Scale-Invariant Random Graphs: a multiscale approach to network modeling

In the last decades, consistent efforts have been spent to cap- ture specific shades of real systems through the development of random graph models, which have been studied extensively either for their practical value as statistical benchmarks and their theoretical appeal, as abstract tools capable of generating synthetic graphs with realistic properties. In particular, establishing a robust representation of a graph at multiple scales of observation would enable considerable progress in the description, modeling, and control of real- world complex systems. Here, by building on the principles of renormalization group theory from statistical mechanics, we derive a random graph model precisely conceived to provide a statistically consistent description of a network for different resolutions of its units and in an exact manner. We explore two interesting facets of the proposed model, which interlace with different branches of network science. On the one hand, it allows complying with empirical networks to provide up-scaled and down-scaled reconstructions according to a chosen hierarchy of partitions of the original nodes. In this sense, the model constitutes solid support for harboring a coarse-graining scheme of real systems without relying on any arbitrary introduction of a metric space. Secondly, this scale-invariant random graph itself turns out to generate net- works with topological properties that are widespread among real-world systems and thus its mathematical sifting has its own theoretical interest.