Degree distribution of large networks generated by the partial duplication model

  • Authors:

  • Si Li;Kwok Pui Choi;Taoyang Wu

  • Affiliations:

  • Department of Mathematics, National University of Singapore, Singapore 119076, Singapore;Department of Mathematics, National University of Singapore, Singapore 119076, Singapore and Department of Statistics and Applied Probability, National University of Singapore, Singapore 119076, S ...;Department of Mathematics, National University of Singapore, Singapore 119076, Singapore

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2013

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Abstract

In this paper, we present a rigorous analysis on the limiting behavior of the degree distribution of the partial duplication model, a random network growth model in the duplication and divergence family that is popular in the study of biological networks. We show that for each non-negative integer k, the expected proportion of nodes of degree k approaches a limit as the network becomes large. This fills in a gap in previous studies. In addition, we prove that p=1/2, where p is the selection probability of the model, is the phase transition for the expected proportion of isolated nodes converging to 1, and hence answer a question raised in Bebek et al. [G. Bebek, P. Berenbrink, C. Cooper, T. Friedetzky, J. Nadeau, S.C. Sahinalp, The degree distribution of the generalized duplication model, Theoret. Comput. Sci. 369 (2006) 239-249]. We also obtain asymptotic bounds on the convergence rates of degree distribution. Since the observed networks typically do not contain isolated nodes, we study the subgraph consisting of all non-isolated nodes contained in the networks generated by the partial duplication model, and show that p=1/2 is again a phase transition for the limiting behavior of its degree distribution.