Communities and classes in symmetric fractals

TL;DR

This paper investigates community and class structures in symmetric fractal networks, specifically Sierpinski triangle and Koch curve, using differential equations and noise to identify overlapping nodes and classify them.

Contribution

It introduces a method to identify overlapping nodes and classify them into classes based on their community spectra in symmetric fractal networks.

Findings

Overlapping nodes are characterized by probability spectra.

Nodes with similar spectra belong to the same class.

The method reveals community structures in fractal networks.

Abstract

Two aspects of fractal networks are considered: the community structure and the class structure, where classes of nodes appear as a consequence of a local symmetry of nodes. The analysed systems are the networks constructed for two selected symmetric fractals: the Sierpinski triangle and the Koch curve. Communities are searched for by means of a set of differential equations. Overlapping nodes which belong to two different communities are identified by adding some noise to the initial connectivity matrix. Then, a node can be characterized by a spectrum of probabilities of belonging to different communities. Our main goal is that the overlapping nodes with the same spectra belong to the same class.