Compressive sensing over strongly connected digraph and its application in traffic monitoring

Compressive sensing over graphs has recently attracted great research attentions, which takes limited number end-to-end measurements along paths (walks) to recover sparse vectors representing link/node properties. Unlike traditional compressive sensing, the along-path measurements rule out the freedom of random sampling, which introduces path constraints to the measurement matrix. The constraint makes explicit analysis of recovery performance difficult Only for undirected graphs, early results showed that O(klog(n)) end-to-end measurements taken by random walks are sufficient to recover k-sparse edge vector. However, the problem becomes more difficult when directed graphs are considered, because of the easy state absorbing and the difficulty of evaluating the stationary distribution. But digraphs inherently model many network systems. In this paper, particularly for strongly connected digraphs with low node degrees, we presents bounds for the stationary distribution of random walks, and present deliberative proofs which put forward that O(klog(n)) path measurements are sufficient to recover k-sparse edge vectors. Further more, because urban road networks are exactly strongly connected, low degree digraphs, we designed efficient recovery methods to estimate road delays by a small number of probing cars. Although the road delay vector is actually not sparse, we leverage the empirical non-congested road delays as references and develop an algorithm which divide the problem to iteratively recover several k-sparse vectors. Simulation results show that when less than 10% edges are congested, more than 90% congestion states can be recovered correctly by 10% measurements.