Evolving networks by merging cliques

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 Erdös-Rényi or regular graphs. Our model becomes the same model as the Barabási-Albert model under a specific condition.

Figure 1

Schematic diagram of growth process of the model network with $a=3$ and $m=1$. Each clique is merged through common node(s) without adding extra edges.

Figure 2

Schematic diagram of coarse-graining method $(a=4)$.

Figure 3

Conditions for increasing of $M_i$ ($a=3$, $m=2$). The existing nodes are filled with black, the new nodes are open circles, and the merged nodes are filled with gray. The thick lines are edges between nearest neighbors of node $i$.

Figure 4

The dependence of $\mathcal{B}(a,\rho ,N)$ on $\rho$ with $N=50000$. With larger $\rho$ and/or $a$ uncertainty to the distribution of clustering coefficient $C\left(k\right)$ becomes larger.

Figure 5

Degree distributions $P\left(k\right)$. Different symbols denote different numerical results and solid lines are depicted by using Eq. (8) with $N=50000$. (A) $a=5$. (B) $a=10$.

Figure 6

Clustering spectra $C\left(k\right)$. Different symbols denote different numerical conditions for $m$ with fixed $N=50000$. (A) $a=5$. (B) $a=10$. The insets show the relationship between $\rho$ and $\alpha$, where $\alpha$ is defined by the exponent from rational distribution $C\left(k\right)\sim k^{-\alpha }$

Figure 7

Comparison of average clustering coefficients $C\left(N\right)$ from two models. $N$ varies from 100 to 50 000 with fixed $a=6$. Inset: dependence of $C$ on $\rho$ with $N=3000$ and $a=6$.

Figure 8

Degree correlations ${\overline{k}}_{nn}\left(k\right)$ with $N=50000$. (A) $0<\rho <0.5$. (B) $\rho =0.5$. (C) $0.5<\rho <1$. Symbols correspond to the numerical results and the solid lines are given by Eqs. (A12, A14, A16), respectively. Note that (A) and (B) are depicted by using a single logarithmic plot, and (C) is a double logarithmic plot.