Assortative and disassortative mixing investigated using the spectra of graphs

We investigate the impact of degree-degree correlations on the spectra of networks. Even though density distributions exhibit drastic changes depending on the (dis)assortative mixing and the network architecture, the short-range correlations in eigenvalues exhibit universal random matrix theory predictions. The long-range correlations turn out to be a measure of randomness in (dis)assortative networks. The analysis further provides insight into the origin of high degeneracy at the zero eigenvalue displayed by a majority of biological networks.

Figure 1

Spectral density for Erdös-Rényi random networks with different values of assortativity coefficient $r$. All graphs are plotted for the networks with size $N=1000$ and connection probability $p=0.01$, averaged over 20 different realizations of the networks.

Figure 2

The NNSD for Erdös-Rényi random networks with different values of assortativity coefficient $r$. All graphs are plotted for the networks with size $N=1000$ and connection probability $p=0.01$. Histograms are from the data points, and the solid line is for fitting with the Brody distribution [Eq. (3)].

Figure 3

The ${\Delta }_3\left(L\right)$ statistic for Erdös-Rényi random networks with different values of assortativity coefficient $r$. All graphs are plotted for the networks with size $N=1000$ and connection probability $p=0.01$. The solid line is the prediction from GOE statistics [Eqs. (4) and (5)], and open circles are calculated from the network.

Figure 4

Parts (a)–(c) represent the NNSD, (d)–(f) represent the ${\Delta }_3\left(L\right)$ statistic, and (g)–(i) depict the spectral density distribution of ER random networks. All graphs are plotted for the networks with size $N=2000,\langle k\rangle =10$ and for the average over 20 different realizations of the network.

Figure 5

Spectral density for scale-free networks with different values of assortativity coefficient $r$. All graphs are plotted for the networks with size $N=1000,\langle k\rangle =10$ for 20 different realizations.

Figure 6

Parts (a), (b), and (c) plot the average NNSD for an ensemble of 20 realizations for different values of $r$, whereas $\left(\text{a}\stackrel{́}{}\right),\left(\text{b}\stackrel{́}{}\right)$, and $\left(\text{c}\stackrel{́}{}\right)$ plot the ensemble having a different number of the network realizations. The histogram is drawn using the data from the networks, and the solid line is the fitted Brody distribution. For all the graphs, $N=1000$ and $\langle k\rangle =10$.