  As network research becomes more sophisticated, it is more common than ever
for researchers to find themselves not studying a single network but needing to
analyze sets of networks. An important task when working with sets of networks
is network comparison, developing a similarity or distance measure between
networks so that meaningful comparisons can be drawn. The best means to
accomplish this task remains an open area of research. Here we introduce a new
measure to compare networks, the Network Portrait Divergence, that is
mathematically principled, incorporates the topological characteristics of
networks at all structural scales, and is general-purpose and applicable to all
types of networks. An important feature of our measure that enables many of its
useful properties is that it is based on a graph invariant, the network
portrait. We test our measure on both synthetic graphs and real world networks
taken from protein interaction data, neuroscience, and computational social
science applications. The Network Portrait Divergence reveals important
characteristics of multilayer and temporal networks extracted from data.
