Horizon Visibility Graphs and Time Series Merge Trees are Dual
HHorizon Visibility Graphs and Time Series Merge Trees are Dual
Colin Stephen
Coventry University, UK ∗ (Dated: June 24, 2019)In this paper we introduce the horizon visibility graph, a simple extension to the popular horizontalvisibility graph representation of a time series, and show that it possesses a rigorous mathematicalfoundation in computational algebraic topology. This fills a longstanding gap in the literature onthe horizontal visibility approach to nonlinear time series analysis which, despite a suite of successfulapplications across multiple domains, lacks a formal setting in which to prove general propertiesand develop natural extensions. The main finding is that horizon visibility graphs are dual to mergetrees arising naturally over a filtered complex associated to a time series, while horizontal visibilitygraphs are weak duals of these trees. Immediate consequences include availability of tree-basedreconstruction theorems, connections to results on the statistics of self-similar trees, and relationsbetween visibility graphs and the emerging field of applied persistent homology. Introduction.
The (directed) horizontal visibility graph[19] or (D)HVG of a time series τ = ( x , . . . , x N ) is anetwork with nodes { , . . . , N } and edges ( i, j ) for eachpair i < j such that i < k < j implies x k < x i , x j . Theundirected version omits the order of i and j . Despiteits structural simplicity, this graph captures much of thegeometry of τ while remaining invariant under (positive)affine transformations.Exact analysis and numerical simulation of HVGsshows that their properties, such as degree distributionsand block entropies, bear an intimate relation to the dy-namic properties of the system generating a time series.For example using the HVG one can determine whetheran observed system is chaotic or stochastic and can esti-mate numerical values of key dynamic parameters includ-ing reversibility, Lyapunov exponents and Hurst indices[22, 32]. This generality has led to successful applica-tions ranging through cardiology, neurophysiology, mete-orology, geophysics, protein dynamics and the financialmarkets [2, 9, 20, 26, 29, 30]. In many cases the statisticsof HVG degree sequences and their subsequence motifsare the main discriminatory feature, and work is ongoingto fully understand why this feature is so effective froma theoretical context [14, 18].Topological data analysis (TDA) for time series fol-lows a seemingly different path [6, 10, 21, 24]. Beginningfrom a piecewise linear interpolation PL τ : R → R of τ , or from a distance or density function on a higherdimensional delay embedding of τ , persistent homologytracks how the connected components of λ -sublevel sets { x : PL τ ( x ) ≤ λ } merge as the threshold λ increasesover R . The resulting merge tree is a rooted metric treewhich has a natural branch decomposition structure sum-marised in a multiset of intervals called the barcode orpersistence diagram of PL τ . This multiset is the centralobject of study in persistent homology, and metrics onthe space of barcodes and individual barcode statisticssuch as entropies are the main discriminatory features ∗ [email protected] in applications of TDA to time series. They detect andquantify many of the same dynamical properties of a sys-tem generating a time series that are captured by HVGs[7, 11, 15, 16, 21, 23, 27]. Contribution.
The wide ranging overlap between prac-tical applications of TDA and HVGs to time series isnot yet reflected in theory, but an intimate connectionexists. It arises from a very simple shift in perspective:given a time series, instead of considering its merge treewith respect to a piecewise interpolation, we study itsmerge tree over a particular weighted graph. After mak-ing this change the branching of the tree exactly reflectsthe hierarchical nesting of the edges in a structure we callthe horizon visibility graph , which extends the standardHVG with two additional vertices representing the pastand future. Establishing this duality involves fixing anappropriate embedding for the merge tree, then proceed-ing recursively on a subtree decomposition of that tree.We show that metric data on the tree imply that its sub-trees correspond to recursively nested subgraphs of thehorizon visibility graph. As a corollary HVGs are weakduals of merge trees, a connection which suggests severaldirections for developing the visibility approach.
HORIZON VISIBILITY GRAPH AND TIMESERIES MERGE TREE DUALITY
Relevant concepts from topology are defined here interms of graph theory. This simplifies the presentationand illustrates the connection to HVGs more clearly. Foradditional details and general topological definitions see[6, 31]. All time series, graphs and trees are finite. With-out loss of generality time series are strictly positive. Webegin with our simple extension to the HVG.
Definition 1.
Given a time series τ = ( x , . . . , x n ) its horizon visibility graph HVG ∞ ( τ ) is defined to be thehorizontal visibility graph of τ ∞ = ( ∞ , x , . . . , x n , ∞ ).The remainder of this section provides a formal foun-dation for the graph HVG ∞ ( τ ) in the framework of 0-dimensional homology over a filtered simplicial complex. a r X i v : . [ phy s i c s . d a t a - a n ] J un Recall that a weighted graph G = ( V, E, f ) is a graph(
V, E ) along with a weight function f : E → R . If theweight function is clear from context we use “graph”.Assume all weights are positive. Definition 2.
Given a graph G = ( V, E, f ) and a ∈ R define the sublevel graph G a to be the subgraph of G whose edges have weight no greater than a : G a :=( V, E a , f ) ⊆ G , where E a := { e ∈ E : f ( e ) ≤ a } .Note that by definition all vertices of G appear in itssublevel graphs and only edges are included or excludeddepending on their weights. Lemma 3.
The weight function f on a finite graph G =( V, E, f ) induces a strictly increasing sequence of sublevelgraphs of G beginning at ( V, ∅ ) and ending at G = ( V, E ) .Proof. Suppose that a < a < . . . < a n are the distinctweights in the image f ( E ) ⊆ R , write G i := G a i , and let a = 0. Then every distinct threshold a i adds at leastone new edge to G i that was not already in G i − for1 ≤ i ≤ n . Moreover since the a i exhaust all the distinctweights, all edges are included upon reaching the upperbound a n . So we have a sequence ( V, ∅ ) = G ⊂ G ⊂ . . . ⊂ G n = ( V, E ) of strictly increasing sublevel graphsof G . Definition 4.
Given an acyclic graph G = ( V, E, f ) and a ∈ R say that two vertices v, w ∈ V are a -connected when any path in G between them contains no weightexceeding a . Additionally say v and w are maximally a -connected when any path in G extending an a -connectedpath between them is not itself a -connected.Being maximally a -connected is clearly an equivalencerelation on V for each a ∈ R . For certain graphs theresulting one parameter family of relations has a treestructure: Lemma 5.
For a connected graph G = ( V, E, f ) the rela-tion of being maximally a -connected induces a refinementof partitions of V . The refinement has the structure of arooted tree called the merge tree T G of G , with V as theroot, {{ v } : v ∈ V } the leaves, and internal vertices beingthe maximally a -connected components of G induced byits edge weights.Proof. By Lemma 3 there is a strictly increasing sequenceof sublevel graphs ( V, ∅ ) = G ⊂ G ⊂ . . . ⊂ G n = ( V, E )corresponding to the distinct weights a = 0 and a <. . . < a n in f ( E ). When G is connected then for all a ≥ a n the only a -equivalence class is the full vertexset V , giving the root of T G . On the other hand the a -equivalence classes are singletons each containing anelement of V , giving the leaves.Between these extremes, each a i -equivalence class atlevel G i is fully contained in some a i +1 -equivalence classat level G i +1 since each a -connected component of G is automatically b -connected for all b ≥ a , and wehave a i +1 > a i for all i by definition. Iterating i over0 , , . . . , n − a i ∈ f ( E )and each edge spans some half open interval [ a i , a j ) for0 ≤ i < j ≤ n = | f ( E ) | corresponding to the heights ofits incident vertices. In what follows a proper subtree istaken to include the (half) root edge above its root vertexcovering this interval. Call a subtree principal when itcontains the descendants of all of its vertices, and withoutloss of generality also assume the root edge of a full mergetree has a fixed finite length r ∈ (0 , ∞ ), say r = 1. Notethat merge trees contain only vertices of degree d = 1 or d ≥ Corollary 6.
The map taking a principal subtree Λ ⊆ T G , whose root vertex is at height a i and whose root edgespans [ a i , a j ) for some i < j , to the subgraph γ Λ of G whose vertices are the leaves of Λ , is a bijection from theset of principal subtrees to the set of all distinct maxi-mally a -connected components of G for a ∈ R . In partic-ular the component γ Λ is maximally a -connected exactlyfor a ∈ [ a i , a j ) under this map. To illustrate this bijection consider the merge tree T G in Figure 1. The line at a = 4 . T G . The associated sublevelgraph components are: the single leftmost vertex ‘born’at G which ‘dies’ at G , the single rightmost vertex bornat G which dies at G , and the six vertex chain born at G which dies at G . Death is always by inclusion in to alarger connected component. The bijection of Corollary FIG. 1. Weighted graph G and its merge tree T G . Verticesof T G are connected components of the sublevel graphs G i .Subtrees at level a = 4 . G i to a (principal) subtree rooted at it inside T G in the sameway. In total there are fifteen subtrees of T G in the figure,including the eight single vertex trees, and there are alsofifteen distinct a -connected components of G as a variesthrough R , including the eight single vertices at G .We are now ready to connect the idea of a merge treeover a graph to time series. Definition 7.
Given a time series τ = ( x , . . . , x N )define the time series weighted path ˇ τ to be the graphˇ τ = ( V, E, f ) where: • V = { , , . . . , N }• E = { e i = ( i − , i ) : 1 ≤ i ≤ N }• f : E → R ; e i (cid:55)→ x i for 1 ≤ i ≤ N If τ is empty then ˇ τ is defined as the graph with a singlevertex and no edges.Since ˇ τ is connected and acyclic the following conceptis well-defined. Definition 8.
The merge tree of a time series τ is themerge tree T ˇ τ of the weighted path ˇ τ .Lemma 5 implies that the leaves of T ˇ τ are exactly thevertices V = { , , . . . , N } of ˇ τ , so T ˇ τ is actually an or-dered tree : we can order the children of any vertex ac-cording to the smallest leaves descended from them. Thisimplies T ˇ τ has an essentially unique plane embedding andso the discussion below is independent of the particularembedding chosen [12].Consider T ˇ τ embedded in the 2-sphere S as follows.Leaves are ordered anti-clockwise around the boundary S of the disk D , and the vertex at the unbounded end ofthe root edge of T ˇ τ , labelled ∞ , is placed on S betweenleaves 0 and n . Then points on S are identified giving anembedding in S . Note that all leaves and the vertex ∞ are identified by this process. The initial D embeddingis shown grey in Figure 2.The key outcome of this paper is that expressing thegeometric relationships between connected regions inside S \ T ˇ τ with respect to the embedding above, and thus toany plane embedding, captures the geometric structureof τ . Such relationships are described by the dual graph,but as S is more difficult to visualise than D we workin D and adjust our definition of duality to compensate.Call points in D external when on S otherwise internal . Definition 9.
Given a graph G embedded in D defineits dual G ∗ as follows. Internal vertices of G ∗ are con-nected regions of D \ ( G ∪ S ) whose boundary does notinclude all of S . External vertices of G ∗ are connectedregions of D \ ( G ∪ S ) whose boundary does include allof S , of which there is at most one. Edges in G ∗ connectvertices whose primal regions share two sides of an edge,including with themselves.In particular the dual is well-defined for merge trees. Definition 10.
The dual to a time series merge treeis the graph dual T ∗ ˇ τ of its merge tree embedded in theclosed disk D as above. The metric dual additionallycopies edge lengths from the primal tree to the edges inthe dual graph. An example of a dual to a time series merge tree isshown in Figure 2, illustrating the general relationshipwe now show: the dual is the horizon visibility graph. FIG. 2. A time series merge tree T ˇ τ embedded in D and itsdual T ∗ ˇ τ . The latter is exactly HVG ∞ ( τ ). Theorem 11.
Given a time series τ = ( x , . . . , x N ) itshorizon visibility graph HVG ∞ ( τ ) is exactly the dual ofits merge tree: HVG ∞ ( τ ) = T ∗ ˇ τ .Proof. Let τ ∞ = ( x −∞ , x , . . . , x n , x ∞ ) where x ±∞ = ∞ ,and consider the embedding of T ˇ τ in D described above.Each value x i for i = 1 , . . . , n corresponds to the intert-erval on S between leaves i − i of T ˇ τ . Similarly thevalues x ±∞ correspond to the intervals on S leading leftand right from the root vertex labelled ∞ . Thus the con-nected regions ˆ i in D \ ( T ˇ τ ∪ S ) for i = −∞ , , . . . , n, ∞ are in bijection with the values x i in τ ∞ .By the definition of duality for time series merge treesit then suffices to show that x i and x j are horizontallyvisible in τ ∞ , written x i ∼ x j , if and only if regions ˆ i andˆ j in D \ ( T ˇ τ ∪ S ) share a unique boundary edge in T ˇ τ . Inother words we want to show that x i ∼ x j ⇐⇒ | ˆ i ∩ E ˆ j | =1 where ∩ E represents intersections of region boundaries,namely along edges. Since each pair of regions boundedby the tree and S share at most one edge, it suffices toshow that ˆ i ∩ E ˆ j (cid:54) = ∅ .Suppose i < j and x i ∼ x j . Then for all k satisfying i < k < j we know that x k < x i , x j . Let a ∗ := max { x k : i < k < j } and a ∗ := min { x i , x j } . Then for any a ∈ [ a ∗ , a ∗ ) no edge on the path from vertex i to vertex j − τ exceeds a , but the edges e i and e j both do. Inother words the pair ( i, j −
1) is maximally a -connectedin ˇ τ over this half open interval. By Corollary 6 thereexists a principal subtree Λ ⊂ T ˇ τ , whose root edge e Λ corresponds to a maximally a -connected component ofˇ τ for a ∈ [ a ∗ , a ∗ ) and whose leaf set is L Λ = { i, i +1 , . . . , j − , j − } ⊂ V . Since Λ is principal it has noother leaves and e Λ must be shared between regions ˆ i andˆ j as illustrated in Figure 3, so ˆ i ∩ E ˆ j (cid:54) = ∅ as required. ... FIG. 3. The principal subtree Λ ⊂ T ˇ τ spanning vertices { i, i +1 , . . . , j − } corresponds to a maximally connected componentin ˇ τ when x i and x j are horizontally visible. In the other direction suppose that i < j and regionsˆ i, ˆ j share an edge ˆ e i,j = ˆ i ∩ E ˆ j in T ˇ τ . Note that everyordered tree can be recursively decomposed into a fan ofnonempty principal subtrees whose roots are the immedi-ate children of the containing tree’s root [4]. Since ˆ i andˆ j share an edge this decomposition implies that there ex-ists a principal subtree Λ ⊂ T ˇ τ whose leaves are exactly L Λ = { i, . . . , j − } between the two regions. By Corol-lary 6 we are back in the situation illustrated in Figure3: Λ corresponds to a connected component of ˇ τ that ismaximally a -connected for a in an interval [ α, β ) where α is the value at the root vertex of Λ and β is the lowest up-per bound of values on the edge emerging upwards fromthe root. But by the construction of the merge tree wemust have α = a ∗ = max { x k : i < k < j } being the max-imum weight on the path between vertices i and j − β = a ∗ = min { x i , x j } the value at which the firstneighbouring edge is added to the connected componentspanning L Λ as a increases. So x i ∼ x j as required.Time series merge trees are metric trees so Theorem11 allows us to extend Definition 1 to the following. Definition 12.
The persistence weighted horizon visibil-ity graph of a time series is the metric dual of its mergetree. In particular it has weights p = β − α on its edgeswhere the half open interval [ α, β ) is spanned by the cor-responding edge in the merge tree.Every rooted tree is naturally directed with all edgesoriented towards, or away from, the root. Therefore The-orem 11 also holds for directed horizon visibility graphswhen a consistent rule for orienting dual edges is appliedthroughout the proof above. Moreover, since horizon vis-ibility graphs are also horizontal visibility graphs withmaximal endpoints we have a similar but weaker resultfor HVGs as follows. Definition 13.
The weak dual T ◦ ˇ τ to a time series mergetree T ˇ τ is the subgraph of its dual T ∗ ˇ τ created by removingvertices (cid:100) ±∞ .The term ‘weak’ is used here because excluding con-nected regions (cid:100) ±∞ respects the intuition that regionswhose boundary includes the infinite root edge are them-selves unbounded, and such regions are omitted from thestandard weak dual. With this intuition formalised thenext result immediately follows. Corollary 14.
Given a time series τ = ( x , . . . , x n ) itshorizontal visibility graph HVG( τ ) is exactly the weakdual of its merge tree: HVG( τ ) = T ◦ ˇ τ . Note however that in general the graph HVG( τ ) is not dual to a tree. DISCUSSION
Reconstruction Results.
The first thing Theorem 11and Corollary 14 imply is that the nesting structure ofedges in a visibility graph carry all of the relevant geo-metric information brought over from a time series. Thishelps explain the widely observed discrimination powerof the degree sequence of an HVG, which by duality is thesequence of counts of internal boundary edges of regionsunder the merge tree. These strongly constrain the possi-ble subtree decompositions a given tree can present. Forexample the next result follows quickly, where a canon-ical time series is one that is in general position exceptits end values are global maxima.
Corollary 15 ([18]) . Canonical horizontal visibilitygraphs are uniquely determined by their degree sequences.Proof.
The largest proper principal subtree of the mergetree of a canonical time series is a binary rooted tree.Ordered binary rooted trees with equal edge lengthsand n + 1 leaves 0 , , . . . , n are uniquely determined bythe n leaf-to-leaf distances between order neighbours: d , , d , , . . . , d n − ,n . This follows by an induction onthe number of leaves and observing that for some i ∈{ , . . . , n } we have d i − ,i = 2, meaning a pair of edgescan always be removed to give a strictly smaller tree withknown distances between its remaining neighbours.Indeed any constraint on a time series forcing thelargest proper principal subtree of its merge tree to bebinary implies the unique reconstruction of its HVG fromits degree sequence in the same way.Unique reconstruction results over arbitrary time se-ries are less straightforward than over subclasses like theone in Corollary 15. However there exist several theo-rems and algorithms for both binary and non-binary treereconstruction developed over many decades for applica-tions to phylogenetic trees and more widely [13]. To takeone example consider the well known neighbour joining algorithm which uniquely reconstructs a tree from is fullpairwise leaf-to-leaf additive distance matrix [25]. Wecan use this to quickly prove the following result. Theorem 16.
The in and out degree sequences of a di-rected horizon visibility graph uniquely determine it.Proof.
Suppose we are given in and out sequences d + =( d +0 , d +1 , . . . , d + n , d + n +1 ) and d − = ( d − , d − , . . . , d − n , d − n +1 )for a horizon visibility graph G , where d ± and d ± n +1 arethe in and out degrees of the regions to the left and rightof the root vertex (labelled n + 1 here) respectively. Con-sider its dual tree G ∗ in D and its n + 2 connected re-gions. The degrees d + and d − fix the number of bound-ary edges each region has on its left and right hand sides,with respect to the root of the smallest principal sub-tree of G ∗ containing the region. Using these values wecan reconstruct the leaf-to-leaf distance matrix D for themerge tree G ∗ as follows. Due to the circular order ofexternal vertices take all leaf labels and subscripts to bemod( n + 2) from now on.Write d i,j for the path length from leaf i to leaf j . Thenwe must have that d i,i +1 = d + i + d − i for 0 ≤ i ≤ n + 1giving the first off-diagonal in D . Writing d + i,i +1 for d + i and d − i,i +1 for d − i it is straightforward to show that weget a recurrence for the second off-diagonal: d i,i +2 = d + i,i +2 + d − i,i +2 for 0 ≤ i ≤ n , where the summands aregiven by d ± i,i +2 = d ± i,i +1 + d ± i +1 ,i +2 − min( d − i,i +1 , d + i +1 ,i +2 ) . Similar recurrences give the remaining off diagonals.Once this is done we can apply the neighbour joining al-gorithm to D to reconstruct the correct unrooted mergetree topology. Since the root edge is already known viaits exterior vertex label n + 1 we have the correct rootedtree as well. Finally compute the oriented dual to givethe horizon visibility graph. Trend Detection.
An interesting practical impact ofmoving to horizon visibility, beyond simpler reconstruc-tion results, is that key geometric information about lead-ing and trailing trends in the data is no longer lost.Horizontal visibility graphs are unable to distinguish be-tween simple trends such as τ = (1 , , τ = (3 , , τ = (1 , , τ = (1 , ,
1) because the weak dualomits edges on the outer boundary of a merge tree. How-ever as shown in Figure 4 horizon visibility graphs detectthe difference. This implies sliding window techniquesover data with statistical trends will be able to detect thetrends using, for example, expected degree sequences.
Connections to TDA.
There is a direct connection be-tween the time series merge trees defined in this pa-per and the merge trees underlying earlier applicationsin TDA, which are typically over continuous domains.The connection is captured by the following propositions,which are straightforward to prove. A single
Hortonpruning of a tree is the operation of cutting off its leavesand their parental edges from the tree, then removingany remaining chains of degree-two vertices. Such opera-tions have been studied in the context of quantifying thefractal dimension of random trees [17, 28].
FIG. 4. Merge trees and their horizontal and horizon visibilitygraphs for simple trends. The root edge is dashed in HVG ∞ . Proposition 17.
The branch structure of the merge treeof a piecewise linear interpolation of a finite time seriesis exactly the first Horton pruning of its horizon visibilitygraph (via duality).
Similarly, when metric data are included we can re-cover the barcodes studied by TDA in full. For details ofhow the
Elder Rule computes barcodes on trees see [3].
Proposition 18.
The Elder Rule on the first Hortonpruning of a persistence weighted horizon visibility graphgives the barcode of its piecewise linear interpolation.
This makes horizon graphs more sensitive to certainfeatures. For example the horizon visibility graph de-tects monotonic subsequences in a time series, which areinvisible to trees over piecewise linear and piecewise con-stant interpolations, as shown in Figure 5. This meansthey can detect changes in frequency more readily, whichcould be useful for topologically aware signal analysis.Similarly, metric data on the persistence weighted graphquantify the scales at which different geometric featuresexist. So their weighted degree sequences, extending thecombinatorial degree sequence, can distinguish betweenfeatures with the same geometry appearing at differentscales.
FIG. 5. A time series τ with monotonic subsequences of differ-ent lengths and its piecewise linear interpolation (top). Thetime series merge tree T ˇ τ and its first Horton pruning (bot-tom). The pruned tree is the merge tree of P L ( τ ). Conclusion.
Horizon visibility graphs simultaneouslyextend and unify HVGs and topological merge trees overpiecewise interpolations of sequences. In doing so theyadd the ability to detect trends and the scale of geomet-ric features, absent from HVGs, and the ability to de-tect monotonic subsequences and thus frequency-basedgeometric features, absent from trees over piecewise lin-ear interpolations. More importantly there exists a widebody of work on the theoretical properties of combina-torial and metric trees in general [5, 8] and topologicalmerge trees applied to data analysis in particular [1], thatapplies to these graphs. This setting offers a number ofdirections to build on the connections established here. Finally, it should be noted that while horizon visibilitygraphs are dual to trees, their graph properties capturefeatures that may not be apparent in the tree representa-tion. In particular the tree analogue of degree sequences,leaf-to-leaf path lengths between ordered neighbours, isnot widely studied in applications of trees, so the hori-zon and horizontal visibility representations continue toexpress useful and complementary features of their own. [1] K. Beketayev, D. Yeliussizov, D. Morozov, G. H. Weber,and B. Hamann. Measuring the distance between mergetrees.
Topological Methods in Data Analysis and Visual-ization III , pages 151–165, 2014.[2] A. Braga, L. Alves, L. Costa, A. Ribeiro, M. de Jesus,A. Tateishi, and H. Ribeiro. Characterization of riverflow fluctuations via horizontal visibility graphs.
PhysicaA: Statistical Mechanics and its Applications , 444:1003–1011, 2016.[3] J. Curry. The fiber of the persistence map for functionson the interval.
Journal of Applied and ComputationalTopology , 2(3):301–321, Dec 2018.[4] N. Dershowitz and S. Zaks. Enumerations of orderedtrees.
Discrete Mathematics , 31(1):9–28, 1980.[5] M. Drmota.
Random trees: an interplay between com-binatorics and probability . Springer Science & BusinessMedia, 2009.[6] H. Edelsbrunner and J. Harer.
Computational topology:An introduction . American Mathematical Society, 2010.[7] S. Emrani, T. Gentimis, and H. Krim. Persistent homol-ogy of delay embeddings and its application to wheezedetection.
IEEE Signal Processing Letters , 21(4):459–463, 2014.[8] S. N. Evans.
Probability and Real Trees . Springer, 2006.[9] R. Flanagan and L. Lacasa. Irreversibility of financialtime series: a graph-theoretical approach.
Physics LettersA , 380(20):1689–1697, 2016.[10] R. Ghrist. Barcodes: the persistent topology of data.
Bulletin of the American Mathematical Society , 45(1):61–75, 2008.[11] M. Gidea and Y. Katz. Topological data analysis of fi-nancial time series: Landscapes of crashes.
Physica A ,491:820–834, 2018.[12] J. L. Gross and T. W. Tucker.
Topological graph theory .Dover Publications, 2001.[13] D. Gusfield.
Algorithms on strings, trees, and sequences:computer science and computational biology . CambridgeUniversity Press, 1997.[14] G. Gutin, T. Mansour, and S. Severini. A characteriza-tion of horizontal visibility graphs and combinatorics onwords.
Physica A , 390(12):2421–2428, 2011.[15] F. A. Khasawneh and E. Munch. Chatter detection inturning using persistent homology.
Mechanical Systemsand Signal Processing , 70:527–541, 2016.[16] F. A. Khasawneh and E. Munch. Topological data anal-ysis for true step detection in periodic piecewise con-stant signals.
Proceedings of the Royal Society of Lon-don A: Mathematical, Physical and Engineering Sciences ,474(2218), 2018.[17] Y. Kovchegov and I. Zaliapin. Horton law in self-similartrees.
Fractals , 24(02):1650017, 2016. [18] B. Luque and L. Lacasa. Canonical horizontal visibil-ity graphs are uniquely determined by their degree se-quence.
The European Physical Journal Special Topics ,226(3):383–389, 2017.[19] B. Luque, L. Lacasa, F. Ballesteros, and J. Luque. Hor-izontal visibility graphs: Exact results for random timeseries.
Physical Review E , 80(4):046103, 2009.[20] T. Madl. Network analysis of heart beat intervals usinghorizontal visibility graphs. In , pages 733–736. IEEE, 2016.[21] K. Mittal and S. Gupta. Topological characterizationand early detection of bifurcations and chaos in complexsystems using persistent homology.
Chaos , 27(5):051102,2017.[22] A. M. Nu˜nez, L. Lacasa, J. P. Gomez, and B. Luque.Visibility algorithms: A short review. In
New Frontiersin Graph Theory . IntechOpen, 2012.[23] J. A. Perea, A. Deckard, S. B. Haase, and J. Harer.SW1PerS: Sliding windows and 1-persistence scoring; dis-covering periodicity in gene expression time series data.
BMC bioinformatics , 16(1):257, 2015.[24] J. A. Perea and J. Harer. Sliding Windows and Persis-tence: An Application of Topological Methods to SignalAnalysis.
Foundations of Computational Mathematics ,15(3):799–838, 2014.[25] N. Saitou and M. Nei. The neighbor-joining method: anew method for reconstructing phylogenetic trees.
Molec-ular Biology and Evolution , 4(4):406–425, 1987.[26] C.-F. Schleussner, D. Divine, J. F. Donges, A. Miettinen,and R. Donner. Indications for a north atlantic oceancirculation regime shift at the onset of the little ice age.
Climate dynamics , 45(11-12):3623–3633, 2015.[27] J. R. Tempelman and F. A. Khasawneh. A look intochaos detection through topological data analysis. arXive-prints , arXiv:1902.05918, 2019.[28] I. Zaliapin and Y. Kovchegov. Tokunaga and Horton self-similarity for level set trees of Markov chains.
Chaos,Solitons & Fractals , 45(3):358–372, 2012.[29] Y.-W. Zhou, J.-L. Liu, Z.-G. Yu, Z.-Q. Zhao, and V. Anh.Fractal and complex network analyses of protein molec-ular dynamics.
Physica A: Statistical Mechanics and itsApplications , 416:21–32, 2014.[30] G. Zhu, Y. Li, and P. P. Wen. Epileptic seizure detec-tion in eegs signals using a fast weighted horizontal vis-ibility algorithm.
Computer methods and programs inbiomedicine , 115(2):64–75, 2014.[31] A. J. Zomorodian.
Topology for computing . CambridgeUniversity Press, 2005.[32] Y. Zou, R. V. Donner, N. Marwan, J. F. Donges, andJ. Kurths. Complex network approaches to nonlineartime series analysis.