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Abstract:
Nodes in real-world networks tend to cluster into densely connected groups, a property captured by the clustering coefficient. This non-random tendency for clustering is notably strong in neuronal networks, where connections are typically directed and were shown to have broad (e.g. lognormal) weight distributions. However, clustering was initially defined for unweighted and undirected networks, which leaves out crucial information about dynamics on weighted graphs. Several generalizations have been proposed for weighted networks but none of them fulfills the continuity condition: graph measures should not be influenced by the addition or deletion of edges having very small weights. This condition means that an edge with infinitesimally small weight should be equivalent to the absence of that edge. We propose here a new definition of the clustering coefficient that tackles this issue while satisfying previously formulated requirements. This new definition is less sensitive to weak spurious connections that are prevalent in inferred brain networks due to noise or statistical biases in measured graphs. Compared to previous methods, it is able to detect topological features more precisely and behaves in a more sensible manner for inferred networks. Finally, we discuss the differences between clustering methods for various brain networks,from C. elegans to the mouse brain. We discuss how our purely weighted continuous definition can differ from both the binary and other weighted definitions, and show how our analysis challenges several preconceptions regarding generic properties of brain networks.
