AI 中文总结
无目标团猜想称非退化组合阈值线性网络稳定不动点支撑集是无目标团，本文给出明确的六神经元反例。还证明在特定参数范围，非退化CTLN稳定不动点支撑集恰是无目标团，特别是在\(\varepsilon\leq\delta/(\delta + n - 2)\)时成立。
AI 中文摘要
无目标团猜想断言，非退化组合阈值线性网络（CTLN）的稳定不动点的支撑集恰好是其无目标团：即没有外部顶点从每个团顶点接收边的双向团。我们给出了一个明确的六神经元反例。对于每个足够小的\(\varepsilon>0\)，在\(\delta = 29\varepsilon/25\)时由一个固定图定义的CTLN是非退化的，并且有一个具有非团满支撑的稳定不动点。其\(q = \delta(1 - \varepsilon)/\varepsilon\)的值趋于\(29/25\)。在相反方向上，对于\(n\geq3\)个顶点上的任何CTLN，我们证明在参数范围\[q\geq n - 2 - \frac{n - 3}{2}\varepsilon\]内，没有非团支撑能同时满足不动点正性和线性稳定性条件。因此，在这个范围内，每个非退化CTLN的稳定不动点支撑集恰好是其无目标团。特别是当\(\varepsilon\leq\delta/(\delta + n - 2)\)时成立。
英文摘要
The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network (CTLN) are exactly its target-free cliques: bidirected cliques for which no outside vertex receives an edge from every clique vertex. We give an explicit six-neuron counterexample. For every sufficiently small $\varepsilon>0$, the CTLN defined by one fixed graph at $δ=29\varepsilon/25$ is nondegenerate and has a stable fixed point with nonclique full support. Its values of $q=δ(1-\varepsilon)/\varepsilon$ tend to $29/25$. In the complementary direction, for any CTLN on $n\geq3$ vertices, we prove that in the parameter range \[ q\geq n-2-\frac{n-3}{2}\varepsilon, \] no nonclique support can satisfy both the fixed-point positivity and linear stability conditions. Consequently, throughout this range, every nondegenerate CTLN has exactly its target-free cliques as supports of stable fixed points. In particular, this holds when $\varepsilon\leqδ/(δ+n-2)$.
