Evolving networks with disadvantaged long-range connections

TL;DR

This paper investigates how introducing a bias against long-range connections in a growing network affects its structure, revealing a transition from scale-free to stretched exponential degree distributions while maintaining small-world properties.

Contribution

It introduces a model where long-range bonds are disadvantaged in preferential attachment, showing how this alters network degree distributions and preserves small-world features.

Findings

Networks with lpha<1 are similar to scale-free networks.

For lpha>1, degree distribution becomes a stretched exponential.

Small-world property persists across all lpha values.

Abstract

We consider a growing network, whose growth algorithm is based on the preferential attachment typical for scale-free constructions, but where the long-range bonds are disadvantaged. Thus, the probability to get connected to a site at distance $d$ is proportional to $d^{-\alpha}$, where $\alpha $ is a tunable parameter of the model. We show that the properties of the networks grown with $\alpha <1$ are close to those of the genuine scale-free construction, while for $\alpha >1$ the structure of the network is vastly different. Thus, in this regime, the node degree distribution is no more a power law, and it is well-represented by a stretched exponential. On the other hand, the small-world property of the growing networks is preserved at all values of $\alpha $.