Semi-metric topology characterizes epidemic spreading on complex networks.

Network sparsification represents an essential tool to extract the core of interactions sustaining both networks dynamics and their connectedness. In the case of infectious diseases, network sparsification methods remove irrelevant connections to unveil the primary subgraph driving the unfolding of epidemic outbreaks in real networks. In this paper, we explore the features determining whether the metric backbone, a subgraph capturing the structure of shortest paths across a network, allows reconstructing epidemic outbreaks. We find that both the relative size of the metric backbone, capturing the fraction of edges kept in such structure, and the distortion of semi-metric edges, quantifying how far those edges not included in the metric backbone are from their associated shortest path, shape the retrieval of Susceptible-Infected (SI) dynamics. We propose a new method to progressively dismantle networks relying on the semi-metric edge distortion, removing first those connections farther from those included in the metric backbone, i.e. those with highest semi-metric distortion values. We apply our method in both synthetic and real networks, finding that semi-metric distortion provides solid ground to preserve spreading dynamics and connectedness while sparsifying networks.