comparing brain networks of different size and connectivity density
Some comments:
"Studying brain networks using measures like the clustering coefficient and the characteristic path length give useful insights within datasets of similar network sizes and link densities, but cause a comparison problem when these requirements are not met. This problem is thoroughly explained in a paper by vanWijk et al. (2010). It stresses the comparison problem between networks, not only because of the differences in network sizes (number of nodes) and degree but also due to arbitrary choices that have to be made (i.e., the threshold for the link density within a weighted network). "
"Even though connectedness in a node level between groups and conditions could be informative, it restricts both the interpretation and the comparison between various graph measures that vary with the degree (Alexander-Bloch et al., 2013). Additionally, absolute thresholding scheme divides the weights of connections into two groups, weak and strong connections emphasizing either the weak or the strong (vanWijk et al., 2010). A number of studies attempted to diminish the effect and the reproducibility of their results due to the adopted thresholding scheme by presenting various network measures over traditional cumulative thresholding. "
"Hence, networks having same number of nodes, will have, after pruning, the same number of links. On the one hand, this ensures that differences between network properties are not merely due to differences in the connection density [18]. On the other hand, ECO does not allow a direct evaluation of neural processes altering the number of links; however it does inform on the possible (re)organizational mechanisms. "
"Thus existing methods for binarizing weighted , there are not widely accepted criteria for thresholding networks. Instead of trying to come up with a proper threshold for network construction that may not work for different clinical populations or cognitive conditions [60], why not use all networks for every possible threshold? Motivated by this question, new multiscale hierarchical network modeling framework based on persistent homology has been developed recently [20,21,38,39,41]. "
"We applied a range of proportional thresholds to each correlation matrix per subject to avoid the confound of discrepant results on network measures, due to their sensitivity to the number of edges (i.e. connections) in a graph (van Wijk et al. 2010). The threshold values ranged from 1 % to 30 % in increments of 1 %. "
"For example, provided the data are of sufficient quality, there is no need to threshold the weight matrix to estimate topological properties like clustering, path length and small-worldness. Indeed, while the binarization procedure was common in early applications of graph theory to neural data [80] , it remains fundamentally agnostic to architectural principles that may be encoded in edge weights [84]. This realization has more generally motivated the field to develop methods that remain sensitive to the patterns of weights on the edges [85] , and to the topologies present in weak versus strong weights [84]. "
"Furthermore, each individual association matrix was transformed into an unweighted binary adjacency matrix by changing the entry into one when the number of tracts was greater than or equal to three (these entries were called edges), and respectively into zero if the number of tracts was less than 3 fibers to avoid false positive connections. Thresholding the connectivity matrices at an absolute threshold results in networks with different number of nodes and degrees (connections) that influences network measures and reduces interpretation of between-group results (van Wijk et al., 2010). Therefore, the constructed adjacency matrices were thresholded at a range of network densities for which the networks were not fragmented (each node had at least one connection with another node in the graph) and displayed small-world properties (non-random graphs). "
"The diagonal elements and negative correlations were set to zero since the biological meaning of negative correlations in structural covariance networks is not clear. Thresholding the association matrices at an absolute correlation threshold results in networks with different number of nodes and degrees (connections) that influences network measures and reduces interpretation of between-group results [van Wijk et al., 2010]. Therefore, association matrices were thresholded at a range of network sparsities. "
"The brain was modeled as a set of nodes and edges using graph theory [8] , where nodes represent the parcels of gray matter , and the edge represents WM tracts interconnecting two nodes. Unbiased methods to find fiber tracks between brain regions have long been debated [21]. Here the presence of an edge connecting two brain regions was defined if there were at least three fiber tracts between them to reduce false-positive connections [22, 23]. "
"Without any correction, the small-world index cannot be used to compare the smallworldness of different empirical networks. However, random surrogates may increase the sensitivity to differences in nodes number and degrees for the commonly used smallworld index [55]. The minimum spanning tree (MST) [56], a mathematically defined and unbiased subnetwork, provides similar information about network topology as conventional graph measures. "
"Because each edge is incident to two vertices, and counts in the degree of both vertices, the average degree of an undirected graph is where |E| and |V| are the cardinality of the set of edges and nodes respectively. The average degree is a way to classify nodes and may influence graph measures (see for example van Wijk et al. 2010). The density of a graph G computes how many edges are in E compared to the maximum possible number of edges between vertices in V. "
"It should be noted that, if the same correlation coefficient level was adopted for two groups to threshold the matrices R, the resulting networks would comprise different numbers of edges, which would lead to the two networks uncomparable [34]. Therefore, we set the matrices R at a range of network densities (D min —D max ), across which the network topologies of the OSA and CON group were compared. "
"Further, note that we could instead consider different baseline models, other than the Erd˝ os- Rényi model, in implementing Anderson et al.'s approach. However, as Van Wijk et al. (2010) points out, this method requires complete trust in the validity of whichever baseline model is used. We believe that a mixture model will allow us to avoid this issue. "
"Hence edge density is one important parameter, but it was typically ignored until recently. When comparing different networks, whether edge density should be matched and how to set the density threshold are still open questions (van Wijk et al., 2010). In some studies, the edge density is matched to the same value for different networks by setting a single threshold (Achard and Bullmore, 2007). "