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Weight-conserving characterization of complex functional brain networks
Mikail Rubinov, Olaf Sporns
TL;DR
Functional brain-network characterization has been limited by sparse or binary representations, incomplete treatment of degenerate modularity partitions, and the lack of a weighted-connectivity null model. The paper develops measures and algorithms for fully connected weighted networks, then applies them to resting-state MRI networks, demonstrating high-modularity degeneracy and strong correlations between complementary centrality measures. The authors conclude that these methods support more comprehensive characterization of functional brain networks, while noting that their degeneracy-search algorithm is not exhaustive.
Problem
Existing functional-network characterizations do not fully handle densely connected weighted networks, degenerate high-modularity partitions, or weighted-connectivity null models.
Method
The study defines weighted modularity and centrality measures, searches for degenerate high-modularity partitions, and introduces a null model preserving positive and negative degrees.
Results
The methods demonstrated degenerate high-modularity partitions and strong correlations between two complementary centrality measures in resting-state functional MRI networks.
Takeaways & Limitations
Characterizing sets of degenerate partitions can capture dynamic regional interactions that single partitions may not detect.
Takeaways & Limitations
The partition-search algorithm uses a conservative threshold, may miss meaningful lower-modularity partitions, and is not exhaustive.
Abstract
from arXiv · showhide
Complex functional brain networks are large networks of brain regions and functional brain connections. Statistical characterizations of these networks aim to quantify global and local properties of brain activity with a small number of network measures. Important functional network measures include measures of modularity (measures of the goodness with which a network is optimally partitioned into functional subgroups) and measures of centrality (measures of the functional influence of individual brain regions). Characterizations of functional networks are increasing in popularity, but are associated with several important methodological problems. These problems include the inability to characterize densely connected and weighted functional networks, the neglect of degenerate topologically distinct high-modularity partitions of these networks, and the absence of a network null model for testing hypotheses of association between observed nontrivial network properties and simple weighted connectivity properties. In this study we describe a set of methods to overcome these problems. Specifically, we generalize measures of modularity and centrality to fully connected and weighted complex networks, describe the detection of degenerate high-modularity partitions of these networks, and introduce a weighted-connectivity null model of these networks. We illustrate our methods by demonstrating degenerate high-modularity partitions and strong correlations between two complementary measures of centrality in resting-state functional magnetic resonance imaging (MRI) networks from the 1000 Functional Connectomes Project, an open-access repository of resting-state functional MRI datasets. Our methods may allow more sound and reliable characterizations and comparisons of functional brain networks across conditions and subjects.
Evaluation of the goodness of modularity partitions
The goodness of a modularity partition can be evaluated with multiple measures that capture within-module weights and triangle structure. These measures assess whether modules contain unusually clustered positive connections and unusually unclustered negative connections.
- Modularity partitions can be evaluated using the proportions of within-module positive and negative weights.
- The geometric mean of balanced and unbalanced within-module triangle weights provides an additional modularity measure.
- A generalized weighted clustering coefficient applies to networks with positive and negative weights and relates to within-module density.
- Good modularity partitions should contain unexpectedly many clustered positive connections within modules and few unclustered negative connections.
Detection of degenerate high-modularity partitions
The study detects degenerate high-modularity partitions in functional networks, introduces a degree-, weight-, and strength-preserving null model, and applies these methods to 18 resting-state MRI networks. Empirical networks show substantial, structured degeneracy and module overlap, while weight-conserving characterizations distinguish this from information-loss artifacts.
- Detection algorithm: The algorithm searches for topologically distinct partitions with similarly high modularity, but its conservative threshold may miss lower-modularity partitions and it is not exhaustive.The authors argue that representative, evenly sampled partitions may suffice even without enumerating every possible partition.
- Weight-conserving null model: The weighted null model preserves positive and negative degrees, connection weights, and closely approximates node strengths.Its null hypothesis associates observed network properties with degree, weight, and strength properties.
- Empirical application: The analysis characterized group-average functional networks from 18 sites in the 1000 Functional Connectomes Project.Networks were constructed from resting-state functional connectivity MRI datasets and included 112 cortical and subcortical regions.
- Empirical application: Increasing negative-weight influence produced fewer modules, larger modules, more within-module positive weights, and lower positive-weight density.It also increased the non-rescaled contribution of negative weights and reduced the contribution of positive weights.
- Degeneracy and null comparison: Empirical networks had many degenerate partitions, whereas null models had very few; ignoring weight information produced substantially more degeneracy.Weight-conserving degeneracy may therefore reflect inherent network structure rather than spurious degeneracy caused by discarded information.
- Degeneracy and network organization: Degenerate empirical partitions were substantially closer within networks than in null models but remained structurally diverse, with between-network distances exceeding within-network distances.Averaging these partitions revealed substantial module overlap and four bilaterally symmetric, broadly reproducible modules.
- Regional centrality: The proposed asymmetric centrality measures showed the highest correlation between regional strength and diversity among the examined measure pairs.Regions high on both measures were predominantly heteromodal, limbic, and paralimbic.
Discussion
The study’s methods reveal that functional brain networks contain multiple distinct high-modularity partitions and support characterization of fully connected, weighted networks without arbitrary sparsification. These analyses also expose interpretive limits of common network measures and motivate broader comparisons across conditions and subjects.
- Methods and principal findings: The study defined modularity, high-modularity degeneracy, centrality, and a null model for complex functional brain networks.These methods were applied to resting-state functional MRI networks, demonstrating degenerate high-modularity partitions and strong correlations between complementary centrality measures.
- Degenerate partitions: Averaging degenerate partitions revealed substantial module overlap, indicating that some regions or connections were not clearly assigned to single modules.The averaged within-module connection likelihood matrices summarize this ambiguity across partitions within each network.
- Weighted-network characterization: The proposed measures retain fully connected positive and negative weights, avoiding arbitrary thresholds and the multiple-comparison problems associated with sparse representations.Partitions computed on increasingly sparse representations became increasingly less optimal.
- Degenerate partitions: Many within-module connections were inconsistent across networks, while the binary likelihood matrix identified connections present inside modules in more than a specified fraction of cases.The resulting consistent modules were reported for 18 examined networks.
- Interpretive scope: The authors restrict characterization to measures with simple neurobiological interpretations because path-based and wiring-cost measures are ambiguous in fully connected functional networks.Functional connections reflect dynamic outcomes of numerous direct and indirect interactions rather than physical anatomical links.