It is often of interest to sample vertices from a graph with a bias towards higher-degree vertices. One well-known method, which we call random neighbor or RN, involves taking a vertex at random and exchanging it for one of its neighbors. Loosely inspired by the friendship paradox, the method is predicated on the fact that the expected degree of the neighbor is greater than or equal to the expected degree of the initial vertex. Another method that is actually perfectly analogous to the friendship paradox is random edge, or RE, where an edge is sampled at random, and then one of the two endpoint vertices is selected at random. Obviously, random sampling is only required when full knowledge of the graph is unattainable. But, while it is true in most cases that knowledge of all vertices’ degrees cannot be obtained, it is often trivial to learn the degree of specific vertices that have already been isolated. In light of this, we suggest a tweak to both RN and RE, inclusive random sampling. In inclusive random neighbor (IRN) the initial vertex and the selected neighbor are considered, in inclusive random edge (IRE) the two endpoint vertices are, and in both cases, we learn the degree of each and select the vertex of higher degree. This paper explores inclusive random sampling through theoretical analysis and experimentation. We establish meaningful bounds on IRN and IRE’s performances, in particular in comparison to each other and to their exclusive counterparts. Our analyses highlight differences of the original, exclusive versions as well. The results provide practical insight for strategizing a random sampling method, and also highlight graph characteristics that impact the question of which methods will perform strongly in which graphs.

从图中采样偏向于更高阶顶点的顶点通常是令人感兴趣的。一种众所周知的方法，我们称之为随机邻居或 RN，涉及随机获取一个顶点并将其交换为其邻居之一。松散地受到友谊悖论的启发，该方法基于邻居的预期度数大于或等于初始顶点的预期度数这一事实。实际上与友谊悖论完全相似的另一种方法是随机边（RE），其中随机对边进行采样，然后随机选择两个端点之一。显然，只有当无法完全了解图表时才需要随机抽样。但是，虽然在大多数情况下确实无法获得所有顶点度数的知识，但了解已隔离的特定顶点的度数通常是微不足道的。有鉴于此，我们建议对 RN 和 RE 进行调整，包括随机抽样。在包容性随机邻居（IRN）中，考虑初始顶点和选定的邻居，在包容性随机边缘（IRE）中，考虑两个端点顶点，在这两种情况下，我们都会了解每个顶点的度数并选择度数较高的顶点。本文通过理论分析和实验探索包容性随机抽样。我们对 IRN 和 IRE 的表现设定了有意义的界限，特别是在相互比较以及与其专有对手进行比较时。我们的分析还强调了原始独家版本的差异。结果为制定随机抽样方法策略提供了实用的见解，并且还突出了影响哪些方法在哪些图中表现出色的问题的图特征。