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Edges in Brain Networks: Contributions to Models of Structure and Function

Joshua Faskowitz, Richard F. Betzel, Olaf Sporns

arXiv:2105.07069v1q-bio.NC

TL;DR

Network neuroscience has often prioritized nodes, although edges encode the relationships through which network organization is expressed. This paper reviews edge definitions and edge-centric analyses, arguing that considering higher-order and dynamic edge information offers complementary insight into brain organization.

  • Problem

    Brain network studies have emphasized node groupings and functions, while edges—the pairwise relationships needed for nodal assessments—have rarely been treated as primary descriptors of organization.

    Source: Introduction · p. 2

  • Method

    The paper reviews how edge definitions, higher-order edge arrangements, communication, and edge dynamics can inform analysis of brain network organization.

    Source: Introduction · p. 2

  • Results

    The review argues that edge-level, higher-order, and dynamic information provides insights complementary to traditional node-centric analyses.

    Source: Introduction · p. 2, Conclusion · p. 20

  • Takeaways & Limitations

    Careful consideration of edge information can disclose underappreciated aspects of structural and functional network organization.

    Source: Introduction · p. 2, Conclusion · p. 20

    Paths over functional similarity edges are less intuitive than paths over connectivity edges, and thresholding dense similarity networks can bias or confound topology.

    Source: Edge-Centric Network Analyses · p. 8

Abstract

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Network models describe the brain as sets of nodes and edges that represent its distributed organization. So far, most discoveries in network neuroscience have prioritized insights that highlight distinct groupings and specialized functional contributions of network nodes. Importantly, these functional contributions are determined and expressed by the web of their interrelationships, formed by network edges. Here, we underscore the important contributions made by brain network edges for understanding distributed brain organization. Different types of edges represent different types of relationships, including connectivity and similarity among nodes. Adopting a specific definition of edges can fundamentally alter how we analyze and interpret a brain network. Furthermore, edges can associate into collectives and higher-order arrangements, describe time series, and form edge communities that provide insights into brain network topology complementary to the traditional node-centric perspective. Focusing on the edges, and the higher-order or dynamic information they can provide, discloses previously underappreciated aspects of structural and functional network organization.

Introduction

Brain networks represent neural elements and their relationships across scales, offering a framework for linking local components to distributed brain structure and function. Although network analyses often emphasize nodes, edges are essential to those assessments and remain less studied as primary descriptors of organization.

Introduction

  • Network models link neural elements and their relationships across scales, from neurons and populations to large-scale brain regions.They integrate local and global perspectives on brain structure and function.Source: Introduction · p. 1
  • Connectomes can be built from inferred axonal pathways, morphometric similarity, or functional correlations, using data from different modalities and scales.These models also enable comparisons across species.Source: Introduction · p. 2
  • Brain network analyses commonly emphasize node properties and groupings, while edges are rarely treated as primary descriptors of organization.Edges record pairwise relationships and are necessary for nodal assessments.Source: Introduction · p. 2
  • The paper reviews how edge definitions, higher-order relationships, and edge-level communication dynamics can inform understanding of brain network organization.It also discusses future directions for interpreting edge-level information.Source: Introduction · p. 2

Network Construction

Brain network construction depends on choices about what counts as a node and how relationships between nodes are represented. Edges may encode connectivity, similarity, or supplementary information, and these distinctions shape their interpretation.

Network Construction

  • Brain network models require choices about node definitions and the mode and metric used to represent relationships; edge definitions are consequential for downstream analysis.Edges may also carry supplementary measurements or weights that fuse modalities.Source: Network Construction · p. 3

Edge Types: Connectivity

  • Connectivity edges represent material linkage or contact, from microscale synapses to macroscale connections estimated with methods such as tractography.They express potential influence between nodes through estimated anatomical linkage; effective connectivity instead estimates causal relationships from functional data or perturbational evidence.Source: Edge Types: Connectivity · pp. 4–5

Edge Types: Similarity

  • Similarity edges compare node-level feature sets to form dense matrices, and their interpretation depends on which features are compared.Features may be anatomical or gene-expression profiles, or functional activity; time-series similarity measures also depend on recording resolution and may capture dynamics distinct from correlation.Source: Edge Types: Similarity · pp. 5–6

Edge-Centric Network Analyses

Edge-centric analyses complement node-level network measures by examining paths, edge relationships, and edge communities. Their interpretation depends on how edges are defined and on choices such as null models and thresholding.

Edge-Centric Network Analyses

  • Paths support edge-level measures and communication models, while removing edges can reveal their effects on global network statistics.Path-based communication models use routing or diffusion to estimate communication potential between nodes.Source: Edge-Centric Network Analyses · p. 6
  • Line graphs and edge-similarity matrices represent relationships among edges, whose clustering yields edge communities and overlapping node affiliations.In a line graph, original-network hubs become dense incidence clusters; structural link communities can overlap at nodes traditionally considered hubs.Source: Edge-Centric Network Analyses · p. 7
  • Network measures should be interpreted in light of the neural data and the meaning assigned to its nodes and edges.The paper recommends incorporating neuroscience expertise when relating a measure to brain organization.Source: Edge-Centric Network Analyses · p. 7
  • Paths over functional similarity edges are less intuitive than paths over connectivity edges, and may mix direct and indirect sources of variance.The paper therefore argues that paths and derived measures should be interpreted differently by edge type.Source: Edge-Centric Network Analyses · pp. 7–8
  • Null models for brain networks should reflect edge properties such as sign, spatial embedding, and transitive relationships; distance-insensitive surrogates yield longer connections than expected.Thresholding dense similarity networks can bias or confound topology, and different thresholds may affect findings.Source: Edge-Centric Network Analyses · p. 8
  • Accounting for noisy edges or imperfect graph observation is identified as a potentially fruitful future direction for network neuroscience.Source: Edge-Centric Network Analyses · p. 9

Edge Constructs: From Motifs to Higher-Order Relations

Edge groupings capture patterns beyond individual pairwise relationships, from local motifs and node fingerprints to community and higher-order organization. These constructs complement node-centric views by linking local relationships to functional capabilities and global network context.

Edge Constructs: From Motifs to Higher-Order Relations

  • Edge groupings capture distributed patterns that mass univariate analyses of independent edges may miss or lack power to detect.The section considers constructs ranging from localized motifs and connectivity fingerprints to global network topology.Source: Edge Constructs: From Motifs to… · p. 9

Motifs

  • Motifs are fixed-size subgraphs distinguished by their edge patterns, and their frequencies reveal local building blocks of network organization.Surrogate networks can assess motif under- or over-expression; some configurations, such as resonance pairs, can induce zero-lag synchrony despite non-zero conduction delays.Source: Motifs · p. 11

Connectional Fingerprints

  • Connectional fingerprints are each node’s complete pattern of edge weights, placing regions in an abstract connectivity space rather than a geometric one.They can help characterize functional roles, predict activation patterns, identify cross-species homologies, and subdivide larger regions by fine-grained connectivity profiles.Source: Connectional Fingerprints · pp. 11–12

Community Structure

  • Edges determine community membership and can characterize integrative hubs and between-community patterns such as modular, core-periphery, or disassortative organization.Summed or averaged edge strengths between communities provide a block-structure summary.Source: Community Structure · p. 12

Higher-Order Relationships

  • Algebraic topology uses cliques and cavities to analyze higher-order relationships, including all-to-all components and topological cycles in brain networks.Applications described include localizing hub-like roles, studying spiking-activity progression, and identifying increased integrative organization after psilocybin administration.Source: Higher-Order Relationships · p. 12
  • Edge-based constructs—from motifs and fingerprints to communities and cliques—connect local relationships with functional capabilities and nodes’ global connectivity context.These approaches complement methods that extract multivariate inter-areal relationships outside a network context.Source: Higher-Order Relationships · p. 13

Edges in Communication and Brain Dynamics

The section examines how structural edges relate to functional organization and how edge-level communication and activity vary over time. It describes communication models and edge time series as ways to study those relationships beyond static, time-averaged connectivity.

Edges in Communication and Brain Dynamics

  • Structural edges can scaffold communication between regions, and communication models use algorithmic rules to relate anatomical and functional network organization.Shortest-path routing follows the structural shortest path; search information quantifies the information needed to navigate it, while path transitivity captures opportunities to return after deviations.Source: Edges in Communication and Brain… · p. 13, Figure 3 · p. 14

Structure-Function Relationships

  • Structure and function show a moderate positive edge-weight association, but higher-order context can better capture their relationship and improve functional-topology prediction.Pairwise comparisons can be confounded by overlap and transitivity or biased by distance; communication models incorporating polysynaptic signaling also improve alignment with empirical functional topology and behavioral prediction.Source: Structure-Function Relationships · pp. 14–15

Time-varying Functional Connectivity

Edge Time Series

  • Edge time series recover frame-by-frame co-fluctuations, revealing intermittent high-amplitude events that account for much of time-averaged functional connectivity.Each frame’s co-fluctuation pattern forms exactly two communities; canonical functional systems are described as transient expressions whose familiar architecture results from superposed bipartitions.Source: Edge Time Series · p. 16
  • Edge time series enable temporally precise study of communication dynamics and could extend to electrophysiology, lagged directionality analyses, synapse-level spike models, and pointwise mutual information.Their networked dynamics and co-fluctuation patterns remain open areas for study, including patterns that evolve intrinsically or during experimental manipulations.Source: Edge Time Series · pp. 16–17

Future Directions

Treating edges as fundamental units makes it possible to compare their information and identify higher-order organization. Edge-based analyses also connect brain network structure to white-matter properties and subject-specific brain–behavior prediction.

Relationships Between Edges

  • Edge-edge matrices compare edge information, particularly over time, while edge-community profiles reveal organization complementary to node-based clustering.Figure 4 illustrates edge-community assignments, node-level entropy, and similarity among community profiles.Source: Relationships Between Edges · p. 17, Figure 4 · p. 18
  • Edge functional connectivity clustering reveals overlapping communities that bridge canonical systems and nested edge-level structure within control and default mode networks.Pairwise temporal co-fluctuation profiles can also form task-specific hyperedges; edge functional connectivity may contain sources of individual variation.Source: Relationships Between Edges · p. 18

White Matter Matters

  • White-matter paths have shape, curvature, and volume, and compete for physical space and metabolic resources, potentially shaping network relationships.Activity-dependent myelination and ephaptic coupling are proposed as possible influences on functional similarity.Source: White Matter Matters · p. 19

Subject-Specific Edge Information

  • Connectome predictive modeling filters edges by statistical criteria and sums their weights to predict phenotypes in left-out subjects.The resulting edge sets can be analyzed for system-level organization, including between-system edges in a high-attention model (Rosenberg, Finn et al. 2016).Source: Subject-Specific Edge Information · pp. 19–20
  • Whether higher-order edge constructs can extend these predictive models remains an open question and a possible future direction alongside algebraic topology.Source: Subject-Specific Edge Information · p. 20

Conclusion

The conclusion calls for greater attention to edge-level information alongside established node-centric approaches. It presents edges as a way to investigate brain organization and communication across data modalities and scales.

  • Edge-centric and node-centric perspectives can both benefit network neuroscience because nodes and edges are intertwined ingredients of network models.Source: Conclusion · p. 20
  • The paper reviews how edge construction choices and groups of edges can contribute higher-order relational information for analysis.Source: Conclusion · p. 20
  • Edges are identified as candidates for exploring how communication processes unfold across neural data modalities and spatial and temporal scales.Source: Conclusion · p. 20

Box: Alternative Weighting Strategies

Edges can carry attributes in addition to connectivity or similarity, including values that combine information across modalities or conditions. Such annotations and weights can support computational modeling and data analysis.

  • Edge attributes can annotate connectivity or similarity with features such as tract length, conduction delay, connection cost, or myelination status.These attributes may come from another modality or embedding space and can aid computational modeling or data analysis.Source: Box: Alternative Weighting Strategies · p. 20
  • Morphometric similarity weights summarize correlations among standardized measures of myelination, gray matter, and curvature at nodes.Source: Box: Alternative Weighting Strategies · p. 20
  • Combining rest and task data can increase intrinsic connectivity estimation reliability, while across-session correlation values can form edge-level features for covariance analysis across conditions.Together, these examples show that edge weights can incorporate multiple data sources.Source: Box: Alternative Weighting Strategies · p. 21

Box: Parcellating Nodes or Edges

Brain network nodes can be parcellated in multiple ways, while edge information can also be clustered or segmented as a basis for network models. Treating such edge information as neural elements could prompt further modeling of white-matter organization.

  • Node definitions can be based on tissue properties or functionally coherent elements, and changing the node definition likely requires recomputing edges.Source: Box: Parcellating Nodes or Edges · p. 21
  • White-matter tractography streamlines can be hierarchically clustered into bundles that serve as edge-based building blocks of network models.The resulting tracts can be represented by their intersections on cortical gray-matter nodes.Source: Box: Parcellating Nodes or Edges · p. 21
  • Clustering white-matter BOLD signal can form parcels related to canonical gray-matter systems, and edge-based parcellation may encourage further modeling of white-matter organization.Source: Box: Parcellating Nodes or Edges · pp. 21–22

Box: Spatial Embedding Makes Brain Networks Unique

Spatial embedding makes brain edges costly: material and energetic demands favor short-range connections, yet a few long-range links remain and may add distinctive signals with functional consequences.

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