Beyond ranking nodes: Predicting epidemic outbreak sizes by network centralities

Identifying important nodes for disease spreading is a central topic in network epidemiology. We investigate how well the position of a node, characterized by standard network measures, can predict its epidemiological importance in any graph of a given number of nodes. This is in contrast to other studies that deal with the easier prediction problem of ranking nodes by their epidemic importance in given graphs. As a benchmark for epidemic importance, we calculate the exact expected outbreak size given a node as the source. We study exhaustively all graphs of a given size, so do not restrict ourselves to certain generative models for graphs, nor to graph data sets. Due to the large number of possible nonisomorphic graphs of a fixed size, we are limited to ten-node graphs. We find that combinations of two or more centralities are predictive (R2 scores of 0.91 or higher) even for the most difficult parameter values of the epidemic simulation. Typically, these successful combinations include one normalized spectral centrality (such as PageRank or Katz centrality) and one measure that is sensitive to the number of edges in the graph.Author summary: A central challenge in network epidemiology is to find nodes that are important for disease spreading. Usually, one starts from a certain graph, and tries to rank the nodes in a way that correlates as strongly as possible with measures of importance estimated using simulations of outbreaks. A more challenging prediction task, and the one we take, is to ask if one can guess, from measures of the network structure alone, the values of quantities describing the outbreak. Having this predictive power is important: one can then target nodes that are more important than a certain threshold, rather than just a top fraction of nodes. By exhaustively studying all small graphs, we show that such prediction is possible to achieve with high accuracy by combining standard network measures.