从Robinson图子（graphon）中采样的图序列
Graph sequences sampled from Robinson graphons
AI总结：
该研究证明图子可被Robinson图子按割范数逼近，刻画了收敛图序列极限为Robinson图子的充要条件，并揭示了采样图序列的Γ收敛速率与Robinson性质强弱的关联。
摘要：
定义在图子（graphon）空间上的函数$\Gamma$由文献[CGH$^+$15]提出，用于衡量图子$w$满足Robinson性质的程度：对所有$x<y<z$，有$w(x,z)\leq \min\{ w(x,y),w(y,z)\}$。Robinson图子构成了一类具有自然直线嵌入的图模型，其大部分边都是局部边。函数$\Gamma$与割范数$\\|\cdot \\|_\Box$相容，即割范数相近的图子具有相近的$\Gamma$值。本文证明其逆命题也成立：每个图子$w$都可以被一个Robinson图子$R_w$逼近，使得$\\|w-R_w\\|_\Box$可由$\Gamma(w)$给出上界。随后我们利用泛函分析的经典方法证明：收敛图序列$\{G_n\}$收敛到一个Robinson图子当且仅当$\Gamma(G_n)\rightarrow 0$。最后，通过概率方法我们发现，从Robinson图子采样得到的图序列的$\Gamma$收敛速率可能存在显著差异，具体取决于$w$满足Robinson性质的强弱程度。
英文摘要：
The function $Γ$ on the space of graphons, introduced in [CGH$^+$15], aims to measure the extent to which a graphon $w$ exhibits the Robinson property: for all $x<y<z$, $w(x,z)\leq \min\{ w(x,y),w(y,z)\}$. Robinson graphons form a model for graphs with a natural line embedding so that most edges are local. Function $Γ$ is compatible with the cut-norm $\|\cdot \|_\Box$, in the sense that graphons close in cut-norm have similar $Γ$-values. Here we show the converse, by proving that every graphon $w$ can be approximated by a Robinson graphon $R_w$ so that $\|w-R_w\|_\Box$ is bounded in terms of $Γ(w)$. We then use classical techniques from functional analysis to show that a converging graph sequence $\{G_n\}$ converges to a Robinson graphon if and only if $Γ(G_n)\rightarrow 0$. Finally, using probabilistic techniques we show that the rate of convergence of $Γ$ for graph sequences sampled from a Robinson graphon can differ substantially depending on how strongly $w$ exhibits the Robinson property.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
