Random Sequential Renormalization and Agglomerative Percolation in Networks: Application to Erd"os-Renyi and Scale-free Graphs

We study the statistical behavior under random sequential renormalization(RSR) of several network models including Erd"os R’enyi (ER) graphs, scale-free networks and an annealed model (AM) related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [C.Song et.al], RSR allows a more fine-grained analysis of the renormalization group (RG) flow, and unravels new features, that were not discussed in the previous analyses. In particular we find that all networks exhibit a second order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling laws seen there. For critical trees it happens as N/N0 -> 0 in the limit of large systems (where N0 is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N0 in sparse ER graphs and in the annealed model, while it happens for N/N0 -> 1 on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a star-like structure in agreement with the results of Radicchi et. al. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering ‘supernodes’ as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics.

💡 Research Summary

**
The paper introduces a novel renormalization scheme for complex networks called Random Sequential Renormalization (RSR) and applies it to Erdős‑Rényi (ER) random graphs and Barabási‑Albert (BA) scale‑free networks. In RSR, at each renormalization step a single node is chosen uniformly at random and all nodes within a prescribed graph distance b (the “box radius”) are merged into a single super‑node. All external links to the merged neighborhood are rewired to the super‑node, internal links are removed, and the mass of the super‑node is updated to the sum of the masses of the absorbed nodes. This process is repeated until only one node remains.

The authors emphasize two major advantages over earlier box‑covering methods (e.g., Song et al., 2005). First, RSR does not require an optimal tiling of the graph; the coarse‑graining is purely local and stochastic, which eliminates the dependence on box placement and size that plagued previous approaches. Second, because each step removes only a small fraction of the nodes, the renormalization group (RG) flow can be sampled with far higher resolution, providing many more data points for finite‑size scaling analyses.

A key insight is that RSR can be interpreted as a cluster‑growth process: each randomly selected node is a cluster that absorbs all neighboring clusters within distance b. This observation allows the authors to adapt the Newman‑Ziff (NZ) algorithm, originally designed for percolation cluster growth, to efficiently track the distribution of cluster masses (the number of original nodes contained in each super‑node). Using the NZ algorithm, the authors simulate networks up to 10⁷ nodes, far beyond what is feasible when tracking full degree information.

The central result is the discovery of a continuous (second‑order) phase transition that the authors term “agglomerative percolation” (AP). In all examined network families, as the renormalization proceeds the system passes from a regime where many small clusters coexist to a regime dominated by a single giant hub that contains a finite fraction of the original nodes. For sparse ER graphs (average degree ≈2.4) the transition occurs at a finite value of the size ratio x = N/N₀ (where N₀ is the initial number of nodes). Finite‑size scaling analyses reveal critical exponents (β, γ, ν) that satisfy the usual scaling relations of ordinary percolation and appear to be independent of the precise average degree of the ER ensemble.

In contrast, for BA scale‑free networks the transition is pushed to x → 1, i.e., it occurs only when the network is almost completely coarse‑grained. This makes the critical region extremely narrow and hampers precise exponent estimation, but the qualitative picture—emergence of a dominant hub and associated scaling of mass distributions—remains the same.

To rationalize the pre‑transition behavior, the authors develop a mean‑field theory based on generating functions. The theory predicts the evolution of the degree distribution, average degree, and cluster‑mass distribution for ER graphs before the AP transition. Numerical results for an “annealed model” (where edges are randomly rewired after each RSR step) agree very well with the mean‑field predictions, confirming that the theory captures the essential stochastic dynamics in the absence of loops. However, once the giant hub forms, fluctuations become large and loops proliferate, causing the mean‑field description to break down.

The authors also examine the effect of the box radius b. While most of the paper focuses on b = 1, additional simulations for larger b show that the qualitative nature of the transition does not change, although the location of the critical point shifts as expected.

After the AP transition, the network evolves toward a star‑like topology: a central hub connected to many leaf nodes. This late‑time structure matches earlier findings by Radicchi et al. (2008) and provides a clear physical picture of how RSR compresses a heterogeneous network into a single dominant super‑node.

Overall, the study demonstrates that (i) RSR offers a fine‑grained, computationally efficient RG framework for complex networks, (ii) the scaling laws previously reported for network renormalization can be reinterpreted as signatures of an underlying agglomerative percolation transition rather than evidence of intrinsic fractality, and (iii) mass‑based observables are far more tractable than degree‑based ones for large‑scale simulations. The paper suggests several avenues for future work, including applying RSR to networks with community structure, exploring different values of b, and testing the AP transition on empirical data sets to assess its relevance for real‑world systems.