A Weighted Quiver Kernel using Functor Homology
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In this paper, we propose a new homological method to study weighted directed networks. Our model of such networks is a directed graph Q equipped with a weight function w on the set Q_1 of arrows in Q. We require that the range W of our weight function is equipped with an addition or a multiplication, i.e., W is a monoid in the mathematical terminology. When W is equipped with a representation on a vector space M, the standard method of homological algebra allows us to define the homology groups H_*(Q,w;M). It is known that when Q has no oriented cycles, H_n(Q,w;M)=0 for n≥ 2 and H_1(Q,w;M) can be easily computed. This fact allows us to define a new graph kernel for weighted directed graphs. We made two sample computations with real data and found that our method is practically applicable.
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