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BIMSA-HSE Joint Seminar on Data Analytics and Topology
Analysis of brain networks with blurred magnitude homology
Analysis of brain networks with blurred magnitude homology
演讲者
Alexander Kachura
时间
2026年01月19日 20:00 至 21:00
地点
Online
线上
Zoom 468 248 1222
(BIMSA)
摘要
The seminar is scheduled online for Monday from 20:00 to 21:00(Beijing Time)/15:00 to 16:00(Moscow
Time).
Alexander Kachura, joint work with Vsevolod Chernyshev. One of the approaches to analyze multivariate
time series with interdependent components is to construct a complete weighted graph with nodes corresponding to the
components of the time series and edge weights representing the strength of dependence between the pair of components
corresponding to their ends. Neuroscience is among the many fields where this methodology is popular. A prominent
example is the analysis of functional connectomes [1], brain networks whose nodes represent brain regions. These
networks exhibit nontrivial structure, so their topological characteristics are useful for analysis. The level of
interconnection between brain regions is usually measured using non-directional correlations, while the functional
connections between brain regions are inherently directional. Topological characteristics of graphs can be analyzed with
persistent homology [2], an increasingly popular dataanalysis tool that can also be applied to graphs. The essence of
this approach is to track the moments of the appearance and disappearance of topological structures as the scale varies,
using algebraic invariants called homology. When applied to weighted graphs – including functional connectomes –
persistent homology allows us to study how the topological structure of a graph changes as the edgefiltering threshold
varies. This enables identification of noise-robust topological features, which is crucial when the graph structure is
not given a priori but estimated from data. Several homology theories exist. Simplicial homology – historically the
first one used to compute persistent homology – cannot account for the directions of graph edges. The most direct
generalization of this homology theory to digraphs is the homology of directed flag complexes [3]. However, it still
discards some directional information. One of the homology theories that makes it possible to preserve a large
proportion of information about the directions of edges is blurred magnitude homology [4].
In this work we
introduce a method for classifying directed functional connectomes using blurred magnitude homology, specifically its
Betti curves, a popular numerical descriptor of persistent homology. As an example, we apply the developed technique to
study graphs constructed from fMRI scans of individuals with ASD and typically developing controls.
We
experimentally tested the approach on the ABIDE dataset.
This work was supported by the Russian Science
Foundation, Grant No. 24-68-00030.
References
1. Uddin, L. Q., Yeo, B. T., & Spreng, R. N. (2019). Towards a
universal taxonomy of macro-scale functional human brain networks. Brain topography, 32(6), 926-942.
2. Edelsbrunner,
H., Harer, J. (2010). Computational topology: an introduction. American Mathematical Society.
3. Lütgehetmann, D.,
Govc, D., Smith, J. P., Levi, R. (2020). Computing persistent homology of directed flag complexes. Algorithms, 13(1),
19.
4. Otter, N. (2018). Magnitude meets persistence. Homology theories for filtered simplicial sets. arXiv preprint
arXiv:1807.01540.
Time).
Alexander Kachura, joint work with Vsevolod Chernyshev. One of the approaches to analyze multivariate
time series with interdependent components is to construct a complete weighted graph with nodes corresponding to the
components of the time series and edge weights representing the strength of dependence between the pair of components
corresponding to their ends. Neuroscience is among the many fields where this methodology is popular. A prominent
example is the analysis of functional connectomes [1], brain networks whose nodes represent brain regions. These
networks exhibit nontrivial structure, so their topological characteristics are useful for analysis. The level of
interconnection between brain regions is usually measured using non-directional correlations, while the functional
connections between brain regions are inherently directional. Topological characteristics of graphs can be analyzed with
persistent homology [2], an increasingly popular dataanalysis tool that can also be applied to graphs. The essence of
this approach is to track the moments of the appearance and disappearance of topological structures as the scale varies,
using algebraic invariants called homology. When applied to weighted graphs – including functional connectomes –
persistent homology allows us to study how the topological structure of a graph changes as the edgefiltering threshold
varies. This enables identification of noise-robust topological features, which is crucial when the graph structure is
not given a priori but estimated from data. Several homology theories exist. Simplicial homology – historically the
first one used to compute persistent homology – cannot account for the directions of graph edges. The most direct
generalization of this homology theory to digraphs is the homology of directed flag complexes [3]. However, it still
discards some directional information. One of the homology theories that makes it possible to preserve a large
proportion of information about the directions of edges is blurred magnitude homology [4].
In this work we
introduce a method for classifying directed functional connectomes using blurred magnitude homology, specifically its
Betti curves, a popular numerical descriptor of persistent homology. As an example, we apply the developed technique to
study graphs constructed from fMRI scans of individuals with ASD and typically developing controls.
We
experimentally tested the approach on the ABIDE dataset.
This work was supported by the Russian Science
Foundation, Grant No. 24-68-00030.
References
1. Uddin, L. Q., Yeo, B. T., & Spreng, R. N. (2019). Towards a
universal taxonomy of macro-scale functional human brain networks. Brain topography, 32(6), 926-942.
2. Edelsbrunner,
H., Harer, J. (2010). Computational topology: an introduction. American Mathematical Society.
3. Lütgehetmann, D.,
Govc, D., Smith, J. P., Levi, R. (2020). Computing persistent homology of directed flag complexes. Algorithms, 13(1),
19.
4. Otter, N. (2018). Magnitude meets persistence. Homology theories for filtered simplicial sets. arXiv preprint
arXiv:1807.01540.