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We study graphs with nonnegative Bakry-\'Emery curvature or Olliviercurvature outside a finite subset. For such a graph, via introducing thediscrete Gromov-Hausdorff convergence we prove that the space of boundedharmonic functions is finite dimensional, and as a corollary the number ofnon-parabolic ends is finite.

Graphs with nonnegative curvature outside a finite subset, harmonic functions and number of ends