Measures of centrality based on the spectrum of the Laplacian

We introduce a family of new centralities, the $k$-spectral centralities. $k$-spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection, $k$-spectral centralities have various interpretations in terms of spectrally determined information. We explore this centrality in the context of several examples. While for sparse unweighted networks 1-spectral centrality behaves similarly to other standard centralities, for dense weighted networks they show different properties. In summary, the $k$-spectral centralities provide a novel and useful measurement of relevance (for single network elements as well as whole subnetworks) distinct from other known measures.

Figure 1

Comparison of 1-spectral centrality and betweenness centrality in the toy network, the Zachary karate club, and the dolphin social network. In (a) and (b), size of the nodes is proportional to the 1-spectral centrality while color (white is high and black is low) is reflective of the betweenness centrality. In (c) and (d), color indicates 1-spectral centrality (white is high and black is low). In (c), size is proportional to betweenness and in (d) to eigenvector centrality.

Figure 2

1-spectral centralities in example dense weighted networks. (Top) A comparison of degree, betweenness, and spectral centralities of the roll call network of the $111\mathrm{th}$ US House of Representatives. Shading denotes party affiliation [lighter gray (red), Republican; darker gray (blue), Democrat] while size is proportional to the relevant statistic. Nodes are ordered by the Fiedler vector. (Bottom) The S&P 500 network given by its two-dimensional spectral embedding where node size is proportional to spectral centrality. Node color is given by an eight-cluster spectral clustering using two eigenvectors, the edges are the 30 edges with the highest spectral centrality.

Figure 3

1-, 2-, and 3-spectral centralities of the S&P 500 network. Size is proportional to centrality scores. The arrows point to the groupings with high spectral centrality.

Figure 4

A comparison of spectral centrality and other centralities over windows of the S&P 500 data. Darker (bluer) colors indicate lower centrality scores while lighter (redder) colors indicate higher ones.

Figure 5

Comparison between efficiency and 1-spectral centrality for a immune system mediator network.