Local Tomography of Large Networks under the Low-Observability Regime

This article studies the problem of reconstructing the topology of a network of interacting agents via observations of the state-evolution of the agents. We focus on the large-scale network setting with the additional constraint of $partial$ observations, where only a small fraction of the agents can be feasibly observed. The goal is to infer the underlying subnetwork of interactions and we refer to this problem as $local$ $tomography$. In order to study the large-scale setting, we adopt a proper stochastic formulation where the unobserved part of the network is modeled as an Erd\"{o}s-R\'enyi random graph, while the observable subnetwork is left arbitrary. The main result of this work is establishing that, under this setting, local tomography is actually possible with high probability, provided that certain conditions on the network model are met (such as stability and symmetry of the network combination matrix). Remarkably, such conclusion is established under the $low$-$observability$ $regime$, where the cardinality of the observable subnetwork is fixed, while the size of the overall network scales to infinity.Comment: To appear in IEEE Transactions on Information Theor

Dynamic Network Reconstruction in Systems Biology: Methods and Algorithms

Dynamic network reconstruction refers to a class of problems that explore causal interactions between variables operating in dynamical systems. This dissertation focuses on methods and algorithms that reconstruct/infer network topology or dynamics from observations of an unknown system. The essential challenges, compared to system identification, are imposing sparsity on network topology and ensuring network identifiability. This work studies the following cases: multiple experiments with heterogeneity, low sampling frequency and nonlinearity, which are generic in biology that make reconstruction problems particularly challenging. The heterogeneous data sets are measurements in multiple experiments from the underlying dynamical systems that are different in parameters, whereas the network topology is assumed to be consistent. It is particularly common in biological applications. This dissertation proposes a way to deal with multiple data sets together to increase computational robustness. Furthermore, it can also be used to enforce network identifiability by multiple experiments with input perturbations. The necessity to study low-sampling-frequency data is due to the mismatch of network topology of discrete-time and continuous-time models. It is generally assumed that the underlying physical systems are evolving over time continuously. An important concept system aliasing is introduced to manifest whether the continuous system can be uniquely determined from its associated discrete-time model with the specified sampling frequency. A Nyquist-Shannon-like sampling theorem is provided to determine the critical sampling frequency for system aliasing. The reconstruction method integrates the Expectation Maximization (EM) method with a modified Sparse Bayesian Learning (SBL) to deal with reconstruction from output measurements. A tentative study on nonlinear Boolean network reconstruction is provided. The nonlinear Boolean network is considered as a union of local networks of linearized dynamical systems. The reconstruction method extends the algorithm used for heterogeneous data sets to provide an approximated inference but improve computational robustness significantly. The reconstruction algorithms are implemented in MATLAB and wrapped as a package. With considerations on generic signal features in practice, this work contributes to practically useful network reconstruction methods in biological applications