Network analysis using Krylov subspace trajectories

TL;DR

Traditional power iteration on networks typically uses a random starting vector and reports only the converged eigenvector centrality, discarding informative intermediate results. The authors propose Krylov subspace trajectories by applying power iteration to the adjacency matrix $\mathbf{A}$ with a fixed non-random initial vector $\mathbf{v}_0$, treating each node's trajectory as its row in the normalized Krylov matrix $\mathbf{K}_n$. They demonstrate that trajectories encode path-length dependent node similarity, enable structure-focused clustering distinct from proximity-based methods, allow perturbation analysis via nonuniform $\mathbf{v}_0$, and yield a trajectory-based node-importance metric $\delta$ that captures oscillations. This trajectory-based framework offers an interpretable, versatile toolkit for network analysis with potential applications to biological networks and other complex systems.

Abstract

We describe a set of network analysis methods based on the rows of the Krylov subspace matrix computed from a network adjacency matrix via power iteration using a non-random initial vector. We refer to these node-specific row vectors as Krylov subspace trajectories. While power iteration using a random initial starting vector is commonly applied to the network adjacency matrix to compute eigenvector centrality values, this application only uses the final vector generated after numerical convergence. Importantly, use of a random initial vector means that the intermediate results of power iteration are also random and lack a clear interpretation. To the best of our knowledge, use of intermediate power iteration results for network analysis has been limited to techniques that leverage just a single pre-convergence solution, e.g., Power Iteration Clustering. In this paper, we explore methods that apply power iteration with a non-random inital vector to the network adjacency matrix to generate Krylov subspace trajectories for each node. These non-random trajectories provide important information regarding network structure, node importance, and response to perturbations. We have created this short preprint in part to generate feedback from others in the network analysis community who might be aware of similar existing work.

Figures (7)

• Figure 1: Example tree network. The left panel visualizes the network with equal sized nodes. The right panel visualizes with node size based on eigenvector centrality.
• Figure 2: Node-specific Krylov trajectories for the tree network visualized in Figure \ref{['fig:tree']}.
• Figure 3: Clustering of the example tree network generated by A) Louvain clustering, B) hierarchical agglomerative clustering (dendrogram cut at k=4) with correlation distance between the Krylov subspace trajectories, C) hierarchical agglomerative clustering using a distance of 1 minus the LHN regular equivalence between nodes, and D) k-means clustering on eigenvector centrality values for k=4.
• Figure 4: Clustering of a random preferential attachment network generated by A) Louvain clustering, B) hierarchical agglomerative clustering (dendrogram cut at k=7) with correlation distance between the Krylov subspace trajectories, C) hierarchical agglomerative clustering using a distance of 1 minus the LHN regular equivalence between nodes, and D) k-means clustering on eigenvector centrality values for k=7.
• Figure 5: Node-specific Krylov trajectories for the tree network visualized in Figure \ref{['fig:tree']} using a non-uniform $\mathbf{v}_0$ where the entries for nodes 5 and 6 that are five times larger than the entries for other nodes.