A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market

We propose a dynamic network model where two mechanisms control the probability of a link between two nodes: (i) the existence or absence of this link in the past, and (ii) node-specific latent variables (dynamic fitnesses) describing the propensity of each node to create links. Assuming a Markov dynamics for both mechanisms, we propose an Expectation-Maximization algorithm for model estimation and inference of the latent variables. The estimated parameters and fitnesses can be used to forecast the presence of a link in the future. We apply our methodology to the e-MID interbank network for which the two linkage mechanisms are associated with two different trading behaviors in the process of network formation, namely preferential trading and trading driven by node-specific characteristics. The empirical results allow to recognise preferential lending in the interbank market and indicate how a method that does not account for time-varying network topologies tends to overestimate preferential linkage.Comment: 19 pages, 6 figure

Transition to Reconstructibility in Weakly Coupled Networks

Across scientific disciplines, thresholded pairwise measures of statistical dependence between time series are taken as proxies for the interactions between the dynamical units of a network. Yet such correlation measures often fail to reflect the underlying physical interactions accurately. Here we systematically study the problem of reconstructing direct physical interaction networks from thresholding correlations. We explicate how local common cause and relay structures, heterogeneous in-degrees and non-local structural properties of the network generally hinder reconstructibility. However, in the limit of weak coupling strengths we prove that stationary systems with dynamics close to a given operating point transition to universal reconstructiblity across all network topologies.Comment: 15 pages, 4 figures, supplementary material include

Stochastic reaction networks with input processes: Analysis and applications to reporter gene systems

Stochastic reaction network models are widely utilized in biology and chemistry to describe the probabilistic dynamics of biochemical systems in general, and gene interaction networks in particular. Most often, statistical analysis and inference of these systems is addressed by parametric approaches, where the laws governing exogenous input processes, if present, are themselves fixed in advance. Motivated by reporter gene systems, widely utilized in biology to monitor gene activation at the individual cell level, we address the analysis of reaction networks with state-affine reaction rates and arbitrary input processes. We derive a generalization of the so-called moment equations where the dynamics of the network statistics are expressed as a function of the input process statistics. In stationary conditions, we provide a spectral analysis of the system and elaborate on connections with linear filtering. We then apply the theoretical results to develop a method for the reconstruction of input process statistics, namely the gene activation autocovariance function, from reporter gene population snapshot data, and demonstrate its performance on a simulated case study

Reconstructing Dynamical Systems From Stochastic Differential Equations to Machine Learning

Die Modellierung komplexer Systeme mit einer großen Anzahl von Freiheitsgraden ist in den letzten Jahrzehnten zu einer großen Herausforderung geworden. In der Regel werden nur einige wenige Variablen komplexer Systeme in Form von gemessenen Zeitreihen beobachtet, während die meisten von ihnen - die möglicherweise mit den beobachteten Variablen interagieren - verborgen bleiben. In dieser Arbeit befassen wir uns mit dem Problem der Rekonstruktion und Vorhersage der zugrunde liegenden Dynamik komplexer Systeme mit Hilfe verschiedener datengestützter Ansätze. Im ersten Teil befassen wir uns mit dem umgekehrten Problem der Ableitung einer unbekannten Netzwerkstruktur komplexer Systeme, die Ausbreitungsphänomene widerspiegelt, aus beobachteten Ereignisreihen. Wir untersuchen die paarweise statistische Ähnlichkeit zwischen den Sequenzen von Ereigniszeitpunkten an allen Knotenpunkten durch Ereignissynchronisation (ES) und Ereignis-Koinzidenz-Analyse (ECA), wobei wir uns auf die Idee stützen, dass funktionale Konnektivität als Stellvertreter für strukturelle Konnektivität dienen kann. Im zweiten Teil konzentrieren wir uns auf die Rekonstruktion der zugrunde liegenden Dynamik komplexer Systeme anhand ihrer dominanten makroskopischen Variablen unter Verwendung verschiedener stochastischer Differentialgleichungen (SDEs). In dieser Arbeit untersuchen wir die Leistung von drei verschiedenen SDEs - der Langevin-Gleichung (LE), der verallgemeinerten Langevin-Gleichung (GLE) und dem Ansatz der empirischen Modellreduktion (EMR). Unsere Ergebnisse zeigen, dass die LE bessere Ergebnisse für Systeme mit schwachem Gedächtnis zeigt, während sie die zugrunde liegende Dynamik von Systemen mit Gedächtniseffekten und farbigem Rauschen nicht rekonstruieren kann. In diesen Situationen sind GLE und EMR besser geeignet, da die Wechselwirkungen zwischen beobachteten und unbeobachteten Variablen in Form von Speichereffekten berücksichtigt werden. Im letzten Teil dieser Arbeit entwickeln wir ein Modell, das auf dem Echo State Network (ESN) basiert und mit der PNF-Methode (Past Noise Forecasting) kombiniert wird, um komplexe Systeme in der realen Welt vorherzusagen. Unsere Ergebnisse zeigen, dass das vorgeschlagene Modell die entscheidenden Merkmale der zugrunde liegenden Dynamik der Klimavariabilität erfasst.Modeling complex systems with large numbers of degrees of freedom have become a grand challenge over the past decades. Typically, only a few variables of complex systems are observed in terms of measured time series, while the majority of them – which potentially interact with the observed ones - remain hidden. Throughout this thesis, we tackle the problem of reconstructing and predicting the underlying dynamics of complex systems using different data-driven approaches. In the first part, we address the inverse problem of inferring an unknown network structure of complex systems, reflecting spreading phenomena, from observed event series. We study the pairwise statistical similarity between the sequences of event timings at all nodes through event synchronization (ES) and event coincidence analysis (ECA), relying on the idea that functional connectivity can serve as a proxy for structural connectivity. In the second part, we focus on reconstructing the underlying dynamics of complex systems from their dominant macroscopic variables using different Stochastic Differential Equations (SDEs). We investigate the performance of three different SDEs – the Langevin Equation (LE), Generalized Langevin Equation (GLE), and the Empirical Model Reduction (EMR) approach in this thesis. Our results reveal that LE demonstrates better results for systems with weak memory while it fails to reconstruct underlying dynamics of systems with memory effects and colored-noise forcing. In these situations, the GLE and EMR are more suitable candidates since the interactions between observed and unobserved variables are considered in terms of memory effects. In the last part of this thesis, we develop a model based on the Echo State Network (ESN), combined with the past noise forecasting (PNF) method, to predict real-world complex systems. Our results show that the proposed model captures the crucial features of the underlying dynamics of climate variability

A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference