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Null models in network neuroscience

Abstract

Recent advances in imaging and tracing technology provide increasingly detailed reconstructions of brain connectomes. Concomitant analytic advances enable rigorous identification and quantification of functionally important features of brain network architecture. Null models are a flexible tool to statistically benchmark the presence or magnitude of features of interest, by selectively preserving specific architectural properties of brain networks while systematically randomizing others. Here we describe the logic, implementation and interpretation of null models of connectomes. We introduce randomization and generative approaches to constructing null networks, and outline a taxonomy of network methods for statistical inference. We highlight the spectrum of null models — from liberal models that control few network properties, to conservative models that recapitulate multiple properties of empirical networks — that allow us to operationalize and test detailed hypotheses about the structure and function of brain networks. We review emerging scenarios for the application of null models in network neuroscience, including for spatially embedded networks, annotated networks and correlation-derived networks. Finally, we consider the limits of null models, as well as outstanding questions for the field.

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Fig. 1: Generating null distributions for network features.
Fig. 2: Randomization versus generative models.
Fig. 3: A spectrum of null models.
Fig. 4: A sampling space of null models.
Fig. 5: Spatial permutation of network annotations.
Fig. 6: Null models can be implemented at different stages of network construction and analysis.

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Acknowledgements

The authors thank A. Goulas for stimulating discussions during the conceptualizing of this work, and E. Suárez, A. Luppi, V. Bazinet, G. Shafiei, J. Hansen, Z.-Q. Liu, O. Sherwood and R. Moran for constructive comments on the manuscript. F.V. acknowledges support from the Data to Early Diagnosis and Precision Medicine Industrial Strategy Challenge Fund, UK Research and Innovation (UKRI) and the Bill & Melinda Gates Foundation. B.M. acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canadian Institutes of Health Research (CIHR), the Brain Canada Foundation Future Leaders Fund, the Canada Research Chairs Program and the Healthy Brains for Healthy Lives initiative.

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Correspondence to Bratislav Mišić.

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Supplementary information

Glossary

Connectomics

The study of wiring patterns in neural systems.

Graph

A mathematical description of a network, capturing pairwise relationships (edges) among elements (nodes).

Degree distribution

The probability distribution of degrees (the number of connections of a node with other nodes) of all nodes in a network.

Hub

A node with many connections.

Modules

Groups of nodes densely connected with each other, but sparsely connected with the rest of the network.

Null models

Synthetic realizations of brain networks, used to benchmark whether observed network features are statistically unexpected.

Null hypotheses

The premises that observed relationships between network features are due to chance alone.

Density

The proportion of possible edges that exist in a network.

Degree sequence

List of degrees of all nodes in a graph.

Topology

Logical arrangement of nodes and edges in a network (used in this Review specifically to refer to network topology).

Network inference

Testing of hypotheses about properties or mechanisms that give rise to observed network features.

Rich club

A group of high-degree nodes disproportionately densely connected with each other.

Rewiring

Swapping pairs of edges either randomly or with constraints.

Monte Carlo sampling methods

Algorithms that use random sampling to generate numerical estimates of population statistics.

Generative models

Procedures for constructing networks by adding edges and/or nodes according to a set of wiring rules.

Wiring cost

The total physical length of all edges in the network.

Homophilic attachment

The tendency for pairs of nodes to be connected if they have similar characteristics, such as degree, connection profile or microscale attributes.

Transitive property

The dependence among the three edges in triangles of nodes when the edges are estimated by statistical association, such as correlations among time series.

Spatial autocorrelation

The tendency for spatially proximal brain regions to possess similar attributes.

Network morphospace

A space of network configurations that can be realized under a set of constraints.

Geometry

The embedding of nodes and edges in physical space.

Surrogate

Realization of a null model. Mainly used to refer to null time series, but frequently to networks as well.

Matrix

A table in which rows and columns correspond to network nodes, and every element encodes a relationship between two nodes, such as connectivity or physical distance.

Phase coefficients

The offsets or temporal dependencies among sinusoidal components of a time series following a Fourier transform to the frequency domain.

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Váša, F., Mišić, B. Null models in network neuroscience. Nat Rev Neurosci 23, 493–504 (2022). https://doi.org/10.1038/s41583-022-00601-9

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