Network comparison has immediate applications to problems in image recognition, network reconstruction, analysis of time-dependent networks, and fitting of network models. Many of these applications deal with comparison of network ensembles --- both because real networks are rarely uniquely defined and because network models produce sets of possible networks --- such that simple pairwise comparison of networks is often insufficient.
Here, we extend the problem of network comparison to the comparison of network ensembles. We leverage tools from the theory of probabilistic metric spaces (PMS) to explore the surfaces defined by both ensembles of real networks as well as synthetic generative models. In particular, our PMS approach uses a Modified Lévy Metric (MLM) to build surfaces in a compact and complete space for network ensembles characterized by any underlying network comparison approach whether they themselves define a metric or not.
Our framework has many advantages over simple pairwise comparison of networks. Mainly, it captures key features of a network ensemble that are not contained in single instances of the ensemble, such as the diversity of networks therein. In doing so, it naturally allows us to fit generative models, which generate network ensembles by definition, to sets of real networks (e.g. a set of food webs or power grids) or even to interpolate between different ensembles. Moreover, it provides a natural approach to comparing network models which can, for example, identify important structural decisions in different parameterizations of the same model or simply explore the space of possible networks models.