Higher-order interaction model from geometric measurements
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We introduce a higher simplicial generalization of the linear consensus model which shares several common features. The well-known linear consensus model is a gradient flow with a sum of squares of distances between each pair of points. Our newly suggested model is also represented as a gradient flow equipped with total n-dimensional volume functional consisting of n+1 points as a potential. In this manner, the linear consensus model coincides with the case of n=1 where distance is understood as the 1-dimensional volume. From a simple mathematical analysis, one can easily show that the linear consensus model (a gradient flow with 1-dimensional volume functional) collapses to one single point, which can be considered as a 0-complex. By extending this result, we show that a solution to our model converges to an (n-1)-dimensional affine subspace. We also perform several numerical simulations with an efficient algorithm that reduces the computational cost.
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