We construct a strong filtered graph $\Gamma_s(\Lambda)$ dependent on the dominant weight $\Lambda$, and a weak filtered graph $\Gamma_w(\Kcen)$ dependent on the canonical central element $\Kcen$ for an arbitrary Kac-Moody algebra $g$. In our construction, both graphs $(\Gamma_s(\Lambda), \Gamma_w(\Kcen))$ have the vertex set as the Weyl group of $g$, with the grading given by the length function. The edges of the graph $\Gamma_s(\La)$ are labeled versions of the $\lambda$-chain model of K-Chevalley rules for Kac-Moody flag manifolds as developed by Lenart and Shimozono, originally defined by Lenart and Postnikov. Meanwhile, the labels on $\Gamma_w(\Kcen)$ come from the dual multiplication map of K-cohomology of affine Grassmannian $Gr_G$. We conjecture that the strong filtered graph and weak filtered graph are dual, which means we get an identity when we apply the up and down operators on the vertices. We proved this identity except one case that where we call the chain is $j$-present. Our identity is similar to the Möbius construction of the dual filtered graph, as previously studied by Patrias and Pylyavskyy, and in fact, in the limit $n\rightarrow \infty$ of the $A^{(1)}_{n-1}$, our construction recovers their identity. We also expect to recover their combinatorics of Möbius deformation of the shifted Young's lattice in type $C^{(1)}_n$ as $n$ approaches infinity. / Doctor of Philosophy / In this thesis, we introduce a pair of graphs $(\Gamma_s(\La),\Gamma_w(\Kcen))$ motivated by the study of affine Schubert calculus. Affine Schubert calculus emerges as an extension and generalization of classical Schubert calculus, which involves questions such as determining the number of lines intersecting four lines in three-dimensional space. This type of questions can often be translated into computations aimed at finding the structure constants for the Schubert basis in the K-(co)homology of the flag varieties such as affine Grassmannian. These structure constants represent the coefficients of the Schubert basis in the product of the other two Schubert bases, all indexed by the Weyl group of the affine Lie algebra $g$. We define up and down operators on the vertices of graphs $(\Gamma_s(\Lambda), \Gamma_w(\Kcen))$, which are elements in Weyl group of $g$, utilizing the structure constants as essential components. We conjecture that, in general, and prove in certain cases, this approach yields new identities for these operators, leading us to define this pair of graphs as a dual filtered graph.