Preferential and anti-preferential attachment conditioned by structural proximity in fixed-size networks

Anti-preferential attachment is a social network formation mechanism where nodes with a lower degree have more chances of acquiring new connections. Recent findings indicate interesting conditions under which anti-preferential attachment appears: it has been documented that pairs of online users with one or more friends in common are more likely to become friends if they have low degrees. Among pairs of users without common peers, there is a positive correlation between their degrees and the probability that they will eventually become friends. We aim to address this empirical observation by investigating how the common friends factor may affect social network dynamics. We present two agent-based network evolution models, in which preferential and anti-preferential attachment regimes are present, with a balancing parameter trading off between these two competing mechanisms. In the first model, at each iteration, one of the two mechanisms is selected at random. Then the chosen mechanism determines a pair of nodes for connection. The second model works similarly, but in this case, preferential attachment links only pairs with no common friends, while anti-preferential attachment connects pairs with one or more shared friends. Both models remove a randomly selected edge at the end of each turn to maintain the number of edges in the network constant. We study these two models using analytical derivations, numerical computations, and agent-based simulations to determine how the inclusion of the factor of common friends in attachment-based network dynamics affects the properties of simulated networks. Our results indicate that both models exhibit considerable differences moderated by the balancing parameter. When this parameter is set to a small value, the second model leads to substantially less clustered, less degree-heterogeneous, but more connected networks. Furthermore, the effects of varying the balancing parameter on network clustering and average path length are opposite for the two models—negative for the first model and positive for the second. Consequently, when the balancing parameter is high, networks generated by the second model exhibit higher clustering rates and the average path length.