Revising clustering and small-worldness in brain networks

TL;DR

The paper argues that classic brain-network analyses lose critical information when direction and weight are ignored. It introduces fully-weighted directed clustering definitions that satisfy a continuity condition, mitigating thresholding issues and enhancing sensitivity to weight structure. Through artificial networks and four connectomes, it shows motif over-expression and hierarchical organization emerge only with fully-weighted directed measures, and that small-world assessments must be reinterpreted in weighted directed contexts. The work challenges the universality of small-worldness in brain networks and advocates a threshold-free, directionally aware framework for structure–function studies with broad implications for connectomics research.

Abstract

As more connectome data become available, the question of how to best analyse the structure of biological neural networks becomes increasingly pertinent. In brain networks, knowing that two areas are connected is often not sufficient, as the directionality and weight of the connection affect the dynamics in crucial ways. Still, the methods commonly used to estimate network properties, such as clustering and small-worldness, usually disregard features encoded in the directionality and strength of network connections. To address this issue, we propose using fully-weighted and directed clustering measures that provide higher sensitivity to non-random structural features. Using artificial networks, we demonstrate the problems with methods routinely used in the field and how fully-weighted and directed methods can alleviate them. Specifically, we highlight their robustness to noise and their ability to address thresholding issues, particularly in inferred networks. We further apply our method to the connectomes of different species and uncover regularities and correlations between neuronal structures and functions that cannot be detected with traditional clustering metrics. Finally, we extend the notion of small-worldness in brain networks to account for weights and directionality and show that some connectomes can no longer be considered ``small-world''. Overall, our study makes a case for a combined use of fully-weighted and directed measures to deal with the variability of brain networks and suggests the presence of complex patterns in neural connectivity that can only be revealed using such methods.

Figures (11)

• Figure 1: Illustration of the four directed clustering motifs. Each possible version of the cycle, fan-in, fan-out and middleman motifs are shown for the top node (in yellow).
• Figure 2: Using the undirected representation of directed graphs, or undirected measures lead to a significant loss of information. A. Two different directed networks correspond to the same undirected network (differing edges are highlighted in black). Node size gives the total degree, and colour gives the out-degree (low in black to high in yellow). Associated node IDs are shown in grey around the undirected version of the graphs. B. Comparison of the strongly connected components (SCCs, given by node colours) and motifs of the two directed graphs. (d1, dark red) is strongly connected with each node reachable from each other, whereas (d2, orange) falls apart into 4 distinct SCCs. This means that if information is injected in 5 for (d2), it can only reach nodes 0 and 1, whereas it can reach all other nodes in (d1). C. Comparison of the undirected and directed clustering coefficients for nodes 4 and 5 and their evolution (represented by the black line) across graphs d1 (dark red) and d2 (orange). The triangular motifs are represented above (for a chosen node, in yellow); the total clustering considers all possible variations of the directed triangles. Though they seem identical on the undirected graph, nodes 4 and 5 actually differ (either within a single graph or across graphs) for all directed clustering motifs except middleman. D. Mouse brain (same rules as A). E. Linear regression between the undirected clustering ($CC_u$) and two directed motifs ($CC_m$). F. $R^2$ and mean absolute percentage error $\overline{E}$ associated to the regression between the undirected clustering and each directed motif.
• Figure 3: Fully-weighted methods are more sensitive to the weights-related features of the nodes. A. 1000-node spatial network with distance-dependent connectivity and weights drawn from a log-normal distribution. Node size indicates the total degree, node color indicates out-degree. B. Difference between binary and weighted middleman clustering ranks of the nodes from the previous graph. Red: 2nd percentile, yellow: 50th percentile, blue 100th percentile. The theoretical locations of the associated binary percentiles are marked by dashed lines of the corresponding colours). While Barrat and Onnela's methods remain relatively close to the binary ranking, notably for extreme values, the fully weighted methods significantly deviate from it even for the lowest and highest percentiles. C. 20 realizations of the spatial network in A, with weights sampled from the same lognormal distribution, are used to make groups of nodes with similar binary clustering coefficients (in bins of 0.05 width). Normalized $CC_w$ spread $s$ is computed as $s = IQ / M$, with $IQ$ the interquartile-range and $M$ the median. Shaded areas give the standard deviation around the mean and are obtained via bootstrapping. The ratio between low and high clustering values is higher for fully weighted clustering methods, meaning that their discriminating power is higher than that of the hybrid methods. The inset is a zoom on the values containing 90% of the measured nodes. See Methods \ref{['methods:analysis']} for details about the network, weights, and analysis.
• Figure 4: Thresholding for the removal of spurious edges strongly affect hybrid, but not fully-weighted clustering measures. A. The measured network (M) is a combination of the ground truth (black) and an Erdős-Rényi noise (red). B. The distribution of true (black, log-normal) and noisy (red, shifted exponential) weights have a non-zero overlap. Thresholds: low: $\theta_l$ - minimum of ground-truth weight, optimal $\theta_o$, and high $\theta_h$ - maximal weight of the noise network. C. Same as B but in linear scale. D. Coefficient of determination $R^2$ (blue) measuring the match between the thresholded and ground-truth total clustering values. The dotted line marks the minimal correlation for the measured network with a fully-weighted method, to be compared with the best values of the binary and hybrid methods.
• Figure 5: Expression of directed clustering motifs can vary significantly between clustering methods and connectomes. A--D. Clustering propensity of four connectomes for each directed motif and each clustering definition. The values are compared to the average of 10 networks with the same binary structure but shuffled weights (see Methods \ref{['methods:analysis']}, for details). The clustering definitions used are Barrat's, "B", and Onnela's, "O" (hybrid measures) as well as continuous, "C", and Zhang--Horvath's, "Z" (fully-weighted measures). A. The central brain of drosophila displays a large increase in all directed motifs according to the fully-weighted measures. B. The Ciona intestinalis tadpole connectome shows almost no effect of weight structure except for a slight increase in middleman motifs and a slight decrease of cycles with the Zhang--Horvath definition. C. In the C. elegans connectome, weights contribute to a reinforcement of middleman and fan-out patterns and a slight reduction of cycles according to the fully-weighted measures. D. Weight distribution in the mouse mesoscale connectome contributes to a notable increase of middleman, fan-in and fan-out motifs but not to cycles, according to the fully-weighted methods. E. Using the continuous clustering, comparison between the actual expression levels of the motifs and the expected value in Erdős-Rényi graphs (with the same number of edges and the same set of weights) reveals that, except for cycle motifs in the mouse connectome, all clustering motifs are over-expressed. F. Level of expression for the middleman, fan-in, and fan-out motifs compared to cycles according to the four clustering definitions. Each dot represents the ratio of the median value of a motif compared to the median cycle clustering. The fully-weighted methods detect significant over-expression of the three motifs compared to cycles for all connectomes except drosophila. G, H. In C. elegans, correlations are significant between the continuous clustering and the Zhang--Horvath and Onnela distributions and strong correlations are also observed in the tadpole and drosophila; however, much weaker correlations are observed for the mouse brain. I. As expected from the low correlations seen in H, the subnetworks formed by the most clustered areas in the mouse brain differ strongly depending on the method used. Node size express total-degree, colors give the total clustering values, from high (yellow) to low (dark green). For the overlays, nodes and edges from the continuous subnetwork are in grey while Zhang/Onnela are in orange. Nodes present in both subnetworks are marked in red.