Spectral Identification of Networks Using Sparse Measurements

Abstract

We propose a new method to recover global information about a network of interconnected dynamical systems based on observations made at a small number (possibly one) of its nodes. In contrast to classical identification of full graph topology, we focus on the identification of the spectral graph-theoretic properties of the network, a framework that we call spectral network identification.The main theoretical results connect the spectral properties of the network to the spectral properties of the dynamics, which are well-defined in the context of the so-called Koopman operator and can be extracted from data through the Dynamic Mode Decomposition algorithm. These results are obtained for networks of diffusively-coupled units that admit a stable equilibrium state. For large networks, a statistical approach is considered, which focuses on spectral moments of the network and is well-suited to the case of heterogeneous populations.Our framework provides efficient numerical methods to infer global information on the network from sparse local measurements at a few nodes. Numerical simulations show for instance the possibility of detecting the mean number of connections or the addition of a new vertex using measurements made at one single node, that need not be representative of the other nodes' properties.Network identification methods developed in the framework of dynamical systems theory are usually not well-suited to the analysis of real networks such as biological networks, social networks, etc. Most of them are invasive, requiring the modification of the network connectivity or dynamics. In addition, some of them cannot be used "offline" for data analysis, since they require to interact dynamically with the network. More importantly, all the methods proposed so far for full network reconstruction require measurements at all the nodes of the network. Partial measurements have been considered in [9] in the context of linear timeinvariant systems for a partial reconstruction of the network between the measured states, and yet the authors showed that the problem cannot be solved without additional information on the system. It can actually be shown that measuring all nodes is necessary for a full network reconstruction, and this is usually out of reach in large real networks. Indeed, the number of sensors is limited and typically (much) smaller that the number of nodes. Some nodes of real networks might also not be accessible, or the only available information might be the averaged activity of a group of nodes lying in a given region of the network (e.g. electrical activity in a region of the brain). All these limitations motivate the network identification framework developed in this paper, which overcomes them.In this work, we take the view that identifying the exact complete topology of large networks is not only practically impossible, as mentioned above, but also often unnecessary. The presence or absence of an edge between two specific nodes is for instance often only marginally relevant when analyzing the global ...

Identifiability of Dynamical Networks With Partial Node Measurements

“…Proof: Given that the number of useful equations resulting from p measurements is at most p(L − s), identifiability of a network with n unknowns and s sinks requires that p(L−s) ≥ n, where p = m + s. This implies (11).…”

Spectral identification of networks with inputs

“…with (x, y) = (ℜ{λ}, ℑ{λ}). We note that this result can be generalized to the case where several clusters of eigenvalues require to consider several convex hulls (see [8] for more details). In the following example, we consider a directed random network of 100 nodes, with a normal distribution of the number of edges at each node (mean: 10, standard deviation: 5).…”

“…Now we infer the Laplacian eigenvalues from the eigenvalues of the matrix K. We rely on previous works, which show that there exists a bijection between the spectra of L and K [8]. In particular, we use the following result.…”

“…Proposition 3: Assume that Q = Q 1 ⊗Q 2 , for some Q 1 ∈ R q1×n and Q 2 ∈ R q2×m (with q = q 1 q 2 ) and consider the eigenvectorsw and v of Γ and L, associated with the eigenvaluesμ = e µT and λ, respectively, where λ and µ are related through (8). Then,…”

Identifiability of dynamical networks: Which nodes need be measured?

“…However, the problem treated in these papers consists of the identification of a subset of the networks transfer functions -typically a single one -and hence is different from the one presented here. Sparse measurements have also been considered in a different context in [12]; the goal there was to recover the network structure under the assumptions that the local dynamics are known, as opposed to re-identifying the dynamics and/or the whole network structure.…”