Abstract
Methods for efficiently controlling dynamics propagated on networks are usually based on identifying the most influential nodes. Knowledge of these nodes can be used for the targeted control of dynamics such as epidemics, or for modifying biochemical pathways relating to diseases. Similarly they are valuable for identifying points of failure to increase network resilience in, for example, social support networks and logistics networks. Many measures, often termed ‘centrality’, have been constructed to achieve these aims. Here we consider Katz centrality and provide a new interpretation as a steady-state solution to continuous-time dynamics. This enables us to implement a sensitivity analysis which is similar to metabolic control analysis used in the analysis of biochemical pathways. The results yield a centrality which quantifies, for each node, the net impact of its absence from the network. It also has the desirable property of requiring a node with a high centrality to play a central role in propagating the dynamics of the system by having the capacity to both receive flux from others and then to pass it on. This new perspective on Katz centrality is important for a more comprehensive analysis of directed networks.
Highlights
Ranking nodes with respect to their degree often provides valuable information on their relative importance, but it can neglect essential factors and, by definition, it cannot distinguish between nodes of the same degree
We introduced a new interpretation of Katz centrality as the unique steady-state solution of an appropriate continuous-time dynamical system
We argued that by removing a node from the network and investigating the net impact on this steady state, both the direct value of the node as well as the impact that the node has on others is determined
Summary
Ranking nodes with respect to their degree often provides valuable information on their relative importance, but it can neglect essential factors and, by definition, it cannot distinguish between nodes of the same degree. Centrality measures that directly address this deficiency include Katz centrality[1], eigenvector centrality[2], and the centrality known as PageRank which partly underpins the Google search engine[3]. Where M = (I − aA)−1 and I is the n by n identity matrix This definition is slightly different to the original paper by Katz[1], but it results in the same ranking of nodes. In the context of eigenvector centrality, Kleinberg[5] proposed a resolution by using two quantities to characterise a directed network These are termed ‘hubs’ and ‘authorities’ where each node has a measure of the extent to which it is behaving like a hub www.nature.com/scientificreports/
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