Characterizing the Dynamical Importance of Network Nodes and Links

The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our characterization can be used to optimize techniques for controlling certain network dynamical processes and apply our results to real networks.

Figure 1

Node dynamical importance $I_k$ and ${\widehat{I}}_k^0$ (solid line) as a function of ${log}_{10}(d_k^{\mathrm{in}}{d}_k^{\mathrm{out}})$ for an assortative network (open circles), and a disassortative network (▪).

Figure 2

Node dynamical importance $I_k$ for the two-community network described in the text (dots) and the uncorrelated reference $I_k^0$ (solid line).

Figure 3

Logarithm of the dynamical importance ${\widehat{I}}_k$ as a function of the logarithm of $\sqrt{d_k^{\mathrm{in}}{d}_k^{\mathrm{out}}}$ for (a) the yeast protein interaction network [13, 14], (b) the Kiel University email network [15], and (c) the internet (AS) network [16]. The solid lines are $I_k^0$. (d) shows ${\widehat{I}}_k$ vs $I_k$ for the AS network. The solid line is the identity.

Figure 4

Largest eigenvalue $\lambda (m)$ of the network resulting from removing $m$ nodes in order of decreasing dynamical importance (solid lines), degree (short dashed lines), and randomly (long dashed lines), for (a) the yeast protein interaction network [13, 14], (b) the internet AS network [16], and (c) the Kiel University email network [15].