MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology

TL;DR

The paper tackles measuring topological autocorrelation in non-hierarchical multiscale clusterings by introducing the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration that encodes cluster intersections across starting scale $s$ and lag $t-s$. Using multiparameter persistent homology, MCbiF yields a finitely presented, block-decomposable module whose Hilbert functions $\mathrm{HF}_k(s,t)$ quantify 0-conflicts (non-hierarchy) and 1-conflicts (higher-order inconsistencies) in the partition sequence. The authors show that MCbiF has a nerve-based equivalent that extends Sankey diagrams to higher order and prove stability of the Hilbert-function invariants. Through synthetic and real-world data (e.g., wild mice social behavior), MCbiF features outperform information-based baselines in regression and classification and offer interpretable, topology-grounded descriptors for non-hierarchical temporal clustering tasks.

Abstract

Datasets often possess an intrinsic multiscale structure with meaningful descriptions at different levels of coarseness. Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily hierarchical sequences of partitions across scales. To analyse and compare such sequences, we use tools from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The MCbiF can be interpreted as a higher-order extension of Sankey diagrams and reduces to a dendrogram for hierarchical sequences. We show that the multiparameter persistent homology (MPH) of the MCbiF yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH captures violations of the refinement order of partitions, whereas at dimension one, the MPH captures higher-order inconsistencies between clusters across scales. We demonstrate through experiments the use of MCbiF Hilbert functions as topological feature maps for downstream machine learning tasks. MCbiF feature maps outperform information-based baseline features on both regression and classification tasks on synthetic sets of non-hierarchical sequences of partitions. We also show an application of MCbiF to real-world data to measure non-hierarchies in wild mice social grouping patterns across time.

Figures (14)

• Figure 1: (a) Illustration of how the MCbiF encodes the structure of a non-hierarchical sequence of partitions $\theta$ as a bifiltration of abstract simplicial complexes $K^{s,t}$. See Example \ref{['ex:toy_example']} for a detailed description. (b) The Hilbert functions $\mathrm{HF}_k(s,t)$ of the MCbiF are invariants that capture the topological autocorrelation of $\theta$: violations of the refinement order at dimension $k=0$, and higher-order cluster inconsistencies at dimension $k=1$. The Hilbert functions can be used as feature maps for downstream machine learning tasks.
• Figure 2: Summary of key theoretical results and their relationships indicated by arrows. Double-headed arrows represent equivalences (iff), whereas single-headed arrows represent implications (if).
• Figure 3: Relationship between different types of conflicts and the crossings in a single-layer Sankey diagram.
• Figure 4: Difference between order-preserving ($y=0$) and non-order-preserving ($y=1$) sequences (**** indicates $p<0.0001$, Mann-Whitney U test).
• Figure 5: (a) Analysis of non-hierarchical sequences of partitions $\theta_{\tau_i}$ compiled from the temporal social interactions of a mice population over a period of 9 weeks. Each $\theta_{\tau_i}$ is formed by a sequence of social groupings $\theta_{\tau_i}(t)$ over week $t$. Different sequences of partitions were computed as a function of the parameter $\tau_i$. We display Sankey diagrams and MCbiF feature maps for $\theta_{\tau_i}$ at three parameters $\tau_i$ ($i=2,4,8$) identified as robust in the original work by bovetFlowStabilityDynamic2022. These three sequences $\theta_{\tau_i}$ exhibit different types of non-hierarchy, as shown by our topological feature maps and our measures of average 0-conflict ($\bar{c}_0$) and average 1-conflict ($\bar{c}_1$). (b) The $\theta_{\tau_i}$ ($i=2,4,8$) found in bovetFlowStabilityDynamic2022 as robust behaviours correspond to distinct topological characteristics of the sequences of partitions, as captured by the block structure in the distance between MCbiF Hilbert functions.

Theorems & Definitions (59)

• Remark 1
• Remark 2
• Definition 3: Multiscale Clustering Bifiltration
• Proposition 4
• Remark 5
• Proposition 6
• Definition 7: Hierarchy
• Definition 8: Nestedness
• Remark 9
• Remark 10