  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.
