Table 1.
Distance measures and complexity.
The size (of the larger) graph is n; the number of edges is m. For the spectral decomposition, k denotes the number of principal eigenvalues we wish to find.
Table 2.
Runtimes for distance various distance measures, for graphs of size n = 100 and n = 300.
Each distance is calculated N = 500 times. Each sample generates two Erdős-Rényi random graphs with parameter p = 0.15, and times the calculation of the distance between the two graphs. All distances are implemented in the NetComp library, which can be found on GitHub at [75].
Fig 1.
Spectral densities λA of the adjacency matrix for a lattice graph and a degree matched configuration model.
Densities are built from an ensemble of 1,000 graphs generated using parameters described in Subsection3.1.5.
Table 3.
Events that punctuate the school day.
Fig 2.
Top to bottom, left to right: Snapshots of the face-to-face contact network at times (shown next to each graph) surrounding significant topological changes.
Table 4.
Table of comparisons performed, and the important structural features therein.
G(n, p) indicates the Erdős-Rényi uncorrelated random graph, SBM is the stochastic blockmodel, PA is the preferential attachment model, CM is the degree matched configuration model, and WS is the Watts-Strogatz model.
Fig 3.
Distance contrast between the stochastic blockmodel and the uncorrelated random graph model (null model).
Fig 4.
Spectral distances contrast —For the three matrices: Adjacency, combinatorial Laplacian, and normalized Laplacian (from left to right)—Between the stochastic blockmodel and the uncorrelated random graph model (null model).
Fig 5.
Distance contrast between the preferential attachment and the uncorrelated random graph model (null model).
Fig 6.
Spectral distance contrast —For the three matrices: Adjacency, combinatorial Laplacian, and normalized Laplacian (from left to right)—Between the preferential attachment and the uncorrelated random graph model (null model).
Fig 7.
Distance contrast between the preferential attachment model and the degree matched configuration model (null model).
Fig 8.
Distance contrast between a small-world graph and the the degree matched configuration model (null model).
Fig 9.
Spectral diastances contrast —For the three matrices: Adjacency, combinatorial Laplacian, and normalized Laplacian (from left to right)—Between the small-world graph and the degree matched configuration model (null model).
Fig 10.
Distance contrast between the 10 × 100 two-dimensional lattice graph and the the degree matched configuration model (null model).
Fig 11.
Spectral distances contrast —For the three matrices: Adjacency, combinatorial Laplacian, and normalized Laplacian (from left to right)—Between the 10 × 10 two-dimensional lattice graph and the degree matched configuration model (null model).
Fig 12.
Primary school data set: Normalized temporal differences for the resistance distance , edit distance
, and DeltaCon distance
.
Fig 13.
Primary school data set: Normalized temporal differences for the NetSimile distance and edit distance
.
Fig 14.
Primary school data set: Normalized temporal differences for the three spectral distances: Combinatorial Laplacian , normalized Laplacian
, and adjacency
.
Fig 15.
EU-emails: Normalized temporal difference, for the edit distance , and absolute value of the changes in the graph volume over time.
Table 5.
Events related to the activity of the European Parliament during 2004 [106].
Fig 16.
EU-emails: Normalized temporal differences for the NetSimile distance and edit distance
.
Fig 17.
EU-emails: Normalized temporal differences for the three spectral distances: Combinatorial Laplacian , normalized Laplacian
, and adjacency
.
Fig 18.
EU-emails: Normalized temporal differences for the resistance distance , edit distance
, and DeltaCon distance
.
Fig 19.
ABIDE data set: Distance contrast between the unweighted ASD and control connectomes for a threshold T = 0.5.
Fig 20.
ABIDE data set: Distance contrast between the unweighted ASD and control connectomes for a threshold T = 0.8.
Fig 21.
ABIDE data set: Distance contrast between the weighted ASD and control connectomes for a threshold T = 0.5.
Fig 22.
ABIDE data set: Distance contrast between the weighted ASD and control connectomes for a threshold T = 0.8.
Fig 23.
Two significant global structures observed in our experiments.
On the left is the community structure typical of the stochastic blockmodel. On the right is the heavy-tailed degree distribution typical of the preferential attachment model.
Fig 24.
Empirical spectral densities λA of the adjacency matrix for the stochastic blockmodel (blue) and the uncorrelated random graph (orange).
Densities are built from an ensemble of 1,000 graphs generated using parameters described in section 3.1.1.
Fig 25.
Empirical spectral densities λL of the combinatorial Laplacian for the preferential attachment model (blue) and the uncorrelated random graph (orange).
Densities are built from an ensemble of 1,000 graphs generated using parameters described in section 3.1.2.
Fig 26.
Empirical spectral densities λA of the adjacency matrix for the Watts-Strogatz model (blue) and the uncorrelated random graph (orange).
Densities are built from an ensemble of 1,000 graphs generated using parameters described in section 3.1.4. The uncorrelated random graph model has a value of p that is smaller than those used in the previous models, creating a sharp peak at λA = 0.
Fig 27.
Contrast for each pair of regions (i, j) in the atlas AAL computed between the ASD subjects and the controls (see (17)).
Table 6.
Regions with anomalous connectivity.
Correspondence between labels and regions is established via the Automated Anatomical Labelling atlas [116].
Fig 28.
Flow chart summarizing the suggested decision process for applying distance measures in empirical data.
Table 7.
Table of commonly used notation.