Curvature in infectious disease network: A metric for vaccination and control.

We investigate the geometric structure of evolving infectious-disease networks using a notion of node curvature, which is based on the heat-kernel expansion that naturally extends to edges. Applying this measure to networks reconstructed from infection counts whose degree and strength distributions transition toward a steady power-law, we observe a corresponding transition in curvature. We relate the latter to an underlying hyperbolic space and reveal a phase transition in the network's effective temperature. We further compare our curvature measure with existing definitions in the literature, demonstrating that all capture the transition consistently. Our results suggest that curvature provides a metric for characterizing structural changes in dynamic networks arising from infectious processes and identifies distinct phases where targeted vaccination strategies may be most effective, when used within a multimetric public health decision-making framework.