Scale Free Projections Arise from Bipartite Random Networks

TL;DR

A bipartite extension to the Randomly Stopped Linking Model for generating networks is developed and it is shown that mixtures of geometric distributions can lead to power laws, an intuition suggested by the Central Limit Theorem for distributions with infinite variance.

Abstract

The degree distribution of a real world network -- the number of links per node -- often follows a power law, with some hubs having many more links than traditional graph generation methods predict. For years, preferential attachment and growth have been the proposed mechanisms that lead to these scale free networks. However, the two sides of bipartite graphs like collaboration networks are usually not scale free, and are therefore not well-explained by these processes. Here we develop a bipartite extension to the Randomly Stopped Linking Model and show that mixtures of geometric distributions lead to power laws according to a Central Limit Theorem for distributions with high variance. The two halves of the actor-movie network are not scale free and can be represented by just 5 geometric distributions, but they combine to form a scale free actor-actor unipartite projection without preferential attachment or growth. This result supports our claim that scale free networks are the natural result of many Bernoulli trials with high variance of which preferential attachment and growth are only one example.

Figures (5)

• Figure 1: Three views of the Hollywood actor-movie network used for this paper, with data from pajek_2004. (a) shows the degree of actor nodes when the bipartite graph is projected into an actor-actor view, with links connecting actors who appeared in the same movie together. The degree of the movie nodes (b) and actor nodes (c) from the underlying bipartite graph are also the number of actors in each movie (b) and the number of movies each actor appears in (c). $\gamma$ is the exponent of the fitted power law and $k_{min}$ is the minimum value for which the power law is the best fit to the data.
• Figure 2: Comparison of the original, real network to one relinked via the Bipartite Configuration Model. Here, movies and actors are linked randomly without regard for preferential attachment according to the degree distribution of the movie and actor nodes from the original. The figure shows the degree distribution of the projected actor-actor network from the resulting bipartite network.
• Figure 3: A geometric distribution fit to the movie node degree distribution. Given that $\mu = 11.5$ in the real network, $p = 0.087$ according to Equation \ref{['eq:p']}.
• Figure 4: A mixture of geometric distributions fit for the actor node degree distribution. The four geometric distributions use $p=\{0.046, 0.184, 0.528, 0.940\}$ and weight coefficient of $a=\{0.094, 0.178, 0.311, 0.562\}$, respectively.
• Figure 5: Comparison of the real actor-actor network to one created with the bipartite Randomly Stopped Linking Network and a total of $5$ different stopping probabilities represented by $5$ geometric distributions.