Persistent Homology of Fractional Gaussian Noise

Among growing applicability of complex networks in many fields and interdisciplinary branches in science Costa et al. (2011), a technique has been suggested which converts a time-series to a network, the so-called visibility graph method Lacasa et al. (2008). Generally, suppose $\{x\}:\{x(t_{i}),i=1,...,N\}$ represents a real-valued time-series. One can construct a network, the so-called visibility graph, denoted by $G=(V,E,w)$, $V\equiv\{v_{i}\}_{i=1}^{N}$ is the node (vertex) set, and $E$ is link (edge) set and $w$ is a map from $E$ to real numbers. The VG is defined by using the bijection as follows,

and the connections are constructed according to the visibility condition between the nodes, i.e., the nodes $v_{i}$ and $v_{j}$ are connected if the node $v_{j}$ is visible from the node $v_{i}$ and vice versa, and therefore the resulting graph is undirected (for more details on the properties of VGs, see Lacasa et al. (2008)). In general, there are two ways to construct a network (graph) from a time-series: the horizontal visibility graph (HVG) Gonçalves et al. (2016); Gao et al. (2016); Xie and Zhou (2011) and the natural visibility graph (NVG) Zheng et al. (2021); Yang et al. (2009); Ahmadlou et al. (2010); the former is more sparse than the latter case and in this work we focus on the NVG. In Fig. 1, we show how an HVG [panel (a)] and an NVG [panel (b)] for a synthetic fGn series can be constructed. In a binary setup, the corresponding visibility graph chooses the range of the weights from a binary set, $w^{(B)}:E\rightarrow\mathbb{Z}_{2}\equiv\{0,1\}$; e.g., for a binary NVG (BNVG) the weight function can be written according to following relation,

where $\Theta$ is the step function, and $s_{ij}\equiv\frac{x(f(v_{j}))-x(f(v_{i}))}{f(v_{j})-f(v_{i})}$. The argument of the $\Theta$ function being positive guarantees that 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 $1$, it is unsuitable for continuous filtering. To take into account the quality of visibility between nodes, we suggest the weighted version of the natural visibility graph (WNVG), by considering the weight function as follows,

There are two factors inside the product. In the second branch of Eq. (5), one is the step function just like the binary graph, and the other is the weight which is proportional to ”how visible is the site $j$ from $i$ and vice versa”; i.e., the more distinguishable the data points are in the original time-series, the higher the corresponding weight is in the constructed network. The term $\frac{1}{j-i-1}$ is necessary to make the weights reasonable numbers for comparison reasons. In the absence of this exponent, the more the distance between the nodes is, the higher the corresponding weights are. For both statistical and topological analysis, we use this weight function which admits continuous filtering.

A conventional approach to quantify the properties of a typical network is determining degree, the number of nodes straightly connected with the underlying node by non-zero weight, and centrality measures such as betweenness centrality, the number of shortest paths between any pair of nodes going through a distinct node, closeness centrality, eigenvector centrality, etc. (see Appendix B for more details). Besides the mentioned approach, other formalisms have been proposed in which not only the pairwise connections (links) are examined but also a systematic pipeline is considered to incorporate higher-order connections ($k$-cliques), including the simplicial complex and hypergraph Bianconi and Rahmede (2016); Kovalenko et al. (2021); Young et al. (2020).

To analyze such higher-order structures, one can generalize the well-known statistical quantities based on higher-order connections. On the other hand, for global analysis of these structures newly born topological tools are proposed. By the term ”global”, we mean the essential features of the object which are not affected by geometric transformations. The well-known topological quantities describing global properties of the space (or any object mapped to a topological space) are Betti numbers. The $k$th Betti number of an object ($\beta_{k}$) is a topological invariant. The dimension of the $k$-homology group of the topological space corresponding to the object counts the number of $k$-dimensional topological holes ($k$-holes) of the object. Intuitively, $\beta_{0}$ indicates the number of connected components, $\beta_{1}$ is the number of topological (unshrinkable) loops, and $\beta_{2}$ counts the number of topological (unshrinkable) voids (see the Appendix C for more details). According to the mentioned properties of the topological measures and weight-based analysis of weighted complex networks, the applied algebraic topology tool, known as the persistent homology (PH) technique, is useful to study global structures of the weighted complex network in higher-dimensions Edelsbrunner and Harer (2010); Rote and Vegter (2006); Zomorodian (2009).

In the PH-based analysis of a weighted complex network, the weighted network is mapped to a weighted clique simplicial complex as a higher-order structure containing any $(k+1)$-clique of the network as the $k$-simplex. Therefore, PH creates a nested sequence of the clique simplicial complex, so-called filtration, by considering the weight as the parameter, such that any link (1-simplex) presents in the complex at a distinct weight if the weight of the link is less than the distinct weight. Accordingly, the $k$-holes of the complex appear and disappear when the weight (threshold) varies. To summarize the evolution of these $k$-dimensional topological features of the system, PH assigns an ordered tuple, the so-called persistence pair (PP), to this global descriptor to visualize the topological variation of the system in the birth-death space. This kind of visualization for the evolution of $k$-holes is called the persistence diagram (PD) for the $k$-homology group. PD for the $k$-homology group includes hidden topological information of the $k$-holes in the weighted complex. For example, one can compute the population of birth, death, and lifetime of $k$-holes using the distribution of PPs in PD for the $k$-homology group. Persistence entropy (PE) for the $k$-homology group is another quantity defined by the Shannon entropy of topological persistence (lifetime) of PPs of the $k$-homology group (see Appendix D for more details).

By local properties, we mean the properties which are node-dependent and are not necessarily globally defined. It has been confirmed that the distribution function of the node degree of VGs of the fBms and fGns is power-law $p(k)\propto k^{-\gamma}$ with the exponent $\gamma(H)=3-2H$ and $\gamma(H)=5-2H$ , respectively Lacasa et al. (2008, 2009). In this subsection, we perform our computation for the WNVG, introduced in this paper.

The probability distribution function of eigenvector ($c^{E}$) and betweenness ($c^{B}$) centralities are indicated in panels (a) and (b) of Fig. 2 in log-log scale, in terms of $H$ for WNVGs. The scaling behavior of mentioned distributions has been checked by the Kolmogorov-Smirnov test (KS test) and the results confirm that the power-law behavior of the computed $p(c^{E})$ and $p(c^{B})$ is highly $H$- and size-dependent [see panels (a) and (b) of Fig. 2]. The probability distribution of closeness centrality and corresponding moments $M_{n}$ for $n=2$ and $n=3$ are illustrated in panels (c) and (d) of Fig. 2, respectively. This part depicts that not only the Hurst dependency is not confirmed but also the overall shape of probability distribution functions of statistical measures depends on the size of the underlying fGn series. Subsequently, such distributions can not discriminate between different fGn series.

The full spectrum of the eigenvalues [Eq. (12)] is illustrated in panel (a) of Fig. 3. We see that the impact of $H$ is changing the range of the spectrum, and by increasing the Hurst exponent, the range of the spectrum for WNVGs becomes tight. This phenomenon can be understood by recalling that correlations (obtained by increasing $H$) smooth the underlying time-series, causing the corresponding network to have more links at low weight. For more smoothed time-series, the typical slopes for the associated WNVG become down, leading to lower weights according to Eq. (5), and equivalently making a shorter range for the distribution of $\lambda$s. Panel (b) of Fig. 3 indicates different moments for $p(\lambda)$. The solid and dashed lines correspond to $M_{2}$ and $M_{3}$, respectively. The $n$th moment of this distribution behaves as an $H$-dependent quantity. The higher the order of moments, the stronger the dependency on sample size is.

For any given value of Hurst exponent, the associated BNVG of fGn is a topological tree; therefore, the BNVGs are topologically equivalent to each other. In other words, various BNVGs are homeomorphic for different $H$ according to the homeomorphism theorem. The looplessness of BNVGs indicates that these networks do not contain some classes of network motifs corresponding to topological loops, so-called topological motifs Xie and Zhou (2011). To make more sense, suppose 4 successive data points as $\{x_{j}\}_{j=i}^{i+3},(j=1,...,N-4)$ from a given time-series $\{x_{i}\}_{i=1}^{N}$ to make a minimal topological loop. From a network perspective, the necessary (not sufficient) condition to construct a topological loop is that $w_{i,i+1}=w_{i,i+3}=w_{i+1,i+2}=w_{i+2,i+3}=1$ and the sufficient condition is $w_{i,i+2}=w_{i+1,i+3}=0$. The weights here are determined by Eq. (4). Geometrically, the necessary condition is equivalent to the case where data points have upward concavity. The visibility condition does not satisfy the sufficient condition, such that either $w_{i,i+2}=1$ (creating two 2-simplices $[v_{i},v_{i+1},v_{i+2}]$ and $[v_{i},v_{i+2},v_{i+3}]$) or $w_{i+1,i+3}=1$ (making two 2-simplices $[v_{i},v_{i+1},v_{i+3}]$ and $[v_{i},v_{i+2},v_{i+3}]$) or even $w_{i,i+2}=w_{i+1,i+3}=1$ (constructing a 3-simplex $[v_{i},v_{i+1},v_{i+2},v_{i+3}]$) killing the topological loop and leads the binary network to be topologically tree. Therefore, the BNVG does not contain a certain class of network motifs equivalent to topological holes, whereas, in the WNVG case these local patterns can be captured as topological loops. Also mentioned loops are extracted through the filtration of the corresponding weighted simplicial complex. According to the PH language, the sufficient condition can be treated as $w_{i,i+1},w_{i+1,i+2},w_{i+2,i+3},w_{i,i+3}<w_{i,i+2},w_{i+1,i+3}$; therefore, the associated topological motifs now remain. Subsequently, we only focus on the topological aspects of WNVGs. Using this terminology, we can analyze local behaviors of a generic fGn using the statistics of corresponding patterns (topological motifs) in WNVGs which are presented as PPs in PD.

Now, we are going to evaluate topological measures for WNVG constructed from a generic time series. Since in this paper, we are interested in examining the capability of the PH technique in determining the Hurst exponent of a typical self-similar process, therefore we take fGn which is an increment of fBm. Classifying the stationary and non-stationary series is in principle possible when the weighted network is constructed from time series in our approach. Intuitively, by increasing the value of the Hurst exponent, the underlying series becomes more smooth and the statistics of weights grow for lower values of weight. For an fBm data set which is the profile set of fGn Taqqu et al. (1995); Kantelhardt et al. (2002), the maximum value of distribution function of weights occurs at lower weight values. Subsequently, the PH of WNVG which is represented by topological curves can recognize the stationary and non-stationary series. The constructed networks according to the visibility graph possess the footprint of Hurst exponent empirically demonstrated in Lacasa et al. (2009). The higher value of $H$ causes making the denser network. Also, the number of $k$-simplices at lower value of threshold increase in the weighted version of the visibility graph; on the other hand, such simplices influence the evolution of $k$-holes in associated simplicial complexes. Therefore, the coefficients of topological curves depend on Hurst exponent. Hereafter, we consider fGn as our case study. Panel (a) of Fig. 4 indicates the $\beta_{0}$ as a function of filtration parameter (threshold) for clique complexes of WNVG produced for various time-series with different values of Hurst exponent. To reduce the effect of sample size, we divide the Betti numbers by the sample size and call these normalized Betti numbers. By increasing the value of the Hurst exponent, the normalized $\beta_{0}$ versus threshold decreases. Namely, the number of connected components in the network decreases with increasing $H$ (the graphs become steeper). Interestingly, the slope of normalized $\beta_{0}$ as a function of $w$ is different for different $H$; consequently, the WNVG of fGn sets reaches to the connected regime (path-connected), $\beta_{0}=1$, [in panel (a) of Fig. 4, we take $N=2^{10}$] by different rates and at different thresholds, $w_{0}$. Panel (b) of Fig. 4 represents the value of $w_{0}$ as a function of Hurst exponent for different network sizes. As depicted in the mentioned panel, the value of $w_{0}$ behaves as a sample size-dependent quantity and by increasing $H$, such dependency becomes negligible.

We define the $\beta_{k}^{\text{(birth)}}(w)$ and $\beta_{k}^{\text{(death)}}(w)$ as the number of $k$-dimensional holes that are born and die at a given threshold, $w$, respectively. It turns out that $\beta_{0}^{\text{(birth)}}=0$ for $w>0$, since at $w=0$ the underlying network has $N$ connected components; therefore, all connected components are born at $w=0$. Panel (c) of Fig. 4 illustrates the $\beta_{0}^{\text{(death)}}/N$ as a function of $w^{2}$. As shown in this plot, one of the proper fitting functions to describe the normalized $\beta_{0}^{\text{(death)}}$ is given by $\beta_{0}^{\text{(death)}}(w)\sim\exp\left[-\alpha_{0}^{\text{(death)}}w^{2}\right]$ for $2\lesssim w^{2}\lesssim w^{2}_{\rm max}$, where the value of $w_{\rm max}$ depends on the $H$ value. The $\alpha_{0}^{\rm(death)}$ depends on the Hurst exponent as increasing function and it behaves as an almost independent function on size of series [panel (d) of Fig. 4].

Panel (a) of Fig. 5 is devoted to $\beta_{1}/N$ for various Hurst exponents. As we expect, for a trivial threshold, $w=0$, we have $N$ connected components ($\beta_{0}=N$) and therefore the number of loops is identically zero ($\beta_{1}=0$). By increasing the threshold, the higher value of the Hurst exponent leads to a more rapidly increasing rate of $\beta_{1}$. On the other hand, for a high enough value of the threshold, again the underlying data set behaves like a topological tree without any topological loops. Therefore, the normalized $\beta_{1}$ goes asymptotically to zero, and such descending is more rapid for higher $H$. We also determine the lowest non-trivial threshold for which there is no loop in the underlying network denoted by $w_{1}$, and depict this threshold versus $H$ for different samples size in panel (b) of Fig. 5. The $w_{1}$ is also a size-dependent quantity. Comparing the $\beta_{0}/N$ and $\beta_{1}/N$ demonstrate that by increasing threshold value, the WNVGs of the fGn series reach the loopless regime ($\beta_{1}=0$) before appearing in the connected regime (for which $\beta_{0}=1$), irrespective of the Hurst exponent, i.e., $w_{0}(H)>w_{1}(H)$. The quantities $\beta_{1}^{\text{(birth)}}\propto\exp\left[-\alpha_{1}^{\text{(birth)}}w\right]$, $\beta_{1}^{\text{(death)}}\propto\exp\left[-\alpha_{1}^{\text{(death)}}w\right]$ versus thresholds and corresponding coefficients are illustrated in panels (c)-(f) of Fig. 5, respectively. The $\alpha_{1}^{\rm(birth)}$ and $\alpha_{1}^{\rm(death)}$ are almost size-independent and they grow by increasing $H$. The local upward concavity behavior of time series satisfying the loop condition and associated topological motifs (1-holes) appears and disappears earlier as the Hurst exponent of the time-series increases.

Another interesting property to assess is the probability distribution of lifetime for 1-holes which is the difference between the death and birth thresholds of a typical measure in a 1-homology class. Panel (a) of Fig. 6 shows the probability distribution of topological 1-dimensional hole lifetime, $\ell_{1}$, for various synthetic data sets with different values for the Hurst exponent when the vertical axis is plotted in log-scale. Our results confirm that $p(\ell_{1})\propto\exp\left[-\alpha_{1}^{\text{(lifetime)}}\ell_{1}\right]$. The $H$-dependency of $\alpha_{1}^{\rm(lifetime)}$ for various systems sizes is depicted in panel (b) of Fig. 6. This result confirms that $\alpha_{1}^{\rm(lifetime)}$ can be considered as a robust measure for determining the Hurst exponent of the fGn signal which is not affected by sample size even compared to $\alpha_{0}^{\rm(death)}$, $\alpha_{1}^{\rm(birth)}$, and $\alpha_{1}^{\rm(death)}$. The increasing behavior of the lifetime coefficient $\alpha_{1}^{\rm(lifetime)}$ versus the Hurst exponent also indicates that the anti-correlated signals ($H<0.5$) contain more persistent topological motifs (1-holes). This behavior also demonstrates a considerable difference between the amount of visibility between the inner and the outer nodes which are corresponding to the local upward concavity behaving data points in these signals.

Figure 7 indicates the persistence diagram (PD) and persistence barcode (PB) for three types of fGn signals for (a) $H=0.2$ (anticorrelated), (b) $H=0.5$ (uncorrelated) and (c) $H=0.8$ (correlated). The open circle and triangle symbols correspond, respectively to the $0$- and $1$-homology groups in a persistence diagram for WNVGs of fGn. Each symbol depicts a pair $(w_{\rm birth},w_{\rm death})$ of a $k$-dimensional hole. As expected, all 0-holes are born in $w_{\rm birth}=0$, and also always $w_{\rm death}>w_{\rm birth}$. In the barcode representation (inset plot), the horizontal lines (blue lines for $k=0$ and orange lines for $k=1$) start and end on the threshold values on which $k$-dimensional holes are born and die, respectively.

The associated persistence entropies (PEs) defined by Eq. (30) are obtained using the persistence diagram. Panels (a) and (b) of Fig. 8 depict the ${PE}_{0}$ and ${PE}_{1}$, as a function of sample size, respectively. We compute persistence entropy for all available Hurst exponent values represented by different symbols. Our results demonstrate that ${PE}_{k}=\mathcal{A}_{k}(H)\log_{10}N$ for $k=0,1$. The behavior of prefactor $\mathcal{A}_{k}$ versus $H$ is represented in panel (c) of Fig. 8. The $\mathcal{A}_{0}$ is almost an increasing function versus Hurst exponent, while the $\mathcal{A}_{1}$ becomes constant.

To model the stochastic fractal processes, Mandelbrot and Van Ness introduced the fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) Mandelbrot and Van Ness (1968). The theory of fBm is a mathematical generalization of the classical random walk and Brownian motion Mandelbrot and Van Ness (1968); Kahane and Kahane (1993). A 1-dimensional fBm is represented by $B\equiv\{B(t):t\geq 0\}$, with power-law variance, for which $Var(B(at))\triangleq Var(a^{H}B(t))=a^{2H}Var(B(t))$, where $H\in(0,1)$ is called the Hurst exponent. For this random force, the Markov property and the stationarity are violated (note that when we have domain Markov property, stationarity, and continuity for a time series, then it should be proportional to a 1-dimensional Brownian motion). A model for generating fBm (denoted by $B_{H}$ to emphasize on its Hurst exponent $H$) is a generalization of the Brownian motion which is non-stationary and non-Markovian Reed et al. (1995), and is given by the Holmgren-Riemann-Liouville fractional integral,

where $\Gamma$ is the Gamma function, $\text{d}B(s)\equiv B(s+\text{d}s)-B(s)$ is the increment of 1-dimensional Brownian motion, and it has the following covariance:

where $\sigma^{2}\equiv\langle B(0)\rangle$ and also $\langle B_{H}\rangle=0$. The increments, $x_{H}(t)\equiv\delta_{t}\left({B_{H}}(t+\delta t)-{B_{H}}(t)\right)$, are known as fractional Gaussian noise (fGn). The power spectrum of fBm and fGn behaves as $S(f)\sim f^{-\xi}$, where $\xi(H)=2H+1$ and $\xi(H)=2H-1$ for fBm and fGn, respectively. For $H>\frac{1}{2}$ ($H<\frac{1}{2}$) the corresponding fGn is positively (negatively) correlated. According to the results provided by detrended fluctuation analysis (DFA), the relation between the scaling exponent derived by the DFA method and associated Hurst exponent of fBm is $H+1$, while constructing the fGn series by making the increment of fBm confirmed that the scaling exponent of fluctuation functions computed by DFA is directly related to the Hurst exponent Taqqu et al. (1995); Kantelhardt et al. (2002).

Suppose that a network (graph) is represented by $G=(V,E,w)$. Here, $V\equiv\{v_{i}\}_{i=1}^{N}$ is a node (vertex) set, $E=V\times V\equiv\Bigr{\{}e_{ij}=(v_{i},v_{j})\Bigr{|}v_{i},v_{j}\in V\Bigr{\}}_{i,j=1}^{N}$ is a link (edge) set, and $w:E\rightarrow\mathbb{R}$ is a weight function (threshold). Subsequently, the degree of $i$th node ($v_{i}$) is the number of nodes straightly connected with the underlying node by non-zero weight, and it is denoted by $k_{i}\equiv\sum_{j}(1-\delta_{0,w_{ij}})$. The degree centrality ($c_{i}^{D}$) is defined by $\frac{k_{i}}{N-1}$, which is apparently related directly to how important the underlying node is, since it is the number of agents that have a connection with it. An important function concerning this quantity is the degree distribution showing the probability distribution function for degree of all nodes in the network:

The eigenvector centrality ($c_{i}^{E}$) is also defined via the eigenvalue equation

where $\lambda$ is the eigenvalue. The maximum value of the $\lambda$ spectrum, i.e., $\lambda_{\rm max}$ plays the dominant role in the network properties, the corresponding eigenvectors of which are denoted by $c_{i,{\rm max}}^{E}$ revealing the importance of the nodes. Let us denote the shortest distance between nodes $v_{i}$ and $v_{j}$ by $d_{ij}$ which is assumed to be $N$ when there is no path connecting them (disconnected graphs). Then the closeness centrality is defined by

and the betweenness centrality is as follows:

where $n_{jk}$ is the number of geodesics from $v_{j}$ to $v_{k}$, and $n_{jk}(i)$ is the number of geodesics from $v_{j}$ to $v_{k}$ which pass through node $v_{i}$.

Topology generally refers to the global features in contrast to the geometrical invariants of underlying objects or sets. We have two spaces represented by $X$ and $Y$, and they have the same local (geometric) features if any relevant features are invariant under congruence, while the mentioned spaces are topologically equivalent if the associated features are invariant under homeomorphisms. In other words, they are homeomorphic. Homology theory plays a crucial role in the mathematical description of the relevant building block of a typical topological space and reveals the connectedness of underlying space Edelsbrunner and Harer (2010); Rote and Vegter (2006); Zomorodian (2009). Based on such properties, for the sake of clarity, we will give a brief review on the building blocks of algebraic topology which are useful to set up homology groups.

Simplex: A $k$-simplex ($\sigma_{k}$) is a convex-hull of any geometrically independent subset, accordingly $\sigma_{k}\equiv[v_{0},v_{1},...,v_{k}]\subseteq\mathbb{R}^{D}$. By this definition, a 0-simplex is a point, a 1-simplex is a segment of a line, a 2-simplex is a filled triangle, a 3-simplex is a filled tetrahedron, and so on (Fig. 10).

Face: An $l$-simplex which is denoted by $\sigma_{l}$ is a subset of the $k$-simplex ($\sigma_{l}\subseteq\sigma_{k}$) and it is called the $l$-face of the $k$-simplex.

Simplicial complex: A simplicial complex ($\psi$) is a collection of simplices such that: any $l$-face of any $k$-simplex of a typical complex ($0<l<k$), is a member of the complex. In addition, the non-empty intersection of any two simplices, $\sigma_{k}$, and $\sigma_{m}$, from the complex is an $l$-face of both simplices. The dimension of a complex is the maximum dimension of all simplices of the complex. According to the definition of the complex, one can define its $k$-ordered subcollection of complexes $\psi$ as follows:

Chain: For a given simplicial complex, a $k$-chain ($k$-dimensional chain) is a linear combination of $k$-simplices of $\psi$, defined by

where $|\Sigma_{k}(\psi)|$ corresponds to the cardinality of the $k$-ordered subcollection of complex and the coefficients, $a_{i}$s, belong to a field, which is usually considered as $\mathbb{F}=\mathbb{Z}_{2}\equiv\{0,1\}$. The collection of all possible $k$-chains in the simplicial complex is called the $k$-chain group as

Boundary operator: For the simplices in any dimension, the boundary operator $\partial_{k}$ is an operator mapping $\sigma_{k}$ to its boundary according to

Boundary: A $k$-chain which is the boundary of a $(k+1)$-chain, is called the $k$-boundary, denoted by $b_{k}$. The $k$-boundary group is the collection of all $k$-boundaries in complex $\psi$,

Cycle: A $k$-chain that has no boundary, is called a $k$-cycle denoted by $z_{k}$ as

The $k$-cycle group is defined as the collection of all $k$-cycles in complex $\psi$,

Since ”boundaries have no boundary”, therefore, we have

hence

Homology group: The $k$-homology group is defined by the quotient group of the $k$-cycles group by the $k$-boundary group,

The $k$th Betti number of a simplicial complex, denoted by $\beta_{k}(\psi)$ , is a topological invariant which counts the number of $k$-homology classes corresponding to the number of $k$-dimensional holes of complex $\psi$ (Fig. 11),

Clique (flag) simplicial complex of an unweighted (binary) network, $G=(V,E,w\in\{0,1\}$), is a simplicial complex, denoted by $\psi^{(G)}$, such that any $k$-simplex of each dimension in the complex corresponds to a $(k+1)$-clique in the network and vice versa (Fig. 12).

A binary network is a topological tree, if and only if, it is topologically holeless in all dimensions. Namely, the associated clique simplicial complex has the following property:

In the context of topological data analysis (TDA), we are interested in statistical analysis of the structure in data. Generally, in TDA-based analysis, data of any type (point cloud data, scalar field, time-series, network, etc.) when mapped to a weighted simplicial complex are worked out topologically in terms of the parameters present inherently in the original data, e.g. the weight of links in a weighted network, or the pairwise distance between data points in point cloud data. More precisely, TDA maps parameter-dependent data, $X(w)$, (where $w$ is a typical parameter, like the threshold value for any weighted network) to a weighted simplicial complex, $\psi^{X}(w)$. Such approach produces the chain group, the cycle group, the boundary group, the homology group and particularly, the $k$th Betti number Nakahara (2003); Edelsbrunner and Harer (2010).

Persistent homology (PH) as a powerful tool of TDA examines the creation (birth) and destruction (death) of topological invariants associated with homology classes during a mathematical process called filtration Edelsbrunner and Harer (2010). Filtration, $\phi$, is a nested sequence of weighted complex $\psi(w)$ in which any complex with a distinct weight is a subcollection of any complex with higher weight,

More precisely, the PH technique enumerates the $k$th Betti number of any subcomplex in $\phi$ and assigns an ordered tuple $w^{(h_{k})}\equiv(w_{{\rm birth}}^{(h_{k})},w_{{\rm death}}^{(h_{k})})$ to the existing $k$-dimensional topological hole. Here $w_{{\rm birth}}^{(h_{k})}$ and $w_{{\rm death}}^{(h_{k})}$ are the thresholds for which $h_{k}$ appears (birth) and disappears (death), respectively. Since $w_{{\rm birth}}^{(h_{k})}<w_{{\rm death}}^{(h_{k})}$, we can define the positive-value quantity $\ell^{(h_{k})}\equiv w_{{\rm death}}^{(h_{k})}-w_{{\rm birth}}^{(h_{k})}$ as persistency (lifetime) of a $k$-dimensional hole. The persistence barcode (PB) and equivalently the persistence diagram (PD) are the famous representations of PH. As an illustration, the $k$-dimensional persistence diagram of weighted complex $\psi(w)$ is a multiset $PD_{k}\Bigr{(}\phi(\psi(w))\Bigr{)}\equiv(\mathcal{M},\mathcal{N})$, where

and $\mathcal{N}:\mathcal{M}\rightarrow\mathbb{N}$ is the count function. Inspired by Shannon entropy for a typical state probability, one can define the persistence entropy (PE) of the $k$th PD (PB). To this end, we construct the probability for lifetime of homology classes as

Therefore, the PE for the $k$-dimensional topological hole is defined by Merelli et al. (2015); Atienza et al. (2019)