dyconnmap.graphs package¶
Submodules¶
dyconnmap.graphs.gdd module¶
Graph Diffusion Distance
The Graph Diffusion Distance (GDD) metric (Hammond2013) is a measure of distance between two (positive) weighted graphs based on the Laplacian exponential diffusion kernel. The notion backing this metric is that two graphs are similar if they emit comparable patterns of information transmission.
This distance is computed by searching for a diffusion time \(t\) that maximizes the value of the Frobenius norm between the two diffusion kernels. The Laplacian operator is defined as \(L = D - A\), where \(A\) is the positive symmetric data matrix and \(D\) is a diagonal degree matrix for the adjacency matrix \(A\). The diffusion process (per vertex) on the adjacency matrix \(A\) is governed by a time-varying vector \(u(t)∈ R^N\). Thus, between each given pair of (vertices’) weights \(i\) and \(j\), their flux is quantified by \(a_{ij} (u_i (t)u_j (t))\). The grand sum of these interactions is given by \(\hat{u}_j(t)=\sum_i{a_{ij}(u_i(t)u_j(t))=-Lu(t)}\). Given the initial condition \(u^0,t=0\) this sum has the following analytic solution \(u(t)=exp(-tL)u^0\). The resulting matrix is known as the Laplacian exponential diffusion kernel. Letting the diffusion process run for \(t\) time we compute and store the diffusion patterns in each column. Finally, the actual distance measure between two adjacency matrices \(A_1\) and \(A_2\), at diffusion time \(t\) is given by:
where \(‖∙‖_F\) is the Frobenious norm.
Notes¶
Based on the code accompanied the original paper. Available at https://www.researchgate.net/publication/259621918_A_Matlab_code_for_computing_the_GDD_presented_in_the_paper
- Hammond2013
Hammond, D. K., Gur, Y., & Johnson, C. R. (2013, December). Graph diffusion distance: A difference measure for weighted graphs based on the graph Laplacian exponential kernel. In Global Conference on Signal and Information Processing (GlobalSIP), 2013 IEEE (pp. 419-422). IEEE.
-
dyconnmap.graphs.gdd.
graph_diffusion_distance
(a, b, threshold=1e-14)[source]¶ Graph Diffusion Distance
- Parameters
a : array-like, shape(N, N)
Weighted matrix.
b : array-like, shape(N, N)
Weighted matrix.
threshold : float
A threshold to filter out the small eigenvalues. If the you get NaN or INFs, try lowering this threshold.
- Returns
gdd : float
The estimated graph diffusion distance.
xopt : float
Parameters (over given interval) which minimize the objective function. (see
scipy.optimize.fmindbound
)
dyconnmap.graphs.imd module¶
Ipsen-Mikhailov Distance
Given two graphs, this method quantifies their difference by comparing their spectral densities. This spectral density is computed as the sum of Lorentz distributions \(\rho(\omega)\):
Where \(\gamma\) is the bandwidth, and \(K\) a normalization constant such that \(\int_{0}^{\infty}\rho(\omega)d\omega=1\). The spectral distance between two graphs \(G\) and \(H\) with densities \(\rho_G(\omega)\) and \(\rho_H(\omega)\) respectively, is defined as:
- Ipsen2004
Ipsen, M. (2004). Evolutionary reconstruction of networks. In Function and Regulation of Cellular Systems (pp. 241-249). Birkhäuser, Basel.
- Donnat2018
Donnat, C., & Holmes, S. (2018). Tracking Network Dynamics: a review of distances and similarity metrics. arXiv preprint arXiv:1801.07351.
dyconnmap.graphs.laplacian_energy module¶
Laplcian Energy
The Laplcian energy (LE) for a graph \(G\) is computed as
Where \(\mu_i\) denote the eigenvalue associated with the node of the Laplcian matrix of \(G\) (Laplcian spectrum) and \(\frac{2m}{n}\) the average vertex degree.
For a details please go through the original work (Gutman2006_).
dyconnmap.graphs.mi module¶
Mutual Information
Normalized Mutual Information (NMI) proposed by [Strehl2002] as an extension to Mutual Information [cover] to enable interpretations and comparisons between two partitions. Given the entropies \(H(P^a)=-\sum_{i=1}^{k_a}{\frac{n_i^a}{n}\log(\frac{n_i^a}{n})}\) where \(n_i^a\) represents the number of patterns in group \(C_i^a \in P^a\) (and computed for \(H(P^b)\) accordingly); the initial matching of these two groups \(P^a\) and \(P^b\) in terms of mutual information is [Fred2005, Strehl2002]:
Where \(n_{ij}^{ab}\) denotes the number of shared patterns between the clusters \(C_i^a\) and \(C_j^b\). By exploiting the definition of mutual information, the following property holds true: \(I(P^a,P^b) \leq \frac{H(P^a)+H(P^b)}{2}\). This leads to the definition of NMI as:
- Fred2005
Fred, A. L., & Jain, A. K. (2005). Combining multiple clusterings using evidence accumulation. IEEE transactions on pattern analysis and machine intelligence, 27(6), 835-850.
- Strehl2002
Strehl, A., & Ghosh, J. (2002). Cluster ensembles—a knowledge reuse framework for combining multiple partitions. Journal of machine learning research, 3(Dec), 583-617.
-
dyconnmap.graphs.mi.
mutual_information
(indices_a, indices_b)[source]¶ Mutual Information
- Parameters
indices_a : array-like, shape(n_samples)
Symbolic time series.
indices_b : array-like, shape(n_samples)
Symbolic time series.
- Returns
MI : float
Mutual information.
NMI : float
Normalized mutual information.
dyconnmap.graphs.mpc module¶
Multilayer Participation Coefficient
- Guillon2016
Guillon, J., Attal, Y., Colliot, O., La Corte, V., Dubois, B., Schwartz, D., … & Fallani, F. D. V. (2017). Loss of brain inter-frequency hubs in Alzheimer’s disease. Scientific reports, 7(1), 10879.
-
dyconnmap.graphs.mpc.
multilayer_pc_degree
(mlgraph)[source]¶ Multilayer Participation Coefficient (Degree)
- Parameters
mlgraph : array-like, shape(n_layers, n_rois, n_rois)
A multilayer (undirected) graph. Each layer consists of a graph.
- Returns
mpc : array-like
Participation coefficient based on the degree of the layers’ nodes.
-
dyconnmap.graphs.mpc.
multilayer_pc_gamma
(mlgraph)[source]¶ Multilayer Participation Coefficient method from Guillon et al.
- Parameters
mlgraph : array-like, shape(n_layers, n_rois, n_rois)
A multilayer graph.
- Returns
gamma : array-like, shape(n_layers*n_rois, n_layers*n_rois)
Returns the original multilayer graph flattened, with the off diagional containing the estimated interlayer multilayer participation coefficient.
-
dyconnmap.graphs.mpc.
multilayer_pc_strength
(mlgraph)[source]¶ Multilayer Participation Coefficient (Strength)
- Parameters
mlgraph : array-like, shape(n_layers, n_rois, n_rois)
A multilayer (undirected) graph. Each layer consists of a graph.
- Returns
mpc : array-like
Participation coefficient based on the strength of the layers’ nodes.
dyconnmap.graphs.nodal module¶
Nodal network features
dyconnmap.graphs.spectral_euclidean_distance module¶
Spectral Euclidean Distance
The spectral distance between graphs is simply the Euclidean distance between the spectra.
Notes¶
The input graphs can be a standard adjency matrix, or a variant of Laplacian.
- Wilson2008
Wilson, R. C., & Zhu, P. (2008). A study of graph spectra for comparing graphs and trees. Pattern Recognition, 41(9), 2833-2841.
dyconnmap.graphs.spectral_k_distance module¶
Spectral-K Distance
Given two graphs \(G\) and \(H\), we can use their \(k\) largest positive eigenvalues of their Laplacian counterparts to compute their distance.
Where \(g\) and \(h\) denote the spectrums of the Laplacian matrices.
This measure is non-negative, separated, symmetric and it satisfies the triangle inequality.
- Jakobson2000
Jakobson, D., & Rivin, I. (2000). Extremal metrics on graphs I. arXiv preprint math/0001169.
- Pincombe2007
Pincombe, B. (2007). Detecting changes in time series of network graphs using minimum mean squared error and cumulative summation. ANZIAM Journal, 48, 450-473.
-
dyconnmap.graphs.spectral_k_distance.
spectral_k_distance
(X, Y, k)[source]¶ Spectral-K Distance
Use the largest \(k\) eigenvalues of the given graphs to compute the distance between them.
- Parameters
X : array-like, shape(N, N)
A weighted matrix.
Y : array-like, shape(N, N)
A weighted matrixY
k : int
Largest k eigenvalues to use.
- Returns
distance : float
Estimated distance based on selected largest eigenvalues.
dyconnmap.graphs.threshold module¶
Thresholding schemes
Notes¶
This is a direct translation from `Data Driven Topological Filtering of Brain Networks via Orthogonal Minimal Spanning Trees <https://github.com/stdimitr/topological_filtering_networks>’
Original author is Stravros Dimitriadis <stidimitriadis@gmail.com>
-
dyconnmap.graphs.threshold.
k_core_decomposition
(mtx, threshold)[source]¶ Threshold a binary graph based on the detected k-cores.
- Alvarez2006
Alvarez-Hamelin, J. I., Dall’Asta, L., Barrat, A., & Vespignani, A. (2006). Large scale networks fingerprinting and visualization using the k-core decomposition. In Advances in neural information processing systems (pp. 41-50).
- Hagman2008
Hagmann, P., Cammoun, L., Gigandet, X., Meuli, R., Honey, C. J., Wedeen, V. J., & Sporns, O. (2008). Mapping the structural core of human cerebral cortex. PLoS biology, 6(7), e159.
- Parameters
mtx : array-like, shape(N, N)
Binary matrix.
threshold : int
Degree threshold.
- Returns
k_cores : array-like, shape(N, 1)
A binary matrix of the decomposed cores.
-
dyconnmap.graphs.threshold.
threshold_eco
(mtx)[source]¶ Ecological Filtering
- Fallani2017
Fallani, F. D. V., Latora, V., & Chavez, M. (2017). A topological criterion for filtering information in complex brain networks. PLoS computational biology, 13(1), e1005305.
-
dyconnmap.graphs.threshold.
threshold_global_cost_efficiency
(mtx, iterations)[source]¶ Threshold a graph based on the Global Efficiency - Cost formula.
- Basset2009
Bassett, D. S., Bullmore, E. T., Meyer-Lindenberg, A., Apud, J. A., Weinberger, D. R., & Coppola, R. (2009). Cognitive fitness of cost-efficient brain functional networks. Proceedings of the National Academy of Sciences, 106(28), 11747-11752.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
iterations : int
Number of steps, as a resolution when search for optima.
- Returns
binary_mtx : array-like, shape(N, N)
A binary mask matrix.
threshold : float
The threshold that maximizes the global cost efficiency.
global_cost_eff_max : float
Global cost efficiency.
efficiency : float
Global efficiency.
cost_max : float
Cost of the network at the maximum global cost efficiency
-
dyconnmap.graphs.threshold.
threshold_mean_degree
(mtx, threshold_mean_degree)[source]¶ Threshold a graph based on the mean degree.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
threshold_mean_degree : int
Mean degree threshold.
- Returns
binary_mtx : array-like, shape(N, N)
A binary mask matrix.
-
dyconnmap.graphs.threshold.
threshold_mst_mean_degree
(mtx, avg_degree)[source]¶ Threshold a graph based on mean using minimum spanning trees.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
avg_degree : float
Mean degree threshold.
- Returns
binary_mtx : array-like, shape(N, N)
A binary mask matrix.
-
dyconnmap.graphs.threshold.
threshold_omst_global_cost_efficiency
(mtx, n_msts=None)[source]¶ Threshold a graph by optimizing the formula GE-C via orthogonal MSTs.
- Dimitriadis2017a
Dimitriadis, S. I., Salis, C., Tarnanas, I., & Linden, D. E. (2017). Topological Filtering of Dynamic Functional Brain Networks Unfolds Informative Chronnectomics: A Novel Data-Driven Thresholding Scheme Based on Orthogonal Minimal Spanning Trees (OMSTs). Frontiers in neuroinformatics, 11.
- Dimitriadis2017n
Dimitriadis, S. I., Antonakakis, M., Simos, P., Fletcher, J. M., & Papanicolaou, A. C. (2017). Data-driven Topological Filtering based on Orthogonal Minimal Spanning Trees: Application to Multi-Group MEG Resting-State Connectivity. Brain Connectivity, (ja).
- Basset2009
Bassett, D. S., Bullmore, E. T., Meyer-Lindenberg, A., Apud, J. A., Weinberger, D. R., & Coppola, R. (2009). Cognitive fitness of cost-efficient brain functional networks. Proceedings of the National Academy of Sciences, 106(28), 11747-11752.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
n_msts : int or None
Maximum number of OMSTs to compute. Default None; an exhaustive computation will be performed.
- Returns
nCIJtree : array-like, shape(n_msts, N, N)
A matrix containing all the orthogonal MSTs.
CIJtree : array-like, shape(N, N)
Resulting graph.
degree : float
The mean degree of the resulting graph.
global_eff : float
Global efficiency of the resulting graph.
global_cost_eff_max : float
The value where global efficiency - cost is maximized.
cost_max : float
Cost of the network at the maximum global cost efficiency.
-
dyconnmap.graphs.threshold.
threshold_shortest_paths
(mtx, treatment=False)[source]¶ Threshold a graph via via shortest path identification using Dijkstra’s algorithm.
- Dimitriadis2010
Dimitriadis, S. I., Laskaris, N. A., Tsirka, V., Vourkas, M., Micheloyannis, S., & Fotopoulos, S. (2010). Tracking brain dynamics via time-dependent network analysis. Journal of neuroscience methods, 193(1), 145-155.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
treatment : boolean
Convert the weights to distances by inversing the matrix. Also, fill the diagonal with zeroes. Default false.
- Returns
binary_mtx : array-like, shape(N, N)
A binary mask matrix.
dyconnmap.graphs.vi module¶
Variation of Information
Variation of Information (VI) [Meilla2007] is an information theoretic criterion for comparing two partitions. It is based on the classic notions of entropy and mutual information. In a nutshell, VI measures the amount of information that is lost or gained in changing from clustering \(A\) to clustering \(B\). VI is a true metric, is always non-negative and symmetric. The following formula is used to compute the VI between two groups:
Where \(H\) denotes the entropy computed for each partition separately, and \(I\) the mutual information between clusterings \(A\) and \(B\).
The resulting distance score can be adjusted to bound it between \([0, 1]\) as follows:
- Meilla2007
Meilă, M. (2007). Comparing clusterings—an information based distance. Journal of multivariate analysis, 98(5), 873-895.
- Dimitriadis2009
Dimitriadis, S. I., Laskaris, N. A., Del Rio-Portilla, Y., & Koudounis, G. C. (2009). Characterizing dynamic functional connectivity across sleep stages from EEG. Brain topography, 22(2), 119-133.
- Dimitriadis2012
Dimitriadis, S. I., Laskaris, N. A., Michael Vourkas, V. T., & Micheloyannis, S. (2012). An EEG study of brain connectivity dynamics at the resting state. Nonlinear Dynamics-Psychology and Life Sciences, 16(1), 5.
Module contents¶
-
dyconnmap.graphs.
graph_diffusion_distance
(a, b, threshold=1e-14)[source]¶ Graph Diffusion Distance
- Parameters
a : array-like, shape(N, N)
Weighted matrix.
b : array-like, shape(N, N)
Weighted matrix.
threshold : float
A threshold to filter out the small eigenvalues. If the you get NaN or INFs, try lowering this threshold.
- Returns
gdd : float
The estimated graph diffusion distance.
xopt : float
Parameters (over given interval) which minimize the objective function. (see
scipy.optimize.fmindbound
)
-
dyconnmap.graphs.
variation_information
(indices_a, indices_b)[source]¶ Variation of Information
- Parameters
indices_a : array-like, shape(n_samples)
Symbolic time series.
indices_b : array-like, shape(n_samples)
Symbolic time series.
- Returns
vi : float
Variation of information.
-
dyconnmap.graphs.
mutual_information
(indices_a, indices_b)[source]¶ Mutual Information
- Parameters
indices_a : array-like, shape(n_samples)
Symbolic time series.
indices_b : array-like, shape(n_samples)
Symbolic time series.
- Returns
MI : float
Mutual information.
NMI : float
Normalized mutual information.
-
dyconnmap.graphs.
threshold_mean_degree
(mtx, threshold_mean_degree)[source]¶ Threshold a graph based on the mean degree.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
threshold_mean_degree : int
Mean degree threshold.
- Returns
binary_mtx : array-like, shape(N, N)
A binary mask matrix.
-
dyconnmap.graphs.
threshold_mst_mean_degree
(mtx, avg_degree)[source]¶ Threshold a graph based on mean using minimum spanning trees.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
avg_degree : float
Mean degree threshold.
- Returns
binary_mtx : array-like, shape(N, N)
A binary mask matrix.
-
dyconnmap.graphs.
threshold_shortest_paths
(mtx, treatment=False)[source]¶ Threshold a graph via via shortest path identification using Dijkstra’s algorithm.
- Dimitriadis2010
Dimitriadis, S. I., Laskaris, N. A., Tsirka, V., Vourkas, M., Micheloyannis, S., & Fotopoulos, S. (2010). Tracking brain dynamics via time-dependent network analysis. Journal of neuroscience methods, 193(1), 145-155.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
treatment : boolean
Convert the weights to distances by inversing the matrix. Also, fill the diagonal with zeroes. Default false.
- Returns
binary_mtx : array-like, shape(N, N)
A binary mask matrix.
-
dyconnmap.graphs.
k_core_decomposition
(mtx, threshold)[source]¶ Threshold a binary graph based on the detected k-cores.
- Alvarez2006
Alvarez-Hamelin, J. I., Dall’Asta, L., Barrat, A., & Vespignani, A. (2006). Large scale networks fingerprinting and visualization using the k-core decomposition. In Advances in neural information processing systems (pp. 41-50).
- Hagman2008
Hagmann, P., Cammoun, L., Gigandet, X., Meuli, R., Honey, C. J., Wedeen, V. J., & Sporns, O. (2008). Mapping the structural core of human cerebral cortex. PLoS biology, 6(7), e159.
- Parameters
mtx : array-like, shape(N, N)
Binary matrix.
threshold : int
Degree threshold.
- Returns
k_cores : array-like, shape(N, 1)
A binary matrix of the decomposed cores.
-
dyconnmap.graphs.
threshold_global_cost_efficiency
(mtx, iterations)[source]¶ Threshold a graph based on the Global Efficiency - Cost formula.
- Basset2009
Bassett, D. S., Bullmore, E. T., Meyer-Lindenberg, A., Apud, J. A., Weinberger, D. R., & Coppola, R. (2009). Cognitive fitness of cost-efficient brain functional networks. Proceedings of the National Academy of Sciences, 106(28), 11747-11752.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
iterations : int
Number of steps, as a resolution when search for optima.
- Returns
binary_mtx : array-like, shape(N, N)
A binary mask matrix.
threshold : float
The threshold that maximizes the global cost efficiency.
global_cost_eff_max : float
Global cost efficiency.
efficiency : float
Global efficiency.
cost_max : float
Cost of the network at the maximum global cost efficiency
-
dyconnmap.graphs.
threshold_omst_global_cost_efficiency
(mtx, n_msts=None)[source]¶ Threshold a graph by optimizing the formula GE-C via orthogonal MSTs.
- Dimitriadis2017a
Dimitriadis, S. I., Salis, C., Tarnanas, I., & Linden, D. E. (2017). Topological Filtering of Dynamic Functional Brain Networks Unfolds Informative Chronnectomics: A Novel Data-Driven Thresholding Scheme Based on Orthogonal Minimal Spanning Trees (OMSTs). Frontiers in neuroinformatics, 11.
- Dimitriadis2017n
Dimitriadis, S. I., Antonakakis, M., Simos, P., Fletcher, J. M., & Papanicolaou, A. C. (2017). Data-driven Topological Filtering based on Orthogonal Minimal Spanning Trees: Application to Multi-Group MEG Resting-State Connectivity. Brain Connectivity, (ja).
- Basset2009
Bassett, D. S., Bullmore, E. T., Meyer-Lindenberg, A., Apud, J. A., Weinberger, D. R., & Coppola, R. (2009). Cognitive fitness of cost-efficient brain functional networks. Proceedings of the National Academy of Sciences, 106(28), 11747-11752.
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
n_msts : int or None
Maximum number of OMSTs to compute. Default None; an exhaustive computation will be performed.
- Returns
nCIJtree : array-like, shape(n_msts, N, N)
A matrix containing all the orthogonal MSTs.
CIJtree : array-like, shape(N, N)
Resulting graph.
degree : float
The mean degree of the resulting graph.
global_eff : float
Global efficiency of the resulting graph.
global_cost_eff_max : float
The value where global efficiency - cost is maximized.
cost_max : float
Cost of the network at the maximum global cost efficiency.
-
dyconnmap.graphs.
threshold_eco
(mtx)[source]¶ Ecological Filtering
- Fallani2017
Fallani, F. D. V., Latora, V., & Chavez, M. (2017). A topological criterion for filtering information in complex brain networks. PLoS computational biology, 13(1), e1005305.
-
dyconnmap.graphs.
nodal_global_efficiency
(mtx)[source]¶ Nodal Global Efficiency
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
- Returns
nodal_ge : array-like, shape(N, 1)
The computed nodal global efficiency.
-
dyconnmap.graphs.
im_distance
(X, Y, bandwidth=1.0)[source]¶ - Parameters
X : array-like, shape(N, N)
A weighted matrix.
Y : array-like, shape(N, N)
A weighted matrix.
bandwidth : float
Bandwidth of the kernel. Default 1.0.
- Returns
distance : float
The estimated Ipsen-Mikhailov distance.
-
dyconnmap.graphs.
spectral_k_distance
(X, Y, k)[source]¶ Spectral-K Distance
Use the largest \(k\) eigenvalues of the given graphs to compute the distance between them.
- Parameters
X : array-like, shape(N, N)
A weighted matrix.
Y : array-like, shape(N, N)
A weighted matrixY
k : int
Largest k eigenvalues to use.
- Returns
distance : float
Estimated distance based on selected largest eigenvalues.
-
dyconnmap.graphs.
spectral_euclidean_distance
(X, Y)[source]¶ - Parameters
X : array-like, shape(N, N)
A weighted matrix.
Y : array-like, shape(N, N)
A weighted matrix:
- Returns
distance : float
The euclidean distance between the two spectrums.
-
dyconnmap.graphs.
laplacian_energy
(mtx)[source]¶ Laplacian Energy
- Parameters
mtx : array-like, shape(N, N)
Symmetric, weighted and undirected connectivity matrix.
- Returns
le : float
The Laplacian Energy.
-
dyconnmap.graphs.
multilayer_pc_strength
(mlgraph)[source]¶ Multilayer Participation Coefficient (Strength)
- Parameters
mlgraph : array-like, shape(n_layers, n_rois, n_rois)
A multilayer (undirected) graph. Each layer consists of a graph.
- Returns
mpc : array-like
Participation coefficient based on the strength of the layers’ nodes.
-
dyconnmap.graphs.
multilayer_pc_degree
(mlgraph)[source]¶ Multilayer Participation Coefficient (Degree)
- Parameters
mlgraph : array-like, shape(n_layers, n_rois, n_rois)
A multilayer (undirected) graph. Each layer consists of a graph.
- Returns
mpc : array-like
Participation coefficient based on the degree of the layers’ nodes.
-
dyconnmap.graphs.
multilayer_pc_gamma
(mlgraph)[source]¶ Multilayer Participation Coefficient method from Guillon et al.
- Parameters
mlgraph : array-like, shape(n_layers, n_rois, n_rois)
A multilayer graph.
- Returns
gamma : array-like, shape(n_layers*n_rois, n_layers*n_rois)
Returns the original multilayer graph flattened, with the off diagional containing the estimated interlayer multilayer participation coefficient.