Networks in the real world are dynamic -- nodes and edges are added and removed over time, and time-varying processes (such as epidemics) run on them.  In this talk, I will describe mathematical aspects of some of my recent work with collaborators on statistical inference and compression problems that involve this time-varying aspect of networks.  I will focus on two related lines of work: (i) network archaeology -- broadly concerning problems of dynamic graph model validation and inference about previous states of a network given a snapshot of its current state, and (ii) structural compression -- for a given graph model, 
exhibit an efficient algorithm for invertibly mapping network structures (i.e., graph isomorphism types) to bit strings of minimum expected length.  For both classes of problems, I give both information-theoretic limits and efficient algorithms for achieving those limits.  Finally, I briefly describe some ongoing projects that continue these lines of work.