Observability in Connected Strongly Regular Graphs and Distance Regular Graphs

International audienceThis paper concerns the study of observability in consensus networks modeled with strongly regular graphs or distance regular graphs. We first give a Kalman-like simple algebraic criterion for observability in distance regular graphs. This criterion consists in evaluating the rank of a matrix built with the components of the Bose-Mesner algebra associated with the considered graph. Then, we define some bipartite graphs that capture the observability properties of the graph to be studied. In particular, we show that necessary and sufficient observability conditions are given by the nullity of the so-called local bipartite observability graph (resp. local unfolded bipartite observability graph) for strongly regular graphs (resp. distance regular graphs). When the nullity cannot be derived directly from the structure of these bipartite graphs, the rank of the associated bi-adjacency matrix allows evaluating observability. Eventually, as a by-product of the main results we show that non-observability can be stated just by comparing the valency of the graph to be studied with a bound computed from the number of vertices of the graph and its diameter. Similarly nonobservability can also be stated by evaluating the size of the maximum matching in the above mentioned bipartite graphs

Node dynamics on graphs: dynamical heterogeneity in consensus, synchronisation and final value approximation for complex networks

Here we consider a range of Laplacian-based dynamics on graphs such as dynamical invariance and coarse-graining, and node-specific properties such as convergence, observability and consensus-value prediction. Firstly, using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterise convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions. We show that the EEP with respect to a node can reveal nodes in the graph with increased rate of asymptotic convergence to the consensus value as characterised by the second smallest eigenvalue of the quotient Laplacian. Secondly, we extend this characterisation of the relationship between the EEP and Laplacian based dynamics to study the synchronisation of coupled oscillator dynamics on networks. We show that the existence of a non-trivial EEP describes partial synchronisation dynamics for nodes within cells of the partition. Considering linearised stability analysis, the existence of a nontrivial EEP with respect to an individual node can imply an increased rate of asymptotic convergence to the synchronisation manifold, or a decreased rate of de-synchronisation, analogous to the linear consensus case. We show that high degree 'hub' nodes in large complex networks such as Erdős-Rényi, scale free and entangled graphs are more likely to exhibit such dynamical heterogeneity under both linear consensus and non-linear coupled oscillator dynamics. Finally, we consider a separate but related problem concerning the ability of a node to compute the final value for discrete consensus dynamics given only a finite number of its own state values. We develop an algorithm to compute an approximation to the consensus value by individual nodes that is ϵ close to the true consensus value, and show that in most cases this is possible for substantially less steps than required for true convergence of the system dynamics. Again considering a variety of complex networks we show that, on average, high degree nodes, and nodes belonging to graphs with fast asymptotic convergence, approximate the consensus value employing fewer steps.Open Acces

Algebraic Characterization of Observability in Distance-Regular Consensus Networks

International audienceIn this paper, we study the observability issue in consensus networks modeled with strongly regular graphs or distance regular graphs. We derive a Kalman-like simple algebraic criterion for observability in distance regular graphs. This criterion consists in evaluating the rank of a matrix built with the components of the Bose-Mesner algebra associated with the considered graph. Then, we state a simple necessary condition of observability based on parameters of the graph, namely the diameter, the degree, and the number of vertices of the graph

A Graph Theory Approach for Regional Controllability of Boolean Cellular Automata

Controllability is one of the central concepts of modern control theory that allows a good understanding of a system's behaviour. It consists in constraining a system to reach the desired state from an initial state within a given time interval. When the desired objective affects only a sub-region of the domain, the control is said to be regional. The purpose of this paper is to study a particular case of regional control using cellular automata models since they are spatially extended systems where spatial properties can be easily defined thanks to their intrinsic locality. We investigate the case of boundary controls on the target region using an original approach based on graph theory. Necessary and sufficient conditions are given based on the Hamiltonian Circuit and strongly connected component. The controls are obtained using a preimage approach

Coordination of passive systems under quantized measurements

In this paper we investigate a passivity approach to collective coordination and synchronization problems in the presence of quantized measurements and show that coordination tasks can be achieved in a practical sense for a large class of passive systems.Comment: 40 pages, 1 figure, submitted to journal, second round of revie