Research seems to converge on the idea that a variety of social networks (e.g., online social networks and scientific collaboration networks) share a distinctive empirical regularity: the degrees of neighboring nodes tend to be correlated. By contrast, other types of networks (e.g., technological and biological ones) are characterized by the opposite correlation pattern. The tendency of nodes with (dis)similar degrees to connect with each other is often referred to as “(dis)assortative mixing by degree”. Assortativity has been regarded as resulting from transitivity or from the underlying community structure of the network. Despite the ubiquity and salience of degree correlations in many networks, the detection of assortativity patterns has been confined primarily to unsigned networks or networks in which the sign of all connections is assumed to be positive. Relatively little attention has been devoted to the emergence of degree correlations in signed network, and in particular in negative social networks, where individuals are connected through links with a negative connotation, such as distrust, enmity, and competition. Whether negative ties tend to be forged between nodes characterized by similar or dissimilar degrees still remains largely unexplored. This paper is an attempt to address this shortcoming.
We analized two online social networks, in which links between individuals can be either positive (trust or friendship) or negative (distrust or enmity). We detect degree correlations by using two measures: the average degree of the first neighbors of nodes with same degree; and the Pearson correlation coefficient evaluated between the degrees of connected nodes. Findings indicate that, when the sign of links is ignored, both networks are assortative. To study the impact of the sign of links on degree correlations, from both networks we extract the positive and negative subnetworks composed only by links of the same sign. The positive subnetworks are assortative, the negative ones are disassortative: high-degree nodes are preferentially connected with low degree ones, and vice versa (see Fig.1). Results seem to indicate that the sign of links has some bearing on the degree correlations observed in social networks. To shed light on this hypothesis, we construct networks with power-law degree distributions. We then assign each node to one of two mutually exclusive groups, and associate a positive sign to connections between nodes of the same group and a negative sign to connections between nodes of different groups. Simulations show that, while the positive subnetwork displays an assortative trend, the negative subnetwork shows a disassortative trend that varies as a function of the difference in size between the two groups. Results indicate that the mixing patterns of the positive and negative subnetworks diverge only when: the corresponding global unsigned network is assortative, the two groups of nodes are of unequal size, and the signed global network is structurally balanced. We investigate the role of each of these conditions, and different combinations of them and extend our analysis by studying the case in which the global unsigned network is disassortative and nodes can be allocated to three or more mutually exclusive groups.
