  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.
