arXiv subjects

Sixtus Dakurah

Publications and source records attributed to Sixtus Dakurah.

7 recordsLinked to original sources

arXiv2025

Topologically Invariant Permutation Test

Functional brain networks exhibit topological structures that reflect neural organization; however, statistical comparison of these networks is challenging for several reasons. This paper introduces a topologically invariant permutation test for detecting topological inequivalence. Under topological equivalence, topological features can be permuted separately between groups without distorting individual network structures. The test statistic uses $2$-Wasserstein distances on persistent diagrams, computed in closed form. To reduce variability in brain connectivities while preserving topology, heat kernel expansion on the Hodge Laplacian is applied with bandwidth $t$ controlling diffusion intensity. Theoretical results guarantee variance reduction through optimal Hilbert space projection. Simulations across diverse network topologies show superior performance compared to conventional two-sample tests and alternative metrics. Applied to resting-state fMRI data from the Multimodal Treatment of ADHD study, the method detects significant topological differences between cannabis users and non-users.

q-bio.NC↗

arXiv2025

Discrete Heat Kernels on Simplicial Complexes and Its Application to Functional Brain Networks

Networks constitute fundamental organizational structures across biological systems, although conventional graph-theoretic analyses capture exclusively pairwise interactions, thereby omitting the intricate higher-order relationships that characterize network complexity. This work proposes a unified framework for heat kernel smoothing on simplicial complexes, extending classical signal processing methodologies from vertices and edges to cycles and higher-dimensional structures. Through Hodge Laplacian, a discrete heat kernel on a finite simplicial complex $\mathcal{K}$ is constructed to smooth signals on $k$-simplices via the boundary operator $\partial_k$. Computationally efficient sparse algorithms for constructing boundary operators are developed to implement linear diffusion processes on $k$-simplices. The methodology generalizes heat kernel smoothing to $k$-simplices, utilizing boundary structure to localize topological features while maintaining homological invariance. Simulation studies demonstrate qualitative signal enhancement across vertex and edge domains following diffusion processes. Application to parcellated human brain functional connectivity networks reveals that simplex-space smoothing attenuates spurious connections while amplifying coherent anatomical architectures, establishing practical significance for computational neuroscience applications.

q-bio.QM↗

arXiv2025

Cycles Communities from the Perspective of Dendrograms and Gradient Sampling

Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for constructing cycle communities through two complementary approaches. First, a dendrogram-based methodology leverages merge-tree algorithms to construct hierarchical representations of homology classes from persistence intervals. The Wasserstein distance on merge trees is introduced as a metric for comparing dendrograms, establishing connections to hierarchical clustering frameworks. Through simulation studies, the discriminative power of dendrogram representations for identifying cycle communities is demonstrated. Second, an extension of Stratified Gradient Sampling simultaneously learns multiple filter functions that yield cycle barycenter functions capable of faithfully reconstructing distinct sets of cycles. The set of cycles each filter function can reconstruct constitutes cycle communities that are non-overlapping and partition the space of all cycles. Together, these approaches transform the problem of cycle matching into both a hierarchical clustering and topological optimization framework, providing principled methods to identify similar topological structures both within and across groups of topological objects.

q-bio.QM↗

arXiv2025

Spanning Tree Basis for Unbiased Averaging of Network Topologies

In recent years there has been a paradigm shift from the study of local task-related activation to the organization and functioning of large-scale functional and structural brain networks. However, a long-standing challenge in this large-scale brain network analysis is how to compare network organizations irrespective of their complexity. The maximum spanning tree (MST) has served as a simple, unbiased, standardized representation of complex brain networks and effectively addressed this long-standing challenge. This tree representation, however, has been limited to individual networks. Group-level trees are always constructed from the average network or through a bootstrap procedure. Constructing the group-level tree from the average network introduces bias from individual subjects with outlying connectivities. The bootstrap method can be computationally prohibitive if a good approximation is desired. To address these issues, we propose a novel spectral representation of trees using the spanning tree basis. This spectral representation enables us to compute the average MST and demonstrate that this average tree captures the global properties of all the MSTs in the group and also overlaps with the union of the shortest paths in the functional brain networks.

q-bio.QM↗

arXiv2025

Brain Networks Flow-Topology via Variance Minimization in the Wasserstein Space

This work introduces a novel framework for testing topological variability in weighted networks by combining Hodge decomposition with Wasserstein variance minimization. Traditional approaches that analyze raw edge weights are susceptible to noise driven perturbations, limiting their ability to detect meaningful structural differences between network populations. Network signals are decomposed into various components using combinatorial Hodge theory, then topological disparity is quantified via the 2-Wasserstein distance between persistence diagrams. The test statistic measures variance reduction when comparing within group to between group dispersions in the Wasserstein space. Simulations demonstrate that the proposed method suppresses small random perturbations while maintaining sensitivity to genuine topological differences, particularly when applied to Hodge decomposed flows rather than raw edge weights. The framework is applied to functional brain networks from the Multimodal Treatment of ADHD dataset, comparing cannabis users and non-users

q-bio.QM↗

arXiv2025

MaxTDA: Robust Statistical Inference for Maximal Persistence in Topological Data Analysis

Persistent homology is an area within topological data analysis (TDA) that can uncover different dimensional holes (connected components, loops, voids, etc.) in data. The holes are characterized, in part, by how long they persist across different scales. Noisy data can result in many additional holes that are not true topological signal. Various robust TDA techniques have been proposed to reduce the number of noisy holes, however, these robust methods have a tendency to also reduce the topological signal. This work introduces Maximal TDA (MaxTDA), a statistical framework addressing a limitation in TDA wherein robust inference techniques systematically underestimate the persistence of significant homological features. MaxTDA combines kernel density estimation with level-set thresholding via rejection sampling to generate consistent estimators for the maximal persistence features that minimizes bias while maintaining robustness to noise and outliers. We establish the consistency of the sampling procedure and the stability of the maximal persistence estimator. The framework also enables statistical inference on topological features through rejection bands, constructed from quantiles that bound the estimator's deviation probability. MaxTDA is particularly valuable in applications where precise quantification of statistically significant topological features is essential for revealing underlying structural properties in complex datasets. Numerical simulations across varied datasets, including an example from exoplanet astronomy, highlight the effectiveness of MaxTDA in recovering true topological signals.

stat.ME↗

arXiv2024

A Subsequence Approach to Topological Data Analysis for Irregularly-Spaced Time Series

A time-delay embedding (TDE), grounded in the framework of Takens's Theorem, provides a mechanism to represent and analyze the inherent dynamics of time-series data. Recently, topological data analysis (TDA) methods have been applied to study this time series representation mainly through the lens of persistent homology. Current literature on the fusion of TDE and TDA are adept at analyzing uniformly-spaced time series observations. This work introduces a novel {\em subsequence} embedding method for irregularly-spaced time-series data. We show that this method preserves the original state space topology while reducing spurious homological features. Theoretical stability results and convergence properties of the proposed method in the presence of noise and varying levels of irregularity in the spacing of the time series are established. Numerical studies and an application to real data illustrates the performance of the proposed method.

stat.ME↗