Densification and Structural Transitions in Networks that Grow by Node Copying

TL;DR

This paper introduces a copying model for growing networks that exhibits a transition from sparse to dense structures at a critical copying probability, revealing complex behaviors like non-universal degree distributions and structural phase transitions.

Contribution

The study presents a new network growth model with a copying mechanism, analyzing its phase transition from sparse to dense networks and uncovering novel structural phenomena.

Findings

Power-law degree distribution with a non-universal exponent

Structural transitions in the number of m-cliques at critical p-values

Networks become effectively complete as N approaches infinity when second neighbor linking is included

Abstract

We introduce a growing network model---the copying model---in which a new node attaches to a randomly selected target node and, in addition, independently to each of the neighbors of the target with copying probability $p$. When $p<\frac{1}{2}$, this algorithm generates sparse networks, in which the average node degree is finite. A power-law degree distribution also arises, with a non-universal exponent whose value is determined by a transcendental equation in $p$. In the sparse regime, the network is "normal", e.g., the relative fluctuations in the number of links are asymptotically negligible. For $p\geq \frac{1}{2}$, the emergent networks are dense (the average degree increases with the number of nodes $N$) and they exhibit intriguing structural behaviors. In particular, the $N$-dependence of the number of $m$-cliques (complete subgraphs of $m$ nodes) undergoes $m-1$ transitions from normal to progressively more anomalous behavior at a $m$-dependent critical values of $p$. Different realizations of the network, which start from the same initial state, exhibit macroscopic fluctuations in the thermodynamic limit---absence of self averaging. When linking to second neighbors of the target node can occur, the number of links asymptotically grows as $N^2$ as $N\to\infty$, so that the network is effectively complete as $N\to \infty$.