论文标题
动作融合图序列的限制$（p，q）$ - 规范
Limits of action convergent graph sequences with unbounded $(p,q)$-norms
论文作者
论文摘要
最近通过Backhausz和Szegedy统一的动作收敛概念统一并概括了图形序列的密集（Graphon）和局部 - 全球（图形）收敛。这是通过将图作为操作员查看并检查其动力学属性来完成的。假设$（a_n）_n^\ infty $是一系列代表图形的操作员，相对于动作度量，库奇。 If $(A_n)_n^\infty$ has uniformly bounded $(p,q)$-norms where $(p,q)$ is any pair in $[1,\infty)\times(1,\infty)$, then Backhausz and Szegedy prove that $(A_n)_n^\infty$ has a limit operator which, moreover, must be self-adjoint and保持积极性。在目前的工作中，我们构建了一大类的图形序列，其仅均匀边界的$（p，q）$ - norm是$（\ infty，1）$ - norm，但仍然会收敛。我们表明，在这种情况下，极限运算符不是独特的，不是自我伴侣，也不需要保持积极性。特别是，在动作收敛语言中，这意味着图形空间并不紧凑。通过识别这些多个限制，我们还证明了$ c $ regularity在弱等效性下并不是不变的，其中$ c $是身份函数的特征值，当身份函数是特征功能时。
The recently developed notion of action convergence by Backhausz and Szegedy unifies and generalises the dense (graphon) and local-global (graphing) convergences of graph sequences. This is done through viewing graphs as operators and examining their dynamical properties. Suppose $(A_n)_n^\infty$ is a sequence of operators representing graphs, Cauchy with respect to the action metric. If $(A_n)_n^\infty$ has uniformly bounded $(p,q)$-norms where $(p,q)$ is any pair in $[1,\infty)\times(1,\infty)$, then Backhausz and Szegedy prove that $(A_n)_n^\infty$ has a limit operator which, moreover, must be self-adjoint and positivity-preserving. In the present work, we construct a large class of graph sequences whose only uniformly bounded $(p,q)$-norm is the $(\infty,1)$-norm, but which converge nonetheless. We show that the limit operators in this case are not unique, not self-adjoint, and need not be positivity-preserving. In particular, in the action convergence language, this means that the space of graphops is not compact. By identifying these multiple limits, we also demonstrate that $c$-regularity is not invariant under weak equivalence, where $c$ is the eigenvalue of the identity function, when the identity function is an eigenfunction.
