P048 From Density to Void: Why Brain Networks Fail to Reveal Complex Higher-Order Structures
Moo K. Chung*1,Anass B El-Yaagoubi2, Anqi Qiu3, Hernando Ombao2
1Department of Biostatistics and Medical informatics, University of Wisconsin, Madison, USA
2Statistics Program, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
3Department of Health Technology and Informatics, Hong Kong, China
*Email:mkchung@wisc.edu
Introduction
In brain network analysis using resting-state fMRI, there is growing interest in modeling higher-order interactions—beyond simple pairwise connectivity—using persistent homology [1]. Despite the promise of these advanced tools, robust and consistently observed time-evolving higher-order interactions remain elusive. In this study, we examine why conventional analyses often fail to reveal complex higher-order structures—such as interactions involving four or five or more nodes —and explore whether higher-order interactions truly exist in functional brain networks.
Methods
We apply persistent homology to analyze correlation networks over a range of thresholds h. A simplicial complex is constructed from the connectivity matrix c(i,j) where nodes (0-simplices) represent individual time series and edges (1-simplices) are included if c(i,j) > h. For triangles (2-simplices), a simplex is formed if all three pairwise connections among a triplet of nodes exceed the threshold h. Higher-order simplices are defined analogously. We then examine the consistency of these higher-order topological features across time and subjects by quantifying the probability of overlap in the persistent features.
Results
Our preliminary analysis based on rs-fMRI of 400 subjects reveals that correlation networks tend to yield either nearly complete graphs or highly fragmented structures, neither of which exhibit robust higher-order topological features. As the number of nodes involved in an interaction increases, the probability that multiple brain regions activate simultaneously decays exponentially, as observed in both empirical data and theoretical models. These findings indicate that resting-state fMRI predominantly reflects pairwise interactions, with only infrequent occurrences of three-node interactions. Nonetheless, even these predominant pairwise interactions are highly intricate, giving rise to complex network dynamics characterized by lower-dimensional topological profiles such as 0D (connected components) and 1D (cycles) features [2].
Discussion
Our results indicate that conventional connectivity analyses are limited in detecting robust higher-order interactions, as they often yield networks that are either overly dense or fragmented, masking subtle connectivity patterns. Alternative metrics, such as mutual information or entropy, may better capture the nonlinear, multiscale dependencies among brain regions [3]. Notably, higher-order interactions are not exclusively defined by multi-node connectivity; even pairwise interactions can become highly complex when organized into cycles or spiral patterns over time. Future work should integrate these alternative measures with persistent homology to reveal hidden connectivity patterns, ultimately enhancing our understanding of functional brain organization.
Figure 1. Left: Graph representation of pairwise interactions between nodes in a brain network. Right: Higher-order interactions depicted with colored simplices—yellow for 3-node (triangle) interactions and blue for 4-node (tetrahedron) interactions.
Acknowledgements
NIH grants EB028753, MH133614 and NSF grant MDS-201077
References
[1] El-Yaagoubi, A.B., Chung, M.K., Ombao. H. (2023). Topological data analysis for multivariate time series data.Entropy,25(11), 1509.
[2]Chung, M.K., Ramos, C.G., De Paiva, F.B., Mathis, J., Prabhakaran, V., Nair, V.A., Meyerand, M.E., Hermann, B.P., Binder, J.R. and Struck, A.F., 2023. Unified topological inference for brain networks in temporal lobe epilepsy using the Wasserstein distance.NeuroImage,284, p.120436.
[3] Li, Q., Steeg, G. V., Yu, S., Malo, J. (2022). Functional connectome of the human brain with total correlation.Entropy,24(12), 1725.
