Persistent Homology of Weighted Visibility Graph from Fractional Gaussian Noise
PPersistent Homology of Weighted Visibility Graph from Fractional Gaussian Noise
H. Masoomy, B. Askari, M. N. Najafi, ∗ and S. M. S. Movahed † Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
In this paper, we utilize persistent homology technique to examine the topological properties ofthe visibility graph constructed from fractional Gaussian noise (fGn). We develop the weightednatural visibility graph algorithm and the standard network in addition to the global properties inthe context of topology, will be examined. Our results demonstrate that the distribution of eigen-vector and betweenness centralities behave as power-law decay. The scaling exponent of eigenvectorcentrality and the moment of eigenvalue distribution, M n , for n ≥ H , containing the sample size effect. We also focus on persistent homology of k -dimensional topological holes incorporating the filtration of simplicial complexes of associated graph.The dimension of homology group represented by Betti numbers demonstrates a strong dependencyon the Hurst exponent. More precisely, the scaling exponent of the number of k -dimensional topo-logical holes appearing and disappearing at a given threshold, depends on H which is almost notaffected by finite sample size. We show that the distribution function of lifetime for k -dimensionaltopological holes decay exponentially and corresponding slope is an increasing function versus H andmore interestingly, the sample size effect is completely disappeared in this quantity. The persistenceentropy logarithmically grows with the size of visibility graph of system with almost H -dependentprefactors. Keywords: Topological Data Analysis, Persistent Homology, Fractional Gaussian Noise, Weighted NaturalVisibility Graph, Topological Persistence, Persistence Entropy
I. INTRODUCTION
A powerful approach to study different types ofdata sets ranging from point cloud data (PCD), scalarfield to complex network (graph), particularly a high-dimensional data is called topological data analysis(TDA) [1–6]. TDA as an application of algebraic topol-ogy [7–9] and a branch of computational topology [10],analyzes the shape of high-dimensional complex data interms of global features (number of connected compo-nents, loops, voids, etc. ) of topological space underlyingthe data set. In the persistent homology (PH) technique,as of a main part of TDA, the topological approximationof phase space of any type of data sets which is called sim-plicial complex is assigned to the associated data, thentopological invariants are computed.The PH aims to capture topological evolution of dataset by varying scale (parameter), and extracts topolog-ical invariants of data set in each scale summarizingthem in different representations, persistence barcode(PB) [11, 12], persistence diagram (PD) [13, 14], per-sistence landscape (PL) [15], persistence image (PI) [16],persistence surface (PS) and β -curve, which reveal topo-logical information of data set. Being robust to noise, PHis able to show us the essential features of the systemswith high internal degrees of freedom and is capable toclassify underlying data sets [17, 18]. The PH techniquehas attracted much attention due to its vast applicationson analyzing complex networks [19–21]. Also it has been ∗ [email protected] † [email protected] used in various systems (see e.g. [22–31] and referencestherein).There are many algorithms to assign a network (graph)to different types of data sets. As an illustration, the Mapper algorithm constructing the Reeb graphs (topo-logical networks) from high-dimensional PCD [32–35].The visibility graph technique makes a network so-calledvisibility graph (VG) for a typical time-series (one-dimensional scalar field). The idea of VG, being a com-plex network constructed by considering the visibility al-gorithm, proposed by Lacasa et al as a novel way toanalyze time-series in terms of complex networks lan-guage [36]. The associate networks can be examined byvarious methods [1, 37]. The advantage of this idea isthat, one can apply many well-known techniques in net-works and even in TDA such as PH for a time-series,helping for classification, discrimination and looking forexotic features hidden in the underlying time-series whichcould be robust in the presence of noise, trends and irreg-ularities [36–40]. The notion of mapping the time-seriesto a network has been applied on different topics (see[41–51] and references therein).As a model containing correlations tuned by one pa-rameter (the Hurst exponent H ), the fractional Gaussiannoise (fGn) time-series has been intensely investigatedby many methods. To quantify the properties of a givenself-similar data set or a generic series whose power spec-trum behaves as power-law in frequency (wavelength) do-main, many methods have been proposed concerning thetrends and noises which may affect the observed time-series. Many preceding methods are implemented ei-ther in time domain or frequency space. A well-studiedmethod is multi-fractal detrended fluctuation analysis(MFDFA) [52, 53], implemented in various areas (see [54– a r X i v : . [ phy s i c s . d a t a - a n ] J a n
74] and references therein).Beyond the weighted auto-correlation notion, thecross-correlation has also been established and utilizedin various disciplines [75–81]. Taking into account thehigher-order detrended covariance, the multi-fractal de-trended cross-correlation analysis (MFDXA) has been in-troduced [82]. In spite of many advantages brought bymentioned methods, but the impact of more complicatedtrends and finite size effect have not been diminishedcompletely in many previous approaches [83–88]. In or-der to eliminate the effect of trend as much as possible,several robust methods have been proposed [64, 86–89].On the other hand, the finite size of time-series impactson the accurate estimation of Hurst exponent by some ofprevious methods [90].The VG approach which is derived by mapping a time-series into a network is able to represent some interestingresults for a typical fGn data [41, 42, 91]. An alternativemethod to estimate H , can be devoted to graph theoret-ical algorithm [41]. Despite of huge literature, very littleattention has been paid to topological properties of VGassociated with a self-similar series.In this regard, knowing a given time-series belongs toa fGn class, some relevant questions can be raised: i)Does the topological aspect of associated VG depend onthe Hurst exponent? ii) What are the effects of samplesize, trends and irregularity of fGn signal on the topo-logical properties of VG? iii) How is the multifractalityexpressed by persistence homology? Motivated by men-tion questions, we focus on the persistent homology ofweighted visibility graph constructed from a typical fGnby using the filtration process and considering higher-order connections ( k -cliques ; k >
2) in all thresholds(weights) [92, 93], to examine the dependency of relevantresults to H and footprint of finite size effect. We showthat, in contrast to local statistical features considered inthis paper which are less-sensitive to auto-correlation offGns, the topological features are sensitive to the value ofcorresponding Hurst exponent and are size-independent.The contribution of trends and irregularity and even de-termining whether an input data is a fGn signal or notare beyond the scope of current study.This paper is organized as follows: In the next sectionwe present the network concepts to be used in the paper.The Sec. III is devoted to how VGs are obtained for atime-series. The weight functions are introduced in thissection. The numerical results are presented in Sec. IV,which contains two subsections: local statistical proper-ties, and topological properties. We close the paper witha summary and conclusion. II. NETWORK ANALYSIS
In this paper, we aim to analyze the complex networkof the visibility graph constructed from a fractional Gaus-sian noise (fGn), with an emphasis on the topological as-pects. Besides this, we also compute some conventional statistical properties of mentioned signal. Therefore, it isworthy to introduce and present a short review of theseanalyzes, referring the interested readers to the relevantreferences such as [94–96].
A. Statistical Analysis
In this subsection, we introduce some conventionalquantities, would be used in the following sections, witha main focus on various centrality measures. Inspired bysocial network science, the centralities play an importantrole in identifying the key elements in a typical network,such as the most effective agents (the degree centrality orthe eigenvector centrality), the easiest access agent (thecloseness centrality), and the betweenness centrality.Suppose that a network (graph) is represented by G = ( V, E, w ). Here, V ≡ { v i } Ni =1 is node (vertex) set, E = V × V ≡ (cid:110) e ij = ( v i , v j ) (cid:12)(cid:12)(cid:12) v i , v j ∈ V (cid:111) Ni,j =1 is link (edge)set and w : E → R is a weight function (threshold). Sub-sequently, the degree of i th node, v i , is the number ofnodes straightly connected with underlying node by non-zero weight, and it is denoted by k i ≡ (cid:80) j (1 − δ ,w ij ).The degree centrality , c Di , is defined by k i N − , which isapparently related directly to how important the under-lying node is, since it is the number of agents that haveconnection with it. An important function concerningthis quantity is the degree distribution, p ( k ), showing theprobability distribution function for degree of all nodesin the network p ( k ) = 1 N N (cid:88) i =1 δ k,k i (1)The eigenvector centrality , c Ei , is also defined via theeigenvalue equation λc Ei = N (cid:88) j =1 w ij c Ej (2)where λ is the eigenvalue. The maximum value of the λ spectrum, i.e. λ max plays the dominant role in thenetwork properties. The corresponding eigenvectors ofwhich are denoted by c Ei, max showing the importance ofthe nodes. To define the closeness centrality , let us de-note the shortest distance between nodes v i and v j by d ij which is assumed to be N when there is no path con-necting them (disconnected graphs). Then the closenesscentrality is defined by c Ci = N − (cid:80) Nj =1 d ij , (3)and the betweenness centrality is as follows c Bi = 1( N − N − N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 ,k (cid:54) = i,j n jk ( i ) n jk , (4) FIG. 1. Low dimensional simplices: 0-simplex (point), 1-simplex (line segment), 2-simplex (filled triangle), 3-simplex(filled tetrahedron). the n jk is the number of geodesics from v j to v k , and n jk ( i ) is the number of geodesics from v j to v k whichpassing through node v i . Finally, another interestingquantity which enables us to asses the statistical prop-erties of a network is known as clustering coefficient ofnode v i . This shows how the first neighbors of node v i are connected together c CCi = 2!( k − k ! N (cid:88) j =1 N (cid:88) k =1 (1 − δ ,w ij )(1 − δ ,w ik )(1 − δ ,w jk ) , (5)which represents the fraction of available triangles in thenetwork. In the upcoming subsection, we will give a briefintroduction about algebraic topology. B. Topological Analysis: Algebraic Topology
In addition to the conventional statistical analysis, in-troduced in the previous subsection, complex networksexhibit some interesting topological properties. Suchproperties enable us in classifying the underlying net-work in a feasible approach. It may utilize to recognizeexotic features in data sets. Topology is generally refersto the global features in contrast to geometrical invari-ants of underlying objects or sets . Having two spacesrepresented by X and Y , and they have same local (ge-ometric) features if any relevant features are invariantunder congruence . While, mentioned spaces are topo-logically equivalent, if associated features are invariantunder homeomorphisms . In other words, they are home-omorphic. Homology theory plays a crucial role in themathematical description of the relevant building blockof a typical topological space, and reveal the connected-ness of underlying space [94–96]. Based on such proper-ties, for the sake of clarity, we will give a brief review onthe building blocks of algebraic topology which are usefulto set up homology groups.
Simplex: A k -simplex, σ k , is a convex-hull of anygeometrically independent subset, accordingly σ k ≡ [ x , x , ..., x k ] ⊆ R D . By this definition, a 0-simplex isa point, a 1-simplex is a segment of a line, a 2-simplexis a filled triangle, a 3-simplex is a filled tetrahedron andso on (Fig. 1). Face: A l -simplex which is denoted by σ l , is a subsetof k -simplex ( σ l ⊆ σ k ) and it is so-called the l -face of k -simplex. Simplicial complex:
A simplicial complex, ψ , is acollection of simplices such that: any l -face of any k -simplex of a typical complex (0 < l < k ), is a member ofcomplex. In addition, the non-empty intersection of anytwo simplices, σ k and σ m , from complex, is a l -face ofboth simplices. Dimension of a complex is the maximumdimension of all simplices of the complex. According tothe definition of complex, one can define k -ordered sub-collection of complex ψ as followsΣ k ( ψ ) ≡ (cid:110) σ ∈ ψ (cid:12)(cid:12)(cid:12) dim ( σ ) = k (cid:111) Chain:
For a given simplicial complex, a k -chain ( k -dimensional chain) is a linear combination of k -simplicesof ψ , defined by c k ≡ | Σ k ( ψ ) | (cid:88) i =1 a i σ ( i ) k ; σ ( i ) k ∈ Σ k ( ψ )where | Σ k ( ψ ) | corresponds to the cardinality of k -orderedsubcollection of complex and the coefficients, a i s, belonga field, which is usually considered as F = Z ≡ { , } .The collection of all possible k -chains in simplicial com-plex is called k -chain group as C k ( ψ ) ≡ (cid:110) c k (cid:12)(cid:12)(cid:12) c k = | Σ k ( ψ ) | (cid:88) i =1 a i σ ( i ) k ; σ ( i ) k ∈ Σ k ( ψ ) , a i ∈ F (cid:111) Boundary operator:
For the simplices in any dimen-sion, the boundary operator ∂ k is an operator mapping σ k to its boundary according to ∂ k ( σ k ) ≡ k (cid:88) j =0 ( − j [ x , x , ..., x j − , x j +1 , ..., x k ] ⊆ σ k Boundary: A k -chain which is the boundary of a ( k +1)-chain, is called k -boundary, denoted by b k . The k -boundary group is the collection of all k -boundaries incomplex ψB k ( ψ ) ≡ (cid:110) c k ∈ C k ( ψ ) (cid:12)(cid:12)(cid:12) ∃ c k +1 ∈ C k +1 ( ψ ); ∂ k +1 ( c k +1 ) = c k (cid:111) ≡ (cid:110) b ( i ) k (cid:111) | B k ( ψ ) | i ⊆ C k ( ψ ) Cycle: A k -chain that has no boundary, is called a k -cycle denoted by z k as ∂ k ( z k ) = (cid:11) The k -cycle group is defined as the collection of all k -cycles in complex ψZ k ( ψ ) ≡ (cid:110) c k ∈ C k ( ψ ) (cid:12)(cid:12)(cid:12) ∂ k ( c k ) = (cid:11) (cid:111) ≡ (cid:110) z ( i ) k (cid:111) | Z k ( ψ ) | i ⊆ C k ( ψ )Since ”Boundaries have no boundary.”, therefore, wehave FIG. 2. Betti numbers of some topological spaces: point,line, circle, sphere, torus, 2-torus.FIG. 3. Clique complex (right) of a typical network (left). ∂ k ( b k ) = ∂ k ( ∂ k +1 ( c k +1 )) = (cid:11) hence B k ( ψ ) ⊆ Z k ( ψ ) ⊆ C k ( ψ ) Homology group:
The k -homology group is definedby the quotient group of the k -cycles group by the k -boundary group H k ( ψ ) ≡ Z k ( ψ ) /B k ( ψ )The k th Betti number of a simplicial complex, denotedby β k ( ψ ) , is a topological invariant which counts numberof k -homology class corresponding to the number of k -dimensional holes of complex ψ (Fig. 2) β k ( ψ ) ≡ dim ( H k ( ψ )) Clique (flag) simplicial complex of an unweighted(binary) network, G = ( V, E, w ∈ { , } ), is a simplicialcomplex, denoted by ψ ( G ) , such that any k -simplex ofeach dimension in complex corresponds to a ( k +1)-cliquein the network and vice versa (Fig. 3).A binary network is a topological tree , if and only if,it is topologically holeless in all dimensions. Namely, theassociated clique simplicial complex has following prop-erty β k ( ψ ( G ) ) = min ( β k ) = (cid:26) k = 10 ; k > etc. ) when mapped to a weighted simplicial complex are worked out topologically in terms of the parame-ters present inherently in the original data, e.g. theweight of links in a weighted network, or the pairwisedistance between data points in point cloud data. Moreprecisely, TDA maps parameter-dependent data, X ( w ),(where w is a typical parameter, like the threshold valuefor any weighted network) to a weighted simplicial com-plex, ψ X ( w ). Such approach produces the chain group,the cycle group, the boundary group, the homology groupand particularly, the k th Betti number [7, 10].Persistent homology (PH) as a powerful tool of TDA,examines the creation (birth) and destruction (death) oftopological invariants associated with homology classesduring a mathematical process called filtration [94]. Infact, filtration, φ , is a nested sequence of weighted com-plex ψ ( w ) which any complex with a distinct weight is asubcollection of any complex with higher weight φ ( ψ ( w )) ≡ (cid:16) ψ ( w ) (cid:12)(cid:12)(cid:12) ∀ w (cid:48) < w (cid:48)(cid:48) : ψ ( w (cid:48) ) ⊆ ψ ( w (cid:48)(cid:48) ) (cid:17) w max w min (6)More precisely, PH technique enumerates k th Betti num-ber of any subcomplex in φ and assigns an ordered tuple w ( h k ) ≡ ( w ( h k )birth , w ( h k )death ) to existed k -dimensional topolog-ical hole. Here w ( h k )birth and w ( h k )death are the thresholds forwhich, h k appears (birth) and disappears (death) respec-tively. Since w ( h k )birth < w ( h k )death , we can define the positive-value quantity (cid:96) ( h k ) ≡ w ( h k )death − w ( h k )birth as persistency (life-time) of k -dimensional hole. Persistence barcode (PB)or equivalently persistence diagram (PD) are the famousrepresentations of PH. As an illustration, k -dimensionalpersistence diagram of weighted complex ψ ( w ) is a multi-set P D k (cid:16) φ ( ψ ( w )) (cid:17) ≡ ( M , N ), where M ≡ (cid:110) w ( h k ) (cid:12)(cid:12)(cid:12) h k ∈ H k ( ψ ( w )) , ψ ( w ) ∈ φ (cid:111) and N : M → N is the countfunction. Inspired by Shannon entropy for a typical stateprobability, one can define the persistence entropy (PE)of k th PD (PB). To this end, we construct the probabilityfor lifetime of homology classes as p ( (cid:96) ( h k ) ) ≡ (cid:96) ( h k ) L ; L ≡ (cid:88) w ( hk ) ∈M ( P D k ) (cid:96) ( h k ) Therefore, the PE for k -dimensional persistent homologyis defined by [97–99]: P E k = − (cid:88) w ( hk ) ∈M ( P D k ) p ( (cid:96) ( h k ) ) log p ( (cid:96) ( h k ) ) (7)Relying on previous quantities, we try to characterize thesynthetic fGn series. III. VISIBILITY GRAPH
Among growing applicability of complex networksin many fields and interdisciplinary branches in sci-ence [100], a technique has been suggested which con-verts a time-series to a network, so-called visibility graph(VG) [36]. Generally, suppose { x } : { x ( t i ) , i = 1 , ..., N } represents a real-valued time-series. One can constructa network, so-called visibility graph, denoted by G =( V, E, w ), the V ≡ { v i } Ni =1 is again node (vertex) set,and E is link (edge) set. The VG is defined by using thebijection as follows f : V ≡ (cid:110) v i (cid:111) Ni =1 ↔ T ≡ (cid:16) t i (cid:17) Ni =1 ; f ( v i ) = t i (8)and the connections are constructed according to visi-bility condition between the nodes, i.e. the nodes v i and v j are connected if the node v j is visible from thenode v i and vice versa, therefore the resulting graph isundirected (for more details on the properties of VGs,see [36]). In general, there are two ways to construct anetwork (graph) from a time-series: the horizontal visi-bility graph (HVG) [39, 101, 102] and the natural visi-bility graph (NVG) [103–105], the former is more sparsethan the latter case and in this work we focus on theNVG. In Fig. 4, we show how a HVG (left panel) and aNVG (right panel) for a synthetic fGn series can be con- structed. In a binary set up, the corresponding visibilitygraph chooses the range of the weights from a binary set, w ( B ) : E → Z ≡ { , } , e.g. for a binary NVG (BNVG)the weight function can be written according to followingrelation w ( BN ) ij ≡ (cid:12)(cid:12)(cid:12) f ( v i ) − f ( v j ) (cid:12)(cid:12)(cid:12) = 1 j − (cid:89) k = i +1 Θ (cid:16) s ij − s ik (cid:17) ; (cid:12)(cid:12)(cid:12) f ( v i ) − f ( v j ) (cid:12)(cid:12)(cid:12) > s ij ≡ x ( f ( v j )) − x ( f ( v i )) f ( v j ) − f ( v i ) .The argument of the Θ function being positive guaranteesthat the node v j is visible from the node v i and vice versa.Since the edge in a BNVG has the weight 0 or 1, con-sequently, making it unsuitable for continuous filtering.Instead, we suggest the weighted version of the natu-ral visibility graph (WNVG), by considering the weightfunction as follows w ( W N ) ij ≡ (cid:12)(cid:12)(cid:12) s ij (cid:12)(cid:12)(cid:12) ; (cid:12)(cid:12)(cid:12) f ( v i ) − f ( v j ) (cid:12)(cid:12)(cid:12) = 1 (cid:16) j − (cid:89) k = i +1 Θ (cid:16) s ij − s ik (cid:17)(cid:104) s ij − s ik (cid:105)(cid:17) / ( j − i − ; (cid:12)(cid:12)(cid:12) f ( v i ) − f ( v j ) (cid:12)(cid:12)(cid:12) > j from i and viceversa”, i.e. the more distinguishable the data points arein the original time-series, the higher the correspondingweight are in the constructed network. The term j − i − is necessary to make the weights reasonable numbers forcomparison reasons. In the absence of this exponent, themore the distance between the nodes are, the higher thecorresponding weights are. For both statistical and thetopological analysis, we use this weight function whichadmits continuous filtering. A. Synthetic Fractional Gaussian Noise
In order to modeling the stochastic fractal processes,Mandelbrot and Van Ness introduced the fractionalBrownian motion (fBm) and fractional Gaussian noise(fGn) [106]. The mathematical generalization of the clas-sical random walk and Brownian motion is given by thetheory of fBm [106–108]. A 1-dimensional fBm is rep-resented by B ≡ { B ( t ) : t ≥ } , with power-law vari-ance is the fractional Brownian motion (fBm), for which V ar ( B ( at )) (cid:44) V ar ( a H B ( t )) = a H V ar ( B ( t )), where H ∈ (0 ,
1) is called Hurst exponent. For this randomforce, the Markov property and the stationarity are vi-olated (note that when we have domain Markov prop-erty, stationarity and continuity for a time-series, then itshould be proportional to a 1-dimensional Brownian mo-tion). A model for generating fBm (denoted by B H toemphasis on its Hurst exponent H ) is a generalization ofthe Brownian motion which is non-stationary and non-Markov [109, 110], is given by the Holmgren-Riemann-Liouville fractional integral B H ( t ) = 1Γ( H + ) t (cid:90) ( t − s ) H − dB ( s ) (11)where Γ is Gamma function, d B ( s ) ≡ B ( s + d s ) − B ( s )is the increment of 1-dimensional Brownian motion andit has the following covariance: (cid:104) B H ( t ) B H ( s ) (cid:105) = σ (cid:16) | t | H + | s | H − | t − s | H (cid:17) , (12)where σ ≡ (cid:104) B (0) (cid:105) and also (cid:104) B H (cid:105) = 0. The com-mon fBm is retrieved by setting H = . The incre-ments, x H ( t ) = δ t ( B H ( t + δt ) − B H ( t )) are known asfractional Gaussian noise (fGn). For H > ( H < ) thecorresponding fGn is positively (negatively) correlated,called the superdiffusive (subdiffusive) regime. Through-out this paper, we simulated various fGn with different t . . . . . . x ( t ) t . . . . . . x ( t ) FIG. 4. The schematic representation of making network for a typical data set. The NVG (right panel) and HVG (left panel)of a synthetic fGn with H = 0 . N = 32. self-similar exponents and size. To reduce any bias innominal Hurst exponent as much as possible, all relevantresults are determined by ensemble average over 10 re-alizations generated by our computational algorithm foreach H . In addition, we have computed the value of H associated with each synthetic data set by our code basedon DFA method [53]. IV. RESULTS
Here we focus on the statistical (next subsection) andtopological (the other subsection) properties of the VGsconstructed from the fGn time-series, and investigatetheir behavior with respect to the Hurst exponent H .The networks of sizes N = 2 , , , , and 2 (that a desktop with 128 GB memory is capable to per-form matrix operations) are considered and the ensembleaverages are performed over 10 realizations. The Pythontoolbox “NetworkX”[111] is employed for the matrix op-erations on the graphs. In the topological analysis weespecially focus on the Betti-0 (represented by the β defined as the number of connected components of thenetwork) and Betti-1 (represented by β defined as thenumber of loops) features, which are extracted by using”Dionysus” Python package [112]. The persistence statis-tics, containing the lifetime (the interval between birthand death) of the topological features, and its Shannonentropy are also analyzed.Each exponent has been estimated by Bayesian statis-tics accordingly, the {D} and { Υ } reveal the data andmodel free parameters, respectively. The posterior func-tion is defined by P (Υ |D ) = L ( D| Υ) P (Υ) (cid:82) L ( D| Υ) P (Υ) d Υ (13) where L is the likelihood and P (Υ) is the prior probabil-ity function containing all information concerning modelparameters. Here we adopt the top-hat function forprior function whose window’s size depends on expectedrange of corresponding exponent. Taking into accountthe central limit theorem, the functional form of like-lihood becomes multivariate Gaussian, i.e. L ( D| Υ) ∼ exp( − χ / χ for determining the best-fit valuefor the scaling exponent reads as χ (Υ) ≡ ∆ † .C − . ∆ (14)where ∆ is a column vector whose elements are deter-mined by difference between computed value and theo-retical form for each measure and C is the correspondingcovariance matrix. Finally, the best fit value of consider-ing exponent is computed by maximizing the likelihoodprobability distribution and associated error-bar is givenby 68 .
3% = (cid:90) + σ Υ − σ Υ L ( D| Υ) d Υ (15)Subsequently, we report the best value of the scaling ex-ponent at a 1 σ confidence interval as Υ + σ Υ − σ Υ . A. Local Statistical Properties
By local properties, we mean the properties thatare node-dependent, and are not necessarily globallydefined. It has been confirmed that, the distributionfunction of the node degree of VGs of the fBms andfGns is power-law P ( k ) ∝ k − γ with the exponent γ ( H ) = 3 − H and γ ( H ) = 5 − H , respectively[36, 41]. In this subsection, we perform our calculationsfor the WNVG, introduced for the first time in this paper. − − − c E − − − − − p ( c E ) H = H = H = H = H = H = H = H = H = . . . . . . . . . H . . . . . . . . . γ c E N = N = N = N = N = N = − − − c B − − − − − − p ( c B ) H = H = H = H = H = H = H = H = H = . . . . . . . H . . . . . . . γ c B N = N = N = N = N = N = FIG. 5. Probability distribution function of local features of WNVGs constructed from fGns for various Hurst exponent. Theupper left panel is p ( c E ) while the lower left panel shows the probability distribution of betweenness centrality for N = 2 .The right panels illustrate the corresponding scaling exponents for the proper range indicated by dashed line as a function ofHurst exponent for various sizes. The probability distribution function of eigenvector ( c E ) and betweenness ( c B ) centralities are indicated inFig. 5, in terms of H for WNVGs. The upper andlower right panels in this plot show the exponents of p ( y ) ∼ y − γ y , where y ≡ c E , c B computed for the properrange in which the probability distribution function re-veals the scaling behavior (noticed by dashed-dot line).The γ c E is almost constant for the negatively correlatedregime (0 < H < ) and is a decreasing function of H for the positively correlated regime ( < H < γ c B is however constant for all H values, showing thatthe betweenness centrality is not affected by changing H .Also Fig. 6 illustrates the probability distributionfunction of closeness centrality ( c C ) and eigenvalue forvarious H exponent for WNVGs with different sizes.As shown in this plot, the p ( c C ) does not depend on H (upper left panel of Fig. 6). The full spectrum ofthe eigenvalues (Eq. (2)) is illustrated in the lower left panel of Fig. 6. It shows the activity, or equivalentlythe strength of the weights of the network [113]. We seethat the impact of H is changing the range of spectrumand by increasing Hurst exponent the range of spectrumfor WNVGs become tight. This phenomenon can beunderstood recalling that, as a well-known fact, correla-tions (obtained by increasing H ) smooths the underlyingtime-series, causing the corresponding network has morelink with low weight. For more smoothed time-series, thetypical slopes for associated WNVG become low, leadingto lower weights according to Eq. (10), and equivalentlymaking shorter range for the distribution of λ s. Theright panels in Fig. 6 indicate different moments for p ( c C ) (upper panel) and p ( λ ) (lower panel). The solidand dashed lines correspond to M and M , respectively.Let us summary the results of this section: - The distribution of the eigenvector and betweennesscentralities behave as power-law. The scaling exponent .
10 0 .
12 0 .
14 0 .
16 0 .
18 0 .
20 0 .
22 0 .
24 0 . c C . . . . . p ( c C ) H = H = H = H = H = H = H = H = H = . . . . . . . H − . . . . . . . M n ( c C ) N = N = N = N = N = N = − − λ − − − − p ( λ ) H = H = H = H = H = H = H = H = H = . . . . . . . . . H . . . . . . . . . M n ( λ ) N = N = N = N = N = N = FIG. 6. Probability distribution function of local features of WNVGs constructed from fGns for various Hurst exponent. Theupper left panel is p ( c C ) while the lower left panel shows the probability distribution of eigenvalue for N = 2 . The rightpanels illustrate the corresponding moments, M n , for n = 2 (solid line) and n = 3 (dashed line) versus H . Different symbolsare taken for various sizes. for p ( c B ) is almost independent of H , while the γ c E behaves as decreasing function in terms of Hurst expo-nent. Precisely, for H < . H and it decreasesby increasing Hurst exponent for H > . - Increasing correlation (i.e. increasing H ) makes thetime-series smooth, leading to more dense networks andshifts the weights to lower values, making the width ofthe distribution function of the eigenvalues smaller. - Fig. 5, confirm that the local observables are weaklydependent on the value of H , but in the next subsection,we will show that the dependence of the topological ob-servables on H are more considerable and more interest-ingly, the impact of sample size is almost diminished. B. Topological Properties
In this subsection, we rely on the statistics of topolog-ical measures to examine the structures embedded in thenetworks constructed by visibility graph mapping. Forany given value of Hurst exponent, the associated BN-VGs of fGns are topological tree, therefore, the BNVGs,are topologically equivalent to each others. In anotherword, various BNVGs are homeomorphic for different H according to homeomorphism theorem. Consequently, weonly focus on topological aspects of WNVGs.Generally, in the homology theory, the computable al-gebraic invariants of topological space are introduced.The Betti numbers are among the topological invariantscounting the number of k -homology class. Such quan-tities represent the number of k -dimensional topologicalholes of associated simplicial complex produced by dif-ferent methods such as clique simplicial complex strat- − − w . . . . . . . . | Σ | / N H = H = H = H = H = H = H = H = H = − w . . . . . . . | Σ | / N H = H = H = H = H = H = H = H = H = FIG. 7. Statistics of weighted clique complex simplices. The number of 1-dimensional (left panel) and 2-dimensional (rightpanel) simplices of weighted simplicial complex of WNVGs associated with fGns computed at the threshold. Different symbolsare devoted to various Hurst exponents. egy. In principle, the
Betti numbers can be modifiedand become as a more simple measures known as
Eulercharacteristics . Our purpose here is that, by computingtopological invariants of underlying sets we attempt todistinguish between various series. To this end, we carryout the filtration approach on the WNVGs produced foreach synthetic fGn and then, we will compute the setof
Betti numbers , β k ( k = 0 corresponds to connectedcomponents, k = 1 is associated with loops and so on).The filtration at (continuous) weight, w , is simply per-formed as follows: we connect nodes v i and v j if w ij < w .It is worth noting that all values reported for relevantquantities have been determined by ensemble average asmentioned before.For the first step, we compute the normalized numberof simplices of weighted clique simplicial complex at thethreshold for the WNVGs associated with fGns. Fig. 7shows the normalized numbers of links (1-simplices), | Σ | , and triangles (2-simplices), | Σ | , for different Hurstexponents. By increasing H , the abundance of links andtriangles grow at low thresholds, on the contrary, by go-ing to the high threshold regime, mentioned statistics de-crease under the same action. This result is compatiblewith our expectation for WNVG of a typical fGn con-structed by Eq.(10). For low H , the underlying signalshas more fluctuations leading to have more roughness,consequently, the abundance of weights effectively movesto high w , while for high value of Hurst exponent, thesignal achieves more smoothed fluctuations yielding toget higher abundance of links with low weights. As de-picted in Fig. 7, the threshold for which the | Σ | and | Σ | reach to their maximum is weakly dependent to Hurstexponent. Here a question arises: which characteristicthreshold can separates “low” weights from the “high”one in the underlying network?. To answer this ques-tion, we look at lines for different H crossing each other more and less in a same point, and we called it as w ∗ .At mentioned threshold, the abundance of links or tri-angles with w < w ∗ increase by increasing H , and onthe contrary, for w > w ∗ regime, the abundance of linksor triangles are reducing when the value of Hurst expo-nent increases. At this threshold, the value of | Σ | ( w ∗ )is nearly independent to H , while for | Σ | , we have asmall range of w , such crossing happens. Nevertheless,we interpret w ∗ as the crossover point, the scale whichseparates the low and high weights.Now, we are going to evaluate topological measures.The upper panel of Fig. 8 indicates the β (the num-ber of connected components) as a function of filtrationparameter (threshold) for clique complexes of WNVGsproduced for various time-series with different values ofHurst exponent. To reduce the effect of sample size, wedivide the Betti numbers by the samples size and calledit as normalized Betti numbers . By increasing the valueof Hurst exponent, the normalized β versus thresholddecrease. Namely, the number of connected componentsin the network decreases with increasing H (the graphsbecome more steep). Interestingly, the slope of normal-ized β as a function of w is different for different H ,consequently, the WNVGs of fGn sets reaches to con-nected regime (path-connected), β = 1, (in the upperleft panel of Fig. 8, we take N = 1024)) by differentrates and at different thresholds, w . The upper rightpanel of Fig. 8 represents the value of w as a function ofHurst exponent for different networks size. As depictedin mentioned panel, the value of w behaves as a sam-ple size dependent quantity and by increasing H , suchdependency becomes negligible.We define the β (birth) k ( w ) and β (death) k ( w ) as the num-ber of k -dimensional holes that born and die at a giventhreshold, w , respectively. It turns out that β (birth)0 = 00 w − − − β / N H = H = H = H = H = H = H = H = H = . . . . . . . . . H w N = N = N = N = N = N = w − − − − − β ( d e a t h ) / N H = H = H = H = H = H = H = H = H = . . . . . . . . . H . . . . . . α ( d e a t h ) N = N = N = N = N = N = FIG. 8. Upper left panel illustrates the normalized β -curve for clique complexes of WNVGs associated with fGns of variousHurst exponent versus threshold. The upper right panel indicates the w as a function of H . Lower left panel corresponds tothe normalized β (death)0 as a function of w in semi-log scale (see the text) for various Hurst exponent as a function of threshold.Lower right panel indicates the value of α (death)0 as a function of Hurst exponent. for w >
0, since at w = 0 the underlying network has N connected components, therefore, all connected com-ponents born at w = 0. The lower left panel of Fig. 8illustrates the β (death)0 /N as a function of w in the semi-log plot. As shown in this plot, one of the proper fittingfunction to describe the normalized β (death)0 is given by β (death)0 ( w ) ∼ exp (cid:16) − α (death)0 w (cid:17) for 2 (cid:46) w (cid:46) w ,where the value of w max depends on the H value. The α (death)0 depends on Hurst exponent as increasing func-tion and it behaves as an almost independent functionon size of series (lower right panel of Fig. 8).The upper left panel of Fig. 9 is devoted to β /N forvarious Hurst exponents. As we expect, for trivial thresh-old, w = 0, we have N connected components ( β = N )and therefore the number of loops is identically zero. Byincreasing the threshold, the higher value of Hurst expo-nent leads to more rapid in the increasing rate of β . On the other hand, for the high enough value of threshold,again the underlying data set behaves as a topologicaltree without any topological loops. Therefore, the nor-malized β goes asymptotically to zero and such descend-ing is more rapid for higher H . We also determine thelowest non-trivial threshold for which, there is no loopin underlying network denoted by w , and depict thisthreshold versus H for different samples size in the upperright panel of Fig. 9. The w is also size-dependent quan-tity. Comparing the β /N and β /N demonstrate thatthe by increasing threshold value, the WNVGs of fGn se-ries reach to loop-less regime ( β = 0) before appearingthe connected regime (for which β = 1), irrespective toHurst exponent, i.e. w ( H ) > w ( H ). The quantities β (birth)1 ∝ exp (cid:104) − α (birth)1 w (cid:105) , β (death)1 ∝ exp (cid:104) − α (death)1 w (cid:105) in semi-log plots versus thresholds and correspondingslopes are illustrated in the middle and the lower pan-els of Fig. 9, respectively. The α (birth)1 and α (death)1 are1 . . . . . . . w . . . . . . . . β / N H = H = H = H = H = H = H = H = H = . . . . . . . . . H . . . . . w N = N = N = N = N = N = . . . . . . w − − − − − β ( b i r t h ) / N H = H = H = H = H = H = H = H = H = . . . . . . . . . H . . . . . . . α ( b i rt h ) N = N = N = N = N = N = . . . . . . . . . w − − − − − β ( d e a t h ) / N H = H = H = H = H = H = H = H = H = . . . . . . . . . H . . . . . . α ( d e a t h ) N = N = N = N = N = N = FIG. 9. Upper left panel shows the normalized β -curve for clique complexes of WNVGs associated with fGns of variousHurst exponent versus threshold, while, upper right panel indicates the w as a function of H for different length of data set.The distribution of persistence diagram versus birth-axis as a function of threshold. The middle right panel represents thecorresponding coefficient of β (birth)1 in terms of threshold known as α (birth)1 as a function of Hurst exponent. The lower panelsare the same as middle just for dying 1-hole statistics. In this plot for computing β , we took N = 1024. . . . . . . . ℓ − − − − − p ( ℓ ) H = H = H = H = H = H = H = H = H = . . . . . . . . . H α ( li f e t i m e ) N = N = N = N = N = N = FIG. 10. The probability distribution of lifetime for 1-dimensional holes is depicted in left panel. Different symbols correspondto various H . This diagram has been obtained by doing ensemble average. Right panel is associated with the α (lifetime)1 exponentfor different sizes represented by different symbols. . . . . . . β /N . . . . . . . . β / N H = H = H = H = H = H = H = H = H = .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . w × | Σ | × | Σ | β × β FIG. 11. Left panel illustrates the trajectory of β -vector of weighted clique simplicial complex derived by filtration process ofWNVG associated with fGn with different values of Hurst exponent. Right panel indicates the number of 1-simplices (links),number of 2-simplices (triangles), number of 0-holes (connected components) and number of 1-holes (loops) for fGn series with H = 0 .
5. We considered N = 1024. For convenient, we multiplied the | Σ | , | Σ | and β by factor 10. almost size-independent and they grow by increasing H .Another interesting property to asses is the probabilitydistribution of lifetime for 1-holes which is difference be-tween the death and birth thresholds of a typical measurein a 1-homology class. The left panel of Fig. 10 showsthe probability distribution of topological 1-dimensionalholes lifetime, (cid:96) , for various synthetic data sets with dif-ferent values for Hurst exponent. Our results confirmthat p ( (cid:96) ) ∝ exp (cid:104) − α (lifetime)1 (cid:96) (cid:105) . The H -dependency of α (lifetime)1 for various systems sizes is depicted in the rightpanel of Fig. 10. This result confirms that, α (lifetime)1 can be considered as a robust measure for determining the Hurst exponent of fGn signal which is not affectedby sample size even compared to α (death)0 , α (birth)1 and α (death)1 .The trajectory of β -vector ( β -curve) of weighted cliquesimplicial complex of WNVGs associated with fGns isalso a sensitive criterion to reveal H -dependency. Theoverall behavior of mentioned trajectory does not dependon sample size during filtration process (the left panel ofFig. 11). By increasing H the peak positions move toleft showing that higher H values favor smaller β s andhigher β s as discussed above.To make more complete our investigation, regardingthe comparison of different topological measures, we de-3 w birth w d e a t h PD PD w PB PB w birth w d e a t h PD PD w PB PB w birth w d e a t h PD PD w PB PB FIG. 12. From left to right are the persistence diagram and persistence barcode (inset) of weighted clique complex of WNVGcorresponding to fGn with H = 0 . H = 0 . H = 0 . pict | Σ | (1-simplex), | Σ | (2-simplex), β (connectedcomponents) and β (loops) as a function of threshold,for H = 0 .
5, in the right panel of Fig. 11. Each symbolshas been computed by ensemble average on different sam-ple size. The maximum value of | Σ | and β take placealmost at the same threshold, while the peak of | Σ | isdifferent, taking place at w which is approximately halfthe one for | Σ | and β . We verify that mentioned treat-ments are almost independent to Hurst exponent.Fig. 12 indicates the persistence diagram (PD) andpersistence barcode (PB) for three type of fGn signals for H = 0 . H = 0 . H = 0 . w birth , w death ) of a k -dimensionalhole. As expected, all 0-holes born in w birth = 0, andalso always w death > w birth . In the barcode representa-tion (inset plot), the horizontal lines (blue lines for k = 0and orange lines for k = 1) start and end on the thresholdvalues that k -dimensional holes born and die respectively.The associated persistence entropies (PEs) defined byEq. (7) are obtained using the persistence diagram. Theupper and middle panels of Fig. 13 depict the P E and P E , as a function of sample size in semi-log scale, re-spectively. We compute persistence entropy for all avail-able Hurts exponent values represented by different sym-bols. Our results demonstrate that P E k = A k ( H ) log N for k = 0 ,
1. The behavior of pre-factor, A k versus H isrepresented in the lower panel of Fig. 13. The A is al-most an increasing function versus Hurst exponent, whilethe A becomes constant. V. SUMMARY AND CONCLUDING REMARKS
In this paper, we used the persistent homology tech-nique to examine the visibility graph (VG) constructedfrom fractional Gaussian noise (fGn), characterized byHurst exponent H . To this end, we developed a methodto derive weighted visibility graph associated with a time-series. The statistical properties of the model were ana-lyzed using the standard network measures. Our resultsrevealed power-law behaviors for probability distributionfunction of eigenvector and betweenness centeralities inthe proper range of corresponding quantities (left panelsof Fig. 5). The scaling exponent of betweenness centeral-ity did not depend on H , while the γ c E depends on Hurtsexponent for H > . closeness cen-trality and associated moments were independent from H . Increasing the correlation results in to have moredense networks and shifts the weights to lower values,making the width of the distribution function of theeigenvalues to be more tight. The M n ( λ ) behaved as H -dependent quantity. The higher order of momentsshowed a weak dependency on sample size (Fig. 6).In the second part, we considered the topological prop-erties of synthetic fGn. The normalized number of sim-plices (1-simplex and 2-simplex) for data sets with dif-ferent H the abundance of links and triangles grow atlow thresholds when we increase the H . The thresh-old for which the | Σ | and | Σ | reach to their maximumis weakly dependent to Hurst exponent. We also de-fined a characteristic threshold for separating “low” from“high” weights. The value of | Σ | ( w ∗ ) was nearly in-dependent on H , while for | Σ | , we has a small rangeof w , such crossing happens. The decreasing rate ofBetti-0 which is a representative for the number of con-nected components with respect to threshold increase4by increasing H . The threshold value for which theunderlying WNVG reaches to connected regime (path-connected), w indicated a H -dependency. Our resultconfirmed that, the value of α (death)0 which is the coef-ficient of β (death)0 ( w ) ∼ exp (cid:16) − α (death)0 w (cid:17) depends onHurts exponent and interestingly it is almost size inde-pendent (Fig. 8). The statistics of 1-holes analyzed andwe showed that the w showing vanishing threshold for β was H -dependent and it contained the sample size ef-fect. The α (birth)1 and α (death)1 revealed the proper criteriafor measuring Hurst exponent (Fig. 9).The probability distribution of lifetime of 1-hole con-firmed that the corresponding coefficient represented by α (lifetime)1 is an increasing function versus H emphasizingthat, the sample size effect is completely diminished inthis quantity. Consequently, for a self-similar time-seriesin the absent of trends, the α (lifetime)1 can be a reliablemeasure for estimating Hurst exponent even for smallsample size irrespective to value of H (Fig. 10).The persistence pairs (PPs) in persistence diagram(PD) and persistence barcode (PB) for weighted cliquecomplex of WNVG which are indicators of persistenthomology have been computed and by increasing thevalue of Hurst exponent are shrink to origin of coordinate(Fig. 12). We also computed persistence entropies (PEs)for 0-homology and 1-homology groups. Both quantitiesdepend on sample size as expected and the correspondingslopes in semi-log scale were almost H -dependent (Fig.13).Finally, we emphasize that, the exponents of the local(statistical) observables depend weakly on H , whereasthe exponents of global (topological) observables are al-most strongly H -dependent. The footprint of sample sizeon the α (lifetime)1 is completely diminished.In this paper, we have not verified that whether thepersistent homology (PH) technique is capable to recog-nize the underlying time-series is a self-similar set or not.Indeed, this purpose is beyond the scope of this paper.In addition, it could be interesting to examine the effectof trends and irregularity which may occur in the widerange of events in the nature in the context of TDA andmore precisely via persistent homology approach and weleft them for future research. Above analysis can be doneon different phenomena ranging from cosmology, astro-physics, economy to biology [73, 114–117] Acknowledgment:
The authors are really gratefulto Marco Piangerelli for his notice on TDA as startingpoint in this research. The numerical simulations werecarried out on the computing cluster of the University ofMohaghegh Ardabili. N . . . . . . . . . P E log( N ) H = H = H = H = H = H = H = H = H = N P E log( N )
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