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We consider stabilities for the weighted length or energy functional of adiscrete map from a finite weighted graph $(X,m_{E})$ into a smooth Riemannianmanifold $(M,g)$. We prove the non-existence of a stable discrete minimalimmersion or a non-constant stable discrete harmonic map from a finite weightedgraph into certain homogeneous spaces, such as K\"ahler $C$-spaces of positiveholomorphic sectional curvature and some simply-connected compact Riemanniansymmetric spaces.

Nonexistence of stable discrete maps into some homogeneous spaces of nonnegative curvature