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. Related in: MedlinePlus |
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Mentions: One may argue that we seek the extreme cases as the N,k-dependence of L and C may differ most between lattices and random networks. Empirical networks often resemble small-world characteristics and, indeed, graph measures will probably have values that lie somewhere between those of a lattice and random network. In fact, since small-world networks are characterized by a path length close to that of random networks even for small rewiring probabilities, effects on normalized L are reduced. By contrast, however, the average clustering coefficient of small-world networks is close to that of a lattice and normalization by random networks introduces a larger N,k-dependence than seen for the non-normalized values. The small-world networks reported here had a rewiring probability of 0.1 (i.e. 10% random connections). In Figure 4 we show the N,k-dependencies of C and L for different rewiring probabilities. As expected, for low rewiring probabilities L is highly dependent on the number of nodes and edges, and much less so for high rewiring probabilities. The opposite is true for C. Remarkably, normalization by random networks only eliminates effects of N and k when the rewiring probability approaches 1, i.e. producing random networks. For lower rewiring probabilities, normalized C and L are still dependent on N and k. Hence, the exact bias introduced by differences in N and k depends on the rewiring probability. |
View Article: PubMed Central - PubMed
Affiliation: Research Institute MOVE, VU University Amsterdam, Amsterdam, The Netherlands. b.vanwijk@fbw.vu.nl