Small-world to fractal transition in complex networks: a renormalization group approach.

We show that renormalization group (RG) theory applied to complex networks is useful to classify network topologies into universality classes in the space of configurations. The RG flow readily identifies a small-world-fractal transition by finding (i) a trivial stable fixed point of a complete graph, (ii) a nontrivial point of a pure fractal topology that is stable or unstable according to the amount of long-range links in the network, and (iii) another stable point of a fractal with shortcuts that exist exactly at the small-world-fractal transition. As a collateral, the RG technique explains the coexistence of the seemingly contradicting fractal and small-world phases and allows us to extract information on the distribution of shortcuts in real-world networks, a problem of importance for information flow in the system.

Laws of population growth.

An important issue in the study of cities is defining a metropolitan area, because different definitions affect conclusions regarding the statistical distribution of urban activity. A commonly employed method of defining a metropolitan area is the Metropolitan Statistical Areas (MSAs), based on rules attempting to capture the notion of city as a functional economic region, and it is performed by using experience. The construction of MSAs is a time-consuming process and is typically done only for a subset (a few hundreds) of the most highly populated cities.

Percolation in hierarchical scale-free nets.

We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small world (the diameter grows either algebraically or logarithmically with the net size), assortative or disassortative (a measure of the tendency of like-degree nodes to be connected to one another), or possess various degrees of clustering. The percolation phase transition can be analyzed exactly in all these cases, due to the self-similar structure of the hierarchical nets.

Synchronous and asynchronous recursive random scale-free nets.

We investigate the differences between scale-free recursive nets constructed by a synchronous, deterministic updating rule (e.g., Apollonian nets), versus an asynchronous, random sequential updating rule (e.

Designer nets from local strategies.

We propose a local strategy for constructing scale-free networks of arbitrary degree distributions, based on the redirection method of Krapivsky and Redner [Phys. Rev. E 63, 066123 (2001)].