Source-linked AI summary
Self-similarity of complex networks and hidden metric spaces
M. Angeles Serrano, Dmitri Krioukov, Marian Boguna
TL;DR
The paper addresses how scale-free networks can exhibit self-similarity without an explicit physical metric. It introduces hidden metric-space network models and finds that their clustering and renormalization behavior reproduce observed real-network patterns.
Problem
Self-similarity and scale invariance in complex networks lack a proper geometric interpretation because shortest-path metrics provide inadequate length scales.
Method
The paper models nodes as points in hidden metric spaces, assigns degree-related hidden variables, and connects node pairs with distance-dependent probabilities.
Results
The modeled networks reproduce the self-similar clustering effects observed in real networks, while degree-preserving randomized networks do not preserve these effects.
Takeaways & Limitations
Hidden geometries provide a plausible explanation for the observed topologies and degree-renormalization self-similarity of some scale-free networks.
Takeaways & Limitations
The model’s clustering dependence on α is not derived in detail, and the paper states that those details will be published elsewhere.
Abstract
from arXiv · showhide
We demonstrate that the self-similarity of some scale-free networks with respect to a simple degree-thresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization.