A unified model for Sierpinski networks with scale-free scaling and small-world effect

TL;DR

This paper introduces a unified evolutionary Sierpinski network model that combines deterministic and random variants, exhibiting scale-free and small-world properties, and analyzes its structural features and game dynamics.

Contribution

It presents a novel unified framework for Sierpinski networks, providing analytical expressions for key properties and demonstrating their small-world and scale-free characteristics.

Findings

Networks follow a power-law degree distribution with tunable exponent.

Networks exhibit small-world effect with high clustering and short path lengths.

The model's properties are confirmed by numerical simulations.

Abstract

In this paper, we propose an evolving Sierpinski gasket, based on which we establish a model of evolutionary Sierpinski networks (ESNs) that unifies deterministic Sierpinski network [Eur. Phys. J. B {\bf 60}, 259 (2007)] and random Sierpinski network [Eur. Phys. J. B {\bf 65}, 141 (2008)] to the same framework. We suggest an iterative algorithm generating the ESNs. On the basis of the algorithm, some relevant properties of presented networks are calculated or predicted analytically. Analytical solution shows that the networks under consideration follow a power-law degree distribution, with the distribution exponent continuously tuned in a wide range. The obtained accurate expression of clustering coefficient, together with the prediction of average path length reveals that the ESNs possess small-world effect. All our theoretical results are successfully contrasted by numerical simulations. Moreover, the evolutionary prisoner's dilemma game is also studied on some limitations of the ESNs, i.e., deterministic Sierpinski network and random Sierpinski network.