Abstract In Chung–Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution, and two vertices form an edge independently with probability proportional to the product of their weights. Chung–Lu graphs have average distance $O(\log\log n)$ and thus reproduce the small-world phenomenon, a key property of real-world networks. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry. The edge probability of two vertices then depends, say, inversely polynomially on their distance. In this paper we study a generic augmented version of Chung–Lu random graphs. We analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in a Chung–Lu random graph. The resulting class contains Chung–Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases. Our main result is that every random graph model in this general class has the same average distance as a Chung–Lu random graph, up to a factor of $1+o(1)$ . This shows in particular that specific choices, such as taking the underlying geometry to be Euclidean, do not significantly influence the average distance. The proof also shows that every random graph model in our class has a giant component and polylogarithmic diameter with high probability.

Karl Bringmann

Ralph Keusch

Johannes Lengler

Advances in Applied Probability

THE ADVANCES IN APPLIED PROBABILITY TRUST 由 APPLIED PROBABILITY TRUST 出版了 40 多年，是《应用概率杂志》的配套出版物。它包含应用概率学家感兴趣的数学和科学论文，重点是广泛学科的应用，包括生物科学、运筹学、电信、计算机科学、工程、流行病学、金融数学、物理和社会科学以及任何使用随机建模的字段。提交给 APPLIED PROBABILITY 代表的提交内容可能会根据主编的判断出现在 JOURNAL OF APPLIED PROBABILITY 或 ADVANCES IN APPLIED PROBABILITY 中。通常，较短的论文出现在 JOURNAL 中，较长的论文出现在 ADVANCES 中。

Advances in Applied Probability, published by the Applied Probability Trust for over 40 years, serves as a companion publication to the Journal of Applied Probability. It features mathematical and scientific papers of interest to applied probabilists, focusing on a wide range of applications across disciplines including biological sciences, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, physics, and social sciences, as well as any field that utilizes stochastic modeling. Submissions to the Journal of Applied Probability may appear in either the Journal of Applied Probability or Advances in Applied Probability, at the discretion of the editor-in-chief. Typically, shorter papers are published in the Journal, while longer papers are published in Advances.

《Advances in Applied Probability》是由应用概率信托（Applied Probability Trust）出版的期刊，已有40多年的历史，是《应用概率杂志》（Journal of Applied Probability）的配套出版物。该期刊包含应用概率学家感兴趣的数学和科学论文，重点关注广泛学科的应用，包括生物科学、运筹学、电信、计算机科学、工程、流行病学、金融数学、物理和社会科学等领域，以及任何使用随机建模的领域。提交给《应用概率杂志》的稿件，可能会根据主编的判断，出现在《应用概率杂志》或《Advances in Applied Probability》中。通常，较短的论文会发表在《应用概率杂志》中，而较长的论文则会发表在《Advances in Applied Probability》中。

Category	Quartile
数学	4
数学, 统计学与概率论	3

Average distance in a general class of scale-free networks

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