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The basic inverse problem in spectral graph theory consists in determiningthe graph given its eigenvalue spectrum. In this paper, we are interested in anetwork of technological agents whose graph is unknown, communicating by meansof a consensus protocol. Recently, the use of artificial noise added toconsensus signals has been proposed to reconstruct the unknown graph, althougherrors are possible. On the other hand, some methodologies have been devised toestimate the eigenvalue spectrum, but noise could interfere with theelaborations. We combine these two techniques in order to simplify calculationsand avoid topological reconstruction errors, using only one eigenvalue.Moreover, we use an high frequency noise to reconstruct the network, thus it iseasy to filter the control signals after the graph identification. Numericalsimulations of several topologies show an exact and robust reconstruction ofthe graphs.

Exact Network Reconstruction from Consensus Signals and One Eigenvalue