Reshuffling scale-free networks: From random to assortative

To analyze the role of assortativity in networks we introduce an algorithm which produces assortative mixing to a desired degree. This degree is governed by one parameter $p$. Changing this parameter one can construct networks ranging from fully random $(p=0)$ to totally assortative $(p=1)$. We apply the algorithm to a Barabási-Albert scale-free network and show that the degree of assortativity is an important parameter governing the geometrical and transport properties of networks. Thus, the average path length of the network increases dramatically with the degree of assortativity. Moreover, the concentration dependences of the size of the giant component in the node percolation problem for uncorrelated and assortative networks are strongly different. The behavior of the clustering coefficient is also discussed.

Figure 1

Scale-free networks for different degrees of assortativity (see text for details). The nodes of the same degree are grouped together; the degree is nondecreasing from left to right. The panels show (a) $\mathcal{A}=0$ (uncorrelated network), (b) $\mathcal{A}=0.26$, (c) $\mathcal{A}=0.43$, and (d) $\mathcal{A}=0.62$ (maximal assortativity).

Figure 2

The lower curve corresponds to the measured assortativity $\mathcal{A}$ of our simulations, whereas the upper curve corresponds to the theory. We note that both curves coincide for $\mathcal{A}<0.7$. Above this value the finite-size corrections get important, leading to the measured value of $\mathcal{A}<1$ for $p\to 1$.

Figure 3

${\mathcal{E}}_{kk}$ as a function of $k$ for different values of $\mathcal{A}$. From bottom to top: $\mathcal{A}=0$, $\mathcal{A}=0.221$, $\mathcal{A}=0.443$, and $\mathcal{A}=0.640$. The points are the results of the simulations and the curves correspond to the theory.

Figure 4

Average path length $l$ of the network versus coefficient of assortativity. We note that $l$ grows rapidly when $\mathcal{A}$ increases. In the inset the average path length is plotted on double-logarithmic scales as function of $\mathcal{K}-\mathcal{A}$, being $\mathcal{K}=0.913$. The slope of the straight line is $-1.12$.

Figure 5

The average path length $l$ is plotted as function of $N$ for three values of the coefficient of assortativity $\mathcal{A}$. From bottom to top: $\mathcal{A}=0$, $\mathcal{A}=0.221$, and $\mathcal{A}=0.443$. Note the logarithmic scale.

Figure 6

$\overline{C}\left(k\right)$ as a function of the degree of nodes, $k$. Different curves correspond to different values of $\mathcal{A}$. From bottom to top: $\mathcal{A}=0$, $\mathcal{A}=0.069$, $\mathcal{A}=0.221$, $\mathcal{A}=0.443$, $\mathcal{A}=0.640$, $\mathcal{A}=0.777$, $\mathcal{A}=0.856$, and maximal assortativity $\mathcal{A}=0.913$. Inset: clustering coefficient $C$ versus the degree of assortativity, $\mathcal{A}$.

Figure 7

Fraction of nodes, $\mathcal{M}$, in the giant component depending on the fraction of nodes removed from the network. The graph compares the results for different degrees of assortativity. From top to bottom: $\mathcal{A}=0$, $\mathcal{A}=0.069$, $\mathcal{A}=0.221$, $\mathcal{A}=0.443$, $\mathcal{A}=0.640$, $\mathcal{A}=0.777$, $\mathcal{A}=0.856$, and $\mathcal{A}=0.913$ (maximal assortativity).