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Subgraph Detection for Average Detectability of LTI Systems

Observation and detection of network systems aim to reconstruct the evolution of the dynamical system based on the measurement of few nodes. In large-scale networks, reconstructing the exact state of each node becomes harduous and is often superfluous in practice, since reconstructing an aggregated version of the system can be sufficient. In the light of this observation, we consider the notion of average detectability: a system is said to be average detectable if it is possible to reconstruct the average of the subset of its unmeasured nodes. We show here that for a particular type of network systems, that is, negative uniform networks, the average detectability property is satisfied when the subgraph induced by the unmeasured nodes is regular. Next, we introduce the relaxed notion of quasi-regularity, which ensures an approximate reconstruction of the average. Motivated by these results, we design algorithms to detect regular induced subgraphs (RIS) and quasi-regular induced subgraph (q-RIS). We also propose an extension to detect multiple quasi-regular induced subgraphs (mq-RIS) that is meant to reconstruct the average of several subgraphs of the system. Finally, we apply our method to the estimation of a linearized SIS model of epidemic diffusion that takes place over a simulated contact network between the largest cities of France.