Dynamics-based centrality for directed networks

Determining the relative importance of nodes in directed networks is important in, for example, ranking websites, publications, and sports teams, and for understanding signal flows in systems biology. A prevailing centrality measure in this respect is the PageRank. In this work, we focus on another class of centrality derived from the Laplacian of the network. We extend the Laplacian-based centrality, which has mainly been applied to strongly connected networks, to the case of general directed networks such that we can quantitatively compare arbitrary nodes. Toward this end, we adopt the idea used in the PageRank to introduce global connectivity between all the pairs of nodes with a certain strength. Numerical simulations are carried out on some networks. We also offer interpretations of the Laplacian-based centrality for general directed networks in terms of various dynamical and structural properties of networks. Importantly, the Laplacian-based centrality defined as the stationary density of the continuous-time random walk with random jumps is shown to be equivalent to the absorption probability of the random walk with sinks at each node but without random jumps. Similarly, the proposed centrality represents the importance of nodes in dynamics on the original network supplied with sinks but not with random jumps.

Figure 1

A network with two root nodes.

Figure 2

Directed chain with $N$ nodes.

Figure 3

(a) Example of directed network with $N=3$. (b) Network with $2N=6$ nodes obtained by adding source nodes to the network shown in (a).

Figure 4

Values of ${\widehat{v}}_i$ for (a) random graph, (b) neural network, and (c) online social network. We set $q=0.001$, 0.1, 1, 10, 1000 (from steep thick lines to flat thin lines). For each $q$, we have sorted ${\widehat{v}}_i$ in the ascending order for demonstration.

Figure 5

Dependence of ${\widehat{v}}_i$ of some nodes on $q$ for (a) random graph, (b) neural network, and (c) online social network. The nodes with the largest influence values for $q=0.001$ and for $q=10$ correspond to the solid and dashed lines, respectively.

Figure 6

Kendall rank correlation coefficient for the influence and the PageRank. (a)–(c) Random graph, (d)–(f) neural network, and (g)–(i) online social network. (a, d, g) Correlation between the influence for different values of $q$. (b, e, h) Correlation between the PageRank for different values of $q$. (c, f, i) Correlation between the influence and the PageRank for various values of $q$.

Figure 7

Two directed networks with two root nodes.