This work uses an information-based methodology to infer the connectivity of complex systems from observed time-series data. We first derive analytically an expression for the Mutual Information Rate (MIR), namely, the amount of information exchanged per unit of time, that can be used to estimate the MIR between two finite-length low-resolution noisy time-series, and then apply it after a proper normalization for the identification of the connectivity structure of small networks of interacting dynamical systems. In particular, we show that our methodology successfully infers the connectivity for heterogeneous networks, different time-series lengths or coupling strengths, and even in the presence of additive noise. Finally, we show that our methodology based on MIR successfully infers the connectivity of networks composed of nodes with different time-scale dynamics, where inference based on Mutual Information fails.The Mutual Information Rate (MIR) measures the time rate of information exchanged between two non-random and correlated variables. Since variables in complex systems are not purely random, MIR is an appropriate quantity to access the amount of information exchanged in complex systems. However, its calculation requires infinitely long measurements with arbitrary resolution. Having in mind that it is impossible to perform infinitely long measurements with perfect accuracy, this work shows how to estimate MIR taking into consideration this fundamental limitation and how to use it for the characterization and understanding of dynamical and complex systems. Moreover, we introduce a novel normalized form of MIR that successfully infers the structure of small networks of interacting dynamical systems. The proposed inference methodology is robust in the presence of additive noise, different time-series lengths, and heterogeneous node dynamics and coupling strengths. Moreover, it also outperforms inference methods based on Mutual Information when analysing networks formed by nodes possessing different time-scales.

Abstract. This article reviews different kinds of models for the electric power grid that can be used to understand the modern power system, the smart grid. From the physical network to abstract energy markets, we identify in the literature different aspects that co-determine the spatio-temporal multilayer dynamics of power system. We start our review by showing how the generation, transmission and distribution characteristics of the traditional power grids are already subject to complex behaviour appearing as a result of the the interplay between dynamics of the nodes and topology, namely synchronisation and cascade effects. When dealing with smart grids, the system complexity increases even more: on top of the physical network of power lines and controllable sources of electricity, the modernisation brings information networks, renewable intermittent generation, market liberalisation, prosumers, among other aspects. In this case, we forecast a dynamical co-evolution of the smart grid and other kind of networked systems that cannot be understood isolated. This review compiles recent results that model electric power grids as complex systems, going beyond pure technological aspects. From this perspective, we then indicate possible ways to incorporate the diverse co-evolving systems into the smart grid model using, for example, network theory and multi-agent simulation.

The inference of an underlying network topology from local observations of a complex system composed of interacting units is usually attempted by using statistical similarity measures, such as cross-correlation (CC) and mutual information (MI). The possible existence of a direct link between different units is, however, hindered within the time-series measurements. Here we show that, for the class of systems studied, when an abrupt change in the ordered set of CC or MI values exists, it is possible to infer, without errors, the underlying network topology from the time-series measurements, even in the presence of observational noise, non-identical units, and coupling heterogeneity. We find that a necessary condition for the discontinuity to occur is that the dynamics of the coupled units is partially coherent, i.e., neither complete disorder nor globally synchronous patterns are present. We critically compare the inference methods based on CC and MI, in terms of how effective, robust, and reliable they are, and conclude that, in general, MI outperforms CC in robustness and reliability. Our findings could be relevant for the construction and interpretation of functional networks, such as those constructed from brain or climate data.Inferring the underlying topology of a complex system from observed data is currently the object of intense research. However, the limits for the exact inference of direct links in realworld systems composed by interacting dynamical units are still not fully understood. Understanding this limitations is often crucial in many applications in social and natural sciences. In order to infer the underlying network, usually, the observed data comes from timeseries recorded at the different units. Then, a direct link between units is assumed depending on how interdependent these observations are. For example, by recording the activity of different brain regions, one wishes to infer which are the relevant structural or functional brain connections by comparing similarity patterns [1][2][3]. In general, the outcome is a complex network [4,5] that interconnects the individual units and allows for a better understanding of the overall system behavior.The main statistical tools used to determine the interdependence of the units have been the cross-correlation (CC) and the mutual information (MI) between their dynamical trajectories [6][7][8][9][10][11][12][13][14][15][16]. Depending on the field of application, the choice of similarity estimators is wider and includes partial correlations, graphical models, and adapted estimators, such as event synchronization [17] (recently used to analyze the summer monsoon rainfall over the Indian peninsula [18]) or response dynamics [19,20]. However, any similarity measure used to compare two time-series usually results in a non-zero value [21][22][23][24][25]. A reason for this is that, in finite data sets, the presence of persistent trends and/or deterministic recurrent oscillations results in spurious correlations [26][27][28]. Therefore, network reconstruct...