Network degree distribution estimation and its application to the susceptible-infectious-susceptible (SIS) epidemic model

In this study, we consider the classical susceptible–infectious–susceptible (SIS) model on heterogeneous networks and introduce a mixture of truncated zeta (MTZ) distributions to model degree heterogeneity more flexibly. We refer to the resulting framework as the MTZ-SIS model. Based on this distributional specification, we formulate a corresponding SIS model under the heterogeneous mean-field (HMF) approximation and use analytical moment properties of the MTZ distribution to investigate the associated epidemic threshold. As with standard HMF approaches, the framework is exact only for locally tree-like networks and should otherwise be interpreted as an approximation. To enable inference from empirical network data, we further develop a projection-based parameter estimation procedure combined with the L-BFGS-B optimization algorithm. The resulting method, termed Parameter Expansion Search, jointly estimates the power-law exponent and mixture weights while accommodating structural sparsity. Numerical results demonstrate that the proposed estimation procedure achieves high accuracy and stability, and that the MTZ-SIS model provides a more flexible and realistic representation of degree heterogeneity, thereby offering a useful framework for modeling epidemic dynamics on heterogeneous networks.