We consider the preferential attachment model with location-based choice introduced by Haslegrave et al. (Random Struct Algorithms 56(3):775–795, 2020) as a model in which condensation phenomena can occur. In this model, each vertex carries an independent and uniformly distributed location. Starting from an initial tree, the model evolves in discrete time. At every time step, a new vertex is added to the tree by selecting r candidate vertices from the graph with replacement according to a sampling probability proportional to these vertices’ degrees. The new vertex then connects to one of the candidates according to a given probability associated to the ranking of their locations. In this paper, we introduce a function that describes the phase transition when condensation can occur. Considering the noncondensation phase, we use stochastic approximation methods to investigate bounds for the (asymptotic) proportion of vertices inside a given interval of a given maximum degree. We use these bounds to observe a power law for the asymptotic degree distribution described by the aforementioned function. Hence, this function fully characterises the properties we are interested in. The power law exponent takes the critical value one at the phase transition between the condensation–noncondensation phase.

我们考虑了 Haslegrave 等人引入的具有基于位置的选择的优先附件模型。（Random Struct Algorithms 56(3):775–795, 2020）作为可以发生冷凝现象的模型。在这个模型中，每个顶点都带有一个独立且均匀分布的位置。从初始树开始，模型在离散时间演化。在每个时间步，通过选择r将一个新顶点添加到树中根据与这些顶点的度数成正比的采样概率，从图中替换出候选顶点。然后，新顶点根据与其位置排名相关的给定概率连接到候选者之一。在本文中，我们引入了一个函数来描述冷凝发生时的相变。考虑到非凝聚阶段，我们使用随机近似方法来研究给定最大度数的给定区间内顶点的（渐近）比例的边界。我们使用这些界限来观察由上述函数描述的渐近度分布的幂律。因此，这个函数完全表征了我们感兴趣的属性。