371 research outputs found
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
Visibility graphs and symbolic dynamics
Visibility algorithms are a family of geometric and ordering criteria by
which a real-valued time series of N data is mapped into a graph of N nodes.
This graph has been shown to often inherit in its topology non-trivial
properties of the series structure, and can thus be seen as a combinatorial
representation of a dynamical system. Here we explore in some detail the
relation between visibility graphs and symbolic dynamics. To do that, we
consider the degree sequence of horizontal visibility graphs generated by the
one-parameter logistic map, for a range of values of the parameter for which
the map shows chaotic behaviour. Numerically, we observe that in the chaotic
region the block entropies of these sequences systematically converge to the
Lyapunov exponent of the system. Via Pesin identity, this in turn suggests that
these block entropies are converging to the Kolmogorov- Sinai entropy of the
map, which ultimately suggests that the algorithm is implicitly and adaptively
constructing phase space partitions which might have the generating property.
To give analytical insight, we explore the relation k(x), x \in[0,1] that, for
a given datum with value x, assigns in graph space a node with degree k. In the
case of the out-degree sequence, such relation is indeed a piece-wise constant
function. By making use of explicit methods and tools from symbolic dynamics we
are able to analytically show that the algorithm indeed performs an effective
partition of the phase space and that such partition is naturally expressed as
a countable union of subintervals, where the endpoints of each subinterval are
related to the fixed point structure of the iterates of the map and the
subinterval enumeration is associated with particular ordering structures that
we called motifs
The Simplicial Characterisation of TS networks: Theory and applications
We use the visibility algorithm to construct the time series networks
obtained from the time series of different dynamical regimes of the logistic
map. We define the simplicial characterisers of networks which can analyse the
simplicial structure at both the global and local levels. These characterisers
are used to analyse the TS networks obtained in different dynamical regimes of
the logisitic map. It is seen that the simplicial characterisers are able to
distinguish between distinct dynamical regimes. We also apply the simplicial
characterisers to time series networks constructed from fMRI data, where the
preliminary results indicate that the characterisers are able to differentiate
between distinct TS networks.Comment: 11 pages, 2 figures, 4 tables. Accepted for publication in
Proceedings of the 4th International Conference on Applications in Nonlinear
Dynamics (ICAND 2016
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
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
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
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
Network structure of multivariate time series.
Our understanding of a variety of phenomena in physics, biology and economics crucially depends on the analysis of multivariate time series. While a wide range tools and techniques for time series analysis already exist, the increasing availability of massive data structures calls for new approaches for multidimensional signal processing. We present here a non-parametric method to analyse multivariate time series, based on the mapping of a multidimensional time series into a multilayer network, which allows to extract information on a high dimensional dynamical system through the analysis of the structure of the associated multiplex network. The method is simple to implement, general, scalable, does not require ad hoc phase space partitioning, and is thus suitable for the analysis of large, heterogeneous and non-stationary time series. We show that simple structural descriptors of the associated multiplex networks allow to extract and quantify nontrivial properties of coupled chaotic maps, including the transition between different dynamical phases and the onset of various types of synchronization. As a concrete example we then study financial time series, showing that a multiplex network analysis can efficiently discriminate crises from periods of financial stability, where standard methods based on time-series symbolization often fail
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