Densification and structural transitions in networks that grow by node copying

Journal article

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 nonuniversal 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 an 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→∞.

Authors: Bhat, U; Krapivsky, P; Lambiotte, R; Redner, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Physical Society
• Journal: Physical Review E
• Volume: 94
• Issue: 6
• Article number: 062302
• Publication date: 2016-12-08
• Acceptance date: 2016-11-17
• DOI: 10.1103/physreve.94.062302
• EISSN: 2470-0053
• ISSN: 2470-0045
• Pmid: 28085466
• Language: English
• Keywords: physics.data-an; cond-mat.stat-mech