Spectral identification of networks using sparse measurements

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.Comment: 3

Online Machine Learning for Graph Topology Identification from Multiple Time Series

High dimensional time series data are observed in many complex systems. In networked data, some of the time series are influenced by other time series. Identifying these relations encoded in a graph structure or topology among the time series is of paramount interest in certain applications since the identified structure can provide insights about the underlying system and can assist in inference tasks. In practice, the underlying topology is usually sparse, that is, not all the participating time series in influence each other. The goal of this dissertation pertains to study the problem of sparse topology identification under various settings. Topology identification from time series is a challenging task. The first major challenge in topology identification is that the assumption of static topology does not hold always in practice since most of the practical systems are evolving with time. For instance, in econometrics, social networks, etc., the relations among the time series can change over time. Identifying the topologies of such dynamic networks is a major challenge. The second major challenge is that in most practical scenarios, the data is not available at once - it is coming in a streaming fashion. Hence, batch approaches are either not applicable or they become computationally expensive since a batch algorithm is needed to be run when a new datum becomes available. The third challenge is that the multi-dimensional time series data can contain missing values due faulty sensors, privacy and security reasons, or due to saving energy. We address the aforementioned challenges in this dissertation by proposing online/-batch algorithms to solve the problem of time-varying topology identification. A model based on vector autoregressive (VAR) process is adopted initially. The parameters of the VAR model reveal the topology of the underlying network. First, two online algorithms are proposed for the case of streaming data. Next, using the same VAR model, two online algorithms under the framework of online optimization are presented to track the time-varying topologies. To evaluate the performance of propose online algorithms, we show that both the proposed algorithms incur a sublinear static regret. To characterize the performance theoretically in time-varying scenarios, a bound on the dynamic regret for one of the proposed algorithms (TIRSO) is derived. Next, using a structural equation model (SEM) for topology identification, an online algorithm for tracking time-varying topologies is proposed, and a bound on the dynamic regret is also derived for the proposed algorithm. Moreover, using a non-stationary VAR model, an algorithm for dynamic topology identification and breakpoint detection is also proposed, where the notion of local structural breakpoint is introduced to accommodate the concept of breakpoint where instead of the whole topology, only a few edges vary. Finally, the problem of tracking VAR-based time-varying topologies with missing data is investigated. Online algorithms are proposed where the joint signal and topology estimation is carried out. Dynamic regret analysis is also presented for the proposed algorithm. For all the previously mentioned works, simulation tests about the proposed algorithms are also presented and discussed in this dissertation. The numerical results of the proposed algorithms corroborate with the theoretical analysis presented in this dissertation.publishedVersio

Sparse temporally dynamic resting-state functional connectivity networks for early MCI identification

In conventional resting-state functional MRI (R-fMRI) analysis, functional connectivity is assumed to be temporally stationary, overlooking neural activities or interactions that may happen within the scan duration. Dynamic changes of neural interactions can be reflected by variations of topology and correlation strength in temporally correlated functional connectivity networks. These connectivity networks may potentially capture subtle yet short neural connectivity disruptions induced by disease pathologies. Accordingly, we are motivated to utilize disrupted temporal network properties for improving control-patient classification performance. Specifically, a sliding window approach is firstly employed to generate a sequence of overlapping R-fMRI sub-series. Based on these sub-series, sliding window correlations, which characterize the neural interactions between brain regions, are then computed to construct a series of temporal networks. Individual estimation of these temporal networks using conventional network construction approaches fails to take into consideration intrinsic temporal smoothness among successive overlapping R-fMRI subseries. To preserve temporal smoothness of R-fMRI sub-series, we suggest to jointly estimate the temporal networks by maximizing a penalized log likelihood using a fused sparse learning algorithm. This sparse learning algorithm encourages temporally correlated networks to have similar network topology and correlation strengths. We design a disease identification framework based on the estimated temporal networks, and group level network property differences and classification results demonstrate the importance of including temporally dynamic R-fMRI scan information to improve diagnosis accuracy of mild cognitive impairment patients