超图过程中的连续相变和不连续相变
Continuous and discontinuous phase transitions in hypergraph processes
摘要：
设V表示N个顶点的集合。为了构造一个“超图过程”，在泊松过程的每个事件时刻创建一个新的超边;这个超边的基数K是随机的，具有任意的概率母函数r（x），除了我们假设P（K=1）+P（K=2）> 0。给定K=k，出现在新超边中的k个顶点是从V中随机均匀选择的。基数为1的超边称为面片，并用作选择根顶点的一种方式。可识别的顶点是那些从这些根顶点可到达的顶点，在很强的意义上，它推广了图组件的概念。如果超边的所有顶点都是可识别的，则超边也称为可识别的。我们使用“流体极限”标度：超边以N的速率到达，并且我们研究大小为O（1）和O（N）的结构。在除以N之后，可识别的顶点和可约超边的数目表现出相变，其可以是连续的或不连续的，这取决于结构函数-log（1 - x）/r '（x）的形状，其中x在（0，1）中。考虑了P（K=1）> 0和P（K=1）= 0 < P（K=2）两种情况;对于后者，添加单个无关面片来标记根顶点。
英文摘要：
Let V denote a set of N vertices. To construct a "hypergraph process", create a new hyperedge at each event time of a Poisson process; the cardinality K of this hyperedge is random, with arbitrary probability generating function r(x), except that we assume P(K=1) +P(K=2) > 0. Given K=k, the k vertices appearing in the new hyperedge are selected uniformly at random from V. Hyperedges of cardinality 1 are called patches, and serve as a way of selecting root vertices. Identifiable vertices are those which are reachable from these root vertices, in a strong sense which generalizes the notion of graph component. Hyperedges are also called identifiable if all of their vertices are identifiable. We use "fluid limit" scaling: hyperedges arrive at rate N, and we study structures of size O(1) and O(N). After division by N, numbers of identifiable vertices and reducible hyperedges exhibit phase transitions, which may be continuous or discontinuous depending on the shape of the structure function -log(1 - x)/r'(x), for x in (0,1). Both the case P(K=1) > 0 and the case P(K=1) = 0 < P(K=2) are considered; for the latter, a single extraneous patch is added to mark the root vertex.
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