Models of Online Social Networks

Anthony Bonato, Noor Hadi, Paul Horn, Paweł Prałat, and Changping Wang
Source: Internet Math. Volume 6, Number 3 (2009), 285-313.

Abstract

We present a deterministic model for online social networks (OSNs) based on transitivity and local knowledge in social interactions. In the iterated local transitivity (ILT) model, at each time step and for every existing node $x$, a new node appears that joins to the closed neighbor set of $x$. The ILT model provably satisfies a number of both local and global properties that have been observed in OSNs and other real-world complex networks, such as a densification power law, decreasing average distance, and higher clustering than in random graphs with the same average degree. Experimental studies of social networks demonstrate poor expansion properties as a consequence of the existence of communities with low numbers of intercommunity edges. Bounds on the spectral gap for both the adjacency and normalized Laplacian matrices are proved for graphs arising from the ILT model indicating such bad expansion properties. The cop and domination numbers are shown to remain the same as those of the graph from the initial time step $G_0$, and the automorphism group of G0 is a subgroup of the automorphism group of graphs generated at all later time steps. A randomized version of the ILT model is presented that exhibits a tunable densification power-law exponent and maintains several properties of the deterministic model.

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