Reconstruction of Graph Signals Through Percolation from Seeding Nodes

New schemes to recover signals defined in the nodes of a graph are proposed. Our focus is on reconstructing bandlimited graph signals, which are signals that admit a sparse representation in a frequency domain related to the structure of the graph. Most existing formulations focus on estimating an unknown graph signal by observing its value on a subset of nodes. By contrast, in this paper, we study the problem of inducing a known graph signal using as input a graph signal that is nonzero only for a small subset of nodes. The sparse signal is then percolated (interpolated) across the graph using a graph filter. Alternatively, one can interpret graph signals as network states and study graph-signal reconstruction as a network-control problem where the target class of states is represented by bandlimited signals. Three setups are investigated. In the first one, a single simultaneous injection takes place on several nodes in the graph. In the second one, successive value injections take place on a single node. The third one is a generalization where multiple nodes inject multiple signal values. For noiseless settings, conditions under which perfect reconstruction is feasible are given, and the corresponding schemes to recover the desired signal are specified. Scenarios leading to imperfect reconstruction, either due to insufficient or noisy signal value injections, are also analyzed. Moreover, connections with classical interpolation in the time domain are discussed. Specifically, for time-varying signals, where the ideal interpolator after uniform sampling is a (low-pass) filter, our proposed approach and the reconstruction of a sampled signal coincide. Nevertheless, for general graph signals, we show that these two approaches differ. The last part of the paper presents numerical experiments that illustrate the results developed through synthetic and real-world signal reconstruction problems.