# Theoretical results on the topological properties of the limited penetrable horizontal visibility graph family

###### Abstract

The limited penetrable horizontal visibility graph algorithm was recently introduced to map time series in complex networks. We extend this visibility graph and create a directed limited penetrable horizontal visibility graph and an image limited penetrable horizontal visibility graph. We define the two algorithms and provide theoretical results on the topological properties of these graphs associated with different types of real-value series (or matrices). We perform several numerical simulations to further check the accuracy of our theoretical results. Finally we present an application of the directed limited penetrable horizontal visibility graph for measuring real-value time series irreversibility, and an application of the image limited penetrable horizontal visibility graph that discriminates noise from chaos. The empirical results show the effectiveness of our proposed algorithms.

###### pacs:

05.45. Tp, 89.75. Hc, 05.45.-a## I introduction

The complex network analysis of univariate (or multivariate) time series has recently attracted the attention of reseachers working in a wide range of fields 1 (). Over the past decade several methodologies have been proposed for mapping a univariate and multivariate time series in a complex network 2 (); 3 (); 4 (); 5 (); 6 (); 7 (); 8 (); 9 (). These include constructing a complex network from a pseudoperiodic time series 2 (), using a visibility graph (VG) algorithm 3 (), a recurrence network (RN) method 4 (), a stochastic processes method 5 (), a coarse geometry theory 6 (), a nonlinear mutual information method 7 (), event synchronization 8 (), and a phase-space coarse-graining method 9 (). These methods have been widely used to solve problems in a variety of research fields 10 (); 11 (); 12 (); 13 (); 14 (); 15 (); 16 (); 17 (); 18 (); 19 (); 20 ().

Among all these time series complex network analysis algorithms, visibility algorithms 3 (); 21 (); 22 () are the most efficient when constructing a complex network from a time series. Visibility algorithms are a family of rules for mapping a real-value time series on graphs that display several cases. In all cases each time series datum is assigned to a node, but the connection criterion differs. For example, in the natural visibility graph (NVG) two nodes and are connected if the geometrical criterion is fulfilled within the time series 3 (). In the parametric natural visibility graph (PNVG) case there are three steps when using this algorithm to map a time series to a complex network, (i) build an NVG [3] as described above using common NVG criteria in the mapping, (ii) set the direction and angle, for every link of the NVG, and (iii) use the parameter view angle rule , to select links from the directed and weighted graph 21 (). In the horizontal visibility graph (HVG) case, this algorithm is similar to the NVG algorithm but has a modified mapping criterion. Here two nodes and are connected if 22 (). These visibility algorithms have been successfully implemented in a variety of fields 23 (); 24 (); 25 ().

Recently a limited penetrable visibility graph (LPVG) 26 (); 27 () and a multiscale limited penetrable horizontal visibility graph (MLPHVG) 28 () were developed from the visibility graph (VG) and the horizontal visibility graph (HVG) to analyze nonlinear time series. The LPVG and MLPHVG have been successfully used to analyze a variety of real signals across different fields, e.g., experimental flow signals [26-27], EEG signals 28 (); 29 (), and electromechanical signals 30 (). Research has shown that the LPVG and MLPHVG inherit the merits of the VG, but also successfully screen out noise, which makes them particularly useful when analyzing signals polluted by unavoidable noise 26 (); 27 (); 28 (); 29 (); 30 ().

Abundant empirical results have already been obtained using the VG algorithm and its extensions, e.g., the PNVG 21 (), the HVG 22 (), the LPVG 26 (), and the MLPHVG 28 (). Thus far there has been little research focusing on rigorous theoretical results. Recently Lacasa et al. presented topological properties of the horizontal visibility graph associated with random time series 22 (), periodic series 31 (), and other stochastic and chaotic processes 32 (). They extended the family of visibility algorithms to map scalar fields of an arbitrary dimension onto graphs and provided analytical results on the topological properties of the graphs associated with different types of real-value matrices 33 (). Wang et al. 34 () focused on a class of general horizontal visibility algorithms, the limited penetrable horizontal visibility graph (LPHVG), and presented exact results on the topological properties of the limited penetrable horizontal visibility graph associated with a random series. Here we use the previous works 22 (); 31 (); 32 (); 33 (); 34 (), focus our attention on the limited penetrable horizontal visibility graph, and present some analytical properties.

This paper is organized as follows. In Section II of this paper we introduce the limited penetrable horizontal visibility graph family. In Section III we derive the analytical properties of the different versions of associated limited penetrable horizontal visibility graphs of a generic random time series (or a random matrix) and present several numerical simulations to check their accuracy. In Section IV we show some applications of the directed limited penetrable horizontal visibility graph and the image limited penetrable horizontal visibility graph. In Section V we present our conclusions.

## Ii limited penetrable horizontal visibility graph family

The LPHVG algorithm 28 (); 34 () and its extensions are called the LPHVG family. We here present three versions of the LPHVG algorithm, the limited penetrable horizontal visibility graph, LPHVG, the directed limited penetrable horizontal visibility graph, DLPHVG, and the image limited penetrable horizontal visibility graph of order , ILPHVG.

### ii.1 Limited Penetrable Horizontal Visibility Graph [LPHVG]

The limited penetrable horizontal visibility graph [LPHVG] 34 () is a geometrically simpler and analytically solvable version of VG 3 (), LPVG 30 (), and MLPHVG 28 (). To define it we let be a time series of real numbers. We set the limited penetrable distance to , and LPHVG maps the time series on a graph with nodes and an adjacency matrix A. Nodes and are connected through an undirected edge () if and have a limited penetrable horizontal visibility (see Fig. 1), i.e., if intermediate data follows

(1) |

where is the number of . The graph spanned by this mapping is the limited penetrable horizontal visibility graph [LPHVG]. When we set the limited penetrable distance to 0, then LPHVG(0) degenerates into an HVG 22 (), i.e., LPHVG(0) = HVG. When there are more connections between any two LPHVG nodes than in HVG. Fig. 1(b) shows the new established connections (red lines) when we infer the LPHVG(1) using HVG. Note that the LPHVG of a time series has all the properties of its corresponding HVG, e.g., it is connected and invariant under affine transformations of series data 22 ().

### ii.2 Directed limited penetrable horizontal visibility graph [DLPHVG]

The limited penetrable horizontal visibility graph [LPHVG] is undirected, because penetrable visibility does not have a predefined temporal arrow. Directionality can be added by using directed networks. Here we address the directed version and define a directed limited penetrable horizontal visibility graph [DLPHVG], where the degree of the node is split between an ingoing degree and an outgoing degree such that . We define the ingoing degree to be the number of links of node with past nodes associated with data in the series, i.e., nodes with . Conversely, we define the outgoing degree to be the number of links with future nodes, i.e., nodes with . Thus DLPHVG maps the time series into a graph with nodes and an adjacency matrix , where is a lower triangular matrix and is a upper triangular matrix. Nodes and , (or and , ) are connected through a directed edge , i.e., (or , i.e. ) if it satisfies Eq. (1).

Fig. 2 shows a graphical representation of the definition. As in the degree distribution , we use the ingoing and outgoing degree distributions of a DLPHVG to define the probability distributions of and on the graph, which are and , respectively. We see the asymmetry of the resulting graph in a first approximation when we use the invariance of the outgoing (or ingoing) degree series under a time reversal.

### ii.3 Image limited penetrable horizontal visibility graph of order [Ilphvg]

One-dimensional versions of the limited penetrable horizontal visibility graph [LPHVG] and directed limited penetrable horizontal visibility graph [DLPHVG] are used to map landscapes (time series) on complex networks. As in the definition of IVG 33 (), the definition of LPHVG can also be extended and applied to two-dimensional manifolds by extending the LPHVG criteria of Eq. (1) along one-dimensional sections of the manifold. To define the image limited penetrable horizontal visibility graph of order [ILPHVG] we let X be a matrix for an arbitrary entry and partition the plane into directions such that direction is at an angle with the row axis of , where . The image limited penetrable visibility graph of order , ILPHVG, has nodes, each of which is labeled using a duple associated with the entry indices , such that two nodes, and , are linked when (i) belongs to one of the angular partition lines, and (ii) and are linked in the LPHVG defined over the ordered sequence that includes and . For example, in ILPHVG the penetrable visibility between two points and is

(2) |

or

(3) |

Fig. 3(a) shows a sample matrix in which is the central entry, which shows the ILPHVG(1) algorithm evaluated along the vertical and horizontal directions. Fig. 3(b) shows the connectivity pattern associated to the entry of the ILPHVG(1) algorithm. Fig. 3(c) shows the ILPHVG(1) algorithm evaluated along the vertical, horizontal, and diagonal directions. Fig. 3(d) shows the connectivity pattern associated to the entry of the ILPHVG(1) algorithm.

## Iii Theoretical results on the topological properties

Theorem 1. 34 () If we let be a bi-infinite sequence of , a random variable with probability density , then the degree distribution of its associated LPHVG is

The mean degree is

Reference 34 () [Wang et al., 2017] provides a lengthy proof of this theorem. We here propose an alternative shorter proof.

Proof. We let be an arbitrary datum of the random time series. The probability that its limited penetrable horizontal visibility is interrupted by two bounding data, one datum on its left and one on its right. There are penetrable data that are larger than between the two bounding data, penetrable data on the left and data on the right of . These data are independent of , then

(4) |

We define the cumulative probability distribution function of any probability distribution to be

(5) |

Then we rewrite Eq. (4) to be

(6) |

The probability that the datum penetrates no more than time seeing data is

(7) |

where is the probability that datum penetrates no more than time seeing at least data. We can recurrently calculate to be

(8) |

from which we deduce

(9) |

Thus we finally obtain

(10) |

Then the mean degree of the limited penetrable horizontal visibility graph associated to an uncorrelated random process is

(11) |

Theorem 1 shows the exact degree distribution for LPHVG, which indicates that the degree distribution of LPHVG associated to random time series has a unified exponential form, independent of the probability distribution from which the series was generated.

Theorem 2. We let be a bi-infinite sequence of , a random variable with probability density , and consider a limited penetrable horizontal visibility graph associated with . We let be a mean degree of the node associated with a datum of height and define it

Proof. We define to be the conditional probability that a given node has degree when its height is . Using the constructive proof process of in Ref. 34 () [Wang et al., 2017], we calculate to be

(12) |

Then is

(13) |

We let , and deduce

(14) |

Theorem 2 shows the relation between data height and the mean degree of the nodes associated with the data of height . The result indicates that the is a monotonically increasing function of . Thus we conclude that the hubs of LPHVG are the data with largest values. We check the accuracy of the result within finite series. Fig. 4(a) shows a plot of the numerical values of of LPHVG(), associated with the random series of 1000 data extracted from a uniform distribution when . The theoretical results (red lines) show a perfect agreement [Eq. (14)]. To check the finite size effect, Fig. 4(b) shows a plot of the numerical values of of LPHVG(2) associated with random series of 500, 1000, 1500, 2000 data. We use root mean square error (RMSE) to measure the agreement between the numerical and theoretical results. We find that when the size of the time series increases, the RMSE between the numerical and theoretical results decreases, indicating an increase in agreement.

Theorem 3. We let be an infinite periodic series of period with no repeated values within a period. The normalized mean distance of LPHVG associated with is

where

Proof. To calculate we consider an infinite periodic series of period with no repeated values in a period and denote it . We let for the subseries and without losing generality assume that corresponds to the largest value of the subseries, , and corresponds to the 2nd to nd largest value of the subseries. Thus we construct the LPHVG associated with subseries . We assume that LPHVG has links and let be the smallest datum of the subseries . Because no data repetitions are allowed in , the degree of is when constructed from LPHVG. We now remove node and its links from LPHVG. The resulting graph now has links and nodes. We iterate this operation times. The resulting graph has nodes, i.e., . When these nodes are connected by links, the total number of deleted links are . Thus the mean degree of a limited penetrable horizontal visibility graph associated with is

(15) |

We let be the mean distance of LPHVG, be the number of nodes, and the normalized mean distance be . Note that depends on for HVG associated with periodic orbits for [31]. Thus we deduce that for LPHVG. Using Eq. (15) we obtain , and finally obtain

(16) |

This result holds for every periodic or aperiodic series (), independent of the deterministic process that generates them, because the only constraint in its derivation is that data within a period not be repeated. Note that one consequence of Eq. (15) is that each time series has an associated LPHVG with a maximum mean degree (for a aperiodic series) of , which agrees with the previous result in Eq. (11). In Eq. (16) the limiting solution holds for all aperiodic, chaotic, and random series. To check the accuracy of the analytical result, we generate four periodic time series (, 100, 200, and 250) with 2000 data points. The data in each period is from the logistic map in which . We construct the limited penetrable horizontal visibility graphs with penetrable distance associated with this periodic time series. Fig. 5(a) shows a plot of the mean degree of the resulting LPHVG values with different values that indicate a good agreement with the theoretical results in Eq. (15). Fig. 5(b) shows a calculation of the normalized mean distance of LPHVG values with , 1, and 2 associated with the period time series of . Numerical values of the mean normalized distance as a function of mean degree agrees with the theoretical linear relation of Eq. (16).

Theorem 4. 34 () We let be a real value bi-infinite time series of random variables with probability distribution and examine its associated LPHVG. The local clustering coefficient distribution is then

and

Theorem 5. [34] We let be a bi-finite sequence of random variables extracted from a continuous probability density . Then the probability that two data separated by intermediate data are two connected nodes in the LPHVG is

Theorem 4 shows the distribution characteristics of the minimum clustering coefficient and the maximum clustering coefficient of the nodes in LPHVG. Theorem 5 indicates that the limited penetrable visibility probability introduces shortcuts in the LPHVG. With these shortcuts the limited penetrable horizontal visibility graph reveals the presence of small-world phenomena 34 ().

Theorem 6. We let be a bi-infinite sequence of of random variable with a probability density . Then both the in and out degree distribution of its associated DLPHVG is

Proof. Examining the out-distribution we let be an arbitrary datum of the random time series, and the probability that its limited penetrable horizontal visibility is interrupted by one bounding datum on its right. There are penetrable data between and the bounding data . These data are independent of . Then

(17) |

The probability that datum penetrates no more than time seeing data is

(18) |

where is the probability that penetrates no more than time to the right seeing at least data. Then can be recurrently calculated

(19) |

from which, with , we deduce

(20) |

Thus we finally obtain

(21) |

To further check the accuracy of Eq. (21), we perform several numerical simulations. We generate random series of 3000 data from uniform, gaussian, and power law distributions and their associated DLPHVG. Fig. 6 show plots of the degree distributions with penetrable distances , 1, and 2. Circles indicate , diamonds ), and the solid line the theoretical results of Eq. (21). We find that the theoretical results agree with the numerics, placing aside finite size effects. As in the degree distribution of LPHVG 34 (), the deviations between the tails of the in and out degree distributions of DLPHVG associated with random series are caused solely by finite size effects.

Theorem 7. We let be a matrix with entries , where is a random variable sampled from a distribution . Then when and in the limited , the degree distribution of the associated ILPHVG converges to

Proof. To derive general results, we consider the two special cases and .

In the case , we let be an arbitrary datum in where the probability of its image limited penetrable horizontal visibility is interrupted by four bounding datum, i.e., on its right, above it, on its left, and below it. There are penetrable data between and the four bounding data. These data are independent of . Then

(22) |

The probability that the node has a penetrable visibility of exactly nodes is

(23) |

Similarly, when from Eq. (22), then

(24) |

Here the probability that node has a penetrable visibility of exactly nodes is

(25) |

From Eqs. (23) and (25) we deduce a generic that yields

(26) |

Note that when this result reduces to that in Eq. (10). To check the accuracy of Eq. (26), we estimate the degree distribution of ILPHVG associated with random matrices whose entries are uniform random variables . To illustrate the finite size effects, we also define the cutoff value. When all the degree distributions of the numerical results are smaller than the theoretical result in Eq. (26), and is the cutoff value. Figs. 7(a) and 7(c) show semi-log plots of the finite size degree distributions of ILPHVG and ILPHVG with . Note that the distributions agree with Eq. (26) when . To assess the convergence speed of Eq. (26) for finite , we estimate the cutoff value under different finite sizes [see Figs. 7(b) and 7(d)]. Note that the location of the cutoff value scales logarithmically with the system size , i.e., finite size effects only affect the tail of the distribution, which quickly converges logarithmically with .

## Iv Application of DLPHVG and ILPHVG)

We use the analytical results of LPHVG to distinguish between random and chaotic signals [34], and we describe the global evolution of crude oil futures. We also describe applications of DLPHVG and ILPHVG.

Measure real-valued time series irreversibility by DLPHVG. Time series irreversibility is an important topic in basic and applied science 35 (). Over the past decade several methods of measuring time irreversibility have been proposed 36 (); 37 (); 38 (). A recent proposal uses the directed horizontal visibility algorithm 39 (). Here the Kullback-Leibler divergence (KLD) between the out- and in-degree distributions is defined

(27) |

Eq. 27 measures the irreversibility of real-value stationary stochastic series, and we here explore the applicability of DLPHVG. We first select an appropriate parameter , map a time series to a directed limited penetrable horizontal visibility graph, and then use Eq. 27 to estimate the degree of irreversibility of the series. Using Theorem 6 and Eq. 27 we find that the KLD between the in- and out-degree distributions associated with an random infinite series is equal to zero. Using our analysis of finite size effects, we infer that the KLD between the in- and out-degree distributions associated with an random finite series of size tends asymptotically to zero. We set , 1, and 2, and calculate the numerical value of the KLD of the random series of 3000 data from uniform, Gaussian, and power-law distributions (see the upper section of Table 1). All numerical values of KLD are approximately 0, which suggests that the time series is reversible.

We next examine the chaotic logistic and Hénon map series. Figures 8(a) and 8(b) show plots of the in- and out-degree distributions of DLPHVG(), associated with the Logistic map at and the Hénon map at and of 3000 data points. Note that in each case there is a clear distinction between the in- and out-degree distributions, and this differs from the series case [see Fig. 6(b)]. We calculate the values of KLD for each case (bottom section of Table 1). We find that the values of KLD are positive and much larger than those of the series. Figs. 8(c) and 8(d) show a finite size analysis of the chaotic maps. Note that the KLD values associated with the chaos maps converges with series size to a asymptotical nonzero value, which indicates that chaos maps are irreversible.

Series description | |||
---|---|---|---|

Uniform distribution | 0.000950 | 0.007106 | 0.007269 |

Gaussian distribution | 0.002633 | 0.007106 | 0.005507 |

Power law distribution | 0.000226 | 0.004257 | 0.005267 |

Logistic map () | 0.342985 | 0.090773 | 0.081985 |

Hénon map () | 0.158358 | 0.125637 | 0.140270 |

Thus by selecting an appropriate parameter for , the of DLPHVG captures the irreversibility of the time series.

Discriminating between and chaos using ILPHVG. Although chaotic processes display an irregular and unpredictable behavior that is frequently perceived to be random, chaos is a deterministic process that often hides patterns that can be extracted using appropriate techniques. In recent decades research efforts to distinguish between noise and chaos have been widespread 40 (), and applications have been developed in all scientific disciplines involving complex, irregular empirical signals. Lacasa et al. 33 () used visibility graphs to distinguish spatiotemporal chaos from simple randomness. We here also examine spatially extended structures, and we explore whether ILPHVG can distinguish distinguish spatiotemporal chaos from simple randomness.

We define to be a two-dimensional square lattice of diffusively-coupled chaotic maps that evolve in time [33]. In each vertex of this coupled map lattice (CML) we allocate a fully chaotic logistic map , and the system is then spatially coupled,

(28) |

where the sum extends to the Von Neumann neighborhood of (four adjacent neighbors). The update is parallel, we use periodic boundary conditions, and the coupling strength is . Fig. 9(a) shows a semi-log plot for of the degree distribution of ILPHVG associated with a two-dimensional uncorrelated random field of uniform random variables (stars), and a two-dimensional coupled map lattice of diffusively coupled fully chaotic logistic maps for the coupling constants (squares), and (diamonds). Figure 9(b) shows a plot of the degree distribution of ILPHVG associated with the two-dimensional coupled map lattices of diffusively coupled fully chaotic logistic maps with a coupling constant . Eq. (26) shows (green line), (red line), and (pink line).

Figs. 9(a) and 9(b) show that the degree distribution of ILPHVG associated with the uncoupled () and weakly coupled () cases is indistinguishable from the degree distribution associated with the random field. Fig. 9(b) shows that the degree distribution deviates from the theoretical result in Eq. (26) only in the strongly coupled case (). Note that the coupled map lattices from Eq. (28) when spatial correlations settle in and the degree distributions of ILPHVG are statistically different from the theoretical result in Eq. (26), but the degree distribution of ILPHVG associated with the random field, the uncoupled case (), and the weakly coupled case () are well approximated by Eq. (26) in each case. There are deviations for ( for , for , and for ) but they are caused by finite size effects (see Fig. 7). To quantify potential deviations of the uncoupled () and weakly coupled () cases from Eq. (26), we compute

(29) |

where is the degree distribution of the numerical result and the theoretical result from Eq. (26). Here we consider 30 realizations of the random field, the uncoupled map lattices (), and the weakly coupled map lattices (), and in each case we use for , for , and for to compute the statistic that measures the deviation between the empirical degree distribution and the theoretical result. Fig. 9(c) shows the calculated results in a two-dimensional phase space with a time delay . Note that there are clear distinctions between the uncorrelated random field, the uncoupled map lattices (), and the weakly coupled map lattices ( for and ), but when the distinction is no longer clear. We thus select an appropriate parameter and use the degree distribution of ILPHVG to distinguish noise from chaos.

Note that when we increase the coupling constant the spatiotemporal dynamics of the coupled map lattice shows a rich phase diagram. Using the degree distribution of ILPHVG, we show this rich spatiotemporal dynamics process. For each we compute the degree distribution of the associated ILPHVG. We then compute the distance between the degree distribution at and the corresponding result for in Eq. (26),

(30) |

where is the degree distribution of ILPHVG, and is a scalar order parameter that describes the spatial configuration of the CML. Figure 9(d) shows that when the evolutions of with changes from 0 to 1, indicating sharp changes in the different phases—fully developed turbulence with weak spatial correlations (I), periodic structure (II), spatially coherent structure (III), and mixed structure (IV)—between periodic and spatially-coherent structures 33 (). Thus the degree distribution of the ILPHVG can capture the rich spatial structure.

## V Discussions

We have introduced a directed limited penetrable horizontal visibility graph DLPHVG and an image limited penetrable horizontal visibility graph (ILPHVG), both inspired by the limited penetrable horizontal visibility graph LPHVG 34 (). These two algorithms are expansions of the limited penetrable horizontal visibility algorithm. We first derive theoretical results on the topological properties of LPHVG, including degree distribution , mean degree , the relation between the datum height and the mean degree of the nodes associated to data with a height equal to , the normalized mean distance , the local clustering coefficient distribution and , and the probability of long distance visibility . We then deduce the in- and out-degree distributions and of DLPHVG, and the degree distribution of ILPHVG. We perform several numerical simulations to check the accuracy of our analytical results. We then present applications of the directed limited penetrable horizontal visibility graph and the image limited penetrable horizontal visibility graph, including measuring the irreversibility of a real-value time series and discriminating between noise and chaos, and empirical results testify to the efficiency of our methods.

Our theoretical results on topological properties are an extension of previous findings 22 (); 32 (); 33 (); 34 (). In the structure of the limited penetrable horizontal visibility graph family, the limited penetrable parameter is a important and affects the structure of the associated graphs. Under certain parameter values, the exact results of the associated graphs reveals the essential characteristics of the system, e.g., when and , using the degree distribution of ILPHVG we can distinguish between uncorrelated and weakly coupled systems, but when the distinction is no longer clear [see Fig. 9 (c)]. Open problem for future research include how to use real data in selecting an optimal limited penetrable parameter , and how to further apply the limited penetrable horizontal visibility graph family.

## Vi Acknowledgments

The Research was supported by the following foundations: The National Natural Science Foundation of China (71503132, 71690242, 91546118, 11731014, 71403105, 61403171),Qing Lan Project of Jiangsu Province (2017), University Natural Science Foundation of Jiangsu Province (14KJA110001), Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, CNPq, CAPES, FACEPE and UPE.

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