Reconstruction of Graph Signals through Percolation from Seeding Nodes

TL;DR

This paper introduces new methods for reconstructing bandlimited graph signals from sparse seed nodes using graph filters, analyzing conditions for perfect and imperfect recovery, with applications in social networks and brain state modulation.

Contribution

It proposes a novel framework for reconstructing known graph signals from sparse seed injections using graph filters, extending classical interpolation concepts to graph domains.

Findings

Conditions for perfect reconstruction in noiseless settings

Analysis of imperfect reconstruction due to noise or insufficient injections

Numerical validation on synthetic and real-world data

Abstract

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 reconstructing a known graph signal using as input a graph signal that is non-zero only for a small subset of nodes (seeding nodes). The sparse signal is then percolated (interpolated) across the graph using a graph filter. Graph filters are a generalization of classical time-invariant systems and represent linear transformations that can be implemented distributedly across the nodes of the graph. 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. The last part of the paper presents numerical experiments that illustrate the results developed through synthetic graph signals and two real-world signal reconstruction problems: influencing opinions in a social network and inducing a desired brain state in humans.