Simultaneously probing structural resolution and network geometry.

<p>Color plots of the number of non-singleton communities as function of both average connection weight and resolution parameter for the <i>(A)</i> fractal hierarchical, <i>(B)</i> small world, <i>(C)</i> Erdös-Rényi, and <i>(D)</i> ring lattice, and for <i>(E)</i> one representative DSI anatomical network. The window size is 25%. For results of the total number of communities (singletons and non-singletons), see the Supporting Information.</p

Effect of connection density and length on mesoscale diagnostics.

<p><i>(A–B)</i> Modularity , <i>(C–D)</i> laterality and <i>(E–F)</i> bipartivity as a function of the fiber tract density density <i>(A,C,E)</i>, fiber tract arc length <i>(B,D,F, blue curve)</i> and Euclidean distance <i>(B,D,F, orange curve)</i>. Orange curves correspond to the mean diagnostic value over DSI networks; blue curves correspond to the diagnostic value estimated from a single individual. The orange curves in panels <i>B</i>, <i>D</i>, and <i>F</i> represent the Euclidean distance between the nodes; the blue curve represents the average arc length of the fiber tracts. The insets illustrate partitions of one representative data set (see the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003712#s2" target="_blank">Methods</a> Section), indicating that communities tend to span the two hemispheres thus leading to low values of bipartivity. All curves and insets were calculated with a window size of .</p

Multiresolution mesoscale structure in functional networks.

<p>Functional brain networks were extracted from resting state fMRI data acquired from 29 people with schizophrenia and 29 healthy controls <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003712#pcbi.1003712-Bassett4" target="_blank">[15]</a> (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003712#s2" target="_blank">Methods</a> Section). <i>(A)</i> Weighted modularity for healthy controls (left) and people with schizophrenia (right). Box plots indicate range and 25% (75%) quartiles over the individuals in each group. The structural resolution parameter is . <i>(B)</i> MRFs for the weighted modularity as a function of the resolution parameter . <i>(C)</i> MRFs for binary modularity as a function of connection weight. <i>(D)</i> MRFs for bipartivity as a function of connection weight. In panels <i>(C)</i> and <i>(D)</i>, diagnostic values were estimated using a window size of . In all panels, -values for group differences in summary statistics (panel <i>(A)</i>) and MRFs (panels <i>(B–D)</i>) were calculated using a non-parametric permutation test <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003712#pcbi.1003712-Bassett4" target="_blank">[15]</a>; resolutions displaying the strongest group differences are highlighted by gray boxes. In panels <i>(B–D)</i>, error bars show the standard deviation of the mean for healthy controls (blue) and people with schizophrenia (green).</p

Weighted connection matrices for the synthetic benchmark networks and an empirical brain DSI network (N = 1000 nodes).

<p>Because network topologies can be difficult to decipher in large networks, here we illustrate the connections between only 100 of the total 1000 nodes. In each network, the topology changes as a function of edge weight (i.e., color) in the adjacency matrix. The windowed thresholding technique isolates topological characteristics in the subnetworks of nodes of similar weight. We report the initial benchmark results for a window size of 25% but find that results from other window sizes are qualitatively similar (see the Supporting Information).</p

Effect of the resolution parameter on measured community structure.

<p><i>(A)</i> Number of non-singleton communities and <i>(B)</i> mean community radius as a function of the resolution parameter . <i>(C)</i> Mean number of nodes per (non-singleton) community as a function of . Results are presented for networks extracted from the DSI data of six individuals, illustrating consistency across subjects.</p

Pictorial schematic of multi-resolution methods for weighted networks.

<p>We can apply soft or windowed thresholding, and vary the resolution parameter of modularity maximization to uncover multiresolution structure in empirical data that we summarize in the form of MRFs of network diagnostics.</p

Mesoscale diagnostics as a function of connection weight.

<p>(A–C) Modularity as a function of the average connection weight (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003712#s2" target="_blank">Methods</a> Section) for fractal hierarchical, small world (B), Erdös-Rényi random, regular lattice (C) and structural brain network (D). The results shown here are averaged over 20 realizations of the community detection algorithm and over 50 realizations of each model (6 subjects for the brain DSI data). The variance in the measurements is smaller than the line width. (D–F) Bipartivity as function of average connection weight of the fractal hierarchical, small world, Erös-Rényi random model networks and the DSI structural brain networks. We report the initial benchmark results for a window size of 25% but find that results from other window sizes are qualitatively similar (see the Supporting Information).</p

Summary of multi-resolution methods for network diagnostics.

<p>We use soft thresholding, windowed thresholding, and variation in the resolution parameter of modularity maximization to probe network architecture across scales (<i>Column 1</i>). For each approach, we provide a method summary (<i>Column 2</i>), a description of the control parameter (<i>Column 3</i>), a brief synopsis of the strengths and limitations of the approach (<i>Column 4</i>), and a few relevant references (<i>Column 5</i>).</p><p>Summary of multi-resolution methods for network diagnostics.</p

Use of soft thresholding strategies to probe geometric drivers of weighted modularity.

<p>Changes in maximum modularity as a function of the control parameter for soft thresholding. MRFs are presented for synthetic (fractal hierarchical, <i>FR</i>, in blue; Erdös-Rényi, <i>ER</i>, in black) and empirical (Brain fMRI in red) networks. Dots mark the peak value for different curves, which occur at different values of . The vertical dashed line marks the conventional value of obtained for . The single, point summary statistic fails to capture the full structure of the MRF revealed using soft thresholding.</p

Correlation matrices of schizophrenia patients constructed from partial correlation coefficients between MODWT wavelet coefficients

Correlation matrices of schizophrenia patients constructed from partial correlation coefficients between MODWT wavelet coefficient