  We analyze complexity in spatial network ensembles through the lens of graph
entropy. Mathematically, we model a spatial network as a soft random geometric
graph, i.e., a graph with two sources of randomness, namely nodes located
randomly in space and links formed independently between pairs of nodes with
probability given by a specified function (the "pair connection function") of
their mutual distance. We consider the general case where randomness arises in
node positions as well as pairwise connections (i.e., for a given pair
distance, the corresponding edge state is a random variable). Classical random
geometric graph and exponential graph models can be recovered in certain
limits. We derive a simple bound for the entropy of a spatial network ensemble
and calculate the conditional entropy of an ensemble given the node location
distribution for hard and soft (probabilistic) pair connection functions. Under
this formalism, we derive the connection function that yields maximum entropy
under general constraints. Finally, we apply our analytical framework to study
two practical examples: ad hoc wireless networks and the US flight network.
Through the study of these examples, we illustrate that both exhibit properties
that are indicative of nearly maximally entropic ensembles.
