Random Sequential Renormalization of Networks I: Application to Critical Trees

We introduce the concept of Random Sequential Renormalization (RSR) for arbitrary networks. RSR is a graph renormalization procedure that locally aggregates nodes to produce a coarse grained network. It is analogous to the (quasi-)parallel renormalization schemes introduced by C. Song {\it et al.} (Nature {\bf 433}, 392 (2005)) and studied more recently by F. Radicchi {\it et al.} (Phys. Rev. Lett. {\bf 101}, 148701 (2008)), but much simpler and easier to implement. In this first paper we apply RSR to critical trees and derive analytical results consistent with numerical simulations. Critical trees exhibit three regimes in their evolution under RSR: (i) An initial regime $N_0^{\nu}\lesssim N<N_0$, where $N$ is the number of nodes at some step in the renormalization and $N_0$ is the initial size. RSR in this regime is described by a mean field theory and fluctuations from one realization to another are small. The exponent $\nu=1/2$ is derived using random walk arguments. The degree distribution becomes broader under successive renormalization -- reaching a power law, $p_k\sim 1/k^{\gamma}$ with $\gamma=2$ and a variance that diverges as $N_0^{1/2}$ at the end of this regime. Both of these results are derived based on a scaling theory. (ii) An intermediate regime for $N_0^{1/4}\lesssim N \lesssim N_0^{1/2}$, in which hubs develop, and fluctuations between different realizations of the RSR are large. Crossover functions exhibiting finite size scaling, in the critical region $N\sim N_0^{1/2} \to \infty$, connect the behaviors in the first two regimes. (iii) The last regime, for $1 \ll N\lesssim N_0^{1/4}$, is characterized by the appearance of star configurations with a central hub surrounded by many leaves. The distribution of sizes where stars first form is found numerically to be a power law up to a cutoff that scales as $N_0^{\nu_{star}}$ with $\nu_{star}\approx 1/4$.