Geographical Coarsegraining of Complex Networks

TL;DR

This study applies a renormalization-like coarsegraining process to geographically embedded complex networks, revealing that key structural properties are preserved, which has implications for analyzing large-scale networks like the human brain.

Contribution

It introduces a coarsegraining method for geographically embedded networks that preserves their structural characteristics, enabling simplified analysis without loss of essential features.

Findings

Coarsegraining preserves scale-free properties.

Structural features remain unchanged after multiple iterations.

Potential applications in brain network analysis.

Abstract

We perform the renormalization-group-like numerical analysis of geographically embedded complex networks on the two-dimensional square lattice. At each step of coarsegraining procedure, the four vertices on each $2 \times 2$ square box are merged to a single vertex, resulting in the coarsegrained system of the smaller sizes. Repetition of the process leads to the observation that the coarsegraining procedure does not alter the qualitative characteristics of the original scale-free network, which opens the possibility of subtracting a smaller network from the original network without destroying the important structural properties. The implication of the result is also suggested in the context of the recent study of the human brain functional network.