Densification and structural transitions in networks that grow by node copying

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 < 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 ≥ 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 → ∞, so that the network is effectively complete as N → ∞.

General expression for the component size distribution in infinite configuration networks

“…Within the scope of random graph models one finds: Erdős-Rényi model, Barabási-Albert model [1], node copying model [5], small-world network [6], configuration network [7,8], and many others. In the configuration network, N nodes are assigned predefined degrees.…”

Ordinal Preferential Attachment: A Self-Organizing Principle Generating Dense Scale-Free Networks

“…Bhat et al . 17 considered a growing network model based on link copying. In this model, nodes are newly created by randomly choosing an existing node and copying each link from the chosen node to its neighbors with independent probability p .…”

“…However, generating dense scale-free networks is difficult without applying external constraints 16 . Although there are a few models that can generate dense scale-free networks with a particular γ value ( 11,17,18 , 12 ), to the authors’ knowledge, the only models with adjustable γ is a configuration model with externally given explicit cutoffs for the scale-free regime 12 . Some authors have proposed network models that can generate networks whose degree distribution follows a power-law with exponent up to a constant degree and decays exponentially beyond that 15,19,20 .…”

Heterogeneous node copying from hidden network structure

“…Node copying is an important network growth mechanism [1][2][3][4][5][6][7]. In social networks, copying is synonymous with triadic closure, playing an important role in the emergence of high clustering [8,9].…”

“…The local clustering coefficient, cc(α), is defined for node α as the number of edges between the (k O ) α neighbors of α, normalized by the the number of edges in a complete subgraph of size (k O ) α . For the CCM, All results for the UCM are from [2,3] or estimated from simulations. The above comparison is for a single copying probability.…”