Social network growth with assortative mixing

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Abstract

Networks representing social systems display specific features that put them apart from biological and technological ones. In particular, the number of links attached to a node is positively correlated to that of its nearest neighbours. We develop a model that reproduces this feature, starting from microscopical mechanisms of growth. The statistical properties arising from the simulations are in good agreement with those of the real-world social networks of scientists co-authoring papers in condensed matter physics. Moreover, the model highlights the determinant role of correlations in shaping the network's topology.

Section snippets

Social networks

The research developed in the field of networks has revealed their pervasive presence in technology, biology and society [1]. Systems as different as the Internet, the neurons of the brain, food-chains and scientific literature can all be represented as graphs [2], i.e., geometrical objects composed by nodes connected by edges. While the statistical analysis of these networks has revealed some nearly ubiquitous features [1], social networks (where nodes are people and edges are interactions)

Assortativity in networks

A network is called assortative (disassortative) with regards to a certain property if one can observe a positive (negative) correlation in that property when considering adjacent nodes. In this work, we characterize the social networks, in a first approximation, with a scalar discrete property: the degree, i.e., the number of edges a node has. In other words, we ask ourselves if nodes with a certain degree tend to connect with others with similar or different degree. Indeed, it results that

The model

The model we define in this work was thought to reproduce the assortative character of social network as a result of a few elementary mechanisms of formation of the network at a microscopic level. This model can be thought as a generalization of the Barabási–Albert preferential attachment one [5], and as the assortative version of the one presented in Ref. [6]. While in the Barabási–Albert model the only allowed microscopical mechanism was the addition of new nodes, we include as well mixing,

Simulation of the model

We performed extensive simulations of the model for the two considered functional forms, and for a wide range of values of their parameters. The first issue to address is the emergence of assortativity. In Fig. 1 we reported the average nearest neighbours degree versus the degree for a few realizations of the model, while in Table 1 the assortativity coefficients are reported. Both results indicate that the model is able to reproduce assortativity (at a stroger level in the exponential case).

Conclusions

In conclusion, we succeeded in obtaining the macroscopical property of assortativity, a feature specific to social network, from the microscopical mechanism of growth called “mixing”. The qualitative agreement of all trends emerging from simulation with the real-world comparison network (cond-mat) suggests that the model is able to catch some of the most important features of real social graphs. Moreover, when the parameters of the model are driven in the limit of strong assortativity, it comes

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    It has been demonstrated that plenty of real-world networks have scale-free degree distributions [5–8], small-world effects [9–12], and high clustering properties [13,14]. Generally, social and collaboration networks are assortative mixing [15,16], while biological and technological networks are disassortative mixing [17,18]. Scale-free networks are very robust to random attacks, but are fragile to target attacks [19–22].

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