论文标题
从罗宾逊图形采样的图形序列
Graph sequences sampled from Robinson graphons
论文作者
论文摘要
在[CGH $^+$ 15]中引入的图形空间上的功能$γ$旨在衡量Graphon $ W $在其上展示Robinson属性的程度：对于所有$ x <y <z $，$ w（x，z，z，z，z）\ leq \ min \ min \ min \ min \ {w（x，x，y），w（y，z），w（y，z）\} $。罗宾逊图形形成了具有天然线嵌入的图形模型，因此大多数边缘都是局部的。函数$γ$与cut-norm $ \ | \ cdot \ | _ \ box $兼容，因为在剪切中关闭的图形具有相似的$γ$ - 值。在这里，我们通过证明Robinson Graphon $ R_W $可以近似每个Graphon $ w $来显示相反，以便$ \ | w-r_w \ | _ \ box $以$γ（w）$的方式界定。然后，我们使用功能分析中的经典技术表明，当且仅当$γ（g_n）\ rightarrow 0 $时，当融合的图序列$ \ {g_n \} $收敛到Robinson Graphon时。最后，使用概率技术，我们表明，从罗宾逊图形上采样的图形序列的收敛速率可以大大差异，这取决于$ w $的强烈表现出Robinson属性。
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.
