Towards Tight Bounds for Estimating Degree Distribution in Streaming and Query Models

TL;DR

This paper develops tight bounds for estimating the degree distribution and its complementary cumulative histogram in streaming and query models, providing algorithms and lower bounds that nearly resolve the problem's complexity.

Contribution

It introduces an algorithm for approximating the ccdh using vertex and edge samples, and establishes the first lower bounds in streaming and query models for this problem.

Findings

Algorithm for ccdh approximation with sample access

Efficient sampling methods in streaming and query models

First lower bounds for degree distribution estimation in these models

Abstract

The degree distribution of a graph $G=(V,E)$, $|V|=n$, $|E|=m$ is one of the most fundamental objects of study in the analysis of graphs as it embodies relationship among entities. In particular, an important derived distribution from degree distribution is the complementary cumulative degree histogram (ccdh). The ccdh is a fundamental summary of graph structure, capturing, for each threshold $d$, the number of vertices with degree at least $d$. For approximating ccdh, we consider the $(\varepsilon_D,\varepsilon_R)$-BiCriteria Multiplicative Approximation, which allows for controlled multiplicative slack in both the domain and the range. The exact complexity of the problem was not known and had been posed as an open problem in WOLA 2019 [Sublinear.info, Problem 98]. In this work, we first design an algorithm that can approximate ccdh if a suitable vertex sample and an edge sample can be obtained and thus, the algorithm is independent of any sublinear model. Next, we show that in the streaming and query models, these samples can be obtained efficiently. On the other end, we establish the first lower bounds for this problem in both query and streaming models, and (almost) settle the complexity of the problem across both the sublinear models.