Irreversible Aggregation and Network Renormalization

TL;DR

This paper revisits irreversible aggregation in the context of complex network renormalization, revealing that similar scaling laws occur in simple systems, challenging assumptions about self-similarity in network renormalization.

Contribution

It provides an exact solution for aggregation probabilities and shows that similar scaling laws appear in simple and complex systems, questioning their interpretation as signs of self-similarity.

Findings

Exact probability distribution for aggregation states

Scaling laws are identical in simple and complex systems

Challenges the interpretation of scaling laws as evidence of self-similarity

Abstract

Irreversible aggregation is revisited in view of recent work on renormalization of complex networks. Its scaling laws and phase transitions are related to percolation transitions seen in the latter. We illustrate our points by giving the complete solution for the probability to find any given state in an aggregation process $(k+1)X\to X$, given a fixed number of unit mass particles in the initial state. Exactly the same probability distributions and scaling are found in one dimensional systems (a trivial network) and well-mixed solutions. This reveals that scaling laws found in renormalization of complex networks do not prove that they are self-similar.