# A Spectral Framework for Tracking Communities in Evolving Networks

arXiv: https://arxiv.org/abs/2412.07378 · HTML: https://mlsanity.com/paper/2412.07378/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 127 references that are in the map.
- First posted on arXiv 10 Dec 2024.

## Builds on (most cited first, 30 of 36)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel](https://mlsanity.com/paper/1309.4111/) (2013)
- [Clustering and Community Detection in Directed Networks: A Survey](https://mlsanity.com/paper/1308.0971/) (2013)
- [Statistical clustering of temporal networks through a dynamic stochastic block model](https://mlsanity.com/paper/1506.07464/) (2015)
- [Hermitian matrices for clustering directed graphs: insights and applications](https://mlsanity.com/paper/1908.02096/) (2019)
- [Exploiting Social Network Structure for Person-to-Person Sentiment Analysis](https://mlsanity.com/paper/1409.2450/) (2014)
- [SPONGE: A generalized eigenproblem for clustering signed networks](https://mlsanity.com/paper/1904.08575/) (2019)
- [Spectral Clustering of Graphs with the Bethe Hessian](https://mlsanity.com/paper/1406.1880/) (2014)
- [A Comparative Analysis of Community Detection Algorithms on Artificial Networks](https://mlsanity.com/paper/1608.00763/) (2016)
- [Detecting Overlapping Communities in Networks Using Spectral Methods](https://mlsanity.com/paper/1412.3432/) (2014)
- [Community Detection in Multiplex Networks](https://mlsanity.com/paper/1910.07646/) (2019)
- [Network structure from rich but noisy data](https://mlsanity.com/paper/1703.07376/) (2017)
- [Element-centric clustering comparison unifies overlaps and hierarchy](https://mlsanity.com/paper/1706.06136/) (2017)
- [Clustering on Multi-Layer Graphs via Subspace Analysis on Grassmann Manifolds](https://mlsanity.com/paper/1303.2221/) (2013)
- [Clustering Signed Networks with the Geometric Mean of Laplacians](https://mlsanity.com/paper/1701.00757/) (2017)
- [Spectral Clustering of Signed Graphs via Matrix Power Means](https://mlsanity.com/paper/1905.06230/) (2019)
- [Regularized spectral methods for clustering signed networks](https://mlsanity.com/paper/2011.01737/) (2020)
- [Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach](https://mlsanity.com/paper/1308.0723/) (2013)
- [Detectability thresholds and optimal algorithms for community structure in dynamic networks](https://mlsanity.com/paper/1506.06179/) (2015)
- [Spectral Detection of Simplicial Communities via Hodge Laplacians](https://mlsanity.com/paper/2108.06547/) (2021)
- [Nearly-Optimal Hierarchical Clustering for Well-Clustered Graphs](https://mlsanity.com/paper/2306.09950/) (2023)
- [Multi-scale detection of hierarchical community architecture in structural and functional brain networks](https://mlsanity.com/paper/1704.05826/) (2017)
- [The Power Mean Laplacian for Multilayer Graph Clustering](https://mlsanity.com/paper/1803.00491/) (2018)
- [Hierarchical community structure in networks](https://mlsanity.com/paper/2009.07196/) (2020)
- [Topological Point Cloud Clustering](https://mlsanity.com/paper/2303.16716/) (2023)
- [Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal](https://mlsanity.com/paper/2311.14427/) (2023)
- [Motif-Based Spectral Clustering of Weighted Directed Networks](https://mlsanity.com/paper/2004.01293/) (2020)
- [Evaluating Community Detection Algorithms for Progressively Evolving Graphs](https://mlsanity.com/paper/2007.08635/) (2020)
- [Convergence and complexity of block majorization-minimization for constrained block-Riemannian optimization](https://mlsanity.com/paper/2312.10330/) (2023)
- [Multiway spectral community detection in networks](https://mlsanity.com/paper/1507.05108/) (2015)

## Abstract (arXiv)

Discovering and tracking communities in time-varying networks is an important task in network science, motivated by applications in fields ranging from neuroscience to sociology. In this work, we characterize the celebrated family of spectral methods for static clustering in terms of the low-rank approximation of high-dimensional node embeddings. From this perspective, it becomes natural to view the evolving community detection problem as one of subspace tracking on the Grassmann manifold. While the resulting optimization problem is nonconvex, we adopt a block majorize-minimize Riemannian optimization scheme to learn the Grassmann geodesic which best fits the data. Our framework generalizes any static spectral community detection approach and leads to algorithms achieving favorable performance on synthetic and real temporal networks, including those that are weighted, signed, directed, mixed-membership, multiview, hierarchical, cocommunity-structured, bipartite, or some combination thereof. We demonstrate how to specifically cast a wide variety of methods into our framework, and demonstrate greatly improved dynamic community detection results in all cases.