  Random network models play a prominent role in modeling, analyzing and
understanding complex phenomena on real-life networks. However, a key property
of networks is often neglected: many real-world networks exhibit spatial
structure, the tendency of a node to select neighbors with a probability
depending on physical distance. Here, we introduce a class of random spatial
networks (RSNs) which generalizes many existing random network models but adds
spatial structure. In these networks, nodes are placed randomly in space and
joined in edges with a probability depending on their distance and their
individual expected degrees, in a manner that crucially remains analytically
tractable. We use this network class to propose a new generalization of
small-world networks, where the average shortest path lengths in the graph are
small, as in classical Watts-Strogatz small-world networks, but with close
spatial proximity of nodes that are neighbors in the network playing the role
of large clustering. Small-world effects are demonstrated on these spatial
small-world networks without clustering. We are able to derive partial
integro-differential equations governing susceptible-infectious-recovered
disease spreading through an RSN, and we demonstrate the existence of traveling
wave solutions. If the distance kernel governing edge placement decays slower
than exponential, the population-scale dynamics are dominated by long-range
hops followed by local spread of traveling waves. This provides a theoretical
modeling framework for recent observations of how epidemics like Ebola evolve
in modern connected societies, with long-range connections seeding new focal
points from which the epidemic locally spreads in a wavelike manner.
