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Dynamic Graph Metrics: Tutorial, Toolbox, and Tale
Ann E. Sizemore, Danielle S. Bassett
TL;DR
Static graph analyses do not adequately represent time-varying connectivity in brain networks. The paper surveys dynamic graph concepts, measures, visualization strategies, and null models, provides a MATLAB toolbox, and illustrates the tools with neuroimaging data. Its supported conclusion is that these resources offer the neuroimaging community practical tools and intuition for studying dynamic graphs.
Problem
Traditional graph representations cannot address the complexity of time-varying connectivity relevant to emotional and cognitive state, task-switching, development, aging, and disease progression.
Method
The paper provides a tutorial and toolbox covering dynamic graph visualization, contact-sequence encoding, temporal measures, null models, and applications to functional and structural graphs.
Results
The paper demonstrates the surveyed dynamic graph measures using a previously published time-varying fMRI functional-graph dataset.
Takeaways & Limitations
The tools and intuition are intended to support analyses of dynamic brain graphs across neuroimaging questions and datasets.
Abstract
from arXiv · showhide
The central nervous system is composed of many individual units -- from cells to areas -- that are connected with one another in a complex pattern of functional interactions that supports perception, action, and cognition. One natural and parsimonious representation of such a system is a graph in which nodes (units) are connected by edges (interactions). While applicable across spatiotemporal scales, species, and cohorts, the traditional graph approach is unable to address the complexity of time-varying connectivity patterns that may be critically important for an understanding of emotional and cognitive state, task-switching, adaptation and development, or aging and disease progression. Here we survey a set of tools from applied mathematics that offer measures to characterize dynamic graphs. Along with this survey, we offer suggestions for visualization and a publicly-available MATLAB toolbox to facilitate the application of these metrics to existing or yet-to-be acquired neuroimaging data. We illustrate the toolbox by applying it to a previously published data set of time-varying functional graphs, but note that the tools can also be applied to time-varying structural graphs or to other sorts of relational data entirely. Our aim is to provide the neuroimaging community with a useful set of tools, and an intuition regarding how to use them, for addressing emerging questions that hinge on accurate and creative analyses of dynamic graphs.
Introduction
The paper motivates dynamic graph analysis because static functional-network representations cannot capture time-varying connectivity relevant to cognition, development, aging, and disease. It introduces a tutorial, visualization guidance, dynamic graph measures, and a MATLAB toolbox for applying these methods.
- Motivation: Graph representations model neural units as nodes connected by functional or structural interactions, supporting holistic analysis across spatial and temporal scales.The framework is intended to complement reductionistic studies of individual units.
- Motivation: Static functional graphs summarize interactions over fixed periods, limiting analyses of network dynamics that support cognition and changing brain states.Dynamic assessments are relevant to emotional and cognitive state, development, aging, and disease progression.
- Scope: The paper surveys mathematical tools for dynamic graphs, including visualization methods, temporal paths, latency, centrality, null models, temporal small-worldness, and modular structure.The paper also outlines scenarios for applying dynamic graphs to brain structure and function.
- Scope: A publicly available MATLAB toolbox enables readers to apply the surveyed measures to their own data and hypotheses.The toolbox is demonstrated on a previously published neuroimaging dataset and is intended as a practical complement to the tutorial.
Visualizing dynamic graphs
Dynamic graphs can be visualized as time-indexed snapshots or as a movie of changing nodes and edges. The paper illustrates these approaches using time-varying functional connectivity derived from fMRI data during motor-skill learning.
- Visualization: A dynamic network may be viewed as a movie in which nodes and edges appear or disappear across time.When movies are impractical, individual snapshots can display the network at each timepoint.
- Visualization: Snapshot visualizations show the network frame by frame, preserving time-indexed structure while providing a static representation for each timepoint.The passage introduces snapshots as an alternative when a movie is not feasible.
- Example dataset: The example uses fMRI data collected while participants learned a sequence of finger movements across four training sessions.The dataset contains activity estimates from 112 cortical and subcortical regions over ten time windows per session.
- Example dataset: Functional connectivity in the example is estimated from the magnitude-squared coherence of wavelet coefficients, producing one coherence matrix for each time window.These matrices provide the time-varying functional graphs used in the illustration.
Basic measures
Dynamic graphs encode network contacts together with their timing, enabling measures of reachability, communication speed, and centrality that preserve temporal dependencies lost in static or time-aggregated representations.
- Encoding dynamic graphs: A dynamic graph can be stored efficiently as a contact sequence listing each edge, its time, and optional weights or traversal delays.This representation replaces separate graph snapshots with contacts represented as tuples such as (i, j, t, w1, w2, …, wk).
- Dynamic measures: Dynamic measures should not simply extend static measures independently across time because that assumes observations at consecutive timepoints are independent.The paper’s toolbox and visualizations are designed to retain the evolution of connectivity across time.
- Time-respecting paths: Time-aggregated graphs can show paths that are invalid when contacts occur in an incompatible temporal order.Time-respecting paths must follow the sequence of contacts, so static connectivity does not guarantee dynamic reachability or transitivity.
- Time-respecting paths: Source sets identify nodes that can reach a target by a specified time, whereas sets of influence identify nodes reachable from a source after a specified time.Together, these forward- and backward-looking sets form light cones describing what could have affected a node or what it may affect.
- Latency and centrality: Latency measures the shortest time required to move between nodes along time-respecting paths, extending path distance to dynamic networks.Temporal centrality measures can then use fastest paths or temporal distances to characterize how nodes communicate through time.
- Latency and centrality: Broadcast and receive centrality capture complementary sides of dynamic communicability, describing how regions distribute information across the network and across time.In the example fMRI graph, broadcast centrality was highest in posterior parietal and posterior temporal fusiform regions, whereas receive centrality was highest in somatomotor and premotor regions.
Null models and additional measures
Dynamic null models preserve different aspects of temporal networks, enabling tests of whether observed temporal structure and small-worldness differ from randomized expectations. Community metrics further characterize how nodes and groups reorganize across time.
- Null models: Dynamic null models test whether observed temporal architecture differs from that expected under an appropriate statistical baseline.The paper focuses on randomized edges and randomly permuted times models.
- Null models: The randomized edges (RE) model rewires edges within each timepoint while preserving contact times.It adapts the degree-preserving configuration-model logic to dynamic graphs.
- Null models: The randomly permuted times (RP) model randomizes when contacts occur, destroying true temporal contact patterns while preserving overall event rates.
- Null models: Both RE and RP produce lower temporal correlation than the empirical network, indicating smoother reconfiguration in the true data than in either null model.Temporal correlation reflects average topological overlap between successive neighbors of a node.
- Temporal small-worldness: Temporal small-worldness compares temporal correlation and temporal path length with corresponding values from RE or RP null models.The resulting measure is based on C/CRE and L/LRP, where C is temporal correlation and L is characteristic temporal path length.
- Temporal small-worldness: Temporal small-worldness decreases with increasing number of trials practiced in the example empirical fMRI graph.
- Dynamic community structure: Dynamic community metrics quantify node flexibility, promiscuity, and cohesion beyond community assignments alone.Flexibility captures community switching, promiscuity captures distribution across communities, and cohesion captures coordinated community changes between nodes.
Contexts for the Application of Dynamic Graph Metrics
Dynamic graph metrics can be applied across scales, species, timescales, and data types to study changing brain organization and behavior. The paper highlights applications to cognition, development and aging, disease and therapy, and non-functional graph data.
- Cross-scale, Cross-species: Dynamic graph tools extend beyond human fMRI to neuronal circuits and network neuroscience across species and spatial scales.
- Cognitive Processes: Cognitive tasks such as memory, attention, and learning involve appreciable changes in human functional network architecture.
- Development and Aging: Development and aging involve network reconfiguration over months, years, and decades, alongside changes in cognitive abilities.The paper proposes dynamic graph metrics as a framework for studying cognitive-control development in fronto-parietal circuitry.
- Disease Processes, Disease Progression, Response to Therapy: Dynamic graph metrics may help test whether disease-related or rehabilitation-related network reconfigurations involve flexibility, promiscuity, or cohesion.The discussion includes stroke rehabilitation and the possibility of a linear or inverted U-shaped relationship with reconfiguration.
- Extensions to Other Sorts of Graphs: Dynamic graphs can represent structural connectivity, symptom relationships, mood-marker covariance, and behavior in addition to functional brain data.These applications include studying skill acquisition and adaptive decision-making.
Conclusion
The paper provides a tutorial and toolbox for defining, visualizing, and characterizing dynamic graphs, supporting their application to neuroscientific questions.
- The tutorial covers dynamic graph concepts, visualization, and characterization measures.Measures include time-respecting paths, latency, centrality, clustering, characteristic temporal path length, and dynamic modular structure.
- The paper discusses null models and measures that depend on them, including temporal small-worldness.
- The authors outline scenarios for constructing and studying dynamic graphs in neuroscience.These scenarios address questions about brain structure and function and the neurophysiological mechanisms of behavior and disease.
- A publicly available MATLAB toolbox enables readers to apply the tools to their own data.
- The paper aims to provide the neuroimaging community with tools and intuition for detailed analyses of dynamic graphs.
Appendix
The appendix defines dynamic-network representations, temporal connectivity concepts, path measures, centrality measures, and community-dynamics measures.
- Initial definitions: A dynamic network uses vertices V and edges across timepoints, represented as a sequence of N × N adjacency matrices.
- Temporal representations: A contact is an edge at a specified time, while a contact sequence lists contacts as tuples (i, j, t).
- Temporal paths: A time-respecting path is an ordered sequence of contacts with strictly increasing times; temporal path length spans its first to last contact.
- Temporal paths: Latency is the temporal path length of the fastest path between two nodes, also called temporal distance.
- Centrality and connectivity: Dynamic-network measures include forward latency, betweenness, closeness, broadcast, receive, and temporal correlation centrality.These measures characterize temporal reachability, shortest paths, communication, and connectivity patterns.
- Network and community measures: Temporal small-worldness compares temporal correlation and characteristic path length with a randomized model, while flexibility, promiscuity, and cohesiveness quantify community changes.Promiscuity uses the number of communities containing a node relative to the total number of communities.