Models of Online Social Networks

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 G0, 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.

Randomised Reproducing Graphs

“…The model differs from duplication graphs, for example those considered in [3,5], in that in those models only one vertex, chosen at random, duplicates at any one time step, whereas in the models considered here and in [2,10,11,12] all vertices simultaneously duplicate.…”

“…(1+2γ+α) n converges to a positive limit, so that the model has a densification power law as defined by [2] with exponent…”

“…The model is a generalisation of the models introduced by Southwell and Cannings [10,11,12] and the Iterated Local Transitivity (ILT) model of [2], introducing stochasticity, which causes the regular structure found in the graphs of [10,11,12] to be lost, and which may make them more suitable for modelling in areas such as social networks; the authors of [2] particularly suggest their model as a model for online social networks, mentioning Facebook and Twitter among other examples.…”

“…In [2], it is shown that the ILT model exhibits a "densification power law", which is defined to mean that, if E n is the number of edges of G n and V n the number of vertices, then E n is proportional to (V n ) a for some a ∈ (1, 2). The following result shows that our model exhibits a phase transition in this respect, with the transition occurring where 2γ + α = 1.…”

“…Additionally the case where α = 0, β = 1, γ = 1 is the ILT model, introduced in [2] as a model for online social networks. The ILT(p) model introduced in [2] as a stochastic generalisation of the ILT model adds extra random edges between the child vertices without regard to whether their parents were connected, and thus cannot be seen as a special case of our model. In addition as defined in [2] the ILT(p) model always has at least the edges found in the basic ILT model, so is not suited to producing relatively sparse graphs.…”