Abstract
This paper proposes a new method for converting a timeseries into a weighted graph (complex network), which builds on electrostatics in physics. The proposed method conceptualizes a timeseries as a series of stationary, electrically charged particles, on which Coulomblike forces can be computed. This allows generating electrostaticlike graphs associated with timeseries that, additionally to the existing transformations, can be also weighted and sometimes disconnected. Within this context, this paper examines the structural similarity between five different types of timeseries and their associated graphs that are generated by the proposed algorithm and the visibility graph, which is currently the most popular algorithm in the literature. The analysis compares the source (original) timeseries with the nodeseries generated by network measures (that are arranged into the nodeordering of the source timeseries), in terms of a linear trend, chaotic behaviour, stationarity, periodicity, and cyclical structure. It is shown that the proposed electrostatic graph algorithm generates graphs with nodemeasures that are more representative of the structure of the source timeseries than the visibility graph. This makes the proposed algorithm more natural rather than algebraic, in comparison with existing physicsdefined methods. The overall approach also suggests a methodological framework for evaluating the structural relevance between the source timeseries and their associated graphs produced by any possible transformation.
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Introduction
The multidisciplinary nature of networks^{1–3} has introduced new directions in the timeseries research that led to the emergence of the complex network analysis of timeseries. This newly established research field showed a remarkable development, at a multidisciplinary level^{4}, when scholars conceptualized^{5–7} that transforming a timeseries into a graph can produce insights that are not visible by current timeseries approaches. In general, studying the topology of a graph instead of the structure of a timeseries promotes timeseries analysis because it enlarges the embedding of the available information, from a firstorder tensor (i.e. the timeseries vector) into a secondorder tensor (i.e. the graph connectivity matrix)^{8}. Within this context, Zhang and Small^{7} were the first who constructed graphs from pseudoperiodic timeseries, and Yang and Yang^{6} applied thresholds to the correlation matrix to convert it into a connectivity matrix. Xu et al.^{9} proposed a transformation for creating graphs from timeseries based on different dynamic systems. Lacasa et al.^{5} built on the intuition of considering a timeseries as a landscape and introduced a connectivity criterion based on visibility from optics physics. Gao and Zin^{10} proposed methods (i.e. flow pattern complex network, dynamic complex network, and fluid–structure complex network) to construct complex networks from experimental flow signals, and Donner et al.^{11} introduced a recurrence method converting graphs from timeseries based on the phasespace of a dynamical system. Amongst the existing methods, the natural visibility graph (NVG) or, in synonym, the visibility graph algorithm (VGA) of Lacasa et al.^{5} seems to prevail in the literature either in terms of citations, or in the number of applications^{12–14}, or the number of derivative methods, such as the horizontal visibility graph of Luque et al.^{15}, and the visibility expansion algorithm of Tsiotas and Charakopoulos^{8},^{16}. The popularity of VGA can be either due to its intuitive conceptualization from optics physics, which makes comprehension and interpretation of results easier, or to its topological consistency to convert periodic timeseries to regular graphs, random timeseries to random graphs, and fractal timeseries to scalefree graphs. However, this method builds on a binary connectivity criterion, which leads to the development of binary connections and thus to unweighted graphs^{5}. Therefore the VGA is by definition restricted in generating visibility graphs that are disassociated from the numerical scale of the source (original) timeseries.
Aiming at serving the demand of promoting a weighted conceptualization in the complex network analysis of timeseries, this paper introduces a method for converting a timeseries into a weighted graph by using an electrostatics transformation algorithm based on Coulomb’s law. The proposed method is driven by a dual motivation: the first builds on the example of the VGA^{5}, which implies that physicsdefined transformations can be more intuitive and easily comprehensive than the algebraic (or computational) ones. The second one is based on the universality of Coulomb’s inverse square law^{17}, which grounded the development of essential research in electromagnetism but also inspired multidisciplinary research in economics^{18}, urban and spatial planning and transport engineering^{19,20}, biology^{21}, geophysics^{22}, computational^{23} and communication sciences^{24}, etc. Within this multidisciplinary context, the proposed method conceptualizes a timeseries as a sequence of stationary and electrically charged particles (nodes) and generates an electrostatic graph based on pairwise calculations of Coulomb’s law across the timeseries nodes. The Coulomblike forces are assigned as weights in the connectivity matrix of the electrostatic graph and can be seen as a measure of relevance between two nodes, in terms of the sign, scale, and spatial proximity. This approach allows quantifying the interaction between the timeseries nodes and thus to conceptualize the dynamics of a timeseries through the effect of electrostatic forces applied between the nodes.
The remainder of the paper is organized as follows: Sect. 2 (methods) describes the proposed ESG algorithm and its modeling context, it introduces the nodeseries of network measures concept in the ESG, and it briefly describes the methods used for testing the performance of the proposed algorithm. Section 3 (Results) shows the results of the multilevel analysis testing the performance of the proposed algorithm in comparison with a wellestablished method of converting a timeseries to a graph. Finally, in Sect. 4, conclusions are given.
Methods
The proposed ESG algorithm
Let us consider a timeseries X = {x_{1}, x_{2},…, x_{n}} with n\(\in \mathbb{N}\) number of nodes i\(\in\) X, where each one has a real numeric value X(i) = x_{i}\(\in {\mathbb{R}}\). If we assume that every node i in the timeseries can be seen as a static particle of electrical charge q(i)≡q_{i} = x_{i}, we can define an (either attractive or repulsive) electrostatic force F_{ij} applied between any pair of nodes i,j (Fig. 1), according to the inversesquare Coulomb’s law expressed by the relation^{17}:
where q_{i} and q_{j} are the electrostatic charges of nodes i and j, d_{ij} is the intermediate discrete distance between nodes i,j that expresses steps of separation and is defined by the difference (i–j), and k_{e} is the Coulomb’s constant.
This assumption allows considering a timeseries X as a series of stationary and electrically charged particles (i.e. timeseries nodes), on which we can compute a square matrix with the Coulomblike forces F(X) = {F_{ij}  i,j = 1, …, n}, according to the relation:
where d_{ij} = (i − j) and k_{e} is the Coulomb’s constant^{17}, which can be considered as a scale factor and in this paper is set to k_{e} = 1.
The square structure of the F(X) matrix (with the Coulomblike forces) can be seen as an electrostatic graphmodel ESG, where each element F_{ij} \(\in {\mathbb{R}}\) expresses the (attractive or repulsive) electrostatic force applied between any pair of nodes i,j. When it is important to note that the ESG is associated with the timeseries X, we can symbolize the electrostatic graph as ESG(X). In terms of graph theory^{25}, F(X) is the weighted connectivity matrix of an undirected graph G_{ESG}(V,E), where V is the nodeset and E is the edgeset. The weights (w_{ij}) in the ESG’s weighted connectivity matrix are equal to the Coulomblike forces (w_{ij} = F_{ij}) and can be seen as a measure of similarity between two nodes, in terms of the sign, scale, and spatial proximity. In particular, positive weights (w_{ij} > 0) indicate that nodes i,j have homogeneous arithmetic signs, where negative cases (w_{ij} < 0) imply that they have heterogeneous signs. Also, high w_{ij} scores may imply either that nodes i,j are close in the timeseries line, in terms of spatial proximity, or that they have relatively high arithmetic values or both. Within the context of the electrostatic conceptualization, the attraction expressed by a negative Coulomblike force (w_{ij} = –F_{ij}) can be seen as a tendency of the nodes to balance their heterogeneity and converge toward the horizontal axis, whereas the repulsion expressed by a positive force (w_{ij} = + F_{ij}) can be seen as a tendency of the nodes to escape from their homonymous electrostatic balance and thus to evolve (either increasingly or decreasingly) through time.
By definition, Coulomb’s law determines a field of infinite distance, where the electrostatic forces are noticed at infinity, although they are negligible. This property makes the ESG by default a fully connected (complete) graph K_{n}, namely a graph where all nodes are linked to any other. Provided that a complete graph K_{n} has a trivial topology, in terms of complexity (since the average degree is always \(\left\langle k \right\rangle\) = n–1 and most of other metrics, such as average path length, network diameter, graph density, and clustering coefficient are equal to one), we filter the set E of the ESG connections, aiming to generate more complex topologies of electrostatic graphs. In particular, we consider a threshold F_{c}, defined within the interval \(F_{c} \in \left( {\min \left\{ {F_{ij} } \right\},\max \left\{ {F_{ij} } \right\}} \right)\), so that the weighted connectivity matrix W_{ESG} include those values that are equal or above F_{c}, as it is expressed by the relation:
where F(X) is the Coulomblike matrix defined in relation (1). This filtering allows considering numerous electrostatic graphs ESG(X), which are expressed as a function W_{ESG} = f(F_{c}) of the thresholdvariable F_{c}. To introduce a reference value to the thresholdvariable F_{c}, we define a typical value f_{z} by the relation:
where n is the number of timeseries nodes, \(\left\langle \cdot \right\rangle\) is the average operator, and sgn(·) is the sign (or signum) function^{26}. In numeric terms, the f_{z} filtering describes that nonzero elements of the weighted ESG’s connectivity matrix are those with values higher than the adjusted meanvalue \(\frac{n}{n  1} \cdot \left\langle x \right\rangle\) of the timeseries X. In physical (electrostatic) terms, f_{z} describes an electrostatic force that is ntimes greater than this applied to a pair of particles with electrical charges q_{i}, q_{j} = \(\sqrt {\left {\left\langle x \right\rangle } \right}\)(i.e. equal to the squareroot of the absolute meanvalue of the timeseries X), which are d_{ij} = \(\sqrt {n  {1}}\) (i.e. equal to the squareroot of the timeseries length) steps of separation distant.
Within this context, the proposed ESG algorithm is implemented in four steps, as it is shown in Fig. 2. First, we compute the matrix F(X) of Coulomblike forces, according to the relations (1) and (2). Secondly, we apply to F(X) the connectivity filter and compute the weighted connectivity matrix W_{ESG}, according to the relations (3) and (4). Thirdly, we manage the disconnected data (i.e. mainly the diagonal element yielding infinite computations due to zero distances included in the denominator) of F(X), by substituting “inf” (infinite) values by zeros. Fourthly, we create the graphlayout of the ESG(X) based on the weighted connectivity matrix W_{ESG}.
According to the first four steps of the algorithm, we can generate the electrostatic graph ESG(X), which is associated with a timeseries X and is an undirected and weighted graph with a nontrivial topology. In this graph model, we can further compute several network measures and metrics and thus reveal the topological properties of the ESG. Therefore, at the fifth and final step of the algorithm, we compute nodeseries of network measures of the ESG(X), and afterward, we compare their structural relevance with this of the source timeseries X. The procedure is described in more detail in the following paragraphs.
Nodeseries of network measures
The electrostatic graph ESG(X) is a graphmodel G_{ESG}(V,E) where each network node v_{i} \(\in\) V is the same with a timeseries node i \(\in\) X, namely v_{i}≡i \(\in\) V,X. Therefore, for every nodemeasure Y (e.g. node degree, local clustering coefficient, closeness, betweenness, and eigenvector centrality, etc.) of the ESG, we can arrange the scores Y(v_{i}) = y_{i} into the timeseries X = {x_{1}, x_{2},…, x_{n}} ordering, and thus to configure nodeseries X(Y) = {y_{1}, y_{2},…, y_{n}} of the ESG network measures that are associated with the source timeseries. This allows comparing the source timeseries X with these of the ESG nodeseries X(Ys) and detecting possible structural similarities that can be seen as a measure of relevance between the timeseries and the ESG. The available network (node) measures that participate in the construction of the nodeseries are shown in Table 1.
In terms of notation, for a (source) timeseries X = {x_{1}, x_{2},…, x_{n}}, where n\(\in {\mathbb{N}}\) and x_{i}\(\in {\mathbb{R}}\), we can write its associated nodeseries for the network measure Y as X(Y) = {Y(x_{1}), Y(x_{2}),…, Y(x_{n})} = {y_{1}, y_{2},…, y_{n}}. We can read that X(Y) is “the nodeseries of the network measure Y, which is computed for the ESG that is associated with the timeseries X” or, in brief, that X(Y) is “the nodeseries of (the measure) Y for the ESG”. Within this context, we can compute the nodeseries for the measures of degree X(Y = k) = {k_{1}, k_{2},…, k_{n}}, strength X(s) = {s_{1}, s_{2},…, s_{n}}, clustering coefficient X(C) = {C_{1}, C_{2},…, C_{n}}, betweenness centrality X(CB) = {CB_{1}, CB_{2},…, CB_{n}}, closeness centrality X(CC) = {CC_{1}, CC_{2},…, CC_{n}}, and eigenvector centrality X(CE) = {CE_{1}, CE_{2},…, CE_{n}}, according to the mathematical formulas shown in Table 1. Provided that we can generate a nodeseries for any graph G(X) that is associated with a timeseries X, we can include a subscript index in the notation X_{G}(Y) when necessary (e.g. X_{ESG}(k)) to denote the type of graph that the timeseries is associated with.
The effect of the connectivity threshold on the ESG topology
The connectivity threshold F_{c} that is applied to the Coulomblike matrix, according to relation (3), is determinative for the configuration of the ESG topology. To illustrate this, let us consider the series X_{1:100} = {1, 2,…, 100} of the first hundred natural numbers. By applying to this series sequentially the connectivity thresholds F_{c} = 0, F_{c} = 1, F_{c} = 5, F_{c} = 10, F_{c} = 25, F_{c} = 50, F_{c} = f_{z}, F_{c} = 75, and F_{c} = 100, we get various ESGs, as it is shown in Fig. 3.
As it can be observed, the ESGs shown in Fig. 3 appear quite different in terms of graph density and node arrangement in the adjacency matrix. In particular, as the F_{c} becomes greater, the connectivity strip toward the main diagonal in the adjacency matrix becomes narrower, expressing each time a separate connectivity pattern. To examine whether and how the network topology is affected by changes in F_{c}, we compute a set of network measures and metrics (average degree \(\left\langle k \right\rangle\), clustering coefficient C, graph density ρ, modularity Q, average path length \(\left\langle l \right\rangle\), network diameter d(G), and the number of components) for a series of ESGs that are generated by applying connectivity thresholds ranging within the interval F_{c}\(\in\)[0, n^{2} = max{ X_{1:100}}^{2} = 10^{4}]. This approach assumes that the network topology is collectively approximated by the set of available network measures, where each measure represents a certain topological aspect. The results of the analysis are shown in Fig. 4, where each network measure is expressed as a function of the connectivity threshold F_{c}.
Also, is evident that all network measures considerably fluctuate as the connectivity threshold F_{c} changes. The cases of average degree \(\left\langle k \right\rangle\)(Fig. 4a), clustering coefficient C (Fig. 4b), and graph density ρ (Fig. 4c) follow a declining pattern to the changes of F_{c}, the cases of average path length \(\left\langle l \right\rangle\)(Fig. 4e) and graph density d(G) (Fig. 4f) follow a bellshaped pattern of negative skew (asymmetry), whereas the number of components (Fig. 4g) follows an increasing pattern. For the case of modularity Q (Fig. 4d), the performance of this measure appears considerably invariant along the biggest part of the F_{c}’s interval. As far as the typical value f_{z} is concerned, we can observe that this value cannot be related to border (i.e. min or max) distribution values, but it can be quite indicative of the average performance of the topological aspects of the ESGs. This indication can support the goodness of the choice of defining the typical f_{z} value within a physical (Coulomblike) context, as it is shown in relation (4).
Overall, this analysis shows that the choice of the connectivity threshold F_{c} can be determinative to the topological features and generally the topology of the resulting ESG. This observation is evident even by the examination of a simple linear series of natural ascending numbers, which can only be considered as an indicative approach for the ESG construction. However, even this simple consideration sufficed to highlight the dependence between the connectivity threshold and the ESG’s network topology and thus to introduce a methodological path for optimally defining the F_{c} value. The examination of the optimum or most representative threshold is a matter of specialized optimization analysis that introduces avenues of further research and falls outside the scope of this paper. However, the physically defined approaches, as this of the Coulomblike definition of the F_{c} shown in relation (4), or others utilizing methods from other disciplines can become insightful toward this optimization direction and are suggested for further research promoting multidisciplinary conceptualization. For instance, further research on this topic can apply to different types of timeseries and more thorough optimization analysis in the choice of F_{c}. For the scope of this paper, the choice of the typical value f_{z} for the connectivity threshold is considered satisfactory to provide a reference value that is representative of the average topological features of the ESGs.
Testing the performance of the ESG algorithm
The analysis examines five different types of timeseries, as it is shown in Fig. 5. The first one (Fig. 5a) was extracted from AirPassengers^{30} and is a timeseries with a linear trend (abbreviated: X_{a}≡AIR), including the monthly totals of US airline passengers for the period 1949 to 1960 (144 cases). The second one (Fig. 5b) was extracted from LorentzTS^{31} and is a typical Lorentz chaotic timeseries (X_{b}≡CHAOS) generated from the Lorenz differential equations, on standard values sigma = 10.0, r = 28.0, and b = 8/3. This timeseries has a length of 1900 cases. The third one (Fig. 5c) was extracted from DEOK.hourly^{32} and is a part (the first 5000 cases) of a broader stationary timeseries (of 57,739 cases) including estimated energy consumption, in Megawatts (MW), for the Duke Energy Ohio/Kentucky (X_{c}≡DEOK). Next, the fourth one (Fig. 5d) was extracted from Wolfersunspotnumbers^{33} and is a periodic timeseries including Wolfer sunspot numbers (X_{d}≡SUNSPOTS), for the period 1770 to 1771 (280 cases). The fifth one (Fig. 5e) was extracted from Dailyminimumtemperaturesinme^{34} and is a cyclical timeseries including daily minimum temperatures in Melbourne, Australia (X_{e}≡TEMP), for the period 1981–1990 (3650 cases). Links to the timeseries databases are available in the reference list.
To examine the effectiveness of the proposed algorithm, we firstly compare the structure of the source timeseries X with its nodeseries X_{ESG}(Ys) of the ESG node measures (Ys). Such comparisons are driven by the rationale that the ESG is a transformation (conversion) of a timeseries to a complex network and therefore possible similarities that can be detected in the structural properties (e.g. data variability, linear trend, chaotic, stationary, periodic, and cyclical structure) between the original timeseries and its associated ESG nodeseries can be seen as aspects of homeomorphism describing this transformation. In general, this approach is expected to illustrate the level at which the topology of the associated electrostatic graph ESG(X) sufficiently incorporates structural information of the source timeseries X. Secondly, we compare the structure of the X_{ESG}(Ys) nodeseries with this of their concordant nodeseries X_{VGA}(Ys) of the node measures (Ys) computed in the visibility graphs defined by Lacasa et al.^{5}. The comparisons between the source timeseries and its associated nodeseries (either of ESG or VGA conversion) build on a multilevel analysis consisting of five tests; the first one detects similarities in datavariability (i.e. whether the original timeseries and the nodeseries have the same fluctuation patterns) based on the Pearson’s bivariate coefficient of correlation^{35,36}, the second one in lineartrend by using the Linear Regression (LSLR) fitting^{36}, the third one in chaoticstructure based on the correlation dimension and embedding dimension diagram^{37}, the fourth one in stationarystructure based on the augmented DickeyFuller test (ADF) for a unit root ^{38}, and the fifth one in periodicstructure based on autocorrelation function^{38}. Each test is briefly described in the following paragraphs.
The visibility graph algorithm
The natural visibility algorithm (NVG) was proposed by Lacasa et al.^{5} and builds on the intuition of considering a timeseries as a path of successive mountains of different height, where each represents the value of the timeseries at a certain time. In this timeseries landscape, an “observer” standing on the top of a mountain can see (either forward or backward) as far as possible, provided that no other top obstructs its visibility field (Fig. 6).
In mathematical terms, each timeseries node (t_{i}, x(t_{i})) corresponds to a graph node i≡(t_{i}, x(t_{i}))\(\in\) V, and thus two nodes i,j \(\in\) V are connected (i,j)\(\in\) E in the visibility graph when the following inequality (NVG connectivity criterion) is satisfied:
where X(t_{i}) and X(t_{j}) are the numerical values of the timeseries nodes (t_{i}, x(t_{i}))≡i and (t_{j}, x(t_{j}))≡j and t_{i}, t_{j} express their time points. In geometric terms, a visibility line can be drawn between two timeseries nodes i,j \(\in\) V, if no other intermediating node (t_{k}, x(t_{k}))≡k obstructs their visibility. That is, two timeseries nodes are connected in the visibility graph whether no other intermediary node is higher so that to intersect the visibility line defined by this pair of nodes (Fig. 6). Therefore, two timeseries nodes can enjoy a connection in the associated visibility graph if they are visible through a visibility line. The visibility algorithm conceptualizes the timeseries as a landscape and generates a visibility graph associated with this landscape. The associated (to the timeseries) visibility graph is a complex network where complex network analysis can be further applied^{8,16}.
Correlation analysis
At the first step of the analysis, we detect linear correlations between the source timeseries X and the available (ESG and VGA) nodeseries. This approach examines whether the original timeseries X and the nodeseries {X_{i}(k), X_{i}(s), X_{i}(C), X_{i}(CB), X_{i}(CC), and X_{i}(CE)  i = ESG,VGA} have the same fluctuation patterns and thus they can be considered as relevant in terms of data variability. In this analysis, the Pearson’s bivariate coefficient of correlation^{35,36} is used, which ranges at the interval r_{X,Y} \(\in\)[–1,1] and detects linear (either positive or negative) correlations when \(\left {r_{XY} } \right \to 1\).
Test of the linear trend
To detect a linear trend, we apply linear fittings to the source timeseries X and to its associated nodeseries { X_{i}(k), X_{i}(s), X_{i}(C), X_{i}(CB), X_{i}(CC), and X_{i}(CE)  i = ESG,VGA}. According to this approach^{36}, a linear curve \(\hat{y} = b \cdot f(x) + c\) is fitted to the available data that bests describes their variability. The curve fitting algorithm estimates the parameters b, c minimizing the square differences \(y_{i}  \hat{y}_{i}\)^{36}, according to the relation:
where y_{i} express the observed and \(\hat{y}_{i}\) the estimated values. The optimization method that is used is the LeastSquares Linear Regression (LSLR) method^{36}, which assumes that the differences \(e = \mathop \sum \limits_{i = 1}^{n} \left[ {y_{i}  \hat{y}_{i} } \right]^{2}\) follow the normal distribution e ~ N(0,\(\sigma_{e}^{2}\)). The goodness of the model fit is measured by the coefficient of determination R^{2}, which is defined by the expression^{35,36}:
where \(\overline{y}\) is the average of the observations and n is the number of cases (i.e. the series length). The coefficient of determination expresses the amount of variability of the response variable that is expressed by the linear model and ranges within the interval [0,1], indicating perfect linear determination when R^{2} = 1^{35,36}. Within this context, amongst the ESG and VGA nodeseries, those being closer to the source timeseries X in determination and model configuration (i.e. values in b and c estimators) are considered as more relevant to X in terms of linear trend.
Detection of chaotic structure
To detect chaotic structure in a timeseries, we examine the patterns of the correlation (v) versus the embedding dimension (m) scatter plots (v,m). According to the Chaos theory^{37}, the correlation dimension (v) is a measure of the dimensionality of the space occupied by a set of random points and thus is used to determine the dimension of the fractal objects, which is often called fractal dimension. For a timeseries X = {x_{i}  i = 1, …, n}, the correlation integral C(ε) is calculated by the expression^{39,40}:
where N(ε) is the total number of pairs of timeseries points (x_{i}, x_{j}) with a distance smaller than ε, namely d(x_{i},x_{j}) = d_{ij} < ε. As the number of points tends to infinity (n → ∞), and therefore as their corresponding distances tend to zero (d_{ij} → 0), the correlation integral tends to the quantity C(ε) ~ ε^{v}, where v is the socalled correlation dimension. Intuitively, the correlation dimension expresses the ways to which the points can be close to each other along different dimensions and is expected to rise faster when the space of embedding is of a higher dimension. Therefore, the correlation (v) versus the embedding dimension (m) diagram (v,m) can provide insights into how the timeseries points are close to each other, as the dimensionality of the space of embedding increases^{39,40}. Within this context, amongst the ESG and VGA nodeseries, those with the (v,m) diagram being closer to the source timeseries X are considered as more relevant to the original timeseries, in terms of chaotic structure.
Detection of stationarity
To detect stationarity in the available series we apply the augmented DickeyFuller test (ADF) for a unit root ^{38}. The ADF algorithm examines the null hypothesis (H_{o}) that a unitroot is present in the model’s timeseries data, which is expressed by the relation:
where Δ is the differencing operator (Δy_{t} = y_{t} − y_{t−1}), p is the number of lagged difference terms (specified by the user), c is a drift term, δ is a deterministic trend coefficient, \(\phi\) is an autoregressive coefficient, β_{i} are the regression coefficients of the lag differences, and ε_{t} is a mean zero innovation process. According to Eq. (10), the unitroot hypothesis testing is expressed as follows^{38}:
and the (lag adjusted) test statistic DF is defined by the expression^{38}:
where the uppercase symbol ‘^’ expresses an estimator. Within this context, amongst the nodeseries of ESG and VGA, first those satisfying the null hypothesis and then those that have more similar DF statistics with the source timeseries X are considered as more relevant to the original timeseries, in terms of stationarity.
Detection of periodicity and cyclical structure
To detect periodicity in the available timeseries we use the autocorrelation function (ACF), which is defined as:
where (s,t) are time points and γ_{x}(s,t) is the autocovariance function of the variable x^{38}. In general, the ACF measures the linear predictability of the series at time t by using only the value x_{s} (at time s) with a timelag dt = t − s. The ACF lies within the interval − 1 ≤ ρ(s,t) ≤ 1, where positive coefficient values imply a positive linear trend and negative values a negative one. Based on the ACF, we construct a set of ACFvariables, where the first refers to the source timeseries X and the others to the nodeseries X_{i}(k), X_{i}(s), X_{i}(C), X_{i}(CB), X_{i}(CC), and X_{i}(CE), where index i can be either i = ESG or VGA. Each variable includes 30 elements corresponding to ACFs of lag dt = 1,2,…,30, respectively, namely:
By constructing these ACFvariables, we compute the Pearson’s bivariate coefficient of correlation^{35,36} to detect linear correlations between the ACF(X) variable of the source timeseries X and the other nodeseries variables. Within this context, amongst the available ESG and VGA nodeseries variables, those being higher correlated with the source timeseries X are considered as more relevant to the original timeseries in terms of periodicity and cyclical (i.e. periodic with a constant oscillation height) structure.
Results
Spy plots and graph layouts
The spy plots and graph layouts of the ESG(X) and VGA(X) graphs associated with the timeseries X are shown in Fig.A1A5 (in the Appendix). The spy plots are matrixplots displaying with dots the nonzero elements of the adjacency matrix and they can thus represent the graph topology within the matrixspace^{3,41}. On the other hand, network visualization is implemented by using the “ForceAtlas” layout, which is available in the opensource software of Bastian et al.^{42}. This layout is generated by a forcedirected algorithm, which applies repulsion strengths between network hubs while it arranges hubs’ connections into surrounding clusters. Graph models that are represented in this layout have therefore their hubs centered and mutually distant (i.e. intermediate distance between hubs is the highest possible), whereas lowerdegree nodes are placed as closely as possible to their hubs^{3}.
As it can be observed in Fig.A1 (Appendix), the ESG(X_{a}) spy plot has a connectivity pattern configuring a tie (along the main diagonal) of increasing width (Fig.A1.a,c, Appendix), which appears indicative of the increasing trend of the source timeseries (X_{a} = AIR). An aspect of such trend is also evident in the chainlike ESG(X_{a}) graph layout (Fig.A1.e, Appendix), where a cluster of hubs appears on the right side that resembles the tie configuration shaped in the spy plot. Also, the sawlike pattern of the source timeseries appears smoother in the pattern of the 2d ESG(X_{a}) spy plot (Fig.A1.a, Appendix), whereas is more evident in the diagonal arrangement of the 3d ESG(X_{a}) spy plot (Fig.A1.c, Appendix). On the other hand, the VGA(X_{a}) spy plot configures a periodic pattern (Fig.A1.b,d, Appendix), where no linear trends are visible. This can be also observed in the VGA(X_{a}) graph layout (Fig.A1.f, Appendix), which shapes an almost symmetric hubandspoke pattern.
In Fig.A2, the ESG(X_{b}) spy plot configures a fractallike tiling (Fig.A2.a, Appendix) illustrating a chaotic structure. Although such structure in the ESG(X_{b}) graph layout (Fig.A2.f, Appendix) is not that clear, we can observe two major components composing the electrostatic graph of X_{b} (Lorentz timeseries). This is a result of the positive and negative values in the structure of the source timeseries (X_{b}), illustrating the ability of the electrostatic graph (ESG) algorithm to generate disconnected graphs^{1,41,27}. Although connectivity is generally a desirable property in complex networks, the ability of the ESG algorithm to generate disconnected graphs can be insightful for removing past or unnecessary information (noise) of the timeseries, therefore proposing avenues for further research. On the other hand, the VGA(X_{b}) graph layout (Fig.A2.f, Appendix) better illustrates a chaotic structure than its spy plot (Fig.A2.b,d Appendix) does, which is more illustrative to a periodic than to chaotic structure.
Next, in Fig.A3 (Appendix) the ESG(X_{c}) spy plot (Fig.A3.a,c, Appendix) configures a tie (along the main diagonal), with an almost constant width, which complies with the stationary structure of the source timeseries (X_{c} = DEOK). Some evidence of stationarity can be also observed in the concentrated (solidlike) pattern of the ESG(X_{c}) graph layout (Fig.A3.e, Appendix). On the other hand, neither the VGA(X_{c}) spy plot (Fig.A3.b,d) nor graph layout (Fig.A3.f, Appendix) are illustrative of a stationary structure describing the original timeseries (X_{c}).
In Fig.A4 (Appendix), the ESG(X_{d}) spy plot also configures a tie (along the main diagonal) with repeated knotconcentrations (Fig.A4.a,c, Appendix), which complies with the periodic structure of the source timeseries (X_{d} = SUNSPOTS). Some insightful indications of such periodicity can be also observed in the clustered (toruslike) pattern that is shown in the ESG(X_{d}) graph layout (Fig.A4.e, Appendix). On the other hand, the VGA(X_{c}) spy plot (Fig.A4.b,d, Appendix) has an interesting periodic pattern, which is slightly mixed by the square areas of the other connections. However, the VGA(X_{d}) graph layout (Fig.A4.f, Appendix) does not appear illustrative of the periodic structure describing the source timeseries (X_{d}).
Finally, the ESG(X_{e}) spy plot configures a tie (along the main diagonal) with repeated slightly thicker segments (Fig.A5.a,c, Appendix), which can relate to the cyclical structure describing the source timeseries (X_{e} = TEMP). However, such cyclical structure is almost hidden in the chain pattern of graph components that have an odd arrangement in the ESG(X_{d}) graph layout (Fig.A5.e, Appendix). Periodicity can become clearer whether the layout will be further stretched to succeed symmetric arrangement similar to this of Fig.A4.e (Appendix). On the other hand, the VGA(X_{c}) spy plot shapes a clearer periodic pattern (Fig.A5.b,d, Appendix), which (although difficult) can be observed in the graph layout (Fig.A5.f, Appendix). Overall, the proposed ESG algorithm appears at least as capable as the VGA is in generating graphs of topologies representative of their source timeseries. This observation will be also quantitatively tested in the following sections.
Correlation analysis
To compare patterns in data variability between the source and the ESG and VGA nodeseries (see Fig. A6A10, Appendix), we apply a Pearson’s bivariate correlation analysis, the results of which are shown in Table 2. Amongst the available correlation coefficients, we compare concordant pairs (r(X,X_{ESG}(z), r(X,X_{VGA}(z)z = k, C, CB, CC, and CE) between ESG and VGA nodeseries and we denote pairwise maxima (max{(r(X,X_{ESG}(z), r(X,X_{VGA}(z)}) in bold font. Cases with the X_{ESG}(s) nodeseries are paired with those of corresponding degree X_{VGA}(k), due to the similarity of the measures of node degree (k) and node strength (s), for the binary and weighted networks. Within this context, according to Table 2, in the case of the X_{a} timeseries, the variability of the ESG nodeseries is overall closer to this of the source timeseries (X_{a}) than the variability of the VGA nodeseries overall is, because the ESG nodeseries count 5 out of 6 maxima, whereas the VGA nodeseries count just one. This observation implies that the ESG transformation generates graphs that better preserve fluctuations with a linear trend of the original timeseries than the VGA does. On the contrary, in the case of the chaotic timeseries (X_{b}), the VGA nodeseries count 5 out of 6 maxima (a double count is given for the k,s pair), whereas the ESG nodeseries count just one. This observation implies that the VGA transformation better preserves chaotic fluctuations of the original timeseries than the ESG does.
In the case of the X_{c} (DEOK), the ESG nodeseries count 4 out of 6 maxima, whereas the VGA nodeseries count 2 out of 6, which implies that the ESG transformation better preserves stationary fluctuations of the original timeseries than the ESG does. In the case of the X_{d} (SUNSPOTS), the ESG node series count 5 out of 6 maxima, whereas the VGA nodeseries count just one, which implies that the ESG transformation better preserves periodical fluctuations of the original timeseries than the VGA does. In the case of the X_{e} (TEMP) both the ESG and the VGA nodeseries count 3 out of 6 maxima, showing a balanced performance. As far as the measure of strength (s) (see Fig. A11, Appendix) is concerned, the analysis shows that, for all types of timeseries except the chaotic one (X_{b}, CHAOS), the ESG nodeseries have higher performance than the VGAs. Overall, this pairwise consideration illustrates that the variability of ESG nodeseries is closer to the source timeseries (X_{i}) than of the VGAs, since the first count 18 out of 30 maximum cases, whereas the latter count 12 out of 30 maxima.
Test of the linear trend
The test of the linear trend was applied to ESG and VGA nodeseries associated with the X_{a} (AIR) timeseries, which is a timeseries with a known linear trend. The results of the analysis are shown in Table 3, where first it can be observed that the source (X_{a}: AIR) timeseries is well described by a linear regression model (R^{2} = 0.8536). However, none of the VGA nodeseries can sufficiently retain this linear structure, as is evident by the low coefficients of determination ranging from R^{2} = 0.0002 to R^{2} = 0.0132.
On the contrary, the ESG nodeseries of degree X_{ESG}(k), strength X_{ESG}(s), and eigenvector centrality X_{ESG}(CE) have a considerable linear structure, as is denoted by their respective coefficients of determination R^{2} = 0.6916, R^{2} = 0.8012, and R^{2} = 0.7579. It should be noted that among these cases, the strength nodeseries X_{ESG}(s) have the highest determination. Overall, this analysis illustrates that the ESG algorithm appears more capable than the VGA in generating graphs that can preserve aspects of the linear trend of the source timeseries.
Detection of chaotic structure
In this part of the analysis, the correlation versus the embedding dimension diagrams (v,m) of the VGA and the ESGs nodeseries are compared for preserving the chaotic structure of the source timeseries X_{b} (CHAOS), which is already known as a chaotic timeseries constructed on the Lorenz equations. The results are shown in Fig. A7 (Appendix), where all (v,m) diagrams of the ESG nodeseries (except this of eigenvector centrality X_{b,ESG}(CE)) illustrate the chaotic structure, but of different characteristics than the source chaotic timeseries X_{b}. However, the (v,m) diagrams of strength X_{b,ESG}(s) and the original timeseries X_{b} almost coincide, a fact that implies a relevant chaotic structure between these timeseries. On the other hand, the degree X_{b,VGA}(k), and possibly the eigenvector centrality X_{b,VGA}(CE) VGA nodeseries illustrate a chaotic structure of high dimensionality, which are also of different characteristics than the original chaotic timeseries X_{b}. Overall, the chaos analysis shows that the ESG is a more capable transformation in incorporating the chaotic structure of the source timeseries in the network topology. Particularly, the measure of strength shows the most relevant chaotic structure that almost coincides with this of source timeseries.
Detection of stationarity
The test of stationarity was applied to the X_{c} (DEOK) timeseries, which is a part of an already known stationary timeseries. The results of the analysis are shown in Table 4, where, first, it can be observed that is 7.03% likely for X_{c} to have a unitroot and thus to be a nonstationary timeseries. This result implies that the nullhypothesis (stating a null unitroot) cannot be rejected, and thus that the source (X_{c}) timeseries cannot be considered as a stationary one. As it can be observed, the results for all VGA nodeseries imply that all cases are statistically safe to be considered as stationary series, which opposes the indication of the original timeseries.
On the other hand, the ESG results imply that 4 out of 5 ESG nodeseries cannot be considered as stationary ones and thus resembling the structure of the original timeseries. An interesting observation here is that the pvalues of the VGA nodeseries are (although insufficient indications to retain the null hypothesis) closer than those of the ESG nodeseries, in terms of distance. These results imply that the nonstationary effects, which are immanent in the source timeseries, probably appear more intensely in the structure of the ESG nodeseries than of the VGA ones.
Detection of periodicity and cyclical structure
This part of analysis builds on bivariate correlations, which are applied to autocorrelation variables ACF(X) that are defined in relation (14) with lag 1,2,…,30, where X = X_{d} (SUNSPOTS timeseries), X_{e} (TEMP timeseries), k (degree nodeseries), C (clustering coefficient nodeseries), CB (betweenness centrality nodeseries), CC (closeness centrality nodeseries), and CE (eigenvector centrality nodeseries). The results of the analysis are shown in Table 5, where the correlation coefficients r_{XY} and their significances are provided, with X\(\in\){ ACF(X_{d}), ACF(X_{e})} and Y\(\in\){ACF_{i}(k), ACF_{i}(s), ACF_{i}(C), ACF_{i}(CB), ACF_{i}(CC), ACF_{i}(CE)  where i = VGA, ESG}.
For the case of X_{d} (SUNSPOTS) timeseries, we can observe that 4 out of 6 VGA nodeseries (k, k≡s, C, CE) and 3 out of 6 ESG nodeseries (k, s, CC) are significantly correlated with the original time series X_{d}. Amongst these significant results, the VGA nodeseries have 2 maxima of concordant pairs, whereas the ESG nodeseries have also 2 maxima. Moreover, the nodeseries of strength (X_{d,ESG}(s)) has the highest correlation coefficient amongst all available nodeseries for the SUNSPOTS (X_{d}) typology, illustrating a better performance of the ESG algorithm to preserve periodicity, probably due to its capability in generating weighted electrostatic networks. For the case of X_{e} (TEMP) timeseries, 1 VGA nodeseries (closeness centrality) is significantly correlated with the source timeseries, where all ESG nodeseries are significantly correlated with the original timeseries. In terms of pairwise comparisons, the VGA nodeseries count 1 (out of 6) maximum case, whereas the ESG nodeseries count 5 out of 6 maxima. However, although is high, the strength does suggest the highest of the maxima of the TEMP (X_{e}) timeseries concordant pairs. Overall, this analysis shows that the ESG nodeseries appear more capable than the VGA ones in preserving periodic and cyclical characteristics of the source timeseries.
Conclusions
This paper proposed a new algorithm, the Electrostatic Graph Algorithm (ESG), for converting a timeseries into a graph (complex network). The ESG builds on the conceptualization of considering a timeseries as a series of stationary and electrically charged particles, on which Coulomblike forces can be computed. The proposed algorithm provides an added value to complex network analysis of timeseries due to its ability to produce weighted graphs, which is currently not applicable. This additional property was quantitatively examined in this paper and was found to produce graphs that are more representative of the structure of the source (original) timeseries, implying that the proposed algorithm suggests a transformation that is more natural rather than algebraic, in comparison with the existing methods. In particular, the analysis showed that the ESG nodeseries can better preserve the linear trend and stationary structural properties of the source timeseries in comparison with the VGA nodeseries and that they appear slightly better in preserving periodical and cyclical structural properties of the original timeseries than the VGA nodeseries can. On the other hand, the VGA nodeseries appeared slightly better in preserving the chaotic structural properties of the original timeseries in comparison with the ESG nodeseries, which complies with the claim of the VGA authors regarding the added value of their algorithm. However, in almost all the parts of the analysis, the ESG nodeseries of the measure of strength outperformed their concordant VGA nodeseries. This result highlighted the added value of the proposed algorithm in generating weighted graphs, in which the measure of node strength can only be computed. Therefore the ESG algorithm attributes to the generated graphs information that is more representative of the source timeseries, due to the weights included in the graph structure. Another property of the proposed ESG algorithm to generate disconnected graphs was indirectly examined by the detection of chaotic and periodic structures, where the ESG algorithm sufficed to provide disconnected graphs, whereas the VGA did not. This analysis showed that insufficient connectivity does not restrict the ESG nodeseries to preserve the structural characteristics of the source timeseries, since the generated electrostatic graphs were representative of the structure of the original timeseries. The authors believe that the property of insufficient connectivity introduces avenues of further research in the field of noise reduction in the timeseries analysis. Other avenues of further research can emerge towards the direction of either choosing the optimum or most representative connectivity threshold to produce the ESGs or examining the applicability of the proposed algorithm to solve problems where standard methods fail to analyze efficiently the timeseries, such as the time evolution of stock price, within the framework of Black Scholes model, and others. The overall approach also suggests a methodological framework for evaluating the structural relevance between the source timeseries and their associated graphs produced by any possible transformation.
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D.T. designed research; D.T. and L.M. performed research; D.T., L.M., and P.A. contributed new reagents/analytic tools; D.T., L.M., and P.A. analyzed data; D.T., L.M., and P.A. wrote the manuscript; All authors reviewed the manuscript.
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Tsiotas, D., Magafas, L. & Argyrakis, P. An electrostatics method for converting a timeseries into a weighted complex network. Sci Rep 11, 11785 (2021). https://doi.org/10.1038/s41598021895522
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DOI: https://doi.org/10.1038/s41598021895522
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