You can't see the roads. You can only count the travelers at each city.
A population of individuals moves on a dynamic random graph — one that resamples its edges at every time step as an Erdős–Rényi graph with some unknown edge probability p. Each person sits at a vertex and decides whether to move: if a vertex has k neighbors, the person jumps to an adjacent vertex with probability k/(k+1) and stays put otherwise. The graph changes every instant. The individuals respond to a network they can sense but you cannot observe.
What you observe: population counts at each vertex over time. What you want to infer: p, the parameter controlling the invisible graph.
Ganguly, Mossel, and Sly show this is possible. They construct two estimators for p — both consistent, both asymptotically normal — from population data alone. The graph is never directly measured. Its fingerprint is embedded in how populations redistribute themselves: denser graphs produce faster mixing, sparser graphs trap individuals at vertices longer. The temporal pattern of occupancy numbers encodes the connectivity that produced it.
The structural insight: hidden architecture betrays itself through aggregate behavior. You don't need to see the network to measure it. You need enough population-level observations to detect the statistical signature that the network imposes on movement. The graph is invisible but not silent — it speaks through the patterns it forces on visible quantities.
This inverts the usual direction of network science, which typically starts with a known graph and asks what dynamics it produces. Here the dynamics are visible and the graph is latent. The population counts are the shadow; the network is the object casting it. But shadows, measured carefully enough, constrain the object completely.
The key condition is that the graph is random and resampling — if it were fixed, population counts would converge to a stationary distribution that could be explained by many different graphs. It's the change in the graph that makes it identifiable: each new random draw creates a new population response, and the ensemble of responses pins down the edge probability. Paradoxically, a less stable network is more identifiable than a stable one. The instability is the information.