Efficient Network Embedding by Approximate Equitable Partitions

TL;DR

This work addresses scalable structural network embedding by introducing ε-BE, an approximate equitable-partition approach that relaxes exact BE with a tunable tolerance $\varepsilon$ and computes embeddings via a partition-refinement algorithm. An iterative scheme progressively increases $\varepsilon$ to balance aggregation and discrimination, achieving embeddings in $ℝ^{n×d}$ where $d$ equals the number of BE blocks. Empirical results on visualization, classification, and regression tasks show competitive or superior performance with runtimes that are orders of magnitude faster than state-of-the-art methods, enabling large-scale network embeddings previously impractical. The approach leverages BE's relation to Markov-lumping and O($m\log n$) refinement, with the overall iterative complexity scaling as $O(mnΔ)$, demonstrating strong scalability and practical impact for complex network analysis.

Abstract

Structural network embedding is a crucial step in enabling effective downstream tasks for complex systems that aims to project a network into a lower-dimensional space while preserving similarities among nodes. We introduce a simple and efficient embedding technique based on approximate variants of equitable partitions. The approximation consists in introducing a user-tunable tolerance parameter relaxing the otherwise strict condition for exact equitable partitions that can be hardly found in real-world networks. We exploit a relationship between equitable partitions and equivalence relations for Markov chains and ordinary differential equations to develop a partition refinement algorithm for computing an approximate equitable partition in polynomial time. We compare our method against state-of-the-art embedding techniques on benchmark networks. We report comparable -- when not superior -- performance for visualization, classification, and regression tasks at a cost between one and three orders of magnitude smaller using a prototype implementation, enabling the embedding of large-scale networks which could not be efficiently handled by most of the competing techniques.

Figures (9)

• Figure 1: Role equivalences (example from axiomatic). Nodes with the same color indicate the same role; white nodes have pairwise distinct roles.
• Figure 2: Running example with equitable partitions.
• Figure 3: Approximate variant of equitable partitions computed with $\varepsilon$-BE on the running example from Figure \ref{['E0']} with $\varepsilon=1$.
• Figure 4: Execution of Algorithm 1. We indicate the current splitter by underlining it. The weight $w^{B'}$ denotes the number of edges connected to elements of the splitter $B'$. The arrays at the bottom show the splitting phase of a certain block $B$ with respect to the current splitter. The red line indicates where the block is split because it does not respect the tolerance $\varepsilon$.
• Figure 5: Execution of the iterative $\varepsilon$-BE on a network composed of a complete clique on the left, and another on the right with missing edges. We show the first and the second iteration with $\varepsilon$ equal to 0 and 1 in panels A and B. For each panel, we indicate the initial partition and the resulting one after the application of $\varepsilon$-BE. The resulting partition of a phase will be employed as the initial partition in the following phase after merging the singleton blocks. The initial partition $\mathcal{H}_{in}$ in panel B corresponds to $\mathcal{H}_{\varepsilon}$ in panel A where the singleton blocks are merged into a single block. We depict the network with the different colors assigned by the resulting partition. Panel C represents the result of the application of Algorithm \ref{['Valm']} without the iterative scheme.

Theorems & Definitions (4)

• Definition 1: Backward equivalence, simplified to graphs from PNAScttv
• Definition 2: Equitable partition, adapted from gupte2017role
• Definition 3: $\varepsilon$-BE
• Definition 4: $\varepsilon$-BE embedding