Problem
Research questions and friction points this paper is trying to address.
Community Detection
Edge Laplacian Dynamics
Ricci Flow
Graph Geometry
Community Detection using the Edge Laplacian Dynamics vis-a-vis Ricci Flow
🤖 AI Summary
This study addresses the high computational complexity and low efficiency of Ricci flow in graph community detection by proposing a simplified approach based on edge Laplacian dynamics. Leveraging the edge Laplacian operator and geometric topology theory, the method replaces traditional Wasserstein distance computations and adjacent edge cycle calculations, thereby effectively circumventing the prohibitive costs associated with Forman and Ollivier-Ricci curvatures. Furthermore, it uncovers the intrinsic dynamic structure of data by exploiting the geometric properties of graphs. Experimental results demonstrate that the proposed approach significantly reduces computational complexity while maintaining community detection accuracy comparable to that of Ricci flow. This work establishes a novel paradigm for the efficient analysis of large-scale graph data.
📝 Abstract
It has been found that utilizing the geometric properties of the graph dynamics can bring out crucial information of the data that a statistical analysis will not. The properties of Ricci flow bring out hidden dynamics of the data in the same way as decomposition of smooth manifolds. In this paper we explore an alternative to Ricci flow, where we find that we can bring out similar properties like it using a simpler and well defined Edge Laplacian to practice community detection in graphs. Moreover, the results obtained by the present approach show that a flow driven by the Edge Laplacian yields results similar to those of the Ricci flow. In contrast, computing the Forman-Ricci curvature requires looping over the adjacent edges, while computing Olivier-Ricci curvature involves the computation of Wasserstein distance and probability distribution associated with the graph nodes, making the Edge Laplacian approach computationally more efficient.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Community Detection
Edge Laplacian
Ricci Flow
Graph Dynamics
Computational Efficiency
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