  The emergence of detailed maps of physical networks, like the brain
connectome, vascular networks, or composite networks in metamaterials, whose
nodes and links are physical entities, have demonstrated the limits of the
current network science toolset. Link physicality imposes a non-crossing
condition that affects both the evolution and the structure of a network, in a
way that the adjacency matrix alone -- the starting point of all graph-based
approaches -- cannot capture. Here, we introduce a meta-graph that helps us
discover an exact mapping between linear physical networks and independent
sets, a central concept in graph theory. The mapping allows us to analytically
derive both the onset of physical effects and the emergence of a jamming
transition, and show that physicality impacts the network structure even when
the total volume of the links is negligible. Finally, we construct the
meta-graphs of several real physical networks, which allows us to predict
functional features, such as synapse formation in the brain connectome, that
agree with empirical data. Overall, our results show that, in order to
understand the evolution and behavior of real complex networks, the role of
physicality must be fully quantified.
