Measuring preferential attachment in evolving networks

Abstract

A key ingredient of current models proposed to capture the topological evolution of complex networks is the hypothesis that highly connected nodes increase their connectivity faster than their less connected peers, a phenomenon called preferential attachment. Measurements on four networks, namely the science citation network, Internet, actor collaboration and science coauthorship network indicate that the rate at which nodes acquire links depends on the node's degree, offering direct quantitative support for the presence of preferential attachment. We find that for the first two systems the attachment rate depends linearly on the node degree, while for the latter two the dependence follows a sublinear power law.PACS numbers: 89.65.-s, 89.75.-k, 05.10.-a Modeling the highly interconnected nature of various social, biological and communication systems as complex networks or graphs has attracted much attention in the last few years. [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. As for a long time these networks were modeled as completely random [15,16], the recent interest is motivated by the increasing evidence that real network display short length-scale clustering [1,2] and obey unexpected scaling laws [3,4], interpreted as signatures of deviation from randomness. Current approaches, using the tools of statistical physics [6,8,9] search for universalities both in the topology of these webs and in the dynamics governing their evolution. These efforts resulted in a class of models that view networks as evolving dynamical systems, rather than static graphs. Most evolving network models [4,6,9] are based on two ingredients [4]: growth and preferential attachment. The growth hypothesis suggests that networks continuously expand through the addition of new nodes and links between the nodes, while the preferential attachment hypothesis states that the rate Π(k) with which a node with k links acquires new links is a monotonically increasing function of k. While most versions of such evolving network models assume that Π(k) is linear in k [4,6,9], recently several authors proposed that Π(k) could follow a power-law [8,10]. Consequently, the time evolution of the degree k i of node i can be obtained from the first order differential equationwhere m is a constant and Π(k) has the formwith α > 0 an unknown scaling exponent. For α = 1 these models reduce to the scale-free model [4], for which the degree distribution P (k), giving the probability that a node has k links, follows P (k) ∝ k −γ with γ = 3. As Krapvisky, Redner and Leyvraz have shown [8], for α < 1 the degree distribution follows a stretched exponential, while for α > 1 a gelation-like phenomenon is expected, where a single site links to nearly all other nodes. On the other hand the hypothesis (2) Here we offer the first direct attempt to answer these questions in quantitative terms by proposing a numerical method that allows us to extract the functional form of Π(k) directly from dynamical data on real evolving networks. Our measurements indicate that Π(k...

Social and semantic coevolution in knowledge networks

“…social links preferentially go to agents having higher degrees both in the social and semantic dimensions. In the very case of social capital in scientific networks, this result also partially corroborates previous works by Jeong et al (2003) (let us mention that an identical phenomenon has also been described in scientific citation networks by de Solla Price (1976); these findings provide a quantitative sketch of capital accumulation dynamics in scientific communities in terms of both interactions and authority attributions). Yet, more interestingly, comparing propensities between the two cases reveals that they are sensibly different: indeed, linking propensities are much flatter for bloggers, and poorer bloggers tend to be less disadvantaged in receiving links.…”

Evolution of the Internet Map and Load Distribution

“…The empirical evidence of the PA in the Internet has been reported [8,9]. As we will see, however, the assumption that the numbers of nodes and links increase linearly in time does not apply to the real situation of the Internet.…”

Hashtags and followers

“…Several models have been built that try to define the behavior of such networks (Kumar et al 2010;Mislove 2009). The most common model is the preferential attachment, created by Barabási and Albert (1999) and tested with positive results by Newman (2001) and Jeong et al (2003). Preferential attachment states that new links tend to form towards already popular links.…”