Classification of quadratic stochastic operators (Q2743043)

Let \(S^{n-1}\) be a finite-dimensional simplex \(S^{n-1}= \{x\in \mathbb{R}^n\mid x_i\geq 0\), \(\sum_{i=1}^nx_i=1\}\) and \(V:S^{n-1}\mapsto S^{n-1}\) be a quadratic stochastic operator \(V(x)_k=\sum_{i,j=1}^nP_{ij,k}x_ix_j\), where \(P_{ij,k}\geq 0\), \(P_{ij,k}=P_{ji,k}\) and \(\sum_{k=1}^n P_{ij,k}=1\) for all \(i,j,k\in\{1,2,\dots,n\}\). The classification of such operators \(V\) constructed on finite graphs is given.