Ethics statement

The ethics approval for using the data was obtained from the local Institutional Review Boards (IRB) of University of Wisconsin-Madison (https://irb.wisc.edu). Informed written consent was obtained from all participants.

2.1 Graphs as simplicial complices

The proposed method for estimating topological state space is based on the topological clustering on a collection of graphs (Fig 1). The initial step involves a birth-death decomposition of a weighted graph, leading to the generation of sorted birth and death sets (section 2.3). The second step entails calculating the topological distance: between birth sets to obtain the 0D topological distance d₀, and between death sets to obtain the 1D topological distance d₁ (section 2.4). The third step involves computing the within-cluster distance l_W among the collection of graphs (section 2.6). Subsequently, we demonstrate the equivalence of topological clustering with k-means clustering in a high-dimensional convex set, employing k-means clustering routines for optimization. To increase the reproducibility, MATLAB codes for performing the methods are provided in https://github.com/laplcebeltrami/PH-STAT.

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Fig 1. Proposed topological clustering pipeline used in estimating the state space.

Given two weighted graphs G₁, G₂, we first perform the birth-death decomposition and partition the edges into sorted birth and death sets (section 2.3). The 0D topological distance D_W0 between birth values quantifies discrepancies in connected components (section 2.4). The 1D topological distance D_W1 between death values quantifies discrepancies in cycles (section 2.4). The combined distance is used in computing the within-cluster distance l_W between graphs. Topological clustering is performed by minimizing l_W over all possible cluster labels C₁, ⋯, C_k (section 2.6).

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A high dimensional object such as a brain network can be modeled as weighted graph consisting of node set V indexed as V = {1, 2, ⋯, p} and edge weights w = (w_ij) between nodes i and j. If we order the edge weights in the increasing order, we have the sorted edge weights: (1) where q ≤ (p² − p)/2. The subscript ₍₎ denotes the order statistic. In terms of sorted edge weight set W = {w₍₁₎, ⋯, w_(q)}, we may also write the graph as . If we connect nodes following some criterion on the edge weights, they will form a simplicial complex which will follow the topological structure of the underlying weighted graph [7, 59]. Note that the k-simplex is the convex hull of k + 1 points in V. A simplicial complex is a finite collection of simplices such as points (0-simplices), lines (1-simplices), triangles (2-simplices) and higher dimensional counter parts.

The Rips complex is a simplicial complex, whose k-simplices are formed by (k + 1) nodes which are pairwise within distance ϵ [60]. While a graph has at most 1-simplices, the Rips complex has at most (p − 1)-simplices. The Rips complex induces a hierarchical nesting structure called the Rips filtration for 0 = ϵ₀ < ϵ₁ < ϵ₂ < ⋯, where the sequence of ϵ-values are called the filtration values. The filtration is quantified through a topological basis called k-cycles. 0-cycles are the connected components, 1-cycles are 1D closed paths or loops while 2-cycles are 3-simplices (tetrahedron) without interior. Any k-cycle can be represented as a linear combination of basis k-cycles. The Betti number β_k counts the number of independent k-cycles. During the Rips filtration, the i-th k-cycle is born at filtration value b_i and dies at d_i. The collection of all the paired filtration values displayed as 1D intervals is called the barcode and displayed as a 2D scatter plot is called the persistent diagram. Since b_i < d_i, the scatter plot in the persistent diagram are displayed above the line y = x line by taking births in the x-axis and deaths in the y-axis.

For a dynamically changing brain network , we assume the node set is fixed while edge weights are changing over time t. If we build persistent homology at each fixed time, the resulting barcode is also time dependent:

2.2 Graph filtrations

As the number of nodes p increases, the resulting Rips complex becomes increasingly dense. Additionally, as the filtration values rise, the number of edges connecting each pair of nodes also increases, leading to a more interconnected structure. At higher filtration values, Rips filtration becomes an ineffective representation of networks. To remedy this problem, graph filtration was introduced [6, 18]. Given weighted graph with edge weight w = (w_ij), the binary network is a graph consisting of the node set V and the binary edge weights w_ϵ = (w_ϵ,ij) given by

Note w_ϵ is the adjacency matrix of , which is a simplicial complex consisting of 0-simplices (nodes) and 1-simplices (edges) [60]. While the binary network has at most 1-simplices, the Rips complex can have at most (p − 1)-simplices. By choosing threshold values at sorted edge weights w₍₁₎, w₍₂₎, ⋯, w_(q), we obtain the sequence of nested graphs [5]: (2)

The sequence of such a nested multiscale graph is called the graph filtration [6, 18]. Note that is the complete weighted graph for any ϵ > 0. On the other hand, is the node set V. By increasing the threshold value, we are thresholding at higher connectivity; thus more edges are removed.

For dynamically changing brain networks (Fig 2), we can similarly build time varying graph filtrations at each time point .

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Fig 2. Dynamically changing correlation matrices computed from rs-fMRI using the sliding window of size 60 for a subject.

The constructed correlation matrices are superimposed on top of the white matter fibers in the MNI space and color coded based on correlation values.

https://doi.org/10.1371/journal.pcbi.1011869.g002

2.3 Birth-death decomposition

Unlike the Rips complex, there are no higher dimensional topological features beyond the 0D and 1D topology in graph filtration. The 0D and 1D persistent diagrams (b_i, d_i) tabulate the life-time of 0-cycles (connected components) and 1-cycles (loops) that are born at the filtration value b_i and die at value d_i, respectively. The 0th Betti number β₀(w_(i)) counts the number of 0-cycles at filtration value w_(i) and can be shown to be non-decreasing over filtration (Fig 3) [21]:

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Fig 3. The birth-death decomposition partitions the edge set into the birth and death edge sets.

The birth set forms the maximum spanning tree (MST) and contains edges that create connected components (0D topology). The death set contains edges that do not belong to the maximum spanning tree (MST) and destroys loops (1D topology).

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On the other hand the 1st Betti number β₁(w_(i)) counts the number of independent loops and can be shown to be non-increasing over filtration [21]:

During the graph filtration, when new components is born, they never die. Thus, 0D persistent diagrams are completely characterized by birth values b_i only. Loops are viewed as already born at −∞. Thus, 1D persistent diagrams are completely characterized by death values d_i only. We can show that the edge weight set W can be partitioned into 0D birth values and 1D death values [61]:

Theorem 1 (Birth-death decomposition). The edge weight set W = {w₍₁₎, ⋯, w_(q)} has the unique decomposition (3) where birth set W_b = {b₍₁₎, b₍₂₎, ⋯, b_(q0)} is the collection of 0D sorted birth values and death set W_d = {d₍₁₎, d₍₂₎, ⋯, d_(q1)} is the collection of 1D sorted death values with q₀ = p − 1 and q₁ = (p − 1)(p − 2)/2. Further W_b forms the 0D persistent diagram while W_d forms the 1D persistent diagram.

In a complete graph with p nodes, there are q = p(p − 1)/2 unique edge weights. There are q₀ = p − 1 number of edges that produce 0-cycles. This is equivalent to the number of edges in the maximum spanning tree (MST) of the graph. Thus, number of edges destroy loops. The 0D persistent diagram is given by . Ignoring ∞, W_b is the 0D persistent diagram. The 1D persistent diagram is given by . Ignoring −∞, W_d is the 1D persistent digram. We can show that the birth set is the MST (Fig 3) [62].

The identification of W_b is based on the modification to Kruskal’s or Prim’s algorithm that identifies the MST [6, 62]. Then W_d is identified as . Fig 4 displays how the birth and death sets change over time for a single subject used in the study. Given edge weight matrix W as input, the Matlab function WS_decompose.m outputs the birth set W_b and the death set W_d.

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Fig 4. The corresponding birth and death sets of dynamically changing correlation matrix shown in Fig 2.

The horizontal axis is the time point. Columns are the sorted birth and death edge values at the time point.

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2.4 Topological distances

Like the majority of clustering methods such as k-means and hierarchical clustering that use geometric distances [6, 63, 64], we propose to develop a topological clustering method using topological distances (Fig 5). For this purpose we use the Wasserstein distance.

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Fig 5. Comparison between geometric distance d_geo and topological distance d_top.

We used the shortest Euclidean distance between objects as the geometric distance. The left (two circles) and middle (circle and arc) objects are topologically different while the left and right (square and circle) objects are topologically equivalent. The geometric distance cannot discriminate topologically different objects (left and middle) and produces false negatives. The geometric distance incorrectly discriminates topologically equivalent objects (left and right) and produces false positive.

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Given two probability distributions X ∼ f₁ and Y ∼ f₂, the r-Wasserstein distance D_W, the probabilistic version of the optimal transport, is defined as (4) where the infimum is taken over every possible joint distribution of X and Y. The Wasserstein distance is the optimal expected cost of transporting points generated from f₁ to those generated from f₂ [35]. There are numerous distances and similarity measures defined between probability distributions such as the Kullback-Leibler (KL) divergence and the mutual information [65]. While the Wasserstein distance is a metric satisfying positive definiteness, symmetry, and triangle inequality, the KL-divergence and the mutual information are not metrics. Although they are easy to compute, the biggest limitation of the KL-divergence and the mutual information is that the two probability distributions must be defined on the same sample space. If the two distributions do not have the same support, it may be difficult to even define the distance between them. If f₁ is discrete while f₂ is continuous, it is difficult to define them. On the other hand, the Wasserstein distance can be computed for any arbitrary distributions that may not have the common sample space, making it extremely versatile.

Consider persistent diagrams P₁ and P₂ given by

Their empirical distributions are given in terms of Dirac-Delta functions

Then we can show that the r-Wasserstein distance on persistent diagrams is given by (5) over every possible bijection ψ, which is a permutation, between P₁ and P₂ [34–36]. Optimization (5) is the standard assignment problem, which is usually solved by the Hungarian algorithm in [66]. However, for graph filtration, the distance can be computed exactly in by simply matching the order statistics on the birth or death values [61, 62, 67]:

Theorem 2. The r-Wasserstein distance between the 0D persistent diagrams for graph filtration is given by (6) where is the i-th smallest birth values in persistent diagram P_j. The r-Wasserstein distance between the 1D persistent diagrams for graph filtration is given by (7) where is the i-th smallest death values in persistent diagram P_j.

The proof is provided in [68]. We can show that the 2-Wasserstein distance is equivalent to the Euclidean distance within a certain convex set. Let be the vector of sorted birth values of persistent diagram P_i. Then is a point in the (q₀ − 1)-simplex given by where x₁ and are bounded below and above respectively. If brith and death values are from correlation matrices, −1 ≤ x₁ and . Hence, the 0D Wasserstein distance is equivalent to Euclidean distance in the q₀-dimensional convex set . Similarly, the vector of sorted death values of persistent diagram P_i is a point in the (q₁ − 1)-simplex given by (8) where x₁ and are bounded below and above respectively. Hence, the 1D Wasserstein distance is equivalent to Euclidean distance in the q₁-dimensional convex set .

2.5 Topological mean and variance

Given a collection of graphs with edge weights , the usual approach for obtaining the average network is simply averaging the edge weight matrices in an element-wise fashion

However, such average is the average of the connectivity strength. Such an approach is usually sensitive to topological outliers [21]. We address the problem through the Wasserstein distance. A similar concept was proposed in the persistent homology literature through the Wasserstein barycenter [69, 70], which is motivated by the Fréchet mean [71–74]. However, the method has not seen many applications in modeling graphs and networks.

To account for both 0D and 1D topological differences in networks, we use the sum of 0D and 1D Wasserstein distances between networks and as the topological distance (9)

The equal weights of the form (9) were used for the following reasons. Through the birth-death decomposition, a weighted graph can be topologically characterized by 0D and 1D features, with no higher-dimensional features present. However, it is unclear which feature contributes the most. Equal weighting of 0D and 1D features ensures a balanced representation without bias towards either type of feature.

Let ⌀ denote a graph with zero edge weights. Then, due to the birth-death decomposition, we have where the squared sums of all the edge weights make the interpretation straightforward. If unequal weighting is used, these relationships do not hold. Further, the Wasserstein distances D_W0 and D_W1 are equivalent to the Euclidean distances in a convex set. Therefore, squared distance is a more natural choice that satisfies the triangle inequality thus qualifying as a metric.

The sum (9) does not uniquely define networks. Like the toy example in Fig 5, we can have many topologically equivalent brain networks that give the identical distance. Thus, the average of two graphs is also not uniquely defined. The situation is analogous to Fréchet mean, which frequently does not result in a unique mean [71–74]. We introduce the concept of the topological mean for networks, defined as the minimizer according to the Wasserstein distance, mirroring how the sample mean minimizes the Euclidean distance. The squared Wasserstein distance is translation invariant such that

If we scale connectivity matrices by c, we have

Definition 1. The topological mean of networks is the graph given by (10)

Unlike the sample mean, we can have many different networks with identical topology that give the minimum. Similarly, we can define the topological variance as follows.

Definition 2. The topological variance of networks is the graph given by (11)

The topological variance can be interpreted as the variability of graphs from the topological mean . To compute the topological mean and variance, we only need to identify a network with identical topology as the topological mean or the topological variance.

Theorem 3. Consider graphs with corresponding birth-death decompositions W_i = W_ib ∪ W_id with birth sets and death sets . Then, there exists topological mean with birth-death decomposition W_b ∪ W_d with and satisfying (12)

2.6 Topological clustering

There are few studies that used the Wasserstein distance for clustering [38, 75]. The existing methods are mainly applied to geometric data without topological consideration or persistence. It is not obvious how to apply such geometric methods to cluster graph or network data. We propose to use the Wasserstein distance to cluster collection of graphs into k clusters C₁, ⋯, C_k such that

Let C = (C₁, ⋯, C_k) be the collection of clusters. Let μ_j be the topological cluster mean within C_j given by

The cluster mean is computed through Theorem 3. Just like Fréchet mean, the cluster mean is not unique in a geometric sense but only unique in a topological sense [71–74]. Let μ = (μ₁, ⋯, μ_k) be the cluster mean vector. The within-cluster distance from the cluster centers is given by (13)

If we let |C_j| to be the number of networks within cluster C_j, (13) can be written as (14) with topological cluster variance within cluster C_j. The optimal cluster is found by minimizing within-cluster distance l_W(C;μ) in (13) over every possible partition of C.

If μ is given and fixed, the identification of clusters C can be done easily by assigning each network to the closest mean. Thus the topological clustering algorithm can be written as the two-step optimization similar to the expectation maximization (EM) algorithm often used in variational inferences and likelihood methods [76]. The first step computes the cluster mean. The second step minimizes the within-cluster distance. Just like k-means clustering, the two-step optimization is then iterated until convergence. Such process converges locally.

Theorem 4. The topological clustering converges locally.

The direct algebraic proof is fairly involving and given in [17]. Here we provide a more intuitive explanation. Note D_W0 and D_W1 are Euclidean distances in convex set . Subsequently, is the Euclidean distance in the Cartesian product . Thus, our topological clustering is equivalent to k-means clustering restricted to the convex set . The convergence of topological clustering is then the direct consequence of the convergence of k-means clustering, which always converges in such a convex space. Numerically we minimize (13) by replacing the Wasserstein distance with the 2-norm between sorted vectors of birth and death values in k-means clustering.

Like k-means clustering algorithm that only converges to local minimum, there is no guarantee the topological clustering converges to the global minimum [77]. This is remedied by repeating the algorithm multiple times with different random seeds and taking the smallest possible minimum. The method is implemented as the Matlab function WS_cluster.m which inputs the collection of networks and outputs the cluster labels and clustering accuracy. Different choice of initial cluster centers may lead to different results. Thus, the algorithm may become stuck in a local minimum and may not converge to the global minimum. Thus, in actual numerical implementation, we used different initializations of centers. Then, we picked the best clustering result with the smallest within cluster distance l_W.

2.7 Validation

We validated the topological clustering in a simulation with the ground truth against k-means and hierarchical clustering [18]. We generated 4 circular patterns of identical topology (Fig 6-top) and different topology (Fig 6-bottom). Along the circles, we uniformly sampled 60 nodes and added Gaussian noise N(0, 0.3²) on the coordinates. We generated 5 random networks per group. The Euclidean distance (L₂-norm) between randomly generated points is used to build connectivity matrices for k-means and hierarchical clustering. Fig 6 shows the superposition of nodes from 20 networks. For k-means and Wasserstein graph clustering, the average result of 100 random seeds is reported.

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Fig 6.

Top: simulation study on topological equivalence. The correct clustering method should not be able to cluster them because they are all topologically equivalent. The pairwise Euclidean distance (L₂-norm) is used in k-means and hierarchical clustering. The Wasserstein distance is used in topological clustering. Bottom: simulation study on topological difference. The correct clustering method should be able to cluster them because they are all topologically different.

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2.8 Weighted Fourier series representation

The predominant method for computing time-varying correlation in time series data, particularly in neuroimaging studies, involves Sliding Windows (SW). This technique entails computing correlations between brain regions across various time windows [77–81]. However, the use of discrete windows in SW can lead to artificially high-frequency fluctuations in dynamic correlations [82]. While tapering methods can occasionally mitigate these effects [78], the correlation computations within these windows remain susceptible to the influence of outliers [83].

To circumvent these limitations, we employed the Weighted Fourier Series (WFS) representation [84, 85]. This approach extends the traditional cosine Fourier transform by incorporating an additional exponential weight. This modification effectively smooths out high-frequency noise and diminishes the Gibbs phenomenon [84, 86]. Crucially, WFS eliminates the need for sliding windows (SW) when computing time-correlated data. Given the necessity for robust signal denoising methods to ensure the efficacy of the persistent homology method across various subjects and time points, such an approach is needed. Consider an arbitrary noise signal f(t), t ∈ [0, 1], which will undergo denoising through the diffusion process.

Theorem 5. The unique solution to 1D heat diffusion: (15) on unit interval [0, 1] with initial condition h(t, s = 0) = f(t) is given by WFS: (16) where ψ₀(t) = 1, are the cosine basis and are the expansion coefficients.

The algebraic derivation is given in [84]. Note the cosine basis is orthonormal where δ_lm is Kroneker-detal taking value 1 if l = m and 0 otherwise. We can rewrite (16) as a more convenient convolution form where heat kernel K_s(t, t′) is given by

The diffusion time s is usually referred to as the kernel bandwidth and controls the amount of smoothing. Heat kernel satisfies for any t′ and s.

To reduce unwanted boundary effects near the data boundary t = 0 and t = 1 [77, 86], we project the data onto the circle with circumference 2 by the mirror reflection:

Then perform WFS on the circle.

Theorem 6. The unique solution to 1D heat diffusion: (17) on the circle with the initial periodic condition h(t, s = 0) = f(t) if t ∈ [0, 1], h(t, s = 0) = f(2 − t) if t ∈ [1, 2] is given by WFS: (18) where ψ₀(t) = 1, are the cosine basis and are the expansion coefficients.

The cosine series coefficients c_fl are estimated using the least squares method by setting up a matrix equation [84]. We set the expansion degree to equate the number of time points, which is 295. The window size of 20 TRs was used in most sliding window methods [77, 78, 87]. We matched the full width at half maximum (FWHM) of heat kernel to the window size numerically. We used the fact that diffusion time s in heat kernel approximately matches to the kernel bandwidth of Gaussian kernel as σ = s²/2 (page 144 in [88]). 20 TRs is approximately equivalent to heat kernel bandwidth of about 4.144 ⋅ 10⁻⁴ in terms of FWHM. Fig 7 displays the WFS representation of rsfMRI with different kernel bandwidths.

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Fig 7.

Left: The original and smoothed fMRI time series using WFS with degree L = 295 and different heat kernel bandwidth s. The bandwidth 4.141 × 10⁻⁴ used in this study approximately matches 20 TRs often used in the sliding window methods. Right: The dotted gray lines are correlations computed over sliding windows. The solid black lines are correlations computed using WFS.

https://doi.org/10.1371/journal.pcbi.1011869.g007

2.9 Dynamic correlation on weighted Fourier series

The weighted Fourier series representation provides a way to compute correlations dynamically without using sliding windows. Consider time series x(t) and y(t) with heat kernel K_s(t, t′). The mean and variance of signals with respect to the heat kernel are given by

Subsequently, the correlation w(t) of x(t) and y(t) is given by (19)

When the kernel is shaped as a sliding window, the correlation w(t) exactly matches the correlation computed over the sliding window. The kernelized correlation generalizes the concept of integral correlations with the additional weighting term [89]. As s → ∞, w(t) converges to the Pearson correlation computed over the whole time points. Thus, the kernel bandwidth behaves like the length of sliding window.

Theorem 7. The correlation w(t) of time series x(t) and y(t) with respect to heat kernel K_s(t, t′) is given by (20) with are the cosine series coefficients. Similarly we expand x(t)y(t), x²(t) and y²(t) using the cosine basis and obtain coefficients c_xyl, c_xxl and c_yyl.

The derivation follows by simply replacing all the terms with the WFS representation. Correlation (20) is the formula we used to compute the dynamic correlation in this study. Fig 7 displays the WFS-based dynamic correlation for different bandwidths. A similar weighted correlation was proposed in [90], where the time varying exponential weights proportional to e^t/θ with exponential decay factor θ were used. However, our exponential weight term is related to the spectral decomposition of heat kernel in the spectral domain and invariant over time. The WFS based correlation is not related to [90].

3.1 Data

The proposed method is applied in the accurate estimation of state spaces in dynamically changing functional brain networks. The 479 subjects resting-state functional magnetic resonance images (rs-fMRI) used in this paper were collected on a 3T MRI scanner (Discovery MR750, General Electric Medical Systems, Milwaukee, WI, USA) with a 32-channel RF head coil array. The 479 healthy subjects consist of 231 males and 248 females ranging in age from 13 to 25 years. The sample contains 132 monozygotic (MZ) twin pairs and 93 same-sex dizygotic (DZ) twin pairs.

The image preprocessing includes motion corrections and image alignment to the MNI template and follows [91, 92]. The resulting rs-fMRI consist of 91 × 109 × 91 isotropic voxels at 295 time points. We further parcellated the brain volume into 116 non-overlapping brain regions using the Automated Anatomical Labelling (AAL) atlas [93]. The fMRI data were averaged across voxels within each brain region, resulting in 116 average fMRI signals with 295 time points for each subject. The rs-fMRI signals were then scaled to fit to unit interval [0, 1] and treated as functional data in [0, 1].

3.2 Topological state space estimation

For p brain regions, we estimated p × p dynamically changing correlation matrices C_i(t) for the i-th subject using WFS. Let C_ij denote the vectorization of the upper triangle of p × p matrix C_i(t_j) at time point t_j into p² × 1 vector. For each fixed i, the collection of C_ij over T = 295 time points is then feed into topological clustering in identifying the recurring brain connectivity at the subject level. We are clustering individual brain networks without putting any constraint on group or twin. We compared the proposed Wasserstein clustering against the k-means clustering, which has been often used as the baseline method in the state space modeling [77, 78, 86]. After clustering, each correlation matrix C_i(t_j) is assigned integers between 1 and k. These discrete states serve as the basis for investigating the dynamic pattern of brain connectivity [94]. For the convergence of both topological clustering and k-means clustering, the clusterings were repeated 10 times with different initial centroids and the best result (smallest within-cluster distance) is reported. Fig 8-left displays the result of the topological clustering against the k-means for three subjects. 295 time points are rescaled to fit into unit interval [0, 1].

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Fig 8.

Left: The time series of estimated state spaces using the topological clustering and k-means clustering for 3 subjects. The time is normalized into unit interval [0, 1]. Right: The ratio of within-cluster to between-cluster distances. Smaller the ratio, better the clustering fit is.

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The optimal number of cluster k was determined by the elbow method [77, 78, 94, 95]. For each value of k, we computed the ratio of the within-cluster to between-cluster distances. The ratio shows the goodness-of-fit of the cluster model. The elbow method gives the largest slope change in the ratio when k = 3 in the both methods (Fig 8-right). At k = 3, the ratio is 0.034 ± 0.012 for 479 subjects for Wasserstein while it is 0.202 ± 0.047 for the k-means. The six times smaller ratio for the topological clustering demonstrates the superior model fit over k-means. Fig 9 shows the results of clustering for both methods. The k-means clustering result is based on averaging correlations of every time point and subject within each state. The resulting states in the k-means clustering are somewhat random without any biologically interpretable pattern. The topological clustering computes the topological mean of every time point and subject within each state.

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Fig 9. The estimated state spaces of dynamically changing brain networks.

The correlations are averaged over every time point and subject within each state for k-means clustering (top) and Wasserstein distance based topological clustering (bottom). In k-means clustering, the connectivity pattern of each state is somewhat random. In topological clustering, the connectivity pattern is highly symmetric even though we did not put any symmetry constraint in the clustering method.

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3.3 Twin correlations over transpositions

Using additional twin information in the data, we further investigated if the state change pattern itself is genetically heritable. As far as we are aware, there is no study on the heritability of the state change pattern itself. This requires computing twin correlations. We assume there are m MZ- and n DZ-twins. For some feature, let x_i = (x_i1, x_i2)^⊤ be the i-th twin pair in MZ-twin and y_i = (y_i1, y_i2)^⊤ be the i-th twin pair in DZ-twin. They are represented as

Let be the j-th row of , i.e., . Similarly let . Then MZ- and DZ-correlations are computed as the sample correlation

However, there is no preference for the order of twins within a pair, and we can transpose the i-th twin pair in MZ-twin such that and obtain another twin correlation [96, 97]. Ignoring symmetry, there are 2^m possible combinations in ordering the twins, which form a permutation group. The size of the permutation group grows exponentially large as the sample size increases. Computing correlations over all permutations is not even computationally feasible for large m beyond 100. Fig 10 illustrates many possible transpositions within twins. Thus, we propose a new fast online computational strategy for computing twin correlations.

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Fig 10. The schematic of transpositions on 3 MZ- and 2 DZ-twins.

a) One possible pairing. b) Transposition within a MZ-twin. c) Transposition within a DZ-twin. d) Transpositions in both MZ- and DZ-twins simultaneously. Any transposition will affect the heritability estimate so it is necessary to account for as many transpositions as possible.

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Over transposition τ_i, the correlation changes (21) incrementally. We will determine the exact increment over the transposition. The sample correlation between and involves the following functions.

The functions μ and ω are updated over transposition τ_i as

Then the MZ-twin correlation after transposition is updated as (22)

The time complexity for correlation computation is 33 operations per transposition, which is substantially lower than the computational complexity of directly computing correlations per permutation. In the numerical implementation, we sequentially apply random transpositions . This results in J different twin correlations, which are averaged. Let

The average correlation of all J transpositions is given by

In each sequential update, the average correlation can be updated iteratively as

If we use enough transpositions, the average correlation converges to the true underlying twin correlation γ^MZ for sufficiently large J. DZ-twin correlation γ^DZ is estimated similarly.

In the widely used ACE genetic model, the heritability index (HI) h, which determines the amount of variation due to genetic difference in a population, is estimated using Falconer’s formula [45, 98, 99]. MZ-twins share 100% of genes while same-sex DZ-twins share 50% of genes on average. Thus, the additive genetic factor A and the common environmental factor C are related as

HI h, which measures the contribution of A, is given by

In numerical implementation, 100 million transpositions can be easily done in 100 seconds in a desktop. Similarly, we update the DZ-correlation over the transposition.

3.4 Heritability of the state space

The heritability estimation of state space is not a trivial task since the estimated state does not synchronize across twins making the task fairly difficult. Fig 11 displays the state visits in randomly selected 3 MZ- and 3 DZ-twins. However, the time series of state changes do not synchronize within twins. This is likely a reason for the lack of reported heritability of the state space in the literature.

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Fig 11. State visits for 3 MZ-twins (left) and 3 DZ-twins (right) obtained from the baseline k-means clustering.

We are interested in determining the heritability of such state changes. Unfortunately, even within a twin, the time series of state change do not synchronize, making the task extremely challenging.

https://doi.org/10.1371/journal.pcbi.1011869.g011

For each subject, we computed the average correlation of each state, where the average is taken over all time points within each state. The correlation matrices are then used as the input to the transposition based twin correlations [98]. Subsequently, we computed the MZ- and DZ-twin correlations within each state (Fig 12). The MZ-twin correlations (Fig 12-top) are densely observed in many connections while there are no DZ-twin correlations (Fig 12-middle) observed above 0.3. We then computed the heritability index (HI) of each state (Fig 12-bottom). The heritability of the first state is characterized by strong lateralization of the hemisphere connections. The heritability of the second state is characterized by front and back connections. We believe the topological approach provides far more accurate and stable heritability index maps for dynamically changing state, which are biologically interpretable.

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Fig 12. MZ-correlation (top) and DZ-correlation (middle) in each state obtained through topological clustering in Fig 9.

There is no DZ-correlation above 0.3 and not displayed. The heritability index (HI) is determined by the twice the difference in twin correlations. HI of each state shows the extensive genetic contribution of dynamically changing states. The first state is characterized by prominent bilateral connections between the left and right hemispheres, whereas the second state primarily features front-back connections.

https://doi.org/10.1371/journal.pcbi.1011869.g012

We reported 10 connections that give the highest HI values in all three states in Tables 1, 2 and 3. Although numerous studies report high heritability for anatomical features such as gray matter density, there are few rs-fMRI studies reporting heritability of rs-fMRI [52, 58]. Most of these studies report low HI compared to our high HI. [52] reported HI of 0.104 in the left cerebellum, 0.304 in the right cerebellum, 0.331 in the left temporal parietal region, 0.420 in the right temporal parietal region. [58] reported HI of 0.41 in the connection between the posterior cingulate cortex and right inferior parietal cortex in the default mode network involving 79 MZ- and 46 same-sex DZ-twins. Other connections are all reporting very low HI below 0.24. We believe our topological method is clustering topologically similar functional network patterns and significantly boost genetic signals.

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Table 1. 10 connections with the highest heritability index for state 1.

Connections are sorted with respect to HI values.

https://doi.org/10.1371/journal.pcbi.1011869.t001

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Table 2. 10 connections with the highest heritability index for state 2.

Connections are sorted with respect to HI values.

https://doi.org/10.1371/journal.pcbi.1011869.t002

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Table 3. 10 connections with the highest heritability index for state 3.

Connections are sorted with respect to HI values.

https://doi.org/10.1371/journal.pcbi.1011869.t003

3.5 Null test on twin study design

Because we are reporting significantly higher diffused heritability compared to existing literature [52, 57, 58], we performed the null test to check the validity of our analysis pipeline further. We generated the null MZ-twin data by randomly pairing each MZ individual with another, excluding their own twin. Such a permutation is generated by derangement, which is a permutation of the elements of a set, such that no element appears in its original position [100]. In other words, if we have a set of distinct items and you rearrange them, a derangement means none of the items are in the spot they started in. The null DZ-twin data is constructed similarly. Such null data should not show any genetic relations beyond random chances. On the null data, we recomputed the twin correlations and the heritability index, following the same pipeline as before. Fig 13 shows an example of one possible derangement out of exponentially many such permutations. For m MZ-twin pairs, there are number of derangements. For the null test, we generated 1000 derangements that followed the proposed pipeline in computing average MZ- and DZ-correlations in each state. We used the Wasserstein distance in measuring the topological discrepancy. Fig 14 displays the normalized histogram of the Wasserstein distance between average MZ- and DZ-twin correlations within each state over 1000 derangements. Because the generated null data has no genetic signal, we are basically computing the Wasserstein distance between two random connectivity matrices. In comparison, the observed Wasserstein distance (red line) between average MZ- and DZ-twin correlation shows huge topological differences. None of the derangements show the large wide spread signals as our observation. We conclude that what we observe is genetic signal and cannot possibly be produced by random chance.

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Fig 13. MZ-correlation (top) and DZ-correlation (middle) in each state obtained through topological clustering in Fig 9.

There is no MZ-correlation above 0.3 and not displayed. The heritability index (HI) is determined by the twice the difference in twin correlations. HI of each state shows extensive genetic contribution of dynamically changing states.

https://doi.org/10.1371/journal.pcbi.1011869.g013

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Fig 14. The normalized histogram of the Wasserstein distance between average MZ- and DZ-twin correlations within each state over 1000 derangements.

Since the generated null data has no genetic signal, we are basically computing the Wasserstein distance between two connectivity matrices with random noises. In comparison, the observed Wasserstein distance (dotted line) between average MZ- and DZ-twin correlation shows huge topological differences.

https://doi.org/10.1371/journal.pcbi.1011869.g014