We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like $\mathbb{R}$ in the sense that the rescaled ball is close to a line segment in the Gromov–Hausdorff metric.

中文翻译：

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大而冗长的图在局部看起来像线

我们应用单模随机根图的理论来研究直径与它们的体积成比例的大型，有限，有界度图的度量几何。我们证明对于这样一个图的顶点的正比例，存在一个介观尺度，该图看起来像$\mathbb{[R}$ 从某种意义上说，重新缩放的球接近Gromov–Hausdorff度量中的线段。