On power-law relationships of the Internet topology
Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication
Small worlds: the dynamics of networks between order and randomness
Small worlds: the dynamics of networks between order and randomness
A random graph model for massive graphs
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
The degree sequence of a scale-free random graph process
Random Structures & Algorithms
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Random Structures & Algorithms
Stochastic models for the Web graph
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Random Evolution in Massive Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
The Diameter of a Scale-Free Random Graph
Combinatorica
Modeling interactome: scale-free or geometric?
Bioinformatics
The web as a graph: measurements, models, and methods
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
The degree distribution of random k-trees
Theoretical Computer Science
A Dynamic Model for On-Line Social Networks
WAW '09 Proceedings of the 6th International Workshop on Algorithms and Models for the Web-Graph
Improved duplication models for proteome network evolution
RECOMB'05 Proceedings of the 2005 joint annual satellite conference on Systems biology and regulatory genomics
Not all scale free networks are Born equal: the role of the seed graph in PPI network emulation
RECOMB'06 Proceedings of the joint 2006 satellite conference on Systems biology and computational proteomics
Degree distribution of large networks generated by the partial duplication model
Theoretical Computer Science
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
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We study and generalize the duplication model of Pastor-Satorras et al. [Evolving protein interaction networks through gene duplication, J. Theor. Biol. 222 (2003) 199-210]. This model generates a graph by iteratively "duplicating" a randomly chosen node as follows: we start at t0 with a fixed graph G(t0) of size t0. At each step t t0 a new node vt is added. The node vt selects an existing node u from V(G(t - 1)) = {v1,...,vt-1} uniformly at random (uar). The node vt then connects to each neighbor of the node u in G(t - 1) independently with probability p. Additionally, vt connects uar to every node of V(G(t - 1)) independently with probability r/t, and parallel edges are merged. Unlike other copy-based models, the degree of the node vt in this model is not fixed in advance; rather it depends strongly on the degree of the original node u it selected. Our main contributions are as follows: we show that (1) the duplication model of Pastor-Satorras et al. does not generate a truncated power-law degree distribution as stated in Pastor-Satorras et al. [Evolving protein interaction networks through gene duplication, J. Theor. Biol. 222 (2003) 199-210]. (2) The special case where r = 0 does not give a power-law degree distribution as stated in Chung et al. [Duplication models for biological networks, J. Comput. Biol. 10 (2003) 677-687]. (3) We generalize the Pastor-Satorras et al. duplication process to ensure (if required) that the minimum degree of all vertices is positive. We prove that this generalized model has a power-law degree distribution.