发表机构
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对有向图on中的线性系统，通过圈覆盖性质$\mathcal N$（必要）和$\mathcal S$（充分）刻画结构稳定性，并给出随机采样拓扑满足这些性质的渐近概率条件，识别稳定动态的可行机制。
AI 中文摘要
结构系统理论研究哪些网络拓扑能够维持规定的系统性质，如可控性或稳定性。当拓扑本身是随机的时，相关问题就变成概率性的：从随机模型中抽取的图有多大的可能性维持该性质？这些概率衡量了该性质在拓扑中的丰度和鲁棒性，并表明需要该性质的系统能否在不确定环境中可靠部署。我们针对有向图on（graphon）设置中线性系统的渐近稳定性来解决这个问题。我们考虑两个图论性质。第一个是$\mathcal N$，它要求对于每个$k\leq n$，$D$的某个$k$顶点诱导子有向图都允许一个圈覆盖；第二个是$\mathcal S$，它要求这些子有向图可以被选择使得它们的节点集形成一个嵌套序列$V_1\subset\cdots\subset V_n=V(D)$，从单个带环的顶点开始。我们已经证明$\mathcal N$是结构稳定性的必要条件，$\mathcal S$是充分条件。我们从有向阶梯图on（step-graphon）$W$中采样$D$。我们的主要结果给出了当$n\to\infty$时$\Pr(\mathcal N)\to 1$和$\Pr(\mathcal S)\to 1$的充分必要条件。更详细地说，对于具有骨架有向图$S$（在$q$个节点上）和浓度向量$x^*$的阶梯图on$W$，我们关联一个圈多胞体$\vec{\mathcal X}(S)\subseteq\Delta_q$。然后条件以$x^*$在$\vec{\mathcal X}(S)$内的位置、多胞体的维度、$W$的环密度以及对于$\mathcal S$，骨架上的圈的一个排序条件来表述。这些结果共同确定了对于有向阶梯图on，采样拓扑极有可能或极不可能维持稳定动态的机制。
英文摘要
Structural system theory studies which network topologies can sustain a prescribed system property such as controllability or stability. When the topology is itself random, the relevant question becomes probabilistic: how likely is a graph drawn from a stochastic model to sustain the property? Such probabilities measure the abundance and robustness of the property across topologies, and indicate whether systems requiring it can be reliably deployed in uncertain environments. We address this question for asymptotic stability of linear systems in the directed graphon setting. We consider two graph-theoretic properties. The first is $\mathcal N$, and it requires that for every $k\leq n$ some $k$-vertex induced subdigraph of $D$ admits a cycle cover, and the second is $\mathcal S$, which requires that these subdigraphs can be chosen so that their node sets form a nested sequence $V_1\subset\cdots\subset V_n=V(D)$ starting from a single vertex with a loop. We have shown that $\mathcal N$ is necessary and $\mathcal S$ is sufficient for structural stability. We sample $D$ from a directed step-graphon $W$. Our main results give necessary and sufficient conditions for $\Pr(\mathcal N)\to 1$ and $\Pr(\mathcal S)\to 1$ as $n\to\infty$. In more detail, to a step-graphon $W$ with skeleton digraph $S$ on $q$ nodes and concentration vector $x^*$ we associate a cycle polytope $\vec{\mathcal X}(S)\subseteqΔ_q$. The conditions are then formulated in terms of the position of $x^*$ within $\vec{\mathcal X}(S)$, the dimension of the polytope, the loop density of $W$ and, for $\mathcal S$, an ordering condition on the cycles of the skeleton. Together these results identify, for directed step-graphons, the regime in which a sampled topology is overwhelmingly likely or unlikely to sustain stable dynamics.
