Donatello Materassi

Topological identification in networks of dynamical systems

The paper deals with the problem of reconstructing the tree-like topological structure of a network of linear dynamical systems. A distance function is defined in order to evaluate the “closeness” of two processes and some useful mathematical properties are derived. Theoretical results to guarantee the correctness of the identification procedure for networked linear systems characterized by a tree topology are provided as well. The paper also suggests the approximation of a complex connected network with a tree in order to detect the most meaningful interconnections. The application of the techniques to the analysis of an actual complex network, i.e., to high frequency time series of the stock market, is extensively illustrated.

On the Problem of Reconstructing an Unknown Topology via Locality Properties of the Wiener Filter

Determining interrelatedness structure of various entities from multiple time series data is of significant interest to many areas. Knowledge of such a structure can aid in identifying cause and effect relationships, clustering of similar entities, identification of representative elements and model reduction. The majority of existing results are based on correlation based indices which effectively assume a static relationship between the time series data and are not suitable for detecting interrelatedness when the time series are dynamically related or when the time series involve loops. In this paper, a methodology for identifying the interrelatedness structure of dynamically related time series data is presented that also allows for the presence of loops in the connectivity structure. A linear dynamic graph model is presented where it is assumed that each time series data is the sum of an independent stochastic noise source and a dynamically weighted sum of other time series data. A link is assumed to be present between two time series if the weight of a time series, which is a linear time-invariant filter, is nonzero in the formation of the other. Reconstruction of the link connectivity structure under various scenarios is considered. It is shown that when the linear dynamic graph is allowed to admit non-causal weights, then the links structure can be recovered with the possibility of identifying spurious connections. However, it is shown that the spurious links remain local, where, a spurious link is restricted to be within one hop of a true link. Furthermore, strategies for exact reconstruction of the link structure when the weights are restricted to be causal are developed. The main tools for determining the network topology are based on variations of Wiener filtering. A significant insight provided by the article is that, in the class of network models considered in the paper, the Wiener filter estimating a stochastic process based on other processes remains local in the sense that the Wiener filter utilizes only measurements local to the node being estimated.

Network reconstruction of dynamical polytrees with unobserved nodes

The paper deals with the problem of unveiling the link structure of a network of linear dynamical systems. A technique is provided guaranteeing an exact detection of the links of a network of dynamical systems with no undirected cycles (Linear Dynamic Polytrees). In particular, the presence of unobserved (latent) nodes is taken into account. Knowledge on the specific number of hidden processes is not required. It is proven that the topology can be consistently reconstructed, as long the degree of each latent node is at least three with outdegree of at least two. The result extends previous work that was limited to a more restricted class of dynamical systems (Rooted Trees).

Relations between structure and estimators in networks of dynamical systems

The article main focus is on the identification of a graphical model from time series data associated with different interconnected entities. The time series are modeled as realizations of stochastic processes (representing nodes of a graph) linked together via transfer functions (representing the edges of the graph). Both the cases of non-causal and causal links are considered. By using only the measurements of the node outputs and without assuming any prior knowledge of the network topology, a method is provided to estimate the graph connectivity. In particular, it is proven that the method determines links to be present only between a node and its “kins”, where kins of a node consist of parents, children and co-parents (other parents of all of its children) in the graph. With the additional hypothesis of strictly casual links, a similar method is provided that allows one to exactly reconstruct the original graph. Main tools for determining the network topology are based on Wiener, Wiener-Hopf and Granger filtering. Analogies with the problem of Compressing Sensing are drawn and two greedy algorithms to address the problem of reducing the complexity of the network structure are also suggested.

Time Scaling of Chaotic Systems: Application to Secure Communications

The paper deals with time-scaling transformations of dynamical systems. Such scaling functions operate a change of coordinates on the time axis of the system trajectories preserving its phase portrait. Exploiting this property, a chaos encryption technique to transmit a binary signal through an analog channel is proposed. The scheme is based on a suitable time-scaling function which plays the role of a private key. The encoded transmitted signal is proved to resist known decryption attacks offering a secure and reliable communication.

Robust and Optimal Consumption Policies for Deadline-Constrained Deferrable Loads

The paper analyzes the optimal response of an individual smart load with deferrable demand for electricity, to exogenous and stochastic price processes. It is assumed that a smart load can delay a time-flexible energy demand up to a fixed deadline. Under mild assumptions on the regularity of the stochastic price process, it is shown that the optimal strategy is to consume only when the price is less than or equal to a certain threshold that depends only on the time left to the deadline and the price statistics. This analysis is performed under both perfect and partial information about the statistics of the price process. Robust policies with performance guarantees are derived for the partial information case. Such performance bounds are also used for deriving upper and lower bounds on the economic value of load-shifting. Numerical simulation results based on price data from wholesale electricity markets suggest that when information must be empirically extracted from data, the robust policies provide various theoretical and practical advantages over the full information policies.

Reduced complexity models in the identification of dynamical networks: Links with sparsification problems

In many applicative scenarios it is important to derive information about the topology and the internal connections of more dynamical systems interacting together. Examples can be found in fields as diverse as economics, neuroscience and biochemistry. The paper deals with the problem of deriving a descriptive model of a network, collecting the node outputs as time series with no use of a priori insight on the topology. We cast the problem as the optimization of a cost function operating a trade-off between accuracy and complexity in the final model. We address the problem of reducing the complexity by fixing a certain degree of sparsity, and trying to find the solution that ¿better¿ satisfies the constraints according to the criterion of approximation.

An open source/real-time atomic force microscope architecture to perform customizable force spectroscopy experiments

We describe the realization of an atomic force microscope architecture designed to perform customizable experiments in a flexible and automatic way. Novel technological contributions are given by the software implementation platform (RTAI-LINUX), which is free and open source, and from a functional point of view, by the implementation of hard real-time control algorithms. Some other technical solutions such as a new way to estimate the optical lever constant are described as well. The adoption of this architecture provides many degrees of freedom in the device behavior and, furthermore, allows one to obtain a flexible experimental instrument at a relatively low cost. In particular, we show how such a system has been employed to obtain measures in sophisticated single-molecule force spectroscopy experiments [Fernandez and Li, Science 303, 1674 (2004)]. Experimental results on proteins already studied using the same methodologies are provided in order to show the reliability of the measure system.

A model for impact dynamics and its application to frequency analysis of tapping-mode atomic force microscopes

The problem of two-body impact dynamics is considered providing a general class of models based on hysteresis functions. The structure of the model and its flexibility allows for a direct application of harmonic balance techniques for the analysis of periodic impacts when the forces involved are repulsive, repulsive-attractive and dissipative. An application to the oscillation analysis of a tapping-mode atomic force microscope (AFM) provides useful analytical results, which give a qualitative explanation of a number of known experimental phenomena.

Exact topology reconstruction of radial dynamical systems with applications to distribution system of the power grid

In this article we present a method to reconstruct the interconnectedness of dynamically related stochastic processes, where the interactions are bi-directional and the underlying topology is a tree. Our approach is based on multivariate Wiener filtering which recovers spurious edges apart from the true edges in the topology reconstruction. The main contribution of this work is to show that all spurious links obtained using Wiener filtering can be eliminated if the underlying topology is a tree based on which we present a three stage network reconstruction procedure for trees. We illustrate the effectiveness of the method developed by applying it on a typical distribution system of the electric grid.