Analytical Framework for Recurrence-Network Analysis of Time Series

Analytical framework for recurrence-network analysis of time series

Jonathan F. Donges,1, 2, ∗ Jobst Heitzig,1 Reik V. Donner,1 and JürgenKurths1, 2, 3 1Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany 2Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany 3Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, United Kingdom (Dated: August 13, 2018) Recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems. While there are already many successful applications ranging from medicine to paleoclima- tology, a solid theoretical foundation of the method has still been missing so far. Here, we interpret an ε-recurrence network as a discrete subnetwork of a “continuous” graph with uncountably many vertices and edges corresponding to the system’s attractor. This step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ε. In particular, we introduce local measures such as the ε-clustering coefficient, mesoscopic measures such as ε-motif density, path-based measures such as ε-betweennesses, and global measures such as ε-efficiency. This new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of ε via a percolation threshold, and it shows that estimation can be improved by so-called node splitting invariant versions of the measures. We finally illustrate the framework for a number of archetypical chaotic attractors such as those of the Bernoulli and logistic maps, periodic and two-dimensional quasi-periodic motions, and for hyperballs and hypercubes, by deriving analytical expressions for the novel measures and com- paring them with data from numerical experiments. More generally, the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints which appear frequently in disciplines ranging from biology to climate science.

PACS numbers: 05.45.Tp, 89.75.Hc, 05.45.-a

I. INTRODUCTION fundamental analogy between the Poincarérecurrence structure [15] of a time series in phase space, which is Analogies are a fundamental motor of innovation in commonly represented by a binary recurrence matrix and physics and other disciplines, since they allow the trans- allows us to recover basic dynamical invariants of the fer of theoretical insights, results and techniques from underlying system [16], and the binary adjacency ma- one field to the other. In the last years, complex network trix describing a complex network. Other methods such theory has been particularly successful in providing uni- as visibility graphs [17] work in the time domain and fying concepts and methods for understanding the struc- focus on studying stochastic properties of time series. ture and dynamics of complex systems in many areas of Aside from these network-based approaches for investi- science, ranging from power grids over social networks to gating (possibly multivariate) time series from isolated neuronal networks [1–4]. Similarly, nonlinear time series dynamical systems, efforts have been spent for develop- analysis aims to gain insights on a wide variety of nat- ing techniques for studying fields of time series such as ural, technological, and experimental dynamical systems functional brain networks [18–20] in the neurosciences or drawing on a generic body of theory and methods [5]. climate networks [21–24] in climatology. In summary, all By exploiting analogies in the structure and descrip- methods mentioned above propose a mapping from the tion of complex networks and dynamical systems, a num- time series to the network domain and then proceed to ber of new network-based techniques for nonlinear time interpret the statistical properties of the resulting (usu- series analysis have been proposed recently [6]. The first ally complex) network in terms of the underlying system’s class of these methods makes use of graph representations dynamical properties. arXiv:1203.4701v1 [physics.data-an] 21 Mar 2012 of certain similarity relationships between state vectors While these interpretations are mostly based on empir- or groups of state vectors (e.g., cycles) in phase space. It ical findings for paradigmatic model systems and heuris- includes transition networks based on a coarse-graining tic arguments, only a few rigorous results are available. of phase space [7], cycle networks [8], correlation net- So far, Lacasa et al. have pointed out a relationship works [9], k-nearest-neighbor [10], and adaptive nearest between the scaling exponent of the degree distribution −γ neighbor networks [11, 12] as well as ε-recurrence net- pk(k) ∝ k in visibility graphs constructed from frac- works [13, 14]. The latter three techniques harness the tional Brownian motion and the Hurst exponent [25]. Furthermore, close relationships between the transitivity properties (network transitivity and local clustering co- efficients) [26] as well as the degree distribution’s power- ∗Electronic address: [email protected] law scaling exponent γ [27] of ε-recurrence networks and 2 the (fractal) global and local dimensionality of the at- Recurrence networks and their statistical properties are tracting set underlying the time series have been found. related, but complementary to the established concepts Constituting random geometric graphs [28], ε-recurrence of recurrence plots (the recurrence matrix is given by networks represent the geometry induced by the time se- Rij(ε) = Aij(ε)+δij) and recurrence quantification anal- ries in phase space in a simple and well-defined way. This ysis (RQA) [16]. In contrast to RQA, which considers enabled Donner et al. [14, 26] to define continuous tran- temporal dependencies between observations in form of sitivity properties depending solely on the geometry of diagonal and vertical line structures in the recurrence the (attracting) set S and the probability density func- plot, recurrence network analysis discards all temporal tion p(x). These can in turn be calculated analytically for information and solely quantifies the geometry of the un- paradigmatic model systems with smooth and self-similar derlying set S (e.g., an attractor) [14, 41]. geometry and are approximated by the corresponding discrete ε-recurrence network measures. Notably, most Given the diverse and successful applications of ε- kinds of time series networks proposed so far are spatial recurrence network analysis reported in the literature, networks [29], since vertices are embedded either in phase it is important to establish a firm theoretical founda- space or on the time axis, implying that general results tion for advancing the understanding of the method. obtained for this class of networks are applicable to time Building on earlier work [26], we propose here an an- series networks as well. alytical framework for ε-recurrence network analysis of Recently, ε-recurrence networks have been demon- time series encompassing neighborhood-based transitiv- strated to be a particularly useful tool in diverse applica- ity measures, mesoscopic measures relying on network tions of nonlinear time series analysis ranging from model motifs [47], path-based network characteristics as well as systems [6, 13, 14, 26, 30–36] via experimental data [37– spectral and random walk-based measures. Specifically, 40] to recent and paleo-climate records [6, 13, 41–43] as our theory describes all graph-theoretical recurrence net- well as financial time series [31]. They allow us to un- work quantifiers that have been used in the literature so cover complex bifurcation scenarios [13, 41] and to re- far [14]. Beyond forming a solid theoretical basis for this liably distinguish between chaotic and non-chaotic dy- modern nonlinear approach to time series analysis and namics [33]. Furthermore, the local and global transi- fostering its detailed understanding in a way compara- tivity characteristics of ε-recurrence networks have been ble to that of standard linear time series analysis [48], shown to enable us to trace unstable periodic orbits [14] our analytical framework opens several avenues for prac- and to define alternative notions of fractal dimension [26] tically improving the method when dealing with finite independently of earlier approaches. An important ad- (real-world) time series: (i) We are able to obtain closed- vantage of nonlinear ε-recurrence-network-based time se- form analytical results for paradigmatic model systems ries analysis is that it performs well with significantly with stochastic (uniform and Gaussian noise) and deter- shorter time series (O(102) data points [13, 33, 41]) than ministic (periodic, quasi-periodic, and chaotic) dynamics. required by classical techniques like estimating the max- These can in turn be harnessed as a benchmark for the imum Lyapunov exponent from data [5, 44]. This ren- discrete standard estimators from complex network the- ders ε-recurrence networks readily applicable to the anal- ory which have been employed so far [14], e.g., for assess- ysis of non-stationary real world data. The method has ing the estimators’ bias and variance. (ii) This bottom- also been applied successfully to time series with irreg- up approach allows us to design improved, weighted sta- ular sampling and/or uncertain timing of observations tistical estimators [49] which may be more appropriate that are commonly found in the geosciences or in astro- in specific situations. (iii) Moreover, our framework en- physics [41, 42]. ables us to derive rigorous bounds for feasible values of An ε-recurrence network is completely defined by its the recurrence threshold ε, the most important param- adjacency matrix Aij(ε) which is obtained from a (mul- eter of the method, the choice of which is critical when tidimensional) time series x(ti), i = 1,...,N, by analyzing finite (experimental) time series [30]. We will argue in Sec. V that the concepts and measures developed Aij(ε) = Θ (ε − kx(ti) − x(tj)k) − δij, (1) in this paper can be readily generalized to describe the structure of a wider class of spatial networks, e.g., ran- where Θ(·) is the Heaviside function, ε a threshold used dom networks with an arbitrary prescribed edge length for defining the neighborhood of a state vector x(ti), k · k distribution pl(l). some norm, and δij denotes Kronecker’s delta introduced to avoid self-loops in the network. Given univariate ob- This paper is organized as follows: We introduce a servational or experimental time series, it is usually nec- continuous framework for recurrence network analysis in essary to reconstruct the corresponding system’s trajec- Sec. II. After reviewing the corresponding discrete esti- tory in some higher dimensional phase space to recover mators (Sec. III), we present examples ranging from peri- its recurrence structure reliably (e.g., by time-delay em- odic and quasi-periodic dynamics and higher dimensional bedding [45, 46]). symmetric sets over chaotic maps to stochastic processes Within the recurrence network, vertices represent ob- and compare some of the results to discrete estimates servations or state vectors in phase space, while edges (Sec. IV). We conclude with a discussion of these achieve- indicate a close proximity between two state vectors. ments (Sec. V). 3

are closely related to a certain notion of the fractal di- S mension of an underlying set S and its associated probability density p(x) with x ∈ S [26]. To capture this y) theoretically, continuous versions of both measures de- g(x, y noted C(x; ε) and T (ε) have been deﬁned together with a continuous degree density ρ(x; ε).

x 1. Local measures FIG. 1: Illustration of a set S (gray), where g(x, y) denotes the geodesic distance between x, y ∈ S. Deﬁnition 1. The continuous ε-degree density Z II. CONTINUOUS FRAMEWORK ρ(x; ε) = dµ(y) (3) Bε(x)

A. General setting measures the probability that a point y randomly drawn according to p lies in an ε-neighborhood Bε(x) = {y ∈ Let us consider a path-connected Lebesgue-measurable S : kx − yk ≤ ε} of x. subset S ⊂ X of an m-dimensional compact smooth manifold X with a non-vanishing continuous probability den- Definition 2. In turn, the continuous local ε-clustering R m sity function p : S → (0, ∞) with S d x p(x) = 1. coefficient of any point x ∈ S, R R m We will use the abbreviation dµ(x) = S d x p(x) throughout the paper, where µ is a probability measure RR dµ(y) dµ(z)Θ(ε − ky − zk) C(x; ε) = Bε(x) , (4) on S (Fig. 1). Then we can define “continuous” equiv- ρ(x; ε)2 alents of all relevant graph-theoretical measures for ε- recurrence networks which may be approximated by cal- is the probability that two points y and z randomly drawn culating their discrete counterparts in the limit ε → 0, according to p are closer than ε given they are both closer N → ∞ (Sec. III). Here ε is the threshold used for net- than ε to x. work construction (Eq. (1)) and N denotes the number of data points (samples, phase space vectors, ... ) consid- Definition 3. The continuous ε-matching index ered. These measures capture the properties of a “con- M(x, y, ε) measures the overlap of the neighborhoods of tinuous” network with uncountably many vertices and x, y ∈ S, edges which may be defined by a continuous analog of the adjacency matrix, the adjacency function R dµ(z) Bε(x)∩Bε(y) M(x, y; ε) = R . (5) A(x, y) = Θ(ε − kx − yk) − δxy (2) dµ(z) Bε(x)∪Bε(y) for all x, y ∈ S. It is important to realize that the framework introduced in this paper is not restricted to It gives the probability that a point z drawn randomly ε-recurrence networks alone, but may be more generally from Bε(x) according to p is also contained in Bε(y) and applied to describe random geometric graphs (aka spatial vice versa. networks) [28, 50, 51] and other types of networks with spatial constraints [29, 52, 53]. For x → y, M(x, y; ε) → 1. Furthermore, M(x, y; ε) = In the following we will formally define the pro- 0 if kx − yk > 2ε. posed continuous recurrence network measures and discuss their properties, interrelationships and interpretations (see Table I for an overview). Statements made for 2. Global measures the limits ε → 0 and x → y for x, y ∈ S should be understood to hold for smooth S and p. We do not consider Definition 4. The continuous ε-edge density them for fractal geometries explicitly. Z ρ(ε) = dµ(x)ρ(x; ε) (6) B. Neighborhood-based measures S Z Z = dµ(x) dµ(y) Among other interesting properties, it has been shown S Bε(x) recently that the local and global transitivity properties of ε-recurrence networks measured by the local clustering is the expectation value of the continuous ε-degree density coefficient Ci and the global transitivity T , respectively, ρ(x; ε). 4

TABLE I: A summary of the continuous geometric properties of the set S and its associated probability density p deﬁned in Sec. II. See main text for formal probabilistic interpretations.

Class Name Heuristic interpretation Neighborhood-based Local Continuous ε-degree density (Eq. (3)) Local density Continuous local ε-clustering (Eq. (4)) Local dimension [26] Continuous ε-matching index (Eq. (5)) Local density gradient Global Continuous ε-edge density (Eq. (6)) Average local density Continuous ε-transitivity (Eq. (7)) Global dimension [26] Continuous global ε-clustering (Eq. (8)) Average local dimension Continuous ε-assortativity (Eq. (9)) Average local density gradient Mesoscopic Continuous ε-motif density (Eq. (10)) Higher-order density structure/ Density anisotropy Path-based Local Continuous ε-closeness (Eq. (12)) Geometric centrality Continuous ε-eﬃciency (Eq. (13)) } Continuous ε-shortest path betweenness (Eq. (14)) Geometric bottleneckishness Continuous ε-shortest path edge betweenness (Eq. (18)) } Global Continuous ε-average path length (Eq. (25)) Average separation Continuous global ε-eﬃciency (Eq. (27)) } ε-diameter (Eq. (32)) Geometric diameter ε-radius (Eq. (33)) Geometric radius

Definition 5. As a global measure of geometric transi- gives the Pearson product-moment correlation coeffi- tivity, we define the continuous ε-transitivity of S as cient [48] of the degree densities ρ(x; ε) and ρ(y; ε) of ZZZ all points x, y that are closer than ε to each other. T (ε) = dµ(x) dµ(y) dµ(z)Θ(ε − kx − yk) × A(ε) can be considered as a measure of the smoothness S of the set S and the probability density p [14]. In the limit ×Θ(ε − ky − zk)Θ(ε − kz − xk) ε → 0 we have A(ε) → 1. ZZZ dµ(x) dµ(y) dµ(z)Θ(ε − kx − yk) × C. Mesoscopic measures S ×Θ(ε − kz − xk) , (7) Motifs of order α are small connected subgraphs of α vertices that are embedded within the topology of a which is the probability that among three points x, y, z complex network [47]. For combinatorial reasons, usu- drawn randomly according to p, y and z are closer than ally α < 5 is considered. Measuring motif densities is ε given they are both closer than ε to x. a useful approach for quantifying higher-order neighbor- Definition 6. Similarly, the continuous global ε- hood relationships in complex geometries [11] and may clustering coefficient be seen as a generalization of the transitivity concepts Z introduced above. C(ε) = dµ(x)C(x; ε) (8) Definition 8. The continuous ε-motif density S α Z ! is the expectation value of the continuous local ε- α Y Y clustering coefficient C(x; ε) (Eq. (4)). Mβ = dµ(xi) Θ(ε − kxj − xkk) (10) i=1 S (j,k)∈Eα Note that the above defined measures of transitivity β have been mainly considered for the supremum norm L∞ quantifies the frequency of occurrence of a certain recur- in [26]. rence motif of order α described by the corresponding edge α set Eβ , where β = 1, . . . , n(α) and n(α) is the total num- Definition 7. Continuous ε-assortativity α ber of distinct motifs of order α. Mβ is the probability that α points drawn randomly according to p are arranged A(ε) = r ρ(x; ε), ρ(y; ε) | kx − yk < ε (9) α according to the recurrence motif described by Eβ . 5

4 For example, the density Mt of the recurrence motif t called a global geodesic or shortest curve [55]. Depending of order 4 is measured by on the geometry of S, there may be none, one, or multiple distinct global geodesics connecting x and y, but in 4 Z ! 4 Y a sufficiently well-behaved set S, there will usually be a Mt = dµ(xi) Θ(ε − kx1 − x2k) × S unique global geodesic for almost every pair x, y (Fig. 1) i=1 and almost every metric (in particular, for the Euclidean ×Θ(ε − kx2 − x3k)Θ(ε − kx3 − x4k). metric), where by “almost every” we mean as usual that In contrast to the study of motifs for other general com- the set of exceptions has zero measure. Note that, how- plex networks [47], it is neither meaningful nor necessary ever, for some pathological metrics global geodesics are to normalize motif densities by their expectation values rarely unique, including the L1 and L∞ metrics. for randomized networks here. The reason is that the To understand the reasoning behind the following def- α initions, one has to note that discrete shortest paths of Mβ already have a natural probability interpretation. To render the results for different α more comparable, one the ε-recurrence network approximate global geodesics may consider to use relative motif densities normalized connecting two points x, y ∈ S for small ε and large N. Pn(α) α Then the shortest path length lij(ε) (the minimum num- by β=1 Mβ . We conjecture that motif densities as generalizations of ber of edges that have to be traversed to reach vertex i the continuous ε-transitivity are like the latter related to from vertex j [1]) approximates g(x(ti), x(tj)), i.e., certain notions of the dimensionality of the set S and its εlij(ε) ≈ g(x(ti), x(tj)) (11) associated probability density p [26]. This would allow us to define and study a new class of motif-based measures (Fig. 1), where x = x(ti) and y = x(tj). In the limit of dimensionality analogously to the sequence of Rényi ε → 0, and if N → ∞ sufficiently fast, we argue in dimensions from dynamical systems theory [54]. Appendix A that indeed εlij(ε) → g(x(ti), x(tj)), independently of which metric is used for constructing the ε-recurrence network. D. Path-based measures

While the neighborhood-based properties defined 1. Local measures above describe the small-scale geometry of the set S and probability density p, path-based measures quantify Definition 9. Given that a point y is drawn randomly their global geometry in terms of global geodesics (see according to p, continuous ε-closeness Fig. 2 for examples). Most of the path-based concepts defined below do not conceptually depend on the thresh- Z g(x, y)−1 old ε. Nevertheless, we introduce the appropriate scal- c(x; ε) = dµ(y) (12) S ε ing with ε into the definitions for consistency with the Z −1 corresponding discrete estimators from complex network = ε dµ(y)g(x, y) theory (Sec. III) and the recurrence network literature. S As in standard topological terminology, a path in S is is the inverse expected geodesic distance of y to another a continuous function f : [0, 1] → S, and its path length Pn chosen point x in units of ε. l(f) ∈ [0, ∞] is the supremum of i=1 d(f(ti−1), f(ti)) over all n > 0 and all tuples 0 = t0 ≤ · · · ≤ tn = 1, where Definition 10. Similarly, continuous local ε-efficiency d(·, ·) is some metric. Note that l(f) can be infinite in −1 which case the path is called non-rectifiable. For points Z g(x, y) e(x; ε) = dµ(y) (13) x, y, the geodesic distance g(x, y) ∈ [0, ∞] is the infimum S ε of l(f) over all paths in S from x to y (i.e., with f(0) = x Z and f(1) = y) 1. A corresponding path of this length is = ε dµ(y)g(x, y)−1 S gives the expected inverse geodesic distance of y to x measured in units of ε. 1 Note that when defined in this way, g(x, y) may change discon- tinuously under continuous changes of the probability density p. Both c(x; ε) and e(x; ε) quantify the geometric close- This is because we require the geodesics to stay within S which ness of x to any other point in S given a probability den- consists of all points x where p(x) 6= 0. When p(x) is continu- sity p. Hence, points in the center of S will carry larger ously changed to zero, the length of geodesics running through x for p(x) > 0 may change abruptly once p(x) = 0 is reached, e.g., values of c(x; ε) and e(x; ε) than those on its boundaries when x constitutes some kind of geometric bottleneck. If this be- (see Fig. 2 and below). havior is undesirable, one may consider generalized p-weighted notions of the geodesic distance. These could be motivated by Definition 11. Continuous ε-shortest path betweenness an analogy to the optical path length in heterogeneous and non- ZZ σ(y, z; x; ε) isotropic media in physics, where the probability density p would b(x; ε) = dµ(y) dµ(z) , (14) play the role of the spatially varying refractive index. S σ(y, z) 6

c(x;ε) e(x;ε) b1(x) b2(x;ε)

FIG. 2: Local path-based measures continuous ε-closeness c(x; ε), -efficiency e(x; ε), and -shortest path betweenness b1(x) (based on Eq. (15)), b2(x; ε) (based on Eq. (16)) in three example sets S with a uniform density p, two convex sets (circle and square), and a nonconvex set. Grayscale indicates the value of the measures (white: small, black: large) obtained by Monte Carlo numerical integration using the Euclidean norm for small ε to avoid boundary effects. Note the more complex structure of the betweenness field, displaying particularly large values (dark) at the inward corners where many shortest paths must pass. In contrast to the path-based measures shown here, continuous ε-degree ρ(x; ε) and local ε-clustering coefficient C(x; ε) are constant in the interior of S due to the uniform p. Variations of these measures due to boundary effects occur only closer than ε to the boundary of S [26]. is the probability that a point x lies on a randomly chosen Both variants of σ(y, z; x; ε) yield different, yet qualita- global geodesic connecting two points y, z drawn randomly tively similar results for b(x; ε) as is illustrated in Fig. (2). from S according to p. Here, σ(y, z; x; ε) denotes the Given convex domains S, σ(y, z) = 1 always holds, number of times x ∈ S lies on a global geodesic between i.e., there is only one straight line connecting y and z, y, z ∈ S and σ(y, z) is the total number of global geodesics parametrized by f(t) = y+t(z−y). For one-dimensional, between y, z (Fig. 3). convex sets S and using σ1(y, z; x), continuous ε-shortest In pathological situations, e.g., for certain open sets path betweenness simplifies to S, σ(y, z) may be zero even when the geodesic distance g(y, z) is well-defined and finite. We ignore these cases ZZ b(x) = 2 dµ(y) dµ(z)Θ(x − y)Θ(z − x). (17) for now. S There are several ways to formally define σ(y, z; x; ε). Using a parametrization fκ(t) of the family of global geodesics connecting y and z, with t ∈ [0, 1] and fκ(0) = Definition 12. Continuous ε-shortest path edge be- y, fκ(1) = z, we may write tweenness σ1(y, z; x; ε) = σ1(y, z; x) σ(y,z) ZZ σ(z, w; x, y; ε) X Z 1 b(x, y; ε) = dµ(z) dµ(w) , (18) = dt δ(fκ(t) − x), (15) S σ(z, w) κ=1 0 where δ(·) is Dirac’s multi-dimensional delta function. is the probability that two points x, y both lie on a ran- Alternatively, we can include the finite ε-effect by domly chosen global geodesic connecting two points z, w counting all shortest paths that pass through the ε- drawn randomly according to p. σ(z, w; x, y; ε) counts neighborhood of x by setting the number of global geodesics between z, w which con- tain x, y. σ(y,z) X Z 1 σ2(y, z; x; ε) = dt Θ(ε − kfκ(t) − xk). (16) κ=1 0 Analogously to continuous ε-shortest path betweenness 7

It should be noted that for general S and p, the center A B y of mass Z X = dµ(x)x (23) x S x z does not necessarily extremize c(x; ε), e(x; ε) or b(x). y However, for convex S the generalized continuous ε- closeness

Z g(x, y)η−1 cη(x; ε) = dµ(y) (24) z S ε Z −1 = εη dµ(y)g(x, y)η FIG. 3: Illustration of the definition of continuous ε-shortest S path betweenness (the set S is indicated by gray shading). (A) There are σ(y, z) = 2 global geodesics connecting y, z ∈ can be shown to assume a global maximum at x = X for S, but only σ(y, z; x; ε) = 1 includes x ∈ S. (B) In this the special case η = 2. In turn, the standard continuous example, x lies on all four global geodesics between x and y, ε-closeness c(x; ε)(η = 1) is maximized at the geometric i.e., σ(y, z) = σ(y, z; x; ε) = 4. median or Fermat-Weber point [57]. b(x; ε), we can define this quantity as 2. Global measures σ1(z, w; x, y; ε) = σ1(z, w; x, y) σ(z,w) X Z 1 Definition 13. The continuous ε-average path length = dt δ(fκ(t) − x) × κ=1 0 ZZ g(x, y) Z 1 L(ε) = dµ(x) dµ(y) (25) S ε × dt δ(fκ(t) − y) . (19) ZZ 0 = ε−1 dµ(x) dµ(y)g(x, y). Further generalizations for including the finite ε-effect S may be deduced as shown above for continuous ε-shortest measures the expected geodesic distance in units of ε be- path betweenness. tween two points x and y drawn randomly according to For one-dimensional convex sets S and using p. σ1(z, w; x, y; ε), Eq. (18) reduces to ZZ From Eq. (25), the equivalence of this formulation of b(x, y) = 2 dµ(z) dµ(w)Θ(x − z)Θ(y − z) × continuous average path length to the intensively stud- S ×Θ(w − x)Θ(w − y). (20) ied problem in probabilistic geometry [58] of finding the expectation value of the distance between two randomly In the limit x → y we always have b(x, y; ε) → b(x; ε). drawn points x, y ∈ S according to the probability dis- We note that b(x, y; ε) does not require the condition tribution p becomes evident. Our definitions imply the Θ(ε−kx−yk) as is the case for the corresponding discrete relationship estimator (Table II). Related generalized concepts of co- and group betweenness have been described for discrete Z L(ε) = dµ(x)c(x; ε)−1. (26) complex networks [56]. S For general non-pathological S we almost surely have σ(z, w) = 1, i.e., the probability that there are more than Definition 14. Similarly, the continuous global ε- one global geodesics connecting z and w drawn randomly efficiency from S according to p is zero. For example, in both !−1 Figs. 3A and B the set of pairs z, w with σ(z, w) = 2 or ZZ g(x, y)−1 σ(z, w) = 4, respectively, is of measure zero. In these E(ε) = dµ(x) dµ(y) (27) ε cases, b(x; ε) and b(x, y; ε) reduce to S ZZ −1 ZZ −1 −1 b(x; ε) = dµ(y) dµ(z)σ(y, z; x; ε) (21) = ε dµ(x) dµ(y)g(x, y) . S S and is the inverse of the expected inverse geodesic distance ZZ between two points x, y drawn randomly according to p b(x, y; ε) = dµ(z) dµ(w)σ(z, w; x, y; ε). (22) S measured in units of ε. 8

Here, we have C ∈ R associated to the eigenvalue λ = 0. This is anal- ogous to the eigenvector (1, 1,..., 1) with eigenvalue 0 Z −1 which is always present for the discrete Laplacian matrix E(ε) = dµ(x)e(x; ε) . (28) of general networks [1]. We can expect more interesting S results for non-uniform p. For example, one may define More generally, let ∆S,p(η) be the expectation value of a continuous analog of the eigenvector centrality of com- a power η of the geodesic distance g(x, y) between two plex network theory [59] by considering the eigenfunction points x, y ∈ S randomly drawn according to p: f˜(x) corresponding to the largest eigenvalue λ˜. ZZ For discrete networks, there are several measures of η betweenness based on random walks rather than short- ∆S,p(η) = dµ(x) dµ(y)g(x, y) . (29) S est paths [60, 61]. Continuous versions of these measures would be based on continuous analogs of random walks Then continuous ε-average path length and global ε- on S that start and end at points y and z randomly cho- efficiency may be expressed as sen from p. Since in a discrete network the limit distribution of a random walk without a sink is proportional to L(ε) = ε−1∆ (1) (30) S,p the degree distribution, a natural choice for a continuous and analog is an It¯odiffusion process, the limit distribution of which is proportional to ρ(x; ε), with a source at y −1 −1 and a sink at z [62]. Such a process can most easily be E(ε) = ε (∆S,p(−1)) . (31) √ defined as a gradient flow dXt = −∇Ψ(Xt) dt + 2T dBt Definition 15. The ε-diameter that combines a Brownian motion B with a local drift coefficient −∇Ψ(Xt) which comes from a potential Ψ(x) D(ε) = ε−1 sup g(x, y) (32) that is the product of a temperature T > 0 and the in- x,y∈S formation corresponding to ρ(x; ε), which is − ln ρ(x; ε). The resulting process and the ε-radius √ −1 ∇ρ(Xt; ε) R(ε) = ε inf sup g(x, y) (33) dXt = T dt + 2T dBt (36) x∈S y∈S ρ(Xt; ε) are global geometric characteristics of the set S that are can then be interpreted as a diffusion that drifts in the independent of p [55]. direction of increasing density. The continuous version of Arenas’ random walk betweenness [60] would then be the expected density of the process at x when the source and E. Further measures sink are drawn from p. Similarly, the continuous version of Newman’s random walk betweenness [61] would be the To illustrate that the proposed framework can be ex- expected absolute value of the resulting flux density at x tended in several directions, we shortly discuss spectral for a random source and sink. and random-walk-based measures in the context of continuous recurrence networks. Motivated by the study of eigenvector centrality in complex networks [1], we can F. Behavior under affine transformations consider spectral properties of the set S and probability density p. All continuous measures defined above are based on neighborhood relationships in S and geodesic distances Definition 16. The linear Laplace operator between points therein. They are therefore invariant with Z respect to the subclass of affine transformations which (Lεf)(x) = dµ(y)[Θ(ε − kx − yk leaves these properties unchanged, i.e., x → Dx + s S for x ∈ S with D being a combination of rotation and −δ(x − y)ρ(y; ε)]f(y) (34) isotropic scaling operations and s a translation. This is to be understood in the sense that for a measure h, is a continuous equivalent of the discrete Laplacian ma- h(Dx + s; aε) = h(x; ε) holds, where a is the scaling fac- trix in network theory [1]. We are interested in its eigen- tor. The measures considered here are generally not in- functions f(x) and eigenvalues λ satisfying variant under non-isotropic scaling and shear operations.

(Lεf)(x) = λf(x) (35) III. DISCRETE ESTIMATORS for all x ∈ S. For example, considering an arbitrary S with uni- Given the continuous framework deﬁned above, we form p, one obtains an eigenfunction f(x) = C for some are able to treat the commonly used recurrence network 9

TABLE II: A summary of standard unweighted network estimators for the continuous geometric properties deﬁned in Sec. II. For a detailed discussion, see [1, 2, 14, 63]. SP abbreviates “shortest path”.

Class Name Definition Comments Neighborhood-based ˆ PN Local Degree ki = j=1 Aij 1 ˆ Degree densityρ î = N−1 ki PN ˆ j,k=1 Aij AjkAki ˆ Local clustering coeff. Ci = ˆ ˆ Ci = 0 iff ki < 2. ki(ki−1) PN l=1 AilAjl Matching indexµ îj = ˆ ˆ PN ki+kj − l=1 AilAjl

1 PN Global Edge densityρ ˆ = N(N−1) i,j=1 Aij PN ˆ i,j,k=1 Aij AjkAki Transitivity T = PN i,j,k=1 AkiAkj ˆ 1 P ˆ Global clustering coeff. C = N i Ci 1 P ˆ ˆ 1 ˆ ˆ 2 kikj Aij −h (ki+kj )i ˆ L j>i 2 i,j P Assortativity A = 2 L = Aij is the number of edges, 1 P 1 (kˆ2+kˆ2)A − 1 (kˆ +kˆ ) j>i L j>i 2 i j ij h 2 i j ii,j D 1 ˆ ˆ E 1 P 1 ˆ ˆ 2 (ki + kj ) = L j>i 2 (ki + kj )Aij [41]. i,j Path-based N−1 Local Closenessc î = PN Set lij = N − 1 iff @ path between i, j [64]. j=1 lij 1 PN −1 Local efficiencye î = N−1 j=1 lij N−1−1 PN σˆjk(i) SP betweenness ˆbi = σˆjk shortest paths connect vertices j, k, 2 j,k6=i σˆjk σˆjk(i) of those include i [64], −1 N−1 PN σˆkl(i,j) SP edge betweenness ˆbij = andσ ˆjk(i, j) include i, j. 2 k,l6=i,j σˆkl ˆ Global Average path length L = hlij ii,j Set lij = N − 1 iff @ path between i, j [1]. −1 Global efficiency Eˆ = l−1 ij i,j

Diameter Dˆ = maxi,j (lij )

Radius Rˆ = mini maxj (lij )

quantifiers [6, 14] taken from standard complex network A. Weighted network statistics and node splitting theory [1, 2] as the most straightforward discrete esti- invariant measures mators of the continuous quantities for a finite number of observations N and finite ε. The discrete estimators We may now ask how the estimation of the above de- will be denoted using hats, e.g., the discrete estimator fined continuous geometric properties from a finite data ˆ of continuous average path length L(ε) is L(ε, N) (we set can be improved with respect to the measures from will in the following omit the estimators’ dependency complex network theory that have been used so far for on ε and N to simplify the notation). Their numer- this purpose. One way to go in line with standard es- ical properties have been elaborated in detail in ear- timation theory is node-weighted network statistics, as lier works [6, 13, 14, 26, 30, 33]. The characteristics proposed by Heitzig et al. [49]. For a full application of of these standard measures for discrete and finite com- that theory, weights wi for all vertices i have to be chosen plex networks have also been studied for random geomet- in a suitable way, which we leave as a subject of future ric graphs and more general network models with strong research. But even with constant weights wi ≡ 1, the ax- spatial contraints [50, 52, 53], e.g., the degree distribu- iomatic theory developed in [49] allows us to improve esti- tion [50], network motifs [53], as well as clustering coef- mation by using so-called node splitting invariant (n.s.i.) ficient and degree correlations [52]. versions of network measures to reduce the estimation bias that results from excluding self-loops from the net- Here, we briefly review the estimator’s definitions (Ta- work. Let us illustrate this for the case of continuous ble II). For some specific examples, the estimators will be ε-degree density, ρ(x; ε) = R dµ(y). If x is a vertex, compared to the results theoretically derived from their Bε(x) continuous counterparts in Sec. IV. This will also allow p is approximately constant in Bε(x), and the latter con- ˆ us to gain certain insights into their bias and variance for tains ki additional vertices (see Table II), then ρ(x; ε) ≈ ˆ ˆ∗ finite data sets. p(x)Vol(Bε(x)) ≈ (ki + 1)/N since Bε(x) contains ki = 10

ˆ ˆ∗ ki + 1 out of N vertices. In other words, ki /N is a bet- (which is possible as dρ(ε)/dε > 0 in non-pathological ˆ ter estimator for ρ(x; ε) than ki/N, since the latter has a situations) yields the associated critical threshold bias of ∼ O(1/N). Likewise, the transitivity measure Tˆ (see Table II) can be improved by using instead the n.s.i. −1 zc(d) ˆ ∗ PN + + + PN + + εc(d) = ρ . (39) transitivity T = i,j,k=1 AijAjkAki/ i,j,k=1 AkiAkj, N − 1 + where Aij = Aij + δij = Rij, showing that this approach is also more in line with recurrence plot analysis. This To our best knowledge this is the most useful result avail- would also reduce the bias in the estimation of the tran- able so far for our aim of choosing the recurrence thresh- sitivity dimension that was observed in [26, Fig. 10A]. old ε. However, one should be aware that the results ˆ∗ ˆ ∗ d The measures ki and T are examples of n.s.i. measures of [28] were obtained for the box S = [0, 1] with uni- with unit weights, which can basically be interpreted as form probability density p which is the most commonly variants of the classical measures in which vertices are studied setting in random geometric graph theory. When considered to be linked to themselves [49]. considering general S and p they may be appropriate as a first educated guess for properly selecting ε in line with the guidelines discussed in [26, 30]. Deriving analytical B. Choice of the recurrence threshold ε bounds on zc for such geometries remains an open problem. A careful choice of the recurrence threshold ε is critical for faithfully estimating the continuous recurrence network properties defined above [30]. For too large ε, i.e., on the order of the diameter of S, boundary effects IV. EXAMPLES dominate, the discrete recurrence network used for estimation becomes too dense and is unable to capture the geometry induced by S and p (see Sec. IV A 4). In con- We illustrate the above defined continuous geometric trast, the network’s giant component breaks down for quantities and their estimators for paradigmatic exam- too small ε with a phase transition at the critical value ples by giving closed-form analytical results and relating εc. This obstructs our ability to properly estimate meso- them to numerical evidence from ε-recurrence networks scopic and path-based measures for ε < εc. Therefore we constructed from time series. The focus will be on exam- expect a good performance of the discrete estimators for ples where all quantities of interest can be calculated ei- thresholds just above the critical εc, where much of the ther analytically or semi-analytically (relying on numer- geometric fine structure is still resolved [26]. ical evaluation of some integrals), i.e., possessing smooth The problem of selecting ε therefore reduces to deriv- sets S and density functions p(x). This implies that when ing the percolation threshold εc which is directly related considering the Euclidean norm (which we will use for all to the critical edge density ρc = ρ(εc) of the theory of examples below) and neglecting boundary effects [14, 26], random geometric graphs [51] via Eq. (6). ρc is linked to we obtain the commonly studied critical mean degree zc by zc dΓ(d/2) 1 1 − d 3 1 ρ = . (37) C(x; ε) = 1 − √ 2F1 , ; ; c N − 1 2 πΓ((d + 1)/2) 2 2 2 4 1 1 − d d + 1 d + 3 1 The Erd˝os-Rényi graph is the simplest random network − F , ; ; model [1]. Since any pair of vertices is linked with d + 1 2 1 2 2 2 4 the same probability ρ independent of their distance, =C(ε) = T (ε) it neglects the effects of spatial embedding. Therefore (40) the Erd˝os-Rényi model is inadequate for describing d- dimensional random geometric graphs, and the corresponding critical mean degree zc = 1 [1] turns out to for all transitivity-based measures, where 2F1(·) is the hy- be too low except for the limiting case d → ∞ [28] (see pergeometric function and d the manifold dimension of S. Sec. IV A 4 for an example). Taking into account the ef- A simpler exponential scaling with d can be found for the fects of clustering of vertices induced by the spatial em- supremum metric. Nontrivial transitivity-based proper- bedding [65] yields improved analytical bounds on the ties for fractal sets S and densities p(x) allowing for non- true zc obtained from numerical simulations [66]. Exact integer d, where an analytical calculation of path-based analytical results for arbitrary d are not available so far, measures is problematic, have been treated exhaustively but Dall and Christensen [28] have empirically found the in [26]. scaling law The results given here hold in the limit ε → 0. For simplicity we ignore boundary effects which have z (d) = z (∞) + Ad−γ (38) c c been treated in [14, 26]. In all examples, we use the from extensive numerical simulations, where zc(∞) = 1, parametrization σ1(y, z; x) to compute continuous ε- γ = 1.74(2) and A = 11.78(5). Inverting ρc = ρ(εc) shortest path betweenness. 11

A. One-dimensional chaotic maps and stochastic A Bernoulli map B Logistic map processes

All examples considered in this section are defined on convex sets S embedded on the real axis. Therefore, the geodesic distance of x, y ∈ R reduces to g(x, y) = |x − y|. R Since for one-dimensional S the integral S dy p(y) |x − y|−1 diverges for all p and all x ∈ S, we get E(ε) = 0 and e(x; ε) = ∞ ∀x in all examples of this section. In contrast, the corresponding integral always converges for FIG. 4: (Color online) Continuous ε-average path length L(ε) for (A) the Bernoulli map and (B) the logistic map. Analyt- non-fractal S with d ≥ 2 and general p. ical results are indicated by solid red lines. Estimates Lˆ(ε) have been obtained from ε-recurrence networks constructed from one realization of N = 1, 000 samples, respectively, for 1. Bernoulli map / uniformly distributed noise each map (black dots). Ensemble mean (dashed-dotted black line) and standard deviation (gray band) for different ε have The Bernoulli map xn+1 = (2xn) mod 1 defined on been obtained from an ensemble of 100 realizations of each the interval S = [0, 1) induces the probability density model with initial conditions uniformly distributed in the in- p(x) = 1. This yields terval [0, 1]. Vertical dashed lines indicate the estimated percolation thresholds εc. 3 T (ε) = C(ε) = C(x; ε) = 4 1 x → x − χ on the right side of the equations for the local L(ε) = ε−1 3 measures given above (see also Sec. II F). 2ε c(x; ε) = 1 − 2x + 2x2 b(x) = 2x(1 − x). 3. Logistic map

The same results hold for uniformly distributed noise on We can also give exact analytical solutions for the logis- the interval [0, 1], since S and p(x) are identical to those tic map in the fully chaotic regime, xn+1 = 4xn(1 − xn), of the Bernoulli map (an exemplary calculation for this defined on the interval S = [0, 1]. Using the probability setting is shown in Appendix B). This equality clearly −1p −1 illustrates that recurrence network analysis is purely density p(x) = π x(1 − x) [67] yields geometric and, hence, by design masks out the auto- 3 dependency structure of dynamical systems. Stochastic T (ε) = C(ε) = C(x; ε) = and deterministic dynamics can be distinguished when 4 4 embedding techniques are used prior to recurrence net- −1 L(ε) = 2 ε work analysis [14]. π c(x; ε) = πε2px(1 − x) + (1 − 2x) × √ √ −1 × arccos x − arcsin x 2. Gaussian noise √ √ 8 Im [arcosh ( x)] arcsin ( x) b(x) = . Considering Gaussian noise with zero mean π2 and standard deviation σ, i.e., p(x) = √ 2 2 2 1/ 2πσ exp −x /(2σ ) , on the real axis 4. Comparison to numerical results S = (−∞, +∞) we obtain 3 Within an intermediate range of ε, the continuous ε- T (ε) = C(ε) = C(x; ε) = 4 average path length L(ε) is approximated well by the es- 2σ timators Lˆ(ε) calculated from ε-recurrence networks for L(ε) = √ ε−1 π both the Bernoulli and logistic map (Fig. 4). For small ε ε, the estimator breaks down due to the finite number c(x; ε) = of samples used (finite size effect) after the network’s gi- q 2 2 σ exp − x + xerf √x π 2σ2 2σ ant component decomposes into smaller and smaller dis- ! connected components. The Erd˝os-Rényi approximation 1 x 2 b(x) = 1 − erf √ , yields a critical percolation threshold εc = 1/(2(N−1)) ≈ 2 2σ 5 · 10−4 for both maps using the parameters of Fig. 4, which is one order of magnitude smaller than the nu- x 2 √2 R −t where erf(x) = π 0 e dt is the error function. The merically observed phase transition point (Fig. 4). As results for mean χ 6= 0 can be derived by substituting explained in Sec. III B, this is because the Erd˝os-Rényi 12 model does not account for the effects of spatial em- A Bernoulli map B Logistic map bedding and clustering. To be able to use the relationship of Eq. (39) based on the empirical results of [28] N = 1000 for m = 1, we approximate ρ(ε) = 2ε (Bernoulli map) and ρ(ε) = 8ε artanh(1 − 2ε)/π2 (logistic map) for small −3 ε. This yields εc ≈ 6.4 · 10 for the Bernoulli map −3 and εc ≈ 6.2 · 10 for the logistic map, which is con- sistent with the phase transition points observed numer- C D ically (Fig. 4). The good agreement of predicted and observed phase transition for the Bernoulli map can be N = 2000 explained by the fact that the latter exactly meets the as- sumptions underlying the theory of Dall and Christensen (Sec. III B). These observations indicate that Eq. (39) is indeed useful to derive an educated guess for the proper choice of ε, even for strongly varying probability densities p. Moreover, the phase transition for the logistic map E F occurs at notably larger ε than for the Bernoulli map. N = 5000 Consistently with the results of [66], this indicates that the increased spatial clustering induced by peaks in the density p leads to larger values of the critical mean degree zc and therefore the associated percolation threshold εc. However, the results of [52] suggest that there may be in fact no true phase transition in giant component size for non-uniform p in the limit N → ∞. FIG. 5: (Color online) Continuous ε-closeness c(x; ε) for For large ε, the approximation in Eq. (11) is not valid (A,C,E) the Bernoulli map and (B,D,F) the logistic map. An- anymore and, hence, the discrete estimator breaks down alytical results are indicated by red solid lines. Estimates in this regime. Note that L(εc0 ) = 1 for a critical εc0 . cˆ(x; ε) have been obtained from ε-recurrence networks at ε = 0.01 constructed from single realizations of (A,B) N = 1, 000, Since L(ε) < 1 whereas Lˆ(ε) = 1 for ε > ε 0 , the defini- c (C,D) N = 2, 000 and (E,F) N = 5, 000 samples (all: black tion of the discrete estimator is not meaningful anymore squares). Ensemble mean (black dashed-dotted lines) and for thresholds larger than the critical threshold. For the standard deviation (gray bands) have been calculated as in Bernoulli map we have εc0 = 1/3 and for the logistic map, Fig. 4. The standard deviation is too small to be visible in 2 εc0 = 4/π follows. the plots for N = 5, 000 (E,F). The continuous ε-closeness c(x; ε) is approximated well by the estimatorc ˆ(x; ε) for both the Bernoulli and logistic maps (Fig. 5). However,c ˆ(x; ε) is notably smaller than between vertices and not in the absolute values. the true theoretical value particularly in the center of S at x = 1/2, implying that shortest paths are longer in the empirical ε-recurrence network than expected theoretically. This is clearly a finite size effect as the bias and B. Periodic and quasi-periodic dynamics variance of the estimator clearly decrease for growing N and fixed ε (Fig. 5). 1. Periodic orbit The shape of continuous ε-shortest path betweenness b(x) is approached well by the estimator ˆb(x; ε) for both We analyze next a periodic orbit (gen- maps (Fig. 6). However, there is a large bias that in- eral closed curve) of curve length l embed- creases with the number of samples N, while the variance ded in an m-dimensional phase space, i.e., decreases with growing N. That the estimator ˆb(x; ε) is S = {x ∈ Rm : x = f(s); s ∈ [0, l]; f(0) = f(l)}, with generally smaller than the theoretical value b(x) for all uniform probability density p(x) = 1/l. The geodesic x can be explained by the skipping of vertices due to distance of two points x(s), x(t) along the curve is then the finite ε in the empirical ε-recurrence network. This given by g(x(s), x(t)) = |s − t|. This yields effect is expected to increase for growing N when ε is 3 fixed, since more and more vertices can be skipped along T (ε) = C(ε) = C(x; ε) = a shortest path for the same recurrence radius ε, which 4 also explains the growing bias in this setting. Accord- l L(ε) = ε−1 ingly, the bias decreases for decreasing ε when N is suffi- 4 ciently large regarding the discussion of suitable choices 4ε c(x; ε) = of ε in Sec. III B (Fig. 6G,H). However, the bias is not l a problem in practical situations, because for local mea- 1 b(x) = . sures we are usually only interested in relative differences 4 13

A Bernoulli map B Logistic map ber. The phase space trajectory ﬁlls a 2-torus S = {x = (s, t): s ∈ [0, 2πR], t ∈ [0, 2πr]} uniformly with N = 1000 p(s, t) = p = 1/(4π2Rr). The radii R, r are related to the oscillation’s amplitudes. With the geodesic distance

g((s, t), (s0, t0)) = min (|s − s0|, 2πR − |s − s0|)2 2 1 + min (|t − t0|, 2πr − |t − t0|) 2 C D we obtain

N = 2000 √ 3 3 T (ε) = C(ε) = C(x; ε) = 1 − ≈ 0.5865 4π πε−1 p R L(ε) = 4rR r2 + R2 + 3r3arsinh 12rR r r E F +2R3artanh √ r2 + R2 N = 5000 R −r3artanh √ r2 + R2 R E(ε) = 2πrRε−1 2rartanh √ r2 + R2 √ ! r + r2 + R2 −1 G H +R ln √ . −r + r2 + R2 N = 10000 Because of symmetry, the local path-based measures do not depend on x as for the periodic orbit discussed above and we have c(x; ε) = L(ε)−1 e(x; ε) = E(ε)−1 FIG. 6: (Color online) Continuous ε-shortest path between- 1 ness b(x) for (A,C,E,G) the Bernoulli map and (B,D,F,H) the b(x) = 2 = p. logistic map. Analytical results are indicated by red solid 4π Rr lines. Estimates ˆb(x; ε) have been obtained from ε-recurrence As expected, in the limit r → 0 the average path length networks at ε = 0.01 constructed from single realizations of converges to the value obtained for a circle of radius R −1 (A,B) N = 1, 000, (C,D) N = 2, 000 and (E,F) N = 5, 000 (see above), i.e., limr→0 L(ε) = (πR/2)ε . samples (all: black squares). Panels (G,H) show results for ε = 0.005 and N = 10, 000. Ensemble mean (black dashed- dotted lines) and standard deviation (gray bands) have been C. Higher-dimensional symmetric sets calculated as in Fig. 4. Note that ˆb(x) is the discrete shortest path betweenness of [14], but normalized by its theoretical maximum value (N − 1)(N − 2)/2. The m-dimensional hyperball and hypercube may be viewed as tractable idealizations of higher dimensional attracting sets of dynamical systems (in this section, we set d = m, since the manifold dimension of S is inte- As the periodic orbit is a one-dimensional set ger). Their study highlights that continuous path-based we have E(ε) = 0 and e(x; ε) = ∞ as above. measures may depend sensitively and non-trivially on the For example, a circular orbit of radius R as global geometry of the set. In contrast, their neighbor- generated by a harmonic oscillator with S = hood and transitivity-based counterparts just depend on x ∈ 2 : x = R sin(s/R), x = R cos(s/R); s ∈ [0, 2πR] R 1 2 the dimension m and are therefore identical for the hy- and p(x) = 1/(2πR) gives the above results with perball and hypercube (Eq. (40)). The sets considered l = 2πR. here are convex, hence, g(x, y) = kx − yk2 holds when using the Euclidean norm.

2. Flat 2-torus 1. m-dimensional hyperball Quasi-periodic dynamics is displayed by a system os- cillating with two incommensurable frequencies ω1 and Here we consider the m-dimensional hyperball S = m ω2, i.e., where the ratio ω1/ω2 is not a rational num- S with the uniform probability density p(x) = p = 14

ΕLHΕL Note that the limit εc(0; ε)−1 → 1 for m → ∞ shows æ æ m æ that almost all of the measure µ(S ) of the unit radius æ m 1.2 æ hyperball S is concentrated at its surface for large m. æ For the special case of m = 2 (unit disk with uniform 1.1 æ p(x)), Lew et al. [69] give a nearly closed-form expression æ for the continuous ε-closeness at x(q) = (q, 0), 1.0

−1 1 2 2 2 2 −1 æ c(x(q); ε) = 16(q − 1)K(q ) + 4(q + 7)E(q ) ε , 0.9 9π

0.8 where 0 ≤ q ≤ 1 and the value for arbitrary x ∈ S2 may be obtained after an appropriate rotation. K(m) m æ 2 4 6 8 10 and E(m) are complete elliptic integrals of the first and second kind (see §17.3 in [70]). FIG. 7: Continuous ε-average path length εL(ε) of the hy- For the continuous local ε-efficiency e(0; ε) of the center m m perball S with uniform probability density, obtained from of S we get for m > 1 Eq. (42). Z q −1 2 2 e(0; ε) = ε dx1 . . . dxm p x1 + ··· + xm R m −1 m m m 1/ dx = Vol ( ) = Γ + 1 /π 2 . Following S S S 2 Z 1 Hammersley [68], the r-th moment of the distribution m−1 −1 m = εΩmp dr r r of point-to-point distances kx − yk2 in S is given by 0 m mΓ(m + 1) Γ 1 m + 1 r + 1 = ε. 2 2 2 m − 1 µmr = 2 1 1 1 . (41) Γ 2 m + 2 (m + r)Γ m + 2 r + 1 A somewhat more involved calculation of the continuous Then the continuous ε-average path length is ε-betweenness b(0) of the center of Sm yields (see Ap- −1 L(ε) = µm1ε pendix C) 1 m Γ(m + 1) Γ m + 1 m 2 −1 1 Γ( + 1) = 2 1 1 3 ε , (42) 2 m + 1 Γ m + Γ m + b(0) = = m . (44) 2 2 2 Ωm mπ 2 and all its higher moments are known via Eq. (41) (see m Fig. 7). Some examples for lower-dimensional spheres The high degree of symmetry of S allows us to derive follow: closed-form results for local path-based measures at its  center for many p(x) = p(r, Ω), as long as the probability 2 m = 1 density separates into a radial and an angular part, i.e.,  3  128 −1 p(r, Ω) = p(r)p(Ω).  45 π m = 2  36 εL(ε) = 35 m = 3 .  16384 π−1 m = 4  4725 2. m-dimensional hypercube  800  693 m = 5 Note that the result for m = 1 agrees with the cor- The hypercube S = Km = [0, 1]m with uniform prob- responding one for the Bernoulli map when consider- ability density p(x) = Vol(Km)−1 = 1 is much harder to ing the stretching of the domain by a factor of 2, since treat analytically than the hyperball Sm. Hence, rigorous 1 results are only available for isolated dimensions m and a S = [−1, 1]. In the limit m →√ ∞ the continuous ε- average path length is L(ε) = 2ε−1 (see Fig. 7). We subset of the continuous measures defined above [58, 71– can also derive in closed form an expression for the con- 73]. Solving the resulting general box integrals remains tinuous ε-closeness c(0; ε) of the center of Sm, taking ad- a largely unsolved problem of applied and experimental vantage of the spherical symmetry: mathematics. The following closed-form expressions for the continu- Z q −1 −1 2 2 ous ε-average path length L(ε) are based on the expecta- c(0; ε) = ε dx1 . . . dxm p x1 + ··· + xm m m S tion values for point-to-point distances ∆K (1) = εL(ε) Z 1 −1 m−1 (see Eq. (29)) listed in [73]: = ε Ωmp dr r r. 0  1 m = 1 With the full solid angle in m dimensions Ω =  3 √ √ m  1 m m  2 + 2 + 5 ln 1 + 2 m = 2 mπ 2 /Γ( + 1) this leads to 15 √ √ 2 εL(ε) = 118 2 34 4 . − 21 − 3 π + 21 2 − 7 3 m = 3 m + 1  √ √ c(0; ε) = ε. (43)  +2 ln(1 + 2) + 8 ln( 1+√ 3 ) m 2 15

ΕLHΕL We can now once again use results from Bailey et al. [73] ìà to give some closed forms for small m: 1.2 ìà ìà  1 à æ m = 1 ì  2 √ √ 1.0 ìà æ −1 1 æ εc(0; ε) = 3 ( 2 + ln(1 + 2)) m = 2 ìà æ √ √  1 3 − 1 π + 1 ln(2 + 3) m = 3 0.8 à æ 4 24 2 ì æ ìà æ and 0.6 æ à  ì ∞ m = 1 æ  √ 0.4 ε−1e(0; ε) = 2 ln(1 + 2) m = 2 . ìæà √  1 2 − 4 π + 3 ln(2 + 3) m = 3 m 0 2 4 6 8 10 Further solutions for m = 4, 5 are given in [73]. FIG. 8: (Color online) Continuous ε-average path length m εL(ε) of the hypercube K with uniform probability density, obtained by numerical Monte Carlo integration using Mathe- V. DISCUSSION matica (yellow diamonds). Analytical lower (blue discs) and upper (red squares) bounds from [71] are also shown. We have shown that the definitions of continuous geometric measures provided in this paper are feasible for describing ε-recurrence networks for time series Some numerical results for m = 1,..., 10 are displayed analysis as well as, more generally, random geometric in Fig. 8. Anderssen et al. [71] proved the bounds graphs [28, 50, 51]. Our theoretical framework may v r u r ! readily be generalized to encompass other classes of ran- 1√ 1 u1 3 m ≤ εL ≤ mt 1 + 2 1 − (45) dom networks with spatial constraints (i.e., spatial net- 3 6 3 5m works) [29, 52, 53], e.g., those with an edge length distribution of the form pl(l) ∝ exp(−l/ξ) describing among implying εL → ∞ for m → ∞. This is in contrast to the others the substrate of climate networks [22, 74] (in con- hyperball, where this limit is finite (see above). Using ex- trast to the sharp cutoff pl(l) ∝ Θ(ε − l) for random ge- pectation values for the inverse point-to-point distances ometric graphs). For the neighborhood-based measures, −1 −1 m ∆K (−1) = ε E(ε) , we are able to give the following this generalization can be achieved by substituting terms expressions for the continuous ε-efficiency E(ε): containing the Heaviside function or Bε(x) with suitably  chosen expressions involving pl(l). One possible appli- ∞ m = 1  √ √ cation to real-world spatial networks is computing ex-  4 (1 − 2) + 4 ln(1 + 2) m = 2  3 √ √ pectation values for the characteristics of an ensemble of  2 − 2 π + 2 2 − 4 3 m = 3 spatial random network surrogates to assess which prop- −1  5 3 5 5 (εE(ε)) = √ . erties of a given empirical network can be explained by  +2 ln(1 + 2)  √ pl(l) alone. Additionally, more general metrics could be  +12 ln 1+√ 3  2 used for measuring the distance l between connected ver-  √  −4 ln(2 + 3) tices. Research along these lines may also help to shed light on the specific topology and dynamics of growing Note that as S = K1 = [0, 1], the results for m = 1 spatial complex networks (cf. [8, 75]). agree with the corresponding ones for the Bernoulli map Furthermore, we have demonstrated that the result- for both continuous ε-average path length and efficiency. ing continuous properties can be approximated by esti- m m Further expressions for ∆K (1) and ∆K (−1) for m = mators calculated from empirical ε-recurrence networks 4, 5 are given in [73]. reasonably well, even for relatively small N and large Another object of interest in the theory of box integrals ε. The continuous framework promotes considerable ad- is the integral vances in the theoretical understanding of ε-recurrence- Z network-based time series analysis. Among others, from ˜ η Bm(η) = dxkxk2, (46) the examples of hyperballs and hypercubes in various di- m K mensions m, the claim that path-based measures depend which is related to the continuous ε-closeness c(0; ε) of explicitly on the global geometry of the set S is theo- the origin x = 0 (and, by symmetry, to that of all the retically justified. This is in contrast to the continuous 2m corners of the hypercube) for η = 1 and to the local notions of local and global transitivity, as at least the efficiency e(0; ε) of the same points for η = −1: continuous local ε-clustering coefficient C(x; ε) depends on the local dimensionality of the set S [26]. Along ˜ −1 c(0; ε) = Bm(1) ε (47) these lines, in the future we may gain an understand- e(0; ε) = B˜m(−1)ε. (48) ing of the differing performance of transitivity-based and 16 path-based measures in classifying qualitatively different cause S is path-connected, l(x, y; ε) is finite. We also put behavior of dynamical systems [13, 33]. For example, l(x, x; ε) = 0. Let lij(ε, N) ≥ l(xi, xj; ε) be the network more complex dynamical systems such as the Lorenz and distance between xi and xj in the ε-recurrence network Rösslermodels or noisy dynamical systems, where no constructed from the first N points of a sequence of in- closed-form expression for the invariant probability den- dependent draws from p. One can then prove that for sity p exists, may be studied by estimatingp ˆ from simu- fixed ε, fixed nodes xi, xj, and N → ∞, it has probabil- lated trajectories. ε-recurrence network measures could ity one that lij(ε, N) = l(xi, xj; ε) eventually (i.e., there then be calculated by numerical integration techniques is some N(i, j, ε) so that lij(ε, N) = l(xi, xj; ε) for all relying onp ˆ and the integral expressions given in this N > N(i, j, ε)). This is because for k = l(xi, xj; ε), there paper. Circumventing the computational limitations of is δ > 0 and z0, . . . , zk ∈ S with z0 = x, zk = y, and discrete ε-recurrence network analysis when N → ∞, this ||zi−1 − zi|| < ε − 2δ for i = 1 . . . k, and with probability approach would in principle allow us to approximate the one, the sequence contains points w0, . . . , wk ∈ S with continuous geometric quantities defined above as closely w0 = xi, wk = xj, and ||wi − zi|| < δ for i = 1 . . . k − 1, as desired. so that also ||wi−1 − wi|| < ε for i = 1 . . . k, implying The examples of hyperballs and hypercubes establish lij(ε, N) ≤ k when N > N(i, j, ε) where N(i, j, ε) is the links to some current research problems in probabilis- index of the last of the wi to occur in the sequence. tic geometry and applied mathematics, among others, to Moreover, l(x, y; ε) ≤ g(x, y)/ε + 1 and εl(x, y; ε) ≤ the theory of box integrals [58, 71–73]. Perhaps these g(x, y) if g(x, y) is not an integer multiple of ε. This is highly symmetric model sets could serve to understand because for all δ > 0 and ε0 < ε, there is a path from x to theoretically some qualitative features of path-based ε- y of length ≤ g(x, y) + δ, hence for k = d(g(x, y) + δ)/ε0e recurrence network measures for strange attractors such (dxe is the smallest integer not less than x), there are as the Lorenz or Rösslerattractors. It remains an open z0, . . . , zk ∈ S with z0 = x, zk = y, and ||zi−1 − zi|| ≤ question as to whether it is possible to solve the inte- ε0 < ε for i = 1 . . . k, so that l(x, y; ε) ≤ k. On the grals for continuous path-based measures in the case of other hand, if S is sufficiently well-behaved, one will also self-similar sets S and more complex, potentially also self- have εl(x, y; ε) % g(x, y) for ε → 0. More precisely, similar densities p. assume S is “locally almost convex” in the sense that for The theoretical framework put forward in this paper all L > 1, there is some ε > 0 so that for all x, y ∈ S enables several practical advances, which are particu- with ||x − y|| < ε, we have g(x, y) < Lε. Then for all larly relevant for applications to time series analysis of L > 1, there is some ε > 0 so that εl(x, y; ε) > g(x, y)/L. real-world data. Analytical solutions for continuous ε- Putting all these facts together, we see that εlij(ε, N) is recurrence network measures allow us to assess the bias a plausible estimate of g(x, y). and variance of the discrete estimators from complex network theory that have been used in the literature so far. These insights led to devising improved discrete es- Appendix B: Continuous ε-average path length for timators based on the concept of node-weighted network Bernoulli map and uniformly distributed noise statistics [49]. Furthermore, for the first time we were able to formulate a theoretically motivated criterion for For illustration, we give the detailed calculation of L(ε) the selection of the recurrence threshold ε based on the for the Bernoulli map and, equivalently, uniformly dis- critical percolation threshold ε which for a given system c tributed noise: can be estimated using our theory. Finally, we should note that we now have a compre- Z 1 Z 1 hensive continuous theory for essentially all relevant mea- εL(ε) = dxdy |x − y| sures of ε-recurrence networks. This foundation will 0 0 Z 1 Z 1 Z x help to further increase our understanding as well as = dx dy |x − y| + dy |x − y| strengthen the general confidence in the method of ε- 0 x 0 recurrence network analysis in practical situations, e.g., Z 1 Z 1 Z x the analysis of real-world time series. Our results sug- = dx dy (y − x) + dy (x − y) gest that ε-recurrence network analysis is the simplest 0 x 0 Z 1 1 x! and best understood network-based approach to nonlin- 1 2 1 2 ear time series analysis available so far. = dx y − xy + xy − y 0 2 x 2 0 Z 1 1 1 1 = dx − x − x2 + x2 + x2 − x2 Appendix A: Sketch of proof of Eq. (11) 0 2 2 2 Z 1 1 = dx − x + x2 For ε > 0, we define the continuous ε-distance l(x, y; ε) 0 2 between x 6= y ∈ S to be the smallest integer k > 0 1 1 3 1 2 1 1 such that there are points z0, . . . , zk ∈ S with z0 = x, = x − x + x = . 3 2 2 3 zk = y, and ||zi−1 − zi|| < ε for i = 1 . . . k. Note that be- 0 17

Appendix C: Continuous ε-betweenness for the Acknowledgments center of a hyperball

ZZ Z 1 b(0) = p2 dy dz dt δ(f(t)) This work has been ﬁnancially supported by the Leib- S 0 niz association (project ECONS), the Federal Ministry ZZ Z 1Z 1 2 0 0 m−1 0m−1 0 for Education and Research (BMBF) via the Pots- = p dΩ dΩ drdr r r δ(Ω − Ω ) dam Research Cluster for Georisk Analysis, Environ- 0 0 mental Change and Sustainability (PROGRESS), and Z Z 1 2 = p2 dΩ dr rm−1 IRTG 1740 (DFG). JFD thanks the German National 0 Academic Foundation for ﬁnancial support. We credit 1 Roger Grzondziel and Ciaron Linstead for help with the = p2Ω m m2 IBM iDataPlex Cluster at the Potsdam Institute for Cli- m mate Impact Research. Complex network measures have Γ 2 + 1 1 = m = been calculated using the software package igraph [76]. mπ 2 Ωm

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