The correlated Erd\"os-R\'enyi random graph ensemble is a probability law on pairs of graphs with $n$ vertices, parametrized by their average degree $\lambda$ and their correlation coefficient $s$. It can be used as a benchmark for the graph alignment problem, in which the labels of the vertices of one of the graphs are reshuffled by an unknown permutation; the goal is to infer this permutation and thus properly match the pairs of vertices in both graphs. A series of recent works has unveiled the role of Otter's constant $\alpha$ (that controls the exponential rate of growth of the number of unlabeled rooted trees as a function of their sizes) in this problem: for $s>\sqrt{\alpha}$ and $\lambda$ large enough it is possible to recover in a time polynomial in $n$ a positive fraction of the hidden permutation. The exponent of this polynomial growth is however quite large and depends on the other parameters, which limits the range of applications of the algorithm. In this work we present a family of faster algorithms for this task, show through numerical simulations that their accuracy is only slightly reduced with respect to the original one, and conjecture that they undergo, in the large $\lambda$ limit, phase transitions at modified Otter's thresholds $\sqrt{\widehat{\alpha}}>\sqrt{\alpha}$, with $\widehat{\alpha}$ related to the enumeration of a restricted family of trees.

翻译：相关埃尔德什-雷尼随机图系综是一种定义在具有$n$个顶点的图对上的概率分布，由平均度$\lambda$和相关系数$s$参数化。该系综可作为图对齐问题的基准测试：其中一个图的顶点标签被未知置换随机打乱，目标是通过推断该置换来正确匹配两个图中的顶点对。近期一系列研究揭示了奥特常数$\alpha$（该常数控制无标号有根树数量随规模增长的指数速率）在此问题中的作用：当$s>\sqrt{\alpha}$且$\lambda$足够大时，可在$n$的多项式时间内恢复隐藏置换的正比例部分。然而该多项式增长的指数较大且依赖于其他参数，限制了算法的应用范围。本文提出了一系列针对此问题的更快算法，通过数值模拟表明其精度仅比原算法略有降低，并推测在$\lambda$趋于无穷的极限下，这些算法会在修正的奥特阈值$\sqrt{\widehat{\alpha}}>\sqrt{\alpha}$处发生相变，其中$\widehat{\alpha}$与受限树族枚举相关。