In some sources (see for instance Discrete Calculus, Grady (2010) , Discrete Differential Forms for Computational Modeling, in Discrete Differential Geometry 38, M. Desbrun, E. Kanso, Y. Tong (2008)), the inner products between $k$-chains for the different $k$ are obtained using the star Hodge operator. Therefore, the inner-products for the different values of $k$ are not independent.
For instance, in section 2.3.3 Primal and Dual Complexes of Grady's book, it is mentioned that the inner products can be fully determined from the construction of dual complexes. As far as I can tell, this approach is equivalent to the definition of a star Hodge operator when the simplicial complex is a combinatorial manifold.
But I am interested in the field of complex networks where we basically deal with arbitrary graphs which, in general, are not combinatorial manifolds. Hence, I started to look for other references.
In Hodge Laplacians On Graphs, SIAM Review 62 3 L-H. Lim (2020), however, (see Pag. 691), an inner product between arbitrary $k$-chains $f,g\in L_{\wedge}^2(K_{k+1})$ is introduced
$$ \langle f,g\rangle_V = \sum_{i_0<i_1<...<i_k} w_{i_0i_1...i_k} f(i_0,i_1,...,i_k)g(i_0,i_1,...,i_k) $$
As far as I can tell, besides the fact that they should define positive definite inner products, there is no relation between the inner products for the different values of $k$. Although I am not sure, it seems that the inner product between $k$-chains can be independently chosen from the inner product between $q$-chains for any $q\neq k$.
Hence, here I have two questions:
- Can the inner products for the different values of $k$ be independently chosen?
- Can we always define these inner products in terms of a star Hodge operator? Or Hodge star operators can be only used for combinatorial manifolds?
Kind regards