Sparse network identifiability via Compressed Sensing

The problem of identifying sparse solutions for the link structure and dynamics of an unknown linear, time-invariant network is posed as finding sparse solutions xx to Ax=bAx=b. If the matrix AA satisfies a rank condition, this problem has a unique, sparse solution. Here each row of AA comprises one experiment consisting of input/output measurements and cannot be freely chosen. We show that if experiments are poorly designed, the rank condition may never be satisfied, resulting in multiple solutions. We discuss strategies for designing experiments such that AA has the desired properties and the problem is therefore well posed. This formulation allows prior knowledge to be taken into account in the form of known nonzero entries of xx, requiring fewer experiments to be performed. Simulated examples are given to illustrate the approach, which provides a useful strategy commensurate with the type of experiments and measurements available to biologists. We also confirm suggested limitations on the use of convex relaxations for the efficient solution of this problem.