We revisit the problem of designing sublinear algorithms for estimating the average degree of an $n$-vertex graph. The standard access model for graphs allows for the following queries: sampling a uniform random vertex, the degree of a vertex, sampling a uniform random neighbor of a vertex, and ``pair queries'' which determine if a pair of vertices form an edge. In this model, original results [Goldreich-Ron, RSA 2008; Eden-Ron-Seshadhri, SIDMA 2019] on this problem prove that the complexity of getting $(1+\varepsilon)$-multiplicative approximations to the average degree, ignoring $\varepsilon$-dependencies, is $\Theta(\sqrt{n})$. When random edges can be sampled, it is known that the average degree can estimated in $\widetilde{O}(n^{1/3})$ queries, even without pair queries [Motwani-Panigrahy-Xu, ICALP 2007; Beretta-Tetek, TALG 2024]. We give a nearly optimal algorithm in the standard access model with random edge samples. Our algorithm makes $\widetilde{O}(n^{1/4})$ queries exploiting the power of pair queries. We also analyze the ``full neighborhood access" model wherein the entire adjacency list of a vertex can be obtained with a single query; this model is relevant in many practical applications. In a weaker version of this model, we give an algorithm that makes $\widetilde{O}(n^{1/5})$ queries. Both these results underscore the power of {\em structural queries}, such as pair queries and full neighborhood access queries, for estimating the average degree. We give nearly matching lower bounds, ignoring $\varepsilon$-dependencies, for all our results. So far, almost all algorithms for estimating average degree assume that the number of vertices, $n$, is known. Inspired by [Beretta-Tetek, TALG 2024], we study this problem when $n$ is unknown and show that structural queries do not help in estimating average degree in this setting.

翻译：我们重新审视了设计用于估计$n$顶点图平均度的亚线性算法问题。图的标准访问模型支持以下查询：采样均匀随机顶点、获取顶点度数、采样顶点的均匀随机邻居，以及判断顶点对是否构成边的“配对查询”。在该模型中，该问题的原始研究结果[Goldreich-Ron, RSA 2008; Eden-Ron-Seshadhri, SIDMA 2019]证明，在忽略$\varepsilon$依赖项的情况下，获得平均度$(1+\varepsilon)$乘性近似的复杂度为$\Theta(\sqrt{n})$。当允许随机边采样时，即使无需配对查询，已知可在$\widetilde{O}(n^{1/3})$次查询内估计平均度[Motwani-Panigrahy-Xu, ICALP 2007; Beretta-Tetek, TALG 2024]。我们在支持随机边采样的标准访问模型中提出了一种近乎最优的算法。该算法通过利用配对查询的优势，仅需$\widetilde{O}(n^{1/4})$次查询。我们还分析了“全邻域访问”模型，其中单次查询即可获取顶点的完整邻接表；该模型在许多实际应用中具有重要意义。在此模型的弱化版本中，我们提出了一种仅需$\widetilde{O}(n^{1/5})$次查询的算法。这些结果共同凸显了{\em 结构查询}（如配对查询和全邻域访问查询）在估计平均度问题中的重要作用。对于所有结果，我们在忽略$\varepsilon$依赖项的情况下给出了近乎匹配的下界。迄今为止，几乎所有估计平均度的算法都假设顶点数$n$已知。受[Beretta-Tetek, TALG 2024]启发，我们研究了$n$未知时的问题，并证明在此设定下结构查询对估计平均度并无助益。