Compressive sensing over graphs: How many measurements are needed?

In this paper, we study the problem of compressive sensing for sparse signal vectors generated over the graphs. The signal vectors to be recovered are sparse vectors representing the parameters of the links over the graphs. The collective additive measurements we are allowed to take must follow connected paths over the underlying graphs. For a sufficiently connected graph with n nodes, using O(k log(n)) path measurements, it is shown that we are able to recover any sparse link vector with no more than k nonzero elements, even though the measurements have to follow the graph path constraints.