Clustering Time-Evolving Networks Using the Spatio-Temporal Graph Laplacian

TL;DR

The paper tackles clustering in time-evolving graphs by deriving a spatio-temporal graph Laplacian from a multiview canonical correlation analysis framework, grounded in transfer operator theory. By maximizing adjacent-view coherence across $M$ time views, the authors obtain a real, well-structured Laplacian $oldsymbol{L}=I-oldsymbol{C}$ whose spectrum encodes both spatial communities and their temporal evolution, including merges and splits. The main contributions include extending transfer and covariance operators to time-varying graphs, formulating a multiview-CCA-based Laplacian, and demonstrating a parameter-free spectral clustering approach that handles directed graphs without symmetrization and outperforms the tunable supra-Laplacian on benchmark and double-gyre data. Overall, the method broadens spectral clustering's applicability to dynamic networks, offering interpretable insights into evolving community structure with practical impact across domains.

Abstract

Time-evolving graphs arise frequently when modeling complex dynamical systems such as social networks, traffic flow, and biological processes. Developing techniques to identify and analyze communities in these time-varying graph structures is an important challenge. In this work, we generalize existing spectral clustering algorithms from static to dynamic graphs using canonical correlation analysis (CCA) to capture the temporal evolution of clusters. Based on this extended canonical correlation framework, we define the spatio-temporal graph Laplacian and investigate its spectral properties. We connect these concepts to dynamical systems theory via transfer operators, and illustrate the advantages of our method on benchmark graphs by comparison with existing methods. We show that the spatio-temporal graph Laplacian allows for a clear interpretation of cluster structure evolution over time for directed and undirected graphs.

Figures (6)

• Figure 1: Time-evolving random walk on a line graph $\mathpzc{G}{[t]{\mathstrut}}_{}^{(M)}$, where $\mathpzc{V}{[t]{\mathstrut}}_{} = \{\mathpzc{v}{[t]{\mathstrut}}_{1}, \dots, \mathpzc{v}{[t]{\mathstrut}}_{6}\}$ and $M = 4$. The solid lines represent edges with weight 1, dashed lines represent edges with weight 0.1, and dotted lines represent edges with weights 0.01. At $t = 1$, we have 3 clusters of 2 vertices each. Green, red and blue random walkers begin in clusters 1, 2 and 3, respectively. At each time step, the graph updates and the random walkers take one step. In this line graph, we update only the edge between $\mathpzc{v}{[t]{\mathstrut}}_{2}$ and $\mathpzc{v}{[t]{\mathstrut}}_{3}$, increasing the weight at $t = 1, 2, 3$, so that by $t = 4$ we have 2 clusters. That is, the clusters $\{\mathpzc{v}{[t]{\mathstrut}}_{1}, \mathpzc{v}{[t]{\mathstrut}}_{2}\}$ and $\{\mathpzc{v}{[t]{\mathstrut}}_{3}, \mathpzc{v}{[t]{\mathstrut}}_{4}\}$ at $t = 1$ merge to form a single cluster $\{\mathpzc{v}{[t]{\mathstrut}}_{1}, \mathpzc{v}{[t]{\mathstrut}}_{2}, \mathpzc{v}{[t]{\mathstrut}}_{3}, \mathpzc{v}{[t]{\mathstrut}}_{4}\}$ at $t=4$.
• Figure 2: (a) Coupling graph with adjacency matrix $\mathbf{A}$. Vertices are connected to their copies in adjacent time views because of the presence of self-loops in the original graph, and the changing cluster structure of the original graph is visible. (b) Coupling graph associated with the supra-Laplacian. Here, the edge weights between vertices and their copies in adjacent time views have weight $a$, represented by the dashed-dotted line.
• Figure 3: (a) Adjacency matrix snapshots of time-evolving undirected benchmark graph at $t = 1, 3, 5, 7, 9$. The graph comprises 3 clusters of $100$ vertices at $t = 1$, and over time the graph evolves so that the first cluster shrinks to $65$ vertices and the second cluster grows to $135$ vertices. (b) The first five dominant eigenvectors of the matrix $\mathbf{C}$ over time. (c) Ten dominant eigenvalues of $\mathbf{C}$, showing which eigenvalues encode temporal information and which encode spatial information. We show also the $k$-means clustering of the graph with $k = 3$ clusters using the first three spatial eigenvectors of (d) the spatio-temporal graph Laplacian and (e) the supra-Laplacian. In both (d) and (e), denotes cluster 1, denotes cluster 2, and denotes cluster 3.
• Figure 4: (a) Adjacency matrix snapshots of directed benchmark graph at $t = 1, 3, 5, 7, 9$. The graph comprises 3 clusters at $t = 1$, where the first cluster has $200$ vertices and the other 2 clusters each have $100$. Over time the graph evolves so that the first cluster splits into two clusters of equal size, so that at $t = 10$ each cluster has $100$ vertices. (b) Five dominant eigenvectors of the matrix $\mathbf{C}$ over time. (c) Ten dominant eigenvalues of $\mathbf{C}$, showing which eigenvalues encode temporal information and which encode spatial information. We also show the $k$-means clustering of the graph with $k = 3$ clusters using the first three spatial eigenvectors of (d) the spatio-temporal graph Laplacian and (e) the supra-Laplacian. In both, vertices colored by are labeled as cluster 1, whilst denotes cluster 2, denotes cluster 3, and denotes cluster 4.
• Figure 5: Stream lines of the time-dependent double gyre system at times $t=0$, $t=2.5$, $t=5$, and $t=7.5$. The boundary separating the gyres oscillates with amplitude $\epsilon = 0.25$ and period of oscillation $10$, so that at $t=0, 5, 10$ the boundary is located at $x=1$, and moves to $x=1.25$ at $t=2.5$ and to $x=0.75$ at $t=7.5$.

Theorems & Definitions (25)

• Definition 2.1: Weighted graph
• Definition 2.2: Time-evolving graph
• Definition 2.3: Transition matrix
• Definition 2.4: Random walk on static graphs
• Definition 2.5: Random walk on time-evolving graphs
• Example 2.6
• Definition 2.7
• Definition 2.8: Probability density
• Definition 2.9: Perron--Frobenius & Koopman operators
• Definition 2.10: Reweighted Perron--Frobenius operator