You can only see the edges. You need to know the center.
An inverse problem in information diffusion (arXiv:2603.20710): information spreads via random walks on a graph, but you can only observe first passage times at a few boundary vertices. From these partial observations, reconstruct the centrality measures of vertices you can't see.
The boundary control method — originally developed for inverse problems in differential equations — transfers to graphs. The first passage time distribution at an observable vertex encodes information about the entire graph's structure, including the centrality of unreachable vertices. The reconstruction unwraps this encoding: given how quickly information arrives at the boundary, infer how important each interior node is.
The structural insight: centrality is not a local property. An important node affects the entire network's diffusion dynamics, and that influence propagates to the boundary. Even if you can't observe the node directly, its centrality leaves a signature in the timing statistics of information that eventually reaches the edges. The more central a node, the more it shapes the distribution of arrival times at every other node — including the ones you can see.
The practical implication: in networks where you can't instrument every node (social networks, biological networks, infrastructure systems), boundary measurements can reconstruct interior properties. You don't need to see the hub to know it's there. The timing pattern at the periphery contains the topology of the center.