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Network sparsification represents an essential tool to extract the core ofinteractions sustaining both networks dynamics and their connectedness. In thecase of infectious diseases, network sparsification methods remove irrelevantconnections to unveil the primary subgraph driving the unfolding of epidemicoutbreaks in real networks. In this paper, we explore the features determiningwhether the metric backbone, a subgraph capturing the structure of shortestpaths across a network, allows reconstructing epidemic outbreaks. We find thatboth the relative size of the metric backbone, capturing the fraction of edgeskept in such structure, and the distortion of semi-metric edges, quantifyinghow far those edges not included in the metric backbone are from theirassociated shortest path, shape the retrieval of Susceptible-Infected (SI)dynamics. We propose a new method to progressively dismantle networks relyingon the semi-metric edge distortion, removing first those connections fartherfrom those included in the metric backbone, i.e. those with highest semi-metricdistortion values. We apply our method in both synthetic and real networks,finding that semi-metric distortion provides solid ground to preserve spreadingdynamics and connectedness while sparsifying networks.

Semi-metric topology characterizes epidemic spreading on complex networks