论文标题
具有健身的递归树的学位分布
Degree Distributions in Recursive Trees with Fitnesses
论文作者
论文摘要
我们研究了一个递归树木的一般模型，其中顶点配备了独立的权重，并且在每个时间阶段，顶点的概率与其适应性函数成正比（其重量和程度的函数），并连接到$ \ ell $ new-comping oping tertices。在某个技术假设下，应用了碎屑模式 - 贾格尔分支过程的理论，我们得出了具有给定程度和权重的顶点比例的几乎确定限制分布的公式，并且边缘的比例具有具有一定重量的端点。作为该定理的应用，我们证明了与[$ \ Mathit {phys的bianconi {phys。 \，Rev. \，E} \; \ Mathbf {66}，\ text {036116（2002）} $]。当适应性功能是仿射时，技术条件可能会失败时，我们还会研究该过程，我们称之为“具有适应性的广义优先附件”。我们表明，该模型可以表现出凝结，其中的边缘的正比例在最大重量的顶点周围积聚，或者更急剧地具有变性的限制度分布，其中整个边缘在这些顶点周围积聚。最后，我们证明了在与该过程相关的分区函数的强大定律的不同假设下，程度分布的随机收敛。
We study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function (a function of its weight and degree) and connects to $\ell$ new-coming vertices. Under a certain technical assumption, applying the theory of Crump-Mode-Jagers branching processes, we derive formulas for the almost sure limiting distribution of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we prove rigorously observations of Bianconi related to the evolving Cayley tree in [$\mathit{Phys. \, Rev. \, E} \; \mathbf{66}, \text{ 036116 (2002)}$]. We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call "generalised preferential attachment with fitness". We show that this model can exhibit condensation where a positive proportion of edges accumulate around vertices with maximal weight, or, more drastically, have a degenerate limiting degree distribution where the entire proportion of edges accumulate around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process.
