稀疏图左右收敛等价性的一个简短证明
A short proof of the equivalence of left and right convergence for sparse graphs
摘要：
有界度图序列的收敛性有几个概念。一个这样的概念是左收敛，它是基于计数邻域分布。另一个概念是右收敛，基于计算到目标（加权）图的同态。Borgs，Chayes，Kahn和Lovász证明了一个有界度图序列是左收敛的当且仅当它对于某些目标图$H$是右收敛的，并且所有权值（包括环）都接近于$1$。我们给这个声明的一个简短的替代证明。特别是，对于每一个有界度图$G$，我们关联的功能$f_{G,k}$为每一个正整数$k$，我们表明，左收敛的一个序列的图形是等价于这些功能的偏导数在原点的收敛，而右收敛是等价于逐点收敛。利用图的最大度的界，我们可以使图的偏导数在原点处一致有界，并证明了Taylor级数在与图无关的区域上一致收敛，这就意味着等价性.
英文摘要：
There are several notions of convergence for sequences of bounded degree graphs. One such notion is left convergence, which is based on counting neighborhood distributions. Another notion is right convergence, based on counting homomorphisms to a target (weighted) graph. Borgs, Chayes, Kahn and Lovász showed that a sequence of bounded degree graphs is left convergent if and only if it is right convergent for certain target graphs $H$ with all weights (including loops) close to $1$. We give a short alternative proof of this statement. In particular, for each bounded degree graph $G$ we associate functions $f_{G,k}$ for every positive integer $k$, and we show that left convergence of a sequence of graphs is equivalent to the convergence of the partial derivatives of each of these functions at the origin, while right convergence is equivalent to pointwise convergence. Using the bound on the maximum degree of the graphs, we can uniformly bound the partial derivatives at the origin, and show that the Taylor series converges uniformly on a domain independent of the graph, which implies the equivalence.
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