Analysis of connectivity matrices

Tools for the analysis of structural connectivity and functional connectivity matrices.

Functions:

`Laplacian`(C[, asAdjacency])	Laplacian matrix of a graph from the connectivity matrix.
`Transition`(C[, asAdjacency])	Trasition matrix from the connectivity matrix.
`adjacencyMatrix`(C)	Boolean logic in a 2D float array.
`applyMeanDelay`(D, C[, mean_delay])	Apply the mean_delay scaling to the matrix of disatances D.
`booleanDegree`(C)	Sum of the rows of the boolean adjacency matrix.
`constructErdosRenyiConnectome`(No_nodes[, p, ...])	Construct a random connected network
`degreeMatrix`(C)	Diagonal degree matrix.
`eigen`(C[, zero_threshold])	Calculate and sort the eigenvalues and eigenvectors of the matrix C.
`intensities`(C)	Sum of the rows of the connectivity matrix.
`loadConnectome`(No_nodes[, filename, field])	Load a square connectivity matrix
`loadDelays`(No_nodes[, filename, field])	Load the matrix of delays between nodes
`loadLabels`([filename, field])	Load labels of the graph nodes
`orthoMatrix`(eig_value, eig_vector)
`vonNewmanDensity`(L[, beta])
`vonNewmanEntropy`(L[, beta])
`vonNewmanRelativeEntropy`(L, J[, beta])

analysis.connectivityMatrices.Laplacian(C, asAdjacency=False)

Laplacian matrix of a graph from the connectivity matrix.

Parameters

C (2D float array) – Connectivity matrix.
asAdjacency (boolean) – If True, uses the binary adajcency matrix. Then, it matters the connection existence, and it not matters the weight.

Returns

L – Laplacian matrix.

Return type

2D float array

analysis.connectivityMatrices.Transition(C, asAdjacency=False)

Trasition matrix from the connectivity matrix.

Parameters

C (2D float array) – Connectivity matrix.
asAdjacency (boolean) – If True, uses the binary adajcency matrix. Then, it matters the connection existence, and it not matters the weight.

Returns

T – Transition matrix.

Return type

2D float array

analysis.connectivityMatrices.adjacencyMatrix(C)

Boolean logic in a 2D float array.

Parameters: C (2D float array) – Conectome or adjacency matrix.
Returns: A – Boolean adjacency matrix, 1 if c[i,j]!=0.
Return type: 2D int array

analysis.connectivityMatrices.applyMeanDelay(D, C, mean_delay=1.0)

Apply the mean_delay scaling to the matrix of disatances D. Take in account for the mean of D only the nonzero elements of C.

Parameters

D (float, 2D array) – Distances matrix (meters).
C (float, 2D array) – Structural connectivity matrix (a. u.).
mean_delay (float, optional) – The mean_delay . The default is 1.0.

Returns

D – DESCRIPTION.

Return type

TYPE

analysis.connectivityMatrices.booleanDegree(C)

Sum of the rows of the boolean adjacency matrix.

Parameters: C (2D float array) – Conectivity matrix.
Returns: k_i – degree of each node.
Return type: 1D int array

analysis.connectivityMatrices.constructErdosRenyiConnectome(No_nodes, p=1, seed=2)

Construct a random connected network

Parameters

No_nodes (int) – Number of nodes.
p (float, optional) – Degree of connectivity. The default is 1, corresponds to fully-connected network. A value of 0, is a empty connectome (full of zeros).

Returns

C – Adjacency matrix with the defined connectivity degree for each node.

Return type

2D float array (boolean)

analysis.connectivityMatrices.degreeMatrix(C)

Diagonal degree matrix.

Parameters: C (2D float array) – Conectome or Adjacency matrix.
Returns: degree_matrix – Diagonal matrix.
Return type: 2D float array

analysis.connectivityMatrices.eigen(C, zero_threshold=1e-09)

Calculate and sort the eigenvalues and eigenvectors of the matrix C.

Parameters

C (float 2D square array) – Connectivity or Laplacian matrix.
zero_threshold (float. Default value 1e-9) – Threshold for smaller eigenvalues. Any eigenvalue with absoulute value lower than zero_threshold is considered a zero eigenvalue.

Returns

eig_values (1D complex array) – sorted eigenvalues of C.
eig_vectors (2D complex array) – eigenvectors of C sorted by eigenvalues.
algebraic_connectivty (float) – Second eigenvalue of C.
con_comp (int) – Connected components of the network (Quantity of zero eigenvalues)

analysis.connectivityMatrices.intensities(C)

Sum of the rows of the connectivity matrix.

Parameters: C (2D float array) – Conectivity matrix.
Returns: intensities – sum of the rows.
Return type: 1D float array

analysis.connectivityMatrices.loadConnectome(No_nodes, filename='../input_data/AAL_matrices.mat', field='C')

Load a square connectivity matrix

Parameters

No_nodes (int) – Number of nodes to load.
filename (String, optional) – The filename, including the directory, where the connection matrix is stored. The default is the AAL90 filename
field (String, optional) – The file of the filename that contains the connection weights matrix. The default is ‘C’.

Returns

C – Connectome matrix.

Return type

2D float array

analysis.connectivityMatrices.loadDelays(No_nodes, filename='../input_data/AAL_matrices.mat', field='D')

Load the matrix of delays between nodes

Parameters

No_nodes (int) – Number of nodes to load.
filename (String, optional) – The filename, including the directory, where the delay matrix is stored. The default is the AAL90 filename
field (String, optional) – The file of the filename that contains the delays matrix. The default is ‘D’.

Returns

D – Matrix of delays, unit: seconds. (Or could be meters, if the mean_delay parameter has unit second/meter.)

Return type

2D float array

analysis.connectivityMatrices.loadLabels(filename='../input_data/AAL_labels.mat', field='label90')

Load labels of the graph nodes

Parameters

filename (String, optional) – The filename, including the directory, where the labels are stored. The default is ‘../input_data/AAL_labels.mat’.
field (String, optional) – The name of the file inside ‘filename’ that contains the labels. The default is ‘label90’ for the AAL90 connectivity matrix.

Returns

labels – Physiologycall related names of the oscillatory nodes.

Return type

String array or list

analysis.connectivityMatrices.orthoMatrix(eig_value, eig_vector)

analysis.connectivityMatrices.vonNewmanDensity(L, beta=1)

analysis.connectivityMatrices.vonNewmanEntropy(L, beta=1)

analysis.connectivityMatrices.vonNewmanRelativeEntropy(L, J, beta=1)