative mixing by degree, networks in which links have a negative connotation, such as disapproval and distrust, are characterized by disassortative patterns. We introduce a class of simple theoretical models to analyze the interplay between network topology and the superimposed structure based on the sign of links. Results uncover the conditions that underpin the emergence of the patterns observed in the data, namely the assortativity of positive subnetworks and the disassortativity of negative ones. We discuss the implications of our study for the analysis of signed complex networks.

Introduction

Over the last few years, an increasing interest in the study of social networks has prompted physicists, mathematicians and computer scientists to join sociologists in their endeavors to develop network models concerned with the antecedents, structure, and evolution of social interaction  [1], [2], [3]. Recent studies have indicated that social networks across many empirical domains display the typical signature of complex networks, namely the long-tailed distribution of the degrees of nodes  [3]. In addition to this, an attempt has been made to uncover the distinctive structural features and empirical regularities that distinguish social networks from other types of complex networks. While in most real networks degrees of neighboring nodes tend to be anticorrelated, research has suggested that social networks tend to be characterized by the opposite correlation pattern  [4], [5]. The tendency of nodes with similar degree to connect with each other is often referred to as “assortative mixing by degree”, and has been observed in a number of social networks, including very large-scale online social networks such as Facebook and Twitter  [6].

A variety of models have been proposed by physicists, sociologists and computer scientists to explain these distinctive properties of social networks. For instance, assortative mixing has been related to the underlying community structure of social networks  [5]. More recently, assortative mixing has been explained in terms of transitivity  [7], homophily  [8], and unsubstitutability of individuals and resources  [9]. Research has also uncovered distinctive interaction patterns within social signed networks in which relationships can have a positive (e.g., trust and friendship) or negative (e.g., distrust and enmity) connotation  [10]. In particular, the theory of “structural balance” has long suggested that, in undirected signed social networks, individuals embedded within closed triads tend to minimize cognitive tension: an individual tends to befriend a friend’s friend, distrust a friend’s enemy, befriend an enemy’s enemy, and distrust an enemy’s friend  [11], [12].

Here we focus our attention on the emergence of degree correlations in signed networks, and how these correlations can be used to predict the sign of links in cases where it is not known or cannot be assessed directly. Indeed, despite the ubiquity and salience of negative relationships in a wide range of social systems, the detection of mixing patterns by degree has been confined primarily to the domain of unsigned networks or simply networks in which nodes were assumed to be connected through positive links (e.g., scientific collaboration networks and interlocking directorate networks  [5], [9]). However, negative networks may exhibit correlation patterns that differ from those detected in positive networks  [13]. Do individuals who distrust many others tend to distrust each other, or do they channel their negative feelings toward other individuals who distrust only very few others? To address this problem, here we propose a class of simple models that help uncover the relation between the sign of links and the type of degree correlations characterizing a network.

The outline of the paper is as follows. In Section  2, we introduce two signed online social networks, and examine the degree distributions and correlations of the positive and negative subnetworks extracted from the data. In Section  3, we propose a generative model of signed networks that polarize into two mutually exclusive groups of nodes. Section  3.1 focuses on the case of random networks with binomial degree distributions, whereas Section  3.2 deals with more realistic cases of networks with power-law degree distributions. Finally, in Section  4 we extend our modeling framework to networks in which nodes can split into three (or more) hostile groups. In Section  5, we summarize our findings and discuss their implications for research on signed complex networks.