Sequential visibility-graph motifs

Visibility algorithms transform time series into graphs and encode dynamical information in their topology, paving the way for graph-theoretical time series analysis as well as building a bridge between nonlinear dynamics and network science. In this work we introduce and study the concept of sequential visibility graph motifs, smaller substructures of n consecutive nodes that appear with characteristic frequencies. We develop a theory to compute in an exact way the motif profiles associated to general classes of deterministic and stochastic dynamics. We find that this simple property is indeed a highly informative and computationally efficient feature capable to distinguish among different dynamics and robust against noise contamination. We finally confirm that it can be used in practice to perform unsupervised learning, by extracting motif profiles from experimental heart-rate series and being able, accordingly, to disentangle meditative from other relaxation states. Applications of this general theory include the automatic classification and description of physical, biological, and financial time series

Bipartisanship Breakdown, Functional Networks, and Forensic Analysis in Spanish 2015 and 2016 National Elections

In this paper we present a social network and forensic analysis of the vote counts of Spanish national elections that took place in December 2015 and their sequel in June 2016. Vote counts are extracted at the level of municipalities, yielding an unusually high resolution dataset with over 8000 samples. We initially consider the phenomenon of Bipartisanship breakdown by analysing spatial distributions of several Bipartisanship indices. We find that such breakdown is more prominent close to cosmopolite and largely populated areas and less important in rural areas where Bipartisanship still prevails, and its evolution mildly consolidates in the 2016 round, with some evidence of Bipartisanship reinforcement which we hypothesize to be due to psychological mechanisms of risk aversion. On a third step we explore to which extent vote data are faithful by applying forensic techniques to vote statistics. We first explore the conformance of first digit distributions to Benford's law for each of the main political parties. The results and interpretations are mixed and vary across different levels of aggregation, finding a general good quantitative agreement at the national scale for both municipalities and precincts but finding systematic nonconformance at the level of individual precincts. As a complementary metric, we further explore the co-occurring statistics of voteshare and turnout, finding a mild tendency in the clusters of the conservative party to smear out towards the area of high turnout and voteshare, what has been previously interpreted as a possible sign of incremental fraud. In every case results are qualitatively similar between 2015 and 2016 elections.Comment: 23 pages, 21 figures, accepted for publication in Complexit

Analytical estimation of the correlation dimension of integer lattices

In this article we address the concept of correlation dimension which has been recently extended to network theory in order to eciently characterize and estimate the dimensionality and geometry of complex networks [1]. This extension is inspired in the Grassberger-Procaccia method [2{4], originally designed to quantify the fractal dimension of strange attractors in dissipative chaotic dynamical systems. When applied to networks, it proceeds by capturing the trajectory of a random walker di using over a network with well de ned dimensionality. From this trajectory, an estimation of the network correlation dimension is retrieved by looking at the scaling of the walker's correlation integral. Here we give analytical support to this methodology by obtaining the correlation dimension of synthetic networks representing well-de ned limits of real networks. In particular, we explore fully connected networks and integer lattices, these latter being coarsely-equivalent [20] to Euclidean spaces. We show that their correlation dimension coincides with the the Haussdor dimension of the respective coarsely-equivalent Euclidean space

Speech earthquakes: scaling and universality in human voice

Submitted for publicationSubmitted for publicationSpeech is a distinctive complex feature of human capabilities. In order to understand the physics underlying speech production, in this work we empirically analyse the statistics of large human speech datasets ranging several languages. We first show that during speech the energy is unevenly released and power-law distributed, reporting a universal robust Gutenberg-Richter-like law in speech. We further show that such earthquakes in speech show temporal correlations, as the interevent statistics are again power-law distributed. Since this feature takes place in the intra-phoneme range, we conjecture that the responsible for this complex phenomenon is not cognitive, but it resides on the physiological speech production mechanism. Moreover, we show that these waiting time distributions are scale invariant under a renormalisation group transformation, suggesting that the process of speech generation is indeed operating close to a critical point. These results are put in contrast with current paradigms in speech processing, which point towards low dimensional deterministic chaos as the origin of nonlinear traits in speech fluctuations. As these latter fluctuations are indeed the aspects that humanize synthetic speech, these findings may have an impact in future speech synthesis technologies. Results are robust and independent of the communication language or the number of speakers, pointing towards an universal pattern and yet another hint of complexity in human speech

Correlation dimension of complex networks

We propose a new measure to characterize the dimension of complex networks based on the ergodic theory of dynamical systems. This measure is derived from the correlation sum of a trajectory generated by a random walker navigating the network, and extends the classical Grassberger-Procaccia algorithm to the context of complex networks. The method is validated with reliable results for both synthetic networks and real-world networks such as the world air-transportation network or urban networks, and provides a computationally fast way for estimating the dimensionality of networks which only relies on the local information provided by the walkers

Phase transition in the Countdown problem

Here we present a combinatorial decision problem, inspired by the celebrated quiz show called the countdown, that involves the computation of a given target number T from a set of k randomly chosen integers along with a set of arithmetic operations. We find that the probability of winning the game evidences a threshold phenomenon that can be understood in the terms of an algorithmic phase transition as a function of the set size k. Numerical simulations show that such probability sharply transitions from zero to one at some critical value of the control parameter, hence separating the algorithm's parameter space in different phases. We also find that the system is maximally efficient close to the critical point. We then derive analytical expressions that match the numerical results for finite size and permit us to extrapolate the behavior in the thermodynamic limit.Comment: Submitted for publicatio

On the spectral properties of Feigenbaum graphs

A Horizontal Visibility Graph (HVG) is a simple graph extracted from an ordered sequence of real values, and this mapping has been used to provide a combinatorial encryption of time series for the task of performing network based time series analysis. While some properties of the spectrum of these graphs --such as the largest eigenvalue of the adjacency matrix-- have been routinely used as measures to characterise time series complexity, a theoretic understanding of such properties is lacking. In this work we explore some algebraic and spectral properties of these graphs associated to periodic and chaotic time series. We focus on the family of Feigenbaum graphs, which are HVGs constructed in correspondence with the trajectories of one-parameter unimodal maps undergoing a period-doubling route to chaos (Feigenbaum scenario). For the set of values of the map's parameter $\mu$ for which the orbits are periodic with period $2^n$, Feigenbaum graphs are fully characterised by two integers (n,k) and admit an algebraic structure. We explore the spectral properties of these graphs for finite n and k, and among other interesting patterns we find a scaling relation for the maximal eigenvalue and we prove some bounds explaining it. We also provide numerical and rigorous results on a few other properties including the determinant or the number of spanning trees. In a second step, we explore the set of Feigenbaum graphs obtained for the range of values of the map's parameter $\mu$ for which the system displays chaos. We show that in this case, Feigenbaum graphs form an ensemble for each value of $\mu$ and the system is typically weakly self-averaging. Unexpectedly, we find that while the largest eigenvalue can distinguish chaos from an iid process, it is not a good measure to quantify the chaoticity of the process, and that the eigenvalue density does a better job.Comment: 33 page

Detecting series periodicity with horizontal visibility graphs

The horizontal visibility algorithm has been recently introduced as a mapping between time series and networks. The challenge lies in characterizing the structure of time series (and the processes that generated those series) using the powerful tools of graph theory. Recent works have shown that the visibility graphs inherit several degrees of correlations from their associated series, and therefore such graph theoretical characterization is in principle possible. However, both the mathematical grounding of this promising theory and its applications are on its infancy. Following this line, here we address the question of detecting hidden periodicity in series polluted with a certain amount of noise. We first put forward some generic properties of horizontal visibility graphs which allow us to define a (graph theoretical) noise reduction filter. Accordingly, we evaluate its performance for the task of calculating the period of noisy periodic signals, and compare our results with standard time domain (autocorrelation) methods. Finally, potentials, limitations and applications are discussed.Comment: To be published in International Journal of Bifurcation and Chao

Feigenbaum graphs: a complex network perspective of chaos

The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.Comment: Published in PLoS ONE (Sep 2011

Time series irreversibility: a visibility graph approach

We propose a method to measure real-valued time series irreversibility which combines two differ- ent tools: the horizontal visibility algorithm and the Kullback-Leibler divergence. This method maps a time series to a directed network according to a geometric criterion. The degree of irreversibility of the series is then estimated by the Kullback-Leibler divergence (i.e. the distinguishability) between the in and out degree distributions of the associated graph. The method is computationally effi- cient, does not require any ad hoc symbolization process, and naturally takes into account multiple scales. We find that the method correctly distinguishes between reversible and irreversible station- ary time series, including analytical and numerical studies of its performance for: (i) reversible stochastic processes (uncorrelated and Gaussian linearly correlated), (ii) irreversible stochastic pro- cesses (a discrete flashing ratchet in an asymmetric potential), (iii) reversible (conservative) and irreversible (dissipative) chaotic maps, and (iv) dissipative chaotic maps in the presence of noise. Two alternative graph functionals, the degree and the degree-degree distributions, can be used as the Kullback-Leibler divergence argument. The former is simpler and more intuitive and can be used as a benchmark, but in the case of an irreversible process with null net current, the degree-degree distribution has to be considered to identifiy the irreversible nature of the series.Comment: submitted for publicatio