We introduce and study a novel semi‐random multigraph process, described as follows. The process starts with an empty graph on n vertices. In every round of the process, one vertex v of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to v. For various natural monotone increasing graph properties , we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies . Along the way, we show that the process is general enough to approximate (using suitable strategies) several well‐studied random graph models.

中文翻译：

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半随机图过程

我们介绍并研究一种新颖的半随机多图过程，如下所述。该过程以n个顶点上的空图开始。在该过程的每一轮中，均独立于所有前一轮随机地均匀地选择图形的一个顶点v。然后，我们选择另一个顶点（根据我们选择的策略），并通过边与v连接。对于各种自然单调递增图的性质，我们证明了该过程以高概率获得满足条件的图所需的最小轮数（在所有可能策略的集合上扩展）的严格上限和下限。在此过程中，我们表明该过程足够通用，可以近似（使用适当的策略）几个经过充分研究的随机图模型。