A Convergence Theorem for the Graph Shift-type Algorithms

TL;DR

This paper establishes a theoretical foundation for Graph Shift algorithms, proving their convergence to local maxima using Zangwill's theorem, and verifies the framework through experiments.

Contribution

It introduces a generic convergence framework for Graph Shift algorithms, filling a key theoretical gap in their analysis.

Findings

GS algorithms converge to local maxima

The framework applies to various GS-type algorithms

Experimental results support the theoretical claims

Abstract

Graph Shift (GS) algorithms are recently focused as a promising approach for discovering dense subgraphs in noisy data. However, there are no theoretical foundations for proving the convergence of the GS Algorithm. In this paper, we propose a generic theoretical framework consisting of three key GS components: simplex of generated sequence set, monotonic and continuous objective function and closed mapping. We prove that GS algorithms with such components can be transformed to fit the Zangwill's convergence theorem, and the sequence set generated by the GS procedures always terminates at a local maximum, or at worst, contains a subsequence which converges to a local maximum of the similarity measure function. The framework is verified by expanding it to other GS-type algorithms and experimental results.