pynets.statistics.individual package

Submodules

pynets.statistics.individual.algorithms module

Created on Tue Nov 7 10:40:07 2017 Copyright (C) 2017 @author: Derek Pisner

pynets.statistics.individual.algorithms.average_local_efficiency(G, weight='weight', engine='nx')

Return the average local efficiency of all of the nodes in the G

Parameters

GObj

NetworkX graph.

Returns

average_local_efficiencyfloat

Average local efficiency of G.

Notes

Adapted from NetworkX to incorporate weight parameter.

References

1

Latora, V., and Marchiori, M. (2001). Efficient behavior of small-world networks. Physical Review Letters 87.

2

Latora, V., and Marchiori, M. (2003). Economic small-world behavior in weighted networks. Eur Phys J B 32, 249-263.

pynets.statistics.individual.algorithms.average_shortest_path_length_fast(G, weight='weight')

pynets.statistics.individual.algorithms.average_shortest_path_length_for_all(G)

Helper function, in the case of graph disconnectedness, that returns the average shortest path length, calculated iteratively for each distinct subgraph of the G.

Parameters

GObj

NetworkX graph.

Returns

average_shortest_path_lengthfloat

The length of the average shortest path for G.

pynets.statistics.individual.algorithms.collect_pandas_df_make(net_mets_csv_list, ID, subnet, plot_switch, embed=False, create_summary=False)

Summarize list of pickled pandas dataframes of graph metrics unique to each unique combination of metaparameters.

Parameters

net_mets_csv_listlist

List of file paths to pickled pandas dataframes as themselves.

IDstr

A subject id or other unique identifier.

subnetstr

Resting-state subnet based on Yeo-7 and Yeo-17 naming (e.g. ‘Default’) used to filter nodes in the study of brain subgraphs.

plot_switchbool

Activate summary plotting (histograms, central tendency, AUC, etc.)

Returns

combination_completebool

If True, then data integration completed successfully.

References

1

Drakesmith, M., Caeyenberghs, K., Dutt, A., Lewis, G., David, A. S., & Jones, D. K. (2015). Overcoming the effects of false positives and threshold bias in graph theoretical analyses of neuroimaging data. NeuroImage. https://doi.org/10.1016/j.neuroimage.2015.05.011

pynets.statistics.individual.algorithms.community_resolution_selection(G)

pynets.statistics.individual.algorithms.countmotifs(A, N=4)

Counts number of motifs with size N from A.

Parameters

Andarray

M x M Connectivity matrix

Nint

Size of motif type. Default is N=4, only 3 or 4 supported.

Returns

umotifsint

Total count of size N motifs for graph A.

References

1

Sporns, O., & Kötter, R. (2004). Motifs in Brain Networks. PLoS Biology. https://doi.org/10.1371/journal.pbio.0020369

pynets.statistics.individual.algorithms.create_communities(node_comm_aff_mat, node_num)

Create a 1D vector of community assignments from a community affiliation matrix.

Parameters

node_comm_aff_matarray

Community affiliation matrix produced from modularity estimation (e.g. Louvain).

node_numint

Number of total connected nodes in the graph used to estimate node_comm_aff_mat.

Returns

com_assignarray

1D numpy vector of community assignments.

References

1

Newman, M.E.J. & Girvan, M. Finding and evaluating community structure in networks. Physical Review E 69, 26113(2004).

2

Blondel, V.D. et al. Fast unfolding of communities in large networks. J. Stat. Mech 10008, 1-12(2008).

pynets.statistics.individual.algorithms.defragment(G)

pynets.statistics.individual.algorithms.diversity_coef_sign(W, ci)

The Shannon-entropy based diversity coefficient measures the diversity of intermodular connections of individual nodes and ranges from 0 to 1.

Parameters

WNxN np.ndarray

undirected connection matrix with positive and negative weights

ciNx1 np.ndarray

community affiliation vector

Returns

HposNx1 np.ndarray

diversity coefficient based on positive connections

HnegNx1 np.ndarray

diversity coefficient based on negative connections

References

1

Rubinov, M., & Sporns, O. (2010). Complex subnet measures of brain connectivity: Uses and interpretations. NeuroImage, 52, 1059-1069.

pynets.statistics.individual.algorithms.get_betweenness_centrality(G_len, metric_list_names, net_met_val_list_final, engine='nx')

pynets.statistics.individual.algorithms.get_clustering(G, metric_list_names, net_met_val_list_final, engine='nx')

pynets.statistics.individual.algorithms.get_comm_centrality(G, metric_list_names, net_met_val_list_final)

pynets.statistics.individual.algorithms.get_community(G, net_met_val_list_final, metric_list_names)

pynets.statistics.individual.algorithms.get_degree_centrality(G, metric_list_names, net_met_val_list_final)

pynets.statistics.individual.algorithms.get_diversity(in_mat, ci, metric_list_names, net_met_val_list_final)

pynets.statistics.individual.algorithms.get_eigen_centrality(G, metric_list_names, net_met_val_list_final, engine='nx')

pynets.statistics.individual.algorithms.get_local_efficiency(G, metric_list_names, net_met_val_list_final)

pynets.statistics.individual.algorithms.get_participation(in_mat, ci, metric_list_names, net_met_val_list_final)

pynets.statistics.individual.algorithms.get_prop_type(value, key=None)

Performs typing and value conversion for the graph_tool PropertyMap class. If a key is provided, it also ensures the key is in a format that can be used with the PropertyMap. Returns a tuple, (type name, value, key)

pynets.statistics.individual.algorithms.get_rich_club_coeff(G, metric_list_names, net_met_val_list_final, engine='nx')

pynets.statistics.individual.algorithms.global_efficiency(G, weight='weight', engine='nx')

Return the global efficiency of the G

Parameters

GNetworkX graph

Returns

global_efficiencyfloat

Notes

The published definition includes a scale factor based on a completely connected graph. In the case of an unweighted subnet, the scaling factor is 1 and can be ignored. In the case of a weighted graph, calculating the scaling factor requires somehow knowing the weights of the edges required to make a completely connected graph. Since that knowlege may not exist, the scaling factor is not included. If that knowlege exists, construct the corresponding weighted graph and calculate its global_efficiency to scale the weighted graph. Distance between nodes is calculated as the sum of weights. If the graph is defined such that a higher weight represents a stronger connection, distance should be represented by 1/weight. In this case, use the invert weights function to generate a graph where the weights are set to 1/weight and then calculate efficiency

References

1

Latora, V., and Marchiori, M. (2001). Efficient behavior of small-world networks. Physical Review Letters 87.

2

Latora, V., and Marchiori, M. (2003). Economic small-world behavior in weighted networks. Eur Phys J B 32, 249-263.

pynets.statistics.individual.algorithms.iterate_nx_global_measures(G, metric_list_glob)

pynets.statistics.individual.algorithms.link_communities(W, type_clustering='single')

The optimal community structure is a subdivision of the subnet into nonoverlapping groups of nodes which maximizes the number of within-group edges and minimizes the number of between-group edges. This algorithm uncovers overlapping community structure via hierarchical clustering of subnet links. This algorithm is generalized for weighted/directed/fully-connected networks

Parameters

WNxN np.array

directed weighted/binary adjacency matrix

type_clusteringstr

type of hierarchical clustering. ‘single’ for single-linkage, ‘complete’ for complete-linkage. Default value=’single’

Returns

MCxN np.ndarray

nodal community affiliation matrix.

References

1

de Reus, M. A., Saenger, V. M., Kahn, R. S., & van den Heuvel, M. P. (2014). An edge-centric perspective on the human connectome: Link communities in the brain. Philosophical Transactions of the Royal Society B: Biological Sciences. https://doi.org/10.1098/rstb.2013.0527

pynets.statistics.individual.algorithms.local_efficiency(G, weight='weight', engine='nx')

Return the local efficiency of each node in the G

Parameters

GObj

NetworkX graph.

Returns

local_efficiencydict

The keys of the dict are the nodes in the G and the corresponding values are local efficiencies of each node

Notes

The published definition includes a scale factor based on a completely connected graph. In the case of an unweighted subnet, the scaling factor is 1 and can be ignored. In the case of a weighted graph, calculating the scaling factor requires somehow knowing the weights of the edges required to make a completely connected graph. Since that knowlege may not exist, the scaling factor is not included. If that knowlege exists, construct the corresponding weighted graph and calculate its local_efficiency to scale the weighted graph.

References

1

Latora, V., and Marchiori, M. (2001). Efficient behavior of small-world networks. Physical Review Letters 87.

2

Latora, V., and Marchiori, M. (2003). Economic small-world behavior in weighted networks. Eur Phys J B 32, 249-263.

pynets.statistics.individual.algorithms.most_important(G, method='betweenness', sd=1, engine='nx')

Returns a copy of G with hubs as defined by centrality, core topology, or rich-club topology.

Parameters

GObj

NetworkX graph.

methodstr

Determines method for defining hubs. Valid inputs are coreness, richclub, and eigenvector centrality. Default is coreness.

sdint

Number of standard errors as cutoff for low-importance pruning.

Returns

GObj

NetworkX graph with isolated and low-importance nodes pruned.

pruned_nodeslist

List of indices of nodes that were pruned from G.

References

1

Power, J. D., Schlaggar, B. L., Lessov-Schlaggar, C. N., & Petersen, S. E. (2013). Evidence for hubs in human functional brain networks. Neuron. https://doi.org/10.1016/j.neuron.2013.07.035

pynets.statistics.individual.algorithms.np2gt(adj)

pynets.statistics.individual.algorithms.nx2gt(nxG)

Converts a networkx graph to a graph-tool graph.

pynets.statistics.individual.algorithms.participation_coef(W, ci, degree='undirected')

Participation coefficient is a measure of diversity of intermodular connections of individual nodes.

Parameters

WNxN np.ndarray

binary/weighted directed/undirected connection matrix

ciNx1 np.ndarray

community affiliation vector

degreestr

Flag to describe nature of graph ‘undirected’: For undirected graphs

‘in’: Uses the in-degree ‘out’: Uses the out-degree

Returns

——-

PNx1 np.ndarray

Participation coefficient

References

1

Guimera, R., & Amaral, L. A. N. (2005). Functional cartography of complex metabolic networks. Nature, 433, 895-900.

2

Rubinov, M., & Sporns, O. (2010). Complex subnet measures of brain connectivity: Uses and interpretations. NeuroImage, 52, 1059-1069.

pynets.statistics.individual.algorithms.participation_coef_sign(W, ci)

Participation coefficient is a measure of diversity of intermodular connections of individual nodes.

Parameters

WNxN np.ndarray

undirected connection matrix with positive and negative weights

ciNx1 np.ndarray

community affiliation vector

Returns

PposNx1 np.ndarray

participation coefficient from positive weights

PnegNx1 np.ndarray

participation coefficient from negative weights

References

1

Guimera, R., & Amaral, L. A. N. (2005). Functional cartography of complex metabolic networks. Nature, 433, 895-900.

2

Rubinov, M., & Sporns, O. (2010). Complex subnet measures of brain connectivity: Uses and interpretations. NeuroImage, 52, 1059-1069.

pynets.statistics.individual.algorithms.prune_small_components(G, min_nodes)

Returns a recomposed graph of all connected components of a minimum size

Parameters

GObj

NetworkX graph with isolated nodes present.

min_nodes: int

Minimum number of nodes permitted in a connected subgraph

Returns

GObj

NetworkX graph with isolated nodes pruned.

pruned_nodeslist

List of indices of nodes that were pruned from G.

References

1

Hayasaka, S. (2017). Anti-Fragmentation of Resting-State Functional Magnetic Resonance Imaging Connectivity Networks with Node-Wise Thresholding. Brain Connectivity. https://doi.org/10.1089/brain.2017.0523

2

Fornito, A., Zalesky, A., & Bullmore, E. T. (2016). Fundamentals of Brain subnet Analysis. In Fundamentals of Brain subnet Analysis. https://doi.org/10.1016/C2012-0-06036-X

pynets.statistics.individual.algorithms.raw_mets(G, i, engine='nx')

API that iterates across NetworkX algorithms for a G.

Parameters

GObj

NetworkX graph.

istr

Name of the NetworkX algorithm.

Returns

net_met_valfloat

Value of the graph metric i that was calculated from G.

pynets.statistics.individual.algorithms.rich_club_coefficient(G, engine='nx')

pynets.statistics.individual.algorithms.smallworldness(G, niter=5, nrand=10, approach='clustering', reference='lattice', engine='nx')

Returns the small-world coefficient of a graph

The small-world coefficient of a G is:

omega/sigma = Lr/L - C/Cl

where C and L are respectively the average clustering coefficient/ transitivity and average shortest path length of G. Lr is the average shortest path length of an equivalent random graph and Cl is the average clustering coefficient/transitivity of an equivalent lattice/random graph.

Parameters

GNetworkX graph

An undirected graph.

niter: integer (optional, default=5)

Approximate number of rewiring per edge to compute the equivalent random graph.

nrand: integer (optional, default=10)

Number of random graphs generated to compute the average clustering coefficient (Cr) and average shortest path length (Lr).

approachstr

Specifies whether to use clustering coefficient directly clustering or transitivity method of counting weighted triangles. Default is clustering.

referencestr

Specifies whether to use a random random or lattice lattice reference. Default is lattice.

Returns

omega/sigmafloat

The smallworld coefficient

References

1

Telesford, Joyce, Hayasaka, Burdette, and Laurienti (2011). “The Ubiquity of Small-World Networks”. Brain Connectivity. 1 (0038): 367-75. PMC 3604768. PMID 22432451. doi:10.1089/brain.2011.0038.

pynets.statistics.individual.algorithms.subgraph_number_of_cliques_for_all(G)

Helper function, in the case of graph disconnectedness, that returns the number of cliques, calculated iteratively for each distinct subgraph of the G.

Parameters

GObj

NetworkX graph.

Returns

number of cliquesint

The average number of cliques for G.

References

1

Bron, C. and Kerbosch, J. “Algorithm 457: finding all cliques of an undirected graph”. Communications of the ACM 16, 9 (Sep. 1973), 575–577. <http://portal.acm.org/citation.cfm?doid=362342.362367>

2

Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi, “The worst-case time complexity for generating all maximal cliques and computational experiments”, Theoretical Computer Science, Volume 363, Issue 1, Computing and Combinatorics, 10th Annual International Conference on Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28-42 <https://doi.org/10.1016/j.tcs.2006.06.015>

3

F. Cazals, C. Karande, “A note on the problem of reporting maximal cliques”, Theoretical Computer Science, Volume 407, Issues 1–3, 6 November 2008, Pages 564–568, <https://doi.org/10.1016/j.tcs.2008.05.010>

pynets.statistics.individual.algorithms.weighted_transitivity(G)

Compute weighted graph transitivity, the fraction of all possible weighted triangles present in G.

Possible triangles are identified by the number of “triads” (two edges with a shared vertex).

The transitivity is

\[T = 3\frac{\#triangles}{\#triads}.\]

Parameters

Ggraph

Returns

outfloat

Transitivity

References

1

Wasserman, S., and Faust, K. (1994). Social subnet Analysis: Methods and Applications. Cambridge: Cambridge University Press.

2

Alain Barrat, Marc Barthelemy, Romualdo Pastor-Satorras, Alessandro Vespignani: The architecture of complex weighted networks, Proc. Natl. Acad. Sci. USA 101, 3747 (2004)

pynets.statistics.individual.spectral module

Created on Tue Nov 7 10:40:07 2017 Copyright (C) 2017 @author: Derek Pisner

pynets.statistics.individual.spectral.build_asetomes(est_path_iterlist)

Embeds single graphs using the ASE algorithm.

Parameters

est_path_iterlistlist

List of file paths to .npy files, each containing a graph.

pynets.statistics.individual.spectral.build_masetome(est_path_iterlist)

Embeds structural-functional graph pairs into a common invariant subspace.

Parameters

est_path_iterlistlist

List of list of pairs of file paths (.npy) corresponding to structural and functional connectomes matched at a given node resolution.

References

1

Rosenthal, G., Váša, F., Griffa, A., Hagmann, P., Amico, E., Goñi, J., Sporns, O. (2018). Mapping higher-order relations between brain structure and function with embedded vector representations of connectomes. Nature Communications. https://doi.org/10.1038/s41467-018-04614-w

pynets.statistics.individual.spectral.build_omnetome(est_path_iterlist)

Embeds ensemble population of graphs into an embedded ensemble feature vector.

Parameters

est_path_iterlistlist

List of file paths to .npy file containing graph.

References

1

Liu, Y., He, L., Cao, B., Yu, P. S., Ragin, A. B., & Leow, A. D. (2018). Multi-view multi-graph embedding for brain network clustering analysis. 32nd AAAI Conference on Artificial Intelligence, AAAI 2018.

2

Levin, K., Athreya, A., Tang, M., Lyzinski, V., & Priebe, C. E. (2017, November). A central limit theorem for an omnibus embedding of multiple random dot product graphs. In Data Mining Workshops (ICDMW), 2017 IEEE International Conference on (pp. 964-967). IEEE.

pynets.statistics.individual.multiplex module

Created on Tue Nov 7 10:40:07 2017 Copyright (C) 2018 @author: Derek Pisner & James Kunert-Graf

pynets.statistics.individual.multiplex.build_multigraphs(est_path_iterlist)

Constructs a multimodal multigraph for each available resolution of vertices.

Parameters

est_path_iterlistlist

List of file paths to .npy file containing graph.

Returns

multigraph_list_alllist

List of multiplex graph dictionaries corresponding to each unique node resolution.

graph_path_list_toplist

List of lists consisting of pairs of most similar structural and functional connectomes for each unique node resolution.

References

1

Bullmore, E., & Sporns, O. (2009). Complex brain networks: Graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience. https://doi.org/10.1038/nrn2575

2

Vaiana, M., & Muldoon, S. F. (2018). Multilayer Brain Networks. Journal of Nonlinear Science. https://doi.org/10.1007/s00332-017-9436-8

pynets.statistics.individual.multiplex.build_mx_multigraph(func_mat, dwi_mat, name, namer_dir)

It creates a symmetric (undirected) MultilayerGraph object from vertex-aligned structural and functional connectivity matrices.

References

1

R. Amato, N. E Kouvaris, M. San Miguel and A. Diaz-Guilera, Opinion competition dynamics on multiplex networks, New J. Phys. DOI: https://doi.org/10.1088/1367-2630/aa936a

2

N. E. Kouvaris, S. Hata and A. Diaz-Guilera, Pattern formation in multiplex networks, Scientific Reports 5, 10840 (2015). http://www.nature.com/srep/2015/150604/srep10840/full/srep10840.html

3

A. Sole-Ribata, M. De Domenico, N. E. Kouvaris, A. Diaz-Guilera, S. Gomez and A. Arenas, Spectral properties of the Laplacian of a multiplex network, Phys. Rev. E 88, 032807 (2013). http://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.032807

pynets.statistics.individual.multiplex.matching(paths, atlas, namer_dir)

pynets.statistics.individual.multiplex.mutual_information_2d(x, y, sigma=1, normalized=True)

Computes (normalized) mutual information between two 2D variate from a joint histogram.

Parameters

x1D array

first variable

y1D array

second variable

sigma: float

sigma for Gaussian smoothing of the joint histogram

Returns

nmi: float

the computed similariy measure

pynets.statistics.individual.multiplex.optimize_mutual_info(dwi_mat, func_mat, bins=20)

Module contents