Stochastic Stability of Discrete-time Phase-coupled Oscillators over Uncertain and Random Networks

This paper studies stochastic stability of a class of discrete-time phase-coupled oscillators. We introduce the two new notions of stochastic and ultimate stochastic phase-cohesiveness using the concepts of Harris and positive Harris recurrent Markov chains. Stochastic phase-cohesiveness of oscillators in two types of networks are studied. First, oscillators in a network with an underlying connected topology subject to both multiplicative and additive stochastic uncertainties are considered. Second, we study a special case of the former problem by assuming that the multiplicative uncertainties are governed by the Bernoulli process representing the well known Erd{\H o}s R\'enyi network

Synchronization in Complex Oscillator Networks and Smart Grids

The emergence of synchronization in a network of coupled oscillators is a fascinating topic in various scientific disciplines. A coupled oscillator network is characterized by a population of heterogeneous oscillators and a graph describing the interaction among them. It is known that a strongly coupled and sufficiently homogeneous network synchronizes, but the exact threshold from incoherence to synchrony is unknown. Here we present a novel, concise, and closed-form condition for synchronization of the fully nonlinear, non-equilibrium, and dynamic network. Our synchronization condition can be stated elegantly in terms of the network topology and parameters, or equivalently in terms of an intuitive, linear, and static auxiliary system. Our results significantly improve upon the existing conditions advocated thus far, they are provably exact for various interesting network topologies and parameters, they are statistically correct for almost all networks, and they can be applied equally to synchronization phenomena arising in physics and biology as well as in engineered oscillator networks such as electric power networks. We illustrate the validity, the accuracy, and the practical applicability of our results in complex networks scenarios and in smart grid applications

Greedy optimization for growing spatially embedded oscillatory networks

The coupling of some types of oscillators requires the mediation of a physical link between them, rendering the distance between oscillators a critical factor to achieve synchronization. In this paper we propose and explore a greedy algorithm to grow spatially embedded oscillator networks. The algorithm is constructed in such a way that nodes are sequentially added seeking to minimize the cost of the added links' length and optimize the linear stability of the growing network. We show that, for appropriate parameters, the stability of the resulting network, measured in terms of the dynamics of small perturbations and the correlation length of the disturbances, can be significantly improved with a minimal added length cost. In addition, we analyze numerically the topological properties of the resulting networks and find that, while being more stable, their degree distribution is approximately exponential and independent of the algorithm parameters. Moreover, we find that other topological parameters related with network resilience and efficiency are also affected by the proposed algorithm. Finally, we extend our findings to more general classes of networks with different sources of heterogeneity. Our results are a first step in the development of algorithms for the directed growth of oscillatory networks with desirable stability, dynamical and topological properties.Comment: 13 pages, 9 figure

Population models for complex non-linear phenomena in biology: from mitochondrial dynamics to brain networks

The human brain is as much fascinating as complicated: this is the reason why it has always captured scientists’ attention in several fields of research, from biology to medicine, from psychology to engineering. In this context various non-invasive technologies have been optimized in order to allow the measure of signals, able to describe brain activities. These data, derived from measurement methods that largely differ in their nature, have opened the door to new characterizations of this organ, that highlighted the main features of its operating principles. Brain signals indeed have revealed to be fluctuating during time, both during a specific task, and when we are not carrying on any activities. Furthermore, a selective coordination among different regions of the brain has emerged. As engineers, we are particularly attracted by the description of our brain as a graph, whose nodes and edges can be representative of several different elements, at distinct spatial scales (from single neurons to large brain areas). In the last decades, wide attention has been devoted to reproduce and explain the complex dynamics of the brain elements by means of computational models. Graph theory tools, as well as the design of population models, allow the exploitation of many mathematical tools, helpful to enlarge the knowledge of healthy and damaged brains functioning, by means of brain networks. Interestingly, the incapability of human brains to work properly in case of disease, has found to be correlated with dysfunctions in the activity of mitochondria, the organelles that produce large part of the cells’ energy. In particular, specific relationships have been reported among neurological diseases and impairments in mitochondrial dynamics, which refers to the continuous change in shape of mitochondria, by means of fusion and fission processes. Although the existing link between brain and mitochondria is still ambiguous and under debate, the huge amount of energy required by our brain to work properly suggests a larger mitochondrial-dependence of the brain than of the other organs. In this thesis we report the results of our research, aimed to investigate a few aspects of this complex brain-mitochondria relationship. We focus on mitochondrial dynamics and brain network, as well as on suitable mathematical models used to describe them. Specifically, the main topics handled in this work can be summarized as follows. Population models for mitochondrial dynamics. We propose a modified preypredator non-linear population model to simulate the main processes, which take part in the mitochondrial dynamics, and the ones that are strongly related to it, without neglecting the energy production process. We present two possible setups, which differ in the inclusion of a feedback link between the available energy and the formation of new mitochondria. We discuss their dynamics, and their potential in reproducing biological behaviors. Brain signals: comparison of datasets derived through different technologies. We analyze two different datasets of brain signals, recorded with various methods (functional magnetic resonance imaging, fMRI, and magnetoencephalography, MEG), both in condition of no activity and during an attentional task. The aim of the analysis is twofold: the investigation of the spontaneous activity of the brain, and the exploration of possible relationships between the two different techniques. Brain network: a Kuramoto-based description. We analyze empirical brain data by means of their oscillatory features, with the purpose of highlighting the characteristics that a computational phase-model should be able to reproduce. Hence, we use a modified version of the classic Kuramoto model to reproduce the empirical oscillatory characteristics. Analysis and control of Kuramoto networks. Most of the theoretical contribution of this thesis refers to analytical results on Kuramoto networks. We analyze the topological and intrinsic conditions required to achieve a desired pattern of synchronization, represented by fully or clustered synchronized configuration of oscillators

Network Formation and Dynamics under Economic Constraints

Networks describe a broad range of systems across a wide variety of topics from social and economic interactions over technical infrastructures such as power grids and the internet to biological contexts such as food webs or neural networks. A number of large scale failures and events in these interconnected systems in recent years has shown that understanding the behavior of individual units of these networks is not necessarily sufficient to handle the increasing complexity of these systems. Many theoretical models have been studied to understand the fundamental mechanisms underlying the formation and function of networked systems and a general framework was developed to describe and understand networked systems. However, most of these models ignore a constraint that affects almost all realistic systems: limited resources. In this thesis I study the effects of economic constraints, such as a limited budget or cost minimization, both on the control of network formation and dynamics as well as on network formation itself. I introduce and analyze a new coupling scheme for coupled dynamical systems, showing that synchronization of chaotic units can be enhanced by restricting the interactions based on the states of the individual units, thus saving interactions costs. This new interaction scheme guarantees synchronizability of arbitrary networks of coupled chaotic oscillators, independent of the network topology even with strongly limited interactions. I then propose a new order parameter to measure the degree of phase coherence of networks of coupled phase oscillators. This new order parameter accurately describes the phase coherence in all stages of incoherent movement, partial and full phase locking up to full synchrony. Importantly, I analytically relate this order parameter directly to the stability of the phase locked state. In the second part, I consider the formation of networks under economic constraints from two different points of view. First I study the effects of explicitly limited resources on the control of random percolation, showing that optimal control can have undesired side effects. Specifically, maximal delay of percolation with a limited budget results in a discontinuous percolation transition, making the transition itself uncontrollable in the sense that a single link can have a macroscopic effect on the connectivity. Finally, I propose a model where network formation is driven by cost minimization of the individual nodes in the network. Based on a simple economically motivated supply problem, the resulting network structure is given as the solution of a large number of individual but interaction optimization problem. I show that these network states directly correspond to the final states of a local percolation algorithm and analyze the effects of local optimization on the network formation process. Overall, I reveal mechanisms and phenomena introduced by these economic constraints that are typically not considered in the standard models, showing that economic constraints can strongly alter the formation and function of networked systems. Thereby, I extend the theoretical understanding that we have of networked systems to economic considerations. I hope that this thesis enables better prediction and control networked systems in realistic settings

Distributed Real-Time Voltage Regulation in Distribution Networks

The increasing penetration of renewable and distributed energy resources (DERs) in distribution networks call for fast and efficient distributed voltage regulation algorithms. This thesis first studies the existing local Volt/VAR control and designs new local algorithms with less restrictive convergence conditions and better voltage regulation. Meanwhile, unlike the traditional assets owned and managed by utility companies, the customer-owned DERs are not necessarily subject to the control of network operators unless properly incentivized. This thesis then investigates the joint design of distributed control and incentive mechanisms for managing DERs by introducing a market-based voltage regulation framework and extending it to a real-time setting with both continuous and discrete decision variables as well as device dynamics under time-varying operating conditions. The resulting randomized distributed algorithm admits asynchronous implementation in practical systems, and its performance is analytically characterized as well as numerically evaluated

An Initial Framework Assessing the Safety of Complex Systems

Trabajo presentado en la Conference on Complex Systems, celebrada online del 7 al 11 de diciembre de 2020.Atmospheric blocking events, that is large-scale nearly stationary atmospheric pressure patterns, are often associated with extreme weather in the mid-latitudes, such as heat waves and cold spells which have significant consequences on ecosystems, human health and economy. The high impact of blocking events has motivated numerous studies. However, there is not yet a comprehensive theory explaining their onset, maintenance and decay and their numerical prediction remains a challenge. In recent years, a number of studies have successfully employed complex network descriptions of fluid transport to characterize dynamical patterns in geophysical flows. The aim of the current work is to investigate the potential of so called Lagrangian flow networks for the detection and perhaps forecasting of atmospheric blocking events. The network is constructed by associating nodes to regions of the atmosphere and establishing links based on the flux of material between these nodes during a given time interval. One can then use effective tools and metrics developed in the context of graph theory to explore the atmospheric flow properties. In particular, Ser-Giacomi et al. [1] showed how optimal paths in a Lagrangian flow network highlight distinctive circulation patterns associated with atmospheric blocking events. We extend these results by studying the behavior of selected network measures (such as degree, entropy and harmonic closeness centrality)at the onset of and during blocking situations, demonstrating their ability to trace the spatio-temporal characteristics of these events.This research was conducted as part of the CAFE (Climate Advanced Forecasting of sub-seasonal Extremes) Innovative Training Network which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 813844