论文标题
图形拉普拉斯式的新分解和质量成分系统的二项式结构
A new decomposition of the graph Laplacian and the binomial structure of mass-action systems
论文作者
论文摘要
我们提供了Laplacian矩阵的新分解（用于具有牢固连接的组件的标记的有向图），涉及可逆的$ \ textit {core矩阵} $，树木常数的向量，以及辅助图的入口矩阵，代表了在角度上的订单。根据特定顺序，核心矩阵具有其他属性。我们的结果本质上是图理论/代数。 作为第一个应用，我们进一步阐明了由化学反应网络引起的（弱可逆的）质量成分系统的二项式结构。其次，我们扩展了Horn和Jackson对特殊稳态（复杂平衡平衡）的渐近稳定性的经典结果。在这里，图形laplacian的新分解使我们能够以给定的$ \ textIt {单次评估顺序} $（以及对数坐标中的相应的多面体锥）来考虑正矫正区域。事实证明，所有动态系统都是渐近稳定的，可以嵌入到某些$ \ textIt {二项式差分夹杂物} $中。特别是，这适用于复杂平衡的质量成分系统，因此我们还获得了经典结果的多面体几何形状证明。
We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible $\textit{core matrix}$, the vector of tree constants, and the incidence matrix of an auxiliary graph, representing an order on the vertices. Depending on the particular order, the core matrix has additional properties. Our results are graph-theoretic/algebraic in nature. As a first application, we further clarify the binomial structure of (weakly reversible) mass-action systems, arising from chemical reaction networks. Second, we extend a classical result by Horn and Jackson on the asymptotic stability of special steady states (complex-balanced equilibria). Here, the new decomposition of the graph Laplacian allows us to consider regions in the positive orthant with given $\textit{monomial evaluation orders}$ (and corresponding polyhedral cones in logarithmic coordinates). As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain $\textit{binomial differential inclusions}$. In particular, this holds for complex-balanced mass-action systems, and hence we also obtain a polyhedral-geometry proof of the classical result.
