Comparing brain networks of different size and connectivity density using graph theory.
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We list benefits and pitfalls of various approaches that intend to overcome these difficulties.For instance, choosing a threshold to fix N and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-significant (significant) connections.To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to correct for size effects, if successful.
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Affiliation: Research Institute MOVE, VU University Amsterdam, Amsterdam, The Netherlands. b.vanwijk@fbw.vu.nl
ABSTRACT
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Graph theory is a valuable framework to study the organization of functional and anatomical connections in the brain. Its use for comparing network topologies, however, is not without difficulties. Graph measures may be influenced by the number of nodes (N) and the average degree (k) of the network. The explicit form of that influence depends on the type of network topology, which is usually unknown for experimental data. Direct comparisons of graph measures between empirical networks with different N and/or k can therefore yield spurious results. We list benefits and pitfalls of various approaches that intend to overcome these difficulties. We discuss the initial graph definition of unweighted graphs via fixed thresholds, average degrees or edge densities, and the use of weighted graphs. For instance, choosing a threshold to fix N and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-significant (significant) connections. Opposed to fixing N and k, graph measures are often normalized via random surrogates but, in fact, this may even increase the sensitivity to differences in N and k for the commonly used clustering coefficient and small-world index. To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to correct for size effects, if successful. We also add a number of methods used in social sciences that build on statistics of local network structures including exponential random graph models and motif counting. We show that none of the here-investigated methods allows for a reliable and fully unbiased comparison, but some perform better than others. |
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Mentions: We used data of the anatomical connections in the macaque visual cortex consisting of a directed, unweighted graph with 30 nodes, and 311 edges (available at http://www.brain-connectivity-toolbox.net). According to Sporns and Kötter [67], in this network the frequency of only five structural motifs appeared significantly increased when optimizing a random network that was constrained by the number of estimated functional motifs (see Figure 3B and Table 4 in [67]). Since we did not intend to replicate these results to all extent but rather want to highlight differences between methods we here only considered motifs with up to three nodes, which left a single triad, namely triads census 201 ([59], [70], see also Figure 7 and S4), i.e. the mutually connected two-path (ID = 9 in [67]). Interestingly, however, the counts of six other motifs with three nodes where significantly decreased but not further discussed in [67], although the reduced frequency of such mostly directed motifs is certainly as interesting as the increased count of triad census 201. The combination of these in total seven motifs led to the first exponential random graph model that was unfortunately degenerate [71]. We therefore reduced the model a priori towith g1 referring to triad 201 and g2 to 021C (see Figure 7). Optimizations revealed that in this model #1 both parameters were significant and, in agreement with [67] (Table 2, z-scores of the simulation with random networks) θ1 was positive and θ2 negative. We re-analyzed this further aiming for an optimal model fit by means of minimizing Akaike's information criterion (AIC) as heuristic for model selection [72]; the AIC-values were determined via the approximated log-likelihood that the empirical network was drawn from the distribution of the corresponding exponential random graph model with optimized parameters θ. |
View Article: PubMed Central - PubMed
Affiliation: Research Institute MOVE, VU University Amsterdam, Amsterdam, The Netherlands. b.vanwijk@fbw.vu.nl