Source-linked AI summary
Persistent homology of time-dependent functional networks constructed from coupled time series
Bernadette J. Stolz, Heather A. Harrington, Mason A. Porter
TL;DR
The paper asks how persistent homology can reveal structure and change in functional networks constructed from time-series data. It applies weight rank clique filtrations and persistence landscapes to coupled Kuramoto oscillators and task-based fMRI networks, finding differences in synchronization patterns and network loops over time. The results indicate that filtration position can be important alongside persistence, with most motor-learning loop changes occurring on the second of three monitored days.
Problem
The paper examines how topological analysis can provide insight into functional networks built from experimental and synthetic time-series data, including how features should be distinguished from noise.
Method
The study constructs functional networks from time-series similarities and analyzes them with persistent homology using a weight rank clique filtration and persistence landscapes.
Results
Persistent homology detects synchronization-pattern differences, community segregation, and changes in 1-dimensional loops in Kuramoto and task-based fMRI functional networks.
Takeaways & Limitations
Filtration birth and death positions can be more informative than persistence alone, while medium-lived 1-loops can capture functional-brain-network changes without a full filtration.
Abstract
from arXiv · showhide
We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging (fMRI) data that was acquired from human subjects during a simple motor-learning task in which subjects were monitored on three days in a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process.
I. INTRODUCTION
The paper constructs functional networks from coupled time series and applies persistent homology to study higher-order network structure across weighted filtrations. It emphasizes that filtration position may matter alongside persistence when interpreting loops as signal or noise.
- Topological analysis: Computational topology captures higher-order structures such as loops and holes beyond pairwise network connections.These methods provide information about global low-dimensional structures in networks.
- Functional networks: Functional networks represent entities as nodes and use weighted edges to quantify similarity between their time series.The paper applies this construction to coupled oscillators and functional brain data.
- Thresholding and filtrations: A global threshold strongly influences the weighted adjacency matrix and the structure of the resulting graph.The paper therefore considers filtrations generated by multiple thresholds rather than relying only on one threshold.
- Topological features: The analysis focuses on 1-dimensional loops rather than 0-dimensional connected components because loops capture richer topology and have demonstrated importance in network applications.Loops are topological circles formed by at least four connected edges.
- Study scope: The study applies persistent homology to time-series networks from coupled Kuramoto oscillators and task-based fMRI data, with intuitive methodological background provided before the two examples.The paper also discusses prior applications of persistent homology and the unresolved statistical relationship between short persistence and noise in empirical data.
A. Simplicial complexes
The paper builds topological representations from simplices and simplicial complexes, then studies their homology across filtrations created by ranking weighted graph edges. Barcodes record when features are born and die, including connected components and loops.
- A. Simplicial complexes: A k-simplex contains k + 1 independent nodes: points, edges, triangles, and tetrahedra are 0-, 1-, 2-, and 3-simplices.Lower-dimensional simplices form faces of higher-dimensional simplices.
- A. Simplicial complexes: A simplicial complex includes every face of each simplex, and any two simplices may intersect only in common faces.These conditions distinguish valid simplicial complexes from collections whose simplex intersections are incomplete.
- A. Simplicial complexes: Homology detects topological invariants such as holes, with dimension-1 homology elements called loops and divided into homology classes.Different loops surrounding the same hole can represent the same homology class.
- C. Filtrations: A filtration is a nested sequence of simplicial complexes from the empty complex to the entire complex, allowing feature persistence to be tracked across steps.A feature is born at its first appearance and dies when it disappears at a later filtration step.
- 1. Weight rank clique filtration: The weight rank clique filtration ranks edge weights, thresholds the graph at each rank, and converts maximal c-cliques into c-simplices.The resulting clique complex is valid because cliques are closed under intersection and taking subsets.
- 1. Weight rank clique filtration: A barcode represents each topological feature by an interval from birth time b_l to death time d_l, whose length d_l − b_l measures persistence.The paper illustrates both 0-dimensional component barcodes and 1-dimensional loop barcodes.
D. Persistence landscapes
Persistence landscapes transform barcode intervals into layered piecewise-linear functions that preserve barcode information while supporting averaging and statistical comparison. Their layers are formed by ranking interval-associated functions pointwise.
- D. Persistence landscapes: A barcode interval (b, d) is converted into a piecewise-linear function supported on the interval and peaked at its midpoint.The supplied equation defines the function as zero outside (b, d) and triangular within it.
- D. Persistence landscapes: The qth persistence landscape λ_q assigns the qth-largest value among all interval functions at each x.If no qth-largest value exists, λ_q(x) is set to zero.
- D. Persistence landscapes: The persistence landscape is the sequence of its layers, which can be viewed as successively peeling away the highest peaks of the barcode representation.The 0th layer outlines the topmost peaks, while later layers summarize remaining peaks.
- D. Persistence landscapes: Persistence landscapes support unique averaging and Lp distances between average landscapes, enabling statistical comparison across groups of barcodes.Barcodes themselves are not elements of a Banach space, so direct averaging is unavailable in the same form.
E. Computational tools
The study models coupled oscillators with the Kuramoto equations, constructs functional networks from time-series similarity, and analyzes them using persistent homology and persistence landscapes.
- Computational tools: Persistent-homology calculations use javaPlex, maximal clique finding uses Bron–Kerbosch, and persistence landscapes are analyzed with a dedicated toolbox.Average landscapes are formed by taking means of function values layer by layer, unlike barcodes.
- Kuramoto model: The Kuramoto model describes coupled phase oscillators with natural frequencies and coupling strength governing their interactions.The generalized model uses an adjacency matrix to impose a structural network between oscillators.
- Kuramoto model: The oscillator system is divided into 8 communities of 16 oscillators, with 13 within-community connections and 1 external connection per oscillator.The network contains N = 128 oscillators and uses coupling strength κ = 0.2.
- Simulation: The simulation uses ODE45 over [0, Tmax] with Tmax = 10 and produces M = 500 time steps for each oscillator.The time series are partitioned into two regimes of 250 time steps, and a full 500-step regime is also analyzed.
- Functional networks: Functional-network edge weights are defined from pairwise synchrony between oscillator time series across simulations and time regimes.The same coupling matrix is used for 20 simulations before similarity values define edges in fully connected weighted networks.
B. Null models for the Kuramoto data
The Kuramoto analysis uses null models to test whether observed network dynamics reflect meaningful oscillator dynamics or can be explained by randomized time-series processes.
- Null-model design: Two null models assess whether observations from the Kuramoto simulations reflect meaningful dynamics rather than random processes.Both null models are constructed from the time-series output.
- Simple null model: The simple null model randomly reassigns the temporal order of each oscillator’s time series before similarity computation.The randomized series are then analyzed using the same similarity measure.
- Fourier null model: The Fourier null model creates surrogate data by randomizing Fourier-transform phases while preserving selected statistical structure.It preserves the original mean, variance, linear autocorrelations, and cross correlations between time series.
- Null-model interpretation: The two null models had previously been used with coupled Kuramoto time series and exhibit dynamics different from those of the coupled oscillators.
C. Persistent homology applied to the Kuramoto model and null models
Persistent homology and persistence landscapes distinguish Kuramoto-network dynamics across time regimes and from null models. The results show that feature position in the filtration, alongside persistence, identifies community-level synchronization patterns.
- Kuramoto model and filtration: The WRCF analyzes dimension-1 loops in functional networks constructed from Kuramoto time-series data across two time regimes and null-model comparisons.The analyses use filtration steps up to 1800 or 2000 for separate regimes and up to 1100 for single-regime comparisons.
- Two time regimes: In the first regime, early short-lived bars represent intra-community loops, while the second regime includes more early bars and inter-community loops associated with stronger global synchronization.In regime 1, the first 44 bars represent intra-community loops; in regime 2, only 2 of the first 28 do.
- Two time regimes: The second time regime contains more persistent 1-loops, clearer separation between peak groups, and peaks reaching about 100, whereas comparable first-regime peaks are about half as high.The regimes are time steps 1–250 and 251–500; stronger synchronization between communities accompanies the increased persistence.
- Landscape comparison: L2 landscape distances are large between the two time regimes and among the Kuramoto and null-model landscapes, but L2 distance does not encode the community identity of early loops.The reported distances are 27078 between time regimes, 13540 between Kuramoto and Fourier null models, 13263 between null models, and 11703 between Kuramoto and simple null models.
- Null-model comparison: Compared with both null models, the Kuramoto landscape shows stronger intra-community synchronization, clearer peak-group separation, and two higher peaks corresponding to persistent inter-community loops.The null-model landscapes contain less persistent or less separated early peaks; the Fourier null model appears noisier and lacks peak separation.
- Interpretation: The study concludes that barcode feature position can be as important as persistence length for interpreting dynamics, while persistence landscapes alone are insufficient without node information.Combining landscapes with the nodes forming loops links topological features to synchronization and community structure.
IV. EXAMPLE II: TASK-BASED FMRI DATA
The fMRI analysis applies WRCF and persistence landscapes to dimension-1 loops in functional brain networks measured across three motor-learning days. Results indicate changes in community structure and synchronization, concentrated especially on day 2, although landscape-distance differences are not statistically significant.
- Data and network construction: The study applies WRCF to functional networks whose weighted edges quantify similarity between brain-region time series.These functional connections differ from structural connections because they represent time-series similarity rather than physical anatomy.
- Data and network construction: 20 healthy subjects performed a simple motor-learning task while fMRI monitored 112 brain areas across three study days.Each subject and day contributed 2000 time points for every monitored region.
- Persistence landscapes: Dimension-1 landscapes consistently show small peaks during filtration steps 1–200, a pattern appearing in most subjects’ barcodes.The average standard deviations in this interval are 127, 167, and 126 for days 1, 2, and 3, respectively.
- Persistence landscapes: Changes in early peaks suggest altered community structure on day 2, potentially reflecting stronger within-community synchronization or greater between-subject variability.Day 2 loses some peaks during steps 20–60, while more persistent day-3 peaks occur between steps 80 and 200.
- Landscape distances and interpretation: Average landscape distances suggest most network changes occur between days 1 and 2, but permutation tests found no statistically significant differences.The reported p-values are about 0.4 for days 1–2, 0.85 for days 2–3, and 0.6 for days 1–3; standard deviations were about four times larger than distances.
- Landscape distances and interpretation: The primary average-landscape peak shifts left across days, indicating increased edge weights and presumably stronger synchronization in edges supporting persistent loops.The authors suggest this may reflect either stronger synchronization of existing loops or newly appearing loops with more synchronized edges.
- Persistence landscapes: Average-network landscapes show larger persistent features on days 1 and 3 than day 2, with larger peaks appearing around step 400 on day 3 versus step 900 on day 1.Average networks were formed by taking the mean edge weight across all 20 subjects separately for each day.
V. CONCLUSION AND DISCUSSION
The paper shows that persistent homology and persistence landscapes reveal synchronization, community, and loop-structure changes in functional networks. It also argues that filtration position can matter alongside persistence, while noting interpretive and data-scope limitations.
- Interpreting filtration features: Early 1-loops can distinguish weighted-network data even when their persistence is short.The authors recommend considering birth and death times, not persistence alone, when separating signal from noise.
- Interpreting filtration features: Persistence landscapes can show statistically significant differences even when barcodes contain short intervals commonly interpreted as noise.This observation agrees with prior synthetic-barcode and protein-structure findings cited by the authors.
- Network and synchronization findings: Weight rank clique filtration landscapes capture changes in functional brain networks during a simple motor-learning task.This finding holds for both average persistence landscapes and landscapes computed from average networks.
- Network and synchronization findings: Persistent homology detects community segregation and distinguishes strongly structured Kuramoto networks from weakly synchronized null-model communities.The analysis also detects community-structure symptoms in task-based fMRI networks across three learning days.
- Motor-learning dynamics: Most changes in fMRI network loops occur on the second learning day, on average.Brain regions forming 1-loops on days 2 and 3 seem more strongly synchronized on average than those forming loops on day 1.
- Interpretive boundaries: The fMRI conclusions should be revisited with larger data sets, and additional tools such as persistence images could help identify responsible topological features.The paper also suggests studying connections with graph Laplacians and applying filtrations to other synthetic community networks.
- Interpretive boundaries: Biological interpretation requires identifying the specific nodes in topological features, and feature relevance depends on the filtration and dynamical system.The paper contrasts medium-sized landscape features associated with weak synchronization in a Fourier null model with increasing loop synchronization in task-based fMRI.
- Overall conclusion: Persistent homology and persistence landscapes apply to functional networks from experimental data and model-generated time series, revealing communities and changing network structure.The conclusion presents these methods as mathematically principled and generalizable.
Appendix A: Table with often-occurring brain regions in 1-dimensional loops
The appendix introduces a table of brain regions that frequently occur in one-dimensional loops of functional networks.
- Table II: Table II lists brain regions that often occur in one-dimensional loops of functional networks.The table records loops whose edges occur at least 50 times across subjects and marks occurrences by study day.
Appendix B: Topological background and definitions
Appendix B provides a brief introduction to the mathematical concepts underlying persistent homology.
- Background: The appendix adapts and summarizes a prior master’s-thesis discussion to introduce the mathematical concepts behind persistent homology.The supplied passage does not enumerate those concepts further.
1. Simplicial complexes
Simplicial complexes are built from simplices and their faces under closure and intersection rules, providing combinatorial representations of topological spaces.
- Simplices: Affine independence means that the vectors from one point to the others are linearly independent.In R2, two distinct points are affinely independent, as are three non-collinear points.
- Convex hulls: Convex combinations restrict coefficients to nonnegative values summing to one, and their set forms the convex hull.A triangle spanned by three points is an example of such a convex hull.
- Simplices: A k-simplex is the convex hull of k + 1 affinely independent points, with k denoting its dimension.Examples include points, edges, triangles, and tetrahedra as dimensions 0 through 3.
- Simplices and complexes: Lower-dimensional simplices are faces of higher-dimensional simplices because subsets of affinely independent points remain affinely independent.The appendix uses a tetrahedron as an example of a shape assembled from triangular faces and edges.
- Faces and cofaces: A face is the convex hull of a subset of a simplex’s vertices, while the original simplex is its coface.The notation τ ≤ σ denotes a face and τ < σ a proper face.
- Simplicial complexes: A simplicial complex is a finite collection of simplices containing every face of each simplex, with pairwise intersections empty or shared faces.These rules distinguish simplicial complexes from arbitrary collections of simplices.
- Simplicial complexes: A simplicial complex may consist of edges without including the triangle they bound, whereas other examples include connected edges and triangles.The appendix illustrates valid and invalid constructions in Fig. 13.
- Simplicial complexes: The dimension of a simplicial complex is the dimension of its highest-dimensional simplex.A simplicial complex represents a topological space only when a homeomorphism preserves properties such as connectedness.
2. Homology and Betti numbers
Homology detects holes by comparing cycles with boundaries, while Betti numbers quantify the resulting homology groups. In this framework, 1-dimensional loops are represented as homology features after boundaries are removed.
- Over Z/2Z, p-chains can be treated as subsets of simplices and added by retaining simplices appearing in only one summand.Shared simplices occur twice and cancel under addition in Z/2Z.
- A p-cycle lies in ker ∂p, whereas a p-boundary lies in Im ∂p+1; every boundary is therefore a cycle.The inclusion Bp ⊆ Zp follows from the chain-complex property that consecutive boundary maps compose to zero.
- Edges map to endpoint nodes under ∂1, while a 1-loop has zero boundary because each node occurs twice.This distinguishes loop cycles from ordinary individual edges.
- Homology groups detect topological holes by quotienting p-cycles by p-boundaries.Hp = Zp/Bp identifies cycles that differ only by a boundary as the same homology class.
- β0, β1, and β2 represent connected components, 1-dimensional loops, and 2-dimensional holes, respectively.Betti numbers are the ranks, or dimensions, of the corresponding homology groups over Z/2Z.
3. Filtrations
A filtration is a nested sequence of subcomplexes from the empty complex to the full complex. Persistent homology tracks when homology classes appear and disappear along this sequence.
- A subcomplex is a subset of simplices that itself satisfies the properties of a simplicial complex.
- A filtration is a nested sequence ∅= Σ0 ⊆Σ1 ⊆Σ2 ⊆· · · ⊆Σk = Σ of subcomplexes.Natural inclusion maps connect each successive complex.
- Filtrations record when homology features first appear and whether they subsequently disappear.
- A homology class is born at Σm when it is present there but absent from the preceding inclusion image.
- A homology class dies entering Σn when it is present in Hp(Σn−1) but absent from the preceding inclusion image.
- The persistence of a class is defined from its birth step mh and death step nh.