Figure 1.
Illustrative figure of the three different complex network models.
(A) Erdös-Rényi; (B) Scale-free; and (C) Small-world and their respective spectra, degree distributions, and entropies, in this order from top to bottom. The estimated entropies are computed for the respective graph type for the respective parameters (probability for the Erdös-Rényi, scaling exponent
for the scale-free, and probability
for the small-world random graphs). In (A) the entropy values estimated from the simulation data is depicted by a solid line and the theoretical value of the entropy computed using equation 4 is indicated by a dashed line.
Table 1.
Average parameters estimated by minimum distance estimator based on KL divergence for Erdös-Rényi random, scale-free, and small-world graphs.
Figure 2.
Figure illustrating the performance of the model selection approach as a function of number of nodes.
Given a graph belonging to (A) Erdös-Rényi with parameter , (B) scale-free with parameter
, and (C) small-world with parameter
= 0.3, the solid, dashed, and dotted lines represent the proportion of graphs classified as Erdös-Rényi, scale-free, and small-world, respectively. Notice that the larger is the graph, the higher is the proportion of correct hits, showing that the model selection approach is consistent. For each graph size (10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120 nodes), 1,000 repetitions were carried out.
Figure 3.
ROC curve under the alternative hypothesis and p-value distribution under the null hypothesis.
(A) Erdös-Rényi graphs; (B) scale-free graphs, and (C) small-world graphs. For the ROC curves, the x-axis represents the 1-specificity and the y-axis the sensitivity. Both ROC curves and p-value distributions were constructed by analyzing 10,000 experiments. Solid and dashed lines represent the test based on the spectral and degree distributions, respectively.
Figure 4.
The frequency plot for the eigenvalues of the eight species (H. pylori, R. norvegicus, M. musculus, E. coli, C. elegans, S. cerevisiae, H. sapiens, D. melanogaster).
Table 2.
The general characteristics of eight protein-protein interaction networks. For each network we indicate the number of nodes, the number of edges, the average degree, the diameter, the clustering coefficient and the average path length.
Table 3.
The estimated Kullback-Leibler divergence between the eight species and the three random graph models. In bold are the lowest KL divergence values.
Figure 5.
(A) Spectral and (B) degree distributions in the log-scale.
Solid line represents the children with typical development. Dashed line represents children with combined type of ADHD (hyperactive-impulsive and inattentive).
Table 4.
Different metrics to measure graph discrepancy between children with typical development and children with combined type of ADHD (hyperactive-impulsive and inattentive) and their respective p-values.
Table 5.
P-values obtained by testing the Jensen-Shannon divergence in the spectra distributions among different laboratories.