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Coupled Map Lattice (CML) models are particularly suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. Complete synchronisation in CMLs emerges when all maps have their state variables with equal magnitude, forming a spatially-uniform pattern that evolves in time. Here, we derive critical values for the parameters -- coupling strength, maximum Lyapunov exponent, and link density -- that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation symmetries). Our derivations are based on the Laplacian matrix eigenvalues, where we give closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of regular graphs and show that these graphs can be classified into two sets according to a topological condition (derived from the stability analysis). We also make derivations for two classes of cyclic graph: $k$-cycles (i.e., regular lattices of even degree $k$, which can be embedded in $T^k$ tori) and $k$-M\"obius ladders, which we introduce here to generalise the M\"obius ladder of degree $k = 3$. Our results highlight differences in the synchronisation manifold's stability of these graphs -- even for identical node degrees -- in the finite size and infinite size limit., Comment: 12 pages (without Appendix), 5 figures

Heterogeneity characterises real-world networks, where nodes show a broad range of different topological features. However, nodes also tend to organise into communities -- subsets of nodes that are sparsely inter-connected but are densely intra-connected (more than the network's average connectivity). This means that nodes belonging to the same community are close to each other by some distance measure, such as the resistance distance, which is the effective distance between any pair of nodes considering all possible paths. In this work, we present automation (i.e., unsupervised) and missing accuracy tests for a recently proposed semi-supervised community detection algorithm based on the resistance distance. The accuracy testing involves quantifying our algorithm's performance in terms of recovering known synthetic communities from benchmark networks, where we present results for Girvan-Newman and Lancichinetti-Fortunato-Radicchi networks. Our findings show that our algorithm falls into the class of accurate performers., Comment: 9 pages (without references), 4 figures. To be published in the 10th International Conference on Complex Networks and their Applications proceedings

Initially, the logistic map became popular as a simplified model for population growth. In spite of its apparent simplicity, as the population growth-rate is increased the map exhibits a broad range of dynamics, which include bifurcation cascades going from periodic to chaotic solutions. Studying coupled maps allows to identify other qualitative changes in the collective dynamics, such as pattern formations or hysteresis. Particularly, hysteresis is the appearance of different attracting sets, a set when the control parameter is increased and another set when it is decreased -- a multi-stable region. In this work, we present an experimental study on the bifurcations and hysteresis of nearly identical, coupled, logistic maps. Our logistic maps are an electronic system that has a discrete-time evolution with a high signal-to-noise ratio ($\sim10^6$), resulting in simple, precise, and reliable experimental manipulations, which include the design of a modifiable diffusive coupling configuration circuit. We find that the characterisations of the isolated and coupled logistic-maps' dynamics agrees excellently with the theoretical and numerical predictions (such as the critical bifurcation points and Feigenbaum's bifurcation velocity). Here, we report multi-stable regions appearing robustly across configurations, even though our configurations had parameter mismatch (which we measure directly from the components of the circuit and also infer from the resultant dynamics for each map) and were unavoidably affected by electronic noise.

There are different measures to classify a network's data set that, depending on the problem, have different success. For example, the resistance distance and eigenvector centrality measures have been successful in revealing ecological pathways and differentiating between biomedical images of patients with Alzheimer's disease, respectively. The resistance distance measures the effective distance between any two nodes of a network taking into account all possible shortest paths between them and the eigenvector centrality measures the relative importance of each node in the network. However, both measures require knowing the network's eigenvalues and eigenvectors -- eigenvectors being the more computationally demanding task. Here, we show that we can closely approximate these two measures using only the eigenvalue spectra, where we illustrate this by experimenting on elemental resistor circuits and paradigmatic network models -- random and small-world networks. Our results are supported by analytical derivations, showing that the eigenvector centrality can be perfectly matched in all cases whilst the resistance distance can be closely approximated. Our underlying approach is based on the work by Denton, Parke, Tao, and Zhang [arXiv:1908.03795 (2019)], which is unrestricted to these topological measures and can be applied to most problems requiring the calculation of eigenvectors., Comment: 5 pages (without references), 5 figures

A main goal in the analysis of a complex system is to infer its underlying network structure from time-series observations of its behaviour. The inference process is often done by using bi-variate similarity measures, such as the cross-correlation (CC), however, the main factors favouring or hindering its success are still puzzling. Here, we use synthetic neuron models in order to reveal the main topological properties that frustrate or facilitate inferring the underlying network from CC measurements. Specifically, we use pulse-coupled Izhikevich neurons connected as in the Caenorhabditis elegans neural networks as well as in networks with similar randomness and small-worldness. We analyse the effectiveness and robustness of the inference process under different observations and collective dynamics, contrasting the results obtained from using membrane potentials and inter-spike interval time-series. We find that overall, small-worldness favours network inference and degree heterogeneity hinders it. In particular, success rates in C. elegans networks -- that combine small-world properties with degree heterogeneity -- are closer to success rates in Erd\"os-R\'enyi network models rather than those in Watts-Strogatz network models. These results are relevant to understand better the relationship between topological properties and function in different neural networks., Comment: 13 pages, 7 figures

Extracting useful information from data is a fundamental challenge across disciplines as diverse as climate, neuroscience, genetics, and ecology. In the era of ``big data'', data is ubiquitous, but appropriated methods are needed for gaining reliable information from the data. In this work we consider a complex system, composed by interacting units, and aim at inferring which elements influence each other, directly from the observed data. The only assumption about the structure of the system is that it can be modeled by a network composed by a set of $N$ units connected with $L$ un-weighted and un-directed links, however, the structure of the connections is not known. In this situation the inference of the underlying network is usually done by using interdependency measures, computed from the output signals of the units. We show, using experimental data recorded from randomly coupled electronic R{\"o}ssler chaotic oscillators, that the information of the lag times obtained from bivariate cross-correlation analysis can be useful to gain information about the real connectivity of the system.

Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found when a Generating Markov Partition (GMP) is obtained, which is only defined once the unstable and stable manifolds are known with infinite precision and for all times. However, these manifolds usually form highly convoluted Euclidean sets, are \emph{a priori} unknown, and, as it happens in any real-world experiment, measurements are made with finite resolution and over a finite time-span. The task gets even more complicated if the system is a network composed of interacting dynamical units, namely, a high-dimensional complex system. Here, we tackle this task and solve it by defining a method to approximately construct GMPs for any complex system's finite-resolution and finite-time trajectory. We critically test our method on networks of coupled maps, encoding their trajectories into symbolic sequences. We show that these sequences are optimal because they minimise the information loss and also any spurious information added. Consequently, our method allows us to approximately calculate the invariant probability measures of complex systems from observed data. Thus, we can efficiently define complexity measures that are applicable to a wide range of complex phenomena, such as the characterisation of brain activity from EEG signals measured at different brain regions or the characterisation of climate variability from temperature anomalies measured at different Earth regions., Comment: 10 pages, 10 figures

Information Theory concepts and methodologies conform the background of how communication systems are studied and understood. They are mainly focused on the source-channel-receiver problem and on the asymptotic limits of accuracy and communication rates, which are the classical problems studied by Shannon. However, the impact of Information Theory on networks (acting as the channel) is just starting. Here, we present an approach to understand how information flows in any connected complex network. Our approach is based on defining linear conservative flows that travel through the network from source to receiver. This framework allows us to have an analytical description of the problem and also linking the topological invariants of the network, such as the node degree, with the information flow. In particular, our approach is able to deal with information transmission in modular networks (networks containing community structures) or multiplex networks (networks with multiple layers), which are nowadays of paramount importance., Comment: 5 pages, 3 figures, proceedings paper

The logistic map is a paradigmatic dynamical system originally conceived to model the discrete-time demographic growth of a population, which shockingly, shows that discrete chaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we design and characterize a simple, low-cost, easy-to-handle, electronic implementation of the logistic map. In particular, our implementation allows for straightforward circuit-modifications to behave as different one-dimensional discrete-time systems. Also, we design a coupling block in order to address the behavior of two coupled maps, although, our design is unrestricted to the discrete-time system implementation and it can be generalized to handle coupling between many dynamical systems, as in a complex system. Our findings show that the isolated and coupled maps' behavior has a remarkable agreement between the experiments and the simulations, even when fine-tuning the parameters with a resolution of $\sim 10^{-3}$. We support these conclusions by comparing the Lyapunov exponents, periodicity of the orbits, and phase portraits of the numerical and experimental data for a wide range of coupling strengths and map's parameters., Comment: 8 pages, 10 figures

Scientists have been considering the Kuramoto model to understand the mechanism behind the appearance of collective behaviour, such as frequency synchronisation (FS) as a paradigm, in real-world networks with a finite number of oscillators. A major current challenge is to obtain an analytical solution for the phase-angles. Here, we provide an approximate analytical solution for this problem by deriving a master solution for the finite-size Kuramoto model, without imposing any restriction on the distribution of the natural frequencies of the oscillators. The master solution embodies all particular solutions of the finite-size Kuramoto model for any frequency distribution and coupling strength larger than the critical one. Furthermore, we present a criterion to determine the stability of the FS solution. This allows one to analytically infer the relationship between the physical parameters and the stable behaviour of networks.

The role of intermediaries in the synchronization of small groups of light controlled oscillators (LCO) is addressed. A single LCO is a two-time-scale phase oscillator. When pulse-coupling two LCOs, the synchronization time decreases monotonously as the coupling strength increases, independent of the initial conditions and frequency detuning. In this work we study numerically the effects that a third LCO induces to the collective behavior of the system. We analyze the new system by dealing with directed heterogeneous couplings among the units. We report a novel and robust phenomenon, absent when coupling two LCOs, which consists of a discontinuous relationship between the synchronization time and coupling strength or initial conditions. The mechanism responsible for the appearance of such discontinuities is discussed., Comment: 11 pages, 8 figures

In non-linear dynamics there are several model systems to study oscillations. One iconic example is the "Brusselator", which describes the dynamics of the concentration of two chemical species in the non-equilibrium phase. In this work we study the Brusselator dynamics as a stochastic chemical reaction without diffusion analysing the corresponding stochastic differential equations with thermal or multiplicative noise. In both stochastic scenarios we investigate numerically how the Hopf bifurcation of the non-stochastic system is modified. Furthermore, we derive analytical expressions for the noise average orbits and variance of general stochastic dynamics, a general diffusion relationship in the thermal noise framework, and an asymptotic expression for the noise average quadratic deviations. Hence, besides the impact of these results on the noisy Brusselator's dynamics, our findings are also relevant for general stochastic systems., Comment: 7 pages, 10 figures, 6 figure captions

In this work, we present novel general analytical solutions for the currents that are developed in the edges of network-like circuits when some nodes of the network act as sources/sinks of DC or AC current. We assume that Ohm's law is valid at every edge and that charge at every node is conserved (with the exception of the source/sink nodes). The resistive, capacitive, and/or inductive properties of the lines in the circuit define a complex network structure with given impedances for each edge. Our solution for the currents at each edge is derived in terms of the eigenvalues and eigenvectors of the Laplacian matrix of the network defined from the impedances. This derivation also allows us to compute the equivalent impedance between any two nodes of the circuit and relate it to currents in a closed circuit which has a single voltage generator instead of many input/output source/sink nodes. Contrary to solving Kirchhoff's equations, our derivation allows to easily calculate the redistribution of currents that occurs when the location of sources and sinks changes within the network. Finally, we show that our solutions are identical to the ones found from Circuit Theory node analysis., Comment: 9 pages, 6 figures, 3 figure captions

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., Comment: 8 pages, 18 figures, 6 figure captions

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., Comment: Submitted to EPJ-ST Power Grids, May 2015

The ability to design a transport network such that commodities are brought from suppliers to consumers in a steady, optimal, and stable way is of great importance for nowadays distribution systems. In this Letter, by using the circuit laws of Kirchhoff and Ohm, we provide the exact capacities of the edges that an optimal supply-demand network should have to operate stably under perturbations. The perturbations we consider are the evolution of the connecting topology, the decentralisation of hub sources or sinks, and the intermittence of suppliers/consumers characteristics. We analyse these conditions and the impact of our results, both on the current UK power-grid structure and on numerically generated evolving archetypal network topologies., Comment: 5 pages, 5 figures, 3 captions

This Letter presents a unified approach for the fundamental relationship between structure and function in flow networks by solving analytically the voltages in a resistor network, transforming the network structure to an effective all-to-all topology, and then measuring the resultant flows. Moreover, it defines a way to study the structural resilience of the graph and to detect possible communities., Comment: 5 pages, 4 figures