Densification Arising from Sampling Fixed Graphs

During the past decade, a number of different studies have identified several peculiar properties of networks that arise from a diverse universe, ranging from social to computer networks. A recently observed feature is known as network densification, which occurs when the number of edges grows much faster than the number of nodes, as the network evolves over time. This surprising phenomenon has been empirically validated in a variety of networks that emerge in the real world and mathematical models have been recently proposed to explain it. Leveraging on how real data is usually gathered and used, we propose a new model called Edge Sampling to explain how densification can arise. Our model is innovative, as we consider a fixed underlying graph and a process that discovers this graph by probabilistically sampling its edges. We show that this model possesses several interesting features, in particular, that edges and nodes discovered can exhibit densification. Moreover, when the node degree of the fixed underlying graph follows a heavy-tailed distribution, we show that the Edge Sampling model can yield power law densification, establishing an approximate relationship between the degree exponent and the densification exponent. The theoretical findings are supported by numerical evaluations of the model. Finally, we apply our model to real network data to evaluate its performance on capturing the previously observed densification. Our results indicate that edge sampling is indeed a plausible alternative explanation for the densification phenomenon that has been recently observed.