A Spectral Framework for Tracking Communities in Evolving Networks

TL;DR

This work introduces a Grassmann-geometry framework for dynamic spectral clustering, treating time-evolving node embeddings as a low-rank subspace that traces a geodesic on the Grassmann manifold. By formulating geodesic regression and solving via block coordinate descent, the method extends any static spectral clustering approach to time-varying networks through a modeling matrix on the subspace, the so-called modeled clustering matrix $M$. The approach is instantiated for multiple modalities (e.g., USC, NSC, SMM) and validated on synthetic dynamic SBMs and real temporal networks, consistently outperforming static and benchmark dynamic methods. The key contribution is a general, scalable, and principled framework that leverages Grassmann geometry to enforce temporal smoothness in spectral embeddings, enabling robust tracking of communities across diverse network types with broad practical impact.

Abstract

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.

Figures (8)

• Figure 1: Comparison of median $\text{AMI} \in [0,1]$ over time versus $p_{\text{in}}$, medianed over $50$ simulations of the dynamic SBM ($d=T=50$, $k=2$, $p_{\text{out}}=0.2$) as $p_{\text{switch}}$ ranges from low to very high. Each unit circle displays the trajectory of modularity matrix first-eigenvectors as $T$ progresses when $p_{\text{in}}=0.4$. For low (\ref{['subfig:pswitch-1000']}) and medium (\ref{['subfig:pswitch-750']}) values of $p_{\text{switch}}$, said trajectory 'walks along' the unit circle, suggesting (Proposition \ref{['prop:geodesic-assumption']}) that the dynamics satisfy the geodesic assumption. When $p_{\text{switch}}$ is very high (\ref{['subfig:pswitch-500']}), the trajectory 'falls off': the assumption has been violated. G- resp. S- refers to geodesic resp. static algorithm versions.
• Figure 2: Comparison of median (over $50$ simulations and $20$ time steps of the appropriate SBM) AMI/E-cS for various static spectral methods and their dynamic generalization. Each color corresponds to a network modality, each line an SBM setting for that modality, and each symbol a spectral method for detecting communities in that modality (hollow for its dynamic extension, filled for static). By default where applicable, $d=120$, $k=2$, $p_{\text{in}}=0.3$, $p_{\text{out}}=0.2$, and $p_{\text{switch}}=10^{-2}$. Exceptions are in the second SSBM and second DSBM parameter settings, where $p_{\text{in}}=p_{\text{out}}$ to enforce clustering based solely on edge affinity/orientation. Appendix \ref{['sec:instantation-detail']} elaborates upon each individual column.
• Figure 3: Evaluation on the two-day elementary school face-to-face interaction network of stehle_high_2011. LEFT: Included as benchmarks are the Label Smoothing (LS) approach of falkowski_mining_2006, the Smoothed Louvain (SL) algorithm of aynaud_static_2010, and the Graph Smoothing (SG) approach of guo_evolutionary_2014. The three geodesic approaches uniformly outperform the benchmarks. RIGHT: The AMI difference at each time step between Algorithm \ref{['alg:geodesic-dcd']} (fixed $k_c$) and its extension (Appendix \ref{['sec:variable-k-appendix']}) to the variable-$k_c$ case. The data points are scattered, with each trajectory smoothened using a third-order Savitzky-Golay filter savitzky1964smoothing to visually enhance any trends. The results imply that the two algorithms perform very similarly.
• Figure 4: Comparison of geodesic and static signed community detection methods, medianed over $50$ simulations of two settings of a dynamic signed stochastic block model ($d=120$, $T=20$, $k=2$, $p_{\text{switch}}=10^{-2}$, $\eta_{\text{in}}=\eta_{\text{out}}=0.4$). (\ref{['subfig:ssbm-pin-greater-pout']}): $p_{\text{in}}=0.3$, $p_\text{out}=0.2$. (\ref{['subfig:ssbm-pin-equal-pout']}): $p_{\text{in}}=p_{\text{out}}=0.3$: even when the intra- and inter-community connection probabilities are identical, the algorithms are still able to discover community structure based solely on positive and negative edge affinities. We compare algorithms based on the signed ratio Laplacian (SR) kunegis_spectral_2010, the geometric mean Laplacian (GM) mercado_clustering_2016, and the more general matrix power mean Laplacian (SPM) mercado_spectral_2019 with $p=-2$.
• Figure 5: Comparison of geodesic and static methods medianed over $50$ simulations of two settings of a dynamic directed stochastic block model, both with $d=120$, $T=20$, $p_{\text{switch}}=10^{-2}$. (\ref{['subfig:density']}) $k=2$ communities are planted with $\boldsymbol F=0.50.4 ; 0.60.5$, $p_{\text{in}}=0.3$, $p_\text{out}=0.2$ in a 'density-based' parameter setting. (\ref{['subfig:pattern']}) $k=3$ communities are planted with $\boldsymbol{F}=1 / 22 /31 / 31 / 31 / 22 / 32 / 31/31/2$, $p_{\text{in}}=p_{\text{out}}=0.2$ in a 'flow/pattern-based' parameter setting cucuringu_hermitian_2020.

Theorems & Definitions (9)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• proof : Proof of Proposition \ref{['prop:mcm-prop']}
• proof : Proof of Proposition \ref{['prop:prop-low-rank-unnormalized-sc']}
• proof : Proof of Proposition \ref{['prop:signless-lapl']}
• Proposition 5
• proof : Proof of Proposition \ref{['prop:signed-low-rank']}