[StreamGraph] Triangle Law

Evolution of the social network of scientific collaborations, Barabashi, 2001

The co-authorship network of scientists represents a prototype of complex evolving networks. In
addition, it offers one of the most extensive database to date on social networks. By mapping the
electronic database containing all relevant journals in mathematics and neuro-science for an eightyear period (1991-98), we infer the dynamic and the structural mechanisms that govern the evolution and topology of this complex system. Three complementary approaches allow us to obtain a detailed characterization. First, empirical measurements allow us to uncover the topological measures that characterize the network at a given moment, as well as the time evolution of these quantities.
The results indicate that the network is scale-free, and that the network evolution is governed by
preferential attachment, affecting both internal and external links. However, in contrast with most model predictions the average degree increases in time, and the node separation decreases. Second, we propose a simple model that captures the network’s time evolution. In some limits the model can be solved analytically, predicting a two-regime scaling in agreement with the measurements. Third, numerical simulations are used to uncover the behavior of quantities that could not be predicted analytically. The combined numerical and analytical results underline the important role internal links play in determining the observed scaling behavior and network topology. The results and methodologies developed in the context of the co-authorship network could be useful for a systematic study of other complex evolving networks as well, such as the world wide web, Internet,or other social networks.

Structure of a large social network. Barabashi, et.al, 2003

We study a social network consisting of over 104 individuals, with a degree distribution exhibiting
two power scaling regimes separated by a critical degree kcrit, and a power law relation between
degree and local clustering. We introduce a growing random model based on a local interaction
mechanism that reproduces all of the observed scaling features and their exponents. Our results
lend strong support to the idea that several very different networks are simultenously present in the human social network, and these need to be taken into account for successful modeling