  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.
