Geometric networks and graph limits

      Many real-life networks can be modelled by stochastic processes with a spatial embedding. The spatial reality can be used to represent attributes of the vertices which are inaccessible or unknown, but which are assumed to inform link formation. For example, in a social network, vertices may be considered as members of a social space, where the coordinates represent the interests and background of the users. The graph formation is modelled as a stochastic process, where the probability of a link occurring between two vertices decreases as their metric distance increases. A fundamental question is to determine whether a given network is compatible with a spatial model. That is, given a graph how can we judge whether the graph is likely generated by a spatial model, and if so what is the underlying metric space? Using the theory of graph limits, we show how to recognize graph sequences produced by random graph processes with a linear embedding (a natural embedding into real line). This talk is based on a joint work with Chuangpishit, Hurshman, Janssen, and Kalyaniwalia.