Evolving networks by merging cliques

Abstract

We propose a model for evolving networks by merging building blocks represented as complete graphs, reminiscent of modules in biological system or communities in sociology. The model shows power-law degree distributions, power-law clustering spectra and high average clustering coefficients independent of network size. The analytical solutions indicate that a degree exponent is determined by the ratio of the number of merging nodes to that of all nodes in the blocks, demonstrating that the exponent is tunable, and are also applicable when the blocks are classical networks such as Erdős-Rényi or regular graphs. Our model becomes the same model as the Barabási-Albert model under a specific condition.

Modeling for evolving biological networks with scale-free connectivity, hierarchical modularity, and disassortativity

“…Due to the fitness updating and the PA, the degree-degree correlations follow the power law, reflecting the disassortativity of the networks. As previously reported, the disassortativity is not reproduced if we consider the PA mechanism only [27,28].…”

“…In order to fill the gap, we propose a model (hereinafter called MM model) with growth mechanism by merging modules and PA from the BA model. The MM model shows power-law clustering spectra and power-law degree distributions with arbitrary degree exponents, demonstrating that our model can reproduce finer details of biological networks [27].…”

“…m/a is the first control parameter of the model where a and m denote the number of nodes of the complete graph and the number of merged node(s), respectively. Please note that the process develops without adding extra edges [27].…”

“…In general, the properties are called disassortatvity [30]. It is reported that the BA and MM model have no disassortativity [27,28]. Pastor-Satorras et al [28] and Barrat et al [31] respectively argue that disassortativity emerges with competitive dynamics [32] and weight-driven dynamics [33], suggesting that growing mechanism needs to include fitness such as spatial distance [34], aging [35], and so on.…”

“…For the proposal, we start with a modification of the MM model that has the scale-free feature and the hierarchical modularity [27]. We embed a tunable fitness-driven (FD) mechanism into the MM model, and provide the numerical and analytical solutions for the statistical properties of model networks.…”