Leskovec, Kleinberg and Faloutsos (2005) observed that many social networks exhibit properties such as shrinking (i.e., bounded) diameter, densification, and (power-law) heavy tail degree distributions. To explain these phenomena, they introduced a generative model, called the Forest Fire model, and using simulations showed that this model indeed exhibited these properties; however, proving this rigorously was left as an open problem.
In this work, we analyse one of these properties, shrinking diameter. We define a restricted version of their model that incorporates the main features that seem to contribute towards this property, and prove that the graphs generated by this model exhibit shrinking distance to the seed graph. We prove that an even simpler model, the random walk model, already exhibits this phenomenon.
Based on joint work with Reut Levi, Zvi Lotker, Frederik Mallmann-Trenn and Claire Mathieu.