Graph Sample and Hold: A Framework for Big-Graph Analytics (1403.3909v1)

Abstract: Sampling is a standard approach in big-graph analytics; the goal is to efficiently estimate the graph properties by consulting a sample of the whole population. A perfect sample is assumed to mirror every property of the whole population. Unfortunately, such a perfect sample is hard to collect in complex populations such as graphs (e.g. web graphs, social networks etc), where an underlying network connects the units of the population. Therefore, a good sample will be representative in the sense that graph properties of interest can be estimated with a known degree of accuracy. While previous work focused particularly on sampling schemes used to estimate certain graph properties (e.g. triangle count), much less is known for the case when we need to estimate various graph properties with the same sampling scheme. In this paper, we propose a generic stream sampling framework for big-graph analytics, called Graph Sample and Hold (gSH). To begin, the proposed framework samples from massive graphs sequentially in a single pass, one edge at a time, while maintaining a small state. We then show how to produce unbiased estimators for various graph properties from the sample. Given that the graph analysis algorithms will run on a sample instead of the whole population, the runtime complexity of these algorithm is kept under control. Moreover, given that the estimators of graph properties are unbiased, the approximation error is kept under control. Finally, we show the performance of the proposed framework (gSH) on various types of graphs, such as social graphs, among others.

• The paper introduces the Graph Sample and Hold (gSH) framework, which efficiently samples large graphs from a single stream of edges to provide unbiased estimates of key graph properties.
• Experiments demonstrate gSH accurately estimates properties like triangle counts and clustering coefficients with small sample sizes (typically eless than 1% of edges) and maintains low relative errors ( eless than 1%).
• gSH offers a balance of accuracy and computational efficiency compared to other graph sampling methods, making it a scalable approach for analyzing diverse big-data graph structures.

Analyzing Graph Sample and Hold (gSH) for Big-Graph Analytics

The paper "Graph Sample and Hold: A Framework for Big-Graph Analytics" introduces a methodology designed to address the computational challenges of analyzing large-scale graph data. Traditional approaches to graph analysis often struggle with the vast number of nodes and edges present in real-world networks such as social media structures or web graphs. To mitigate this issue, the authors propose the Graph Sample and Hold (gSH) framework, which efficiently samples graph edges in a streaming fashion and provides unbiased estimates of various graph properties.

Framework and Methodology

The core contribution of the paper is the gSH framework, which samples graph edges one at a time in a single pass through the graph, maintaining a minimal state. This process is crucial for reducing computational overhead, allowing for scalable graph analysis without the need to process the entire graph in memory. Edges are sampled based on their adjacency to previously sampled edges, with different probabilities assigned for edges that are already adjacent ('hold' probability) and those that are not ('sample' probability).

The framework leverages the Horvitz-Thompson construction to produce unbiased estimators for graph properties, such as the counts of links, triangles, and connected paths of length two, as well as for estimating the global clustering coefficient. The structure allows for the computation of both the estimates themselves and the variances of these estimates, enabling the derivation of confidence intervals. This approach provides not only efficiency in terms of computational resources but also robustness in analytical accuracy.

Numerical Results and Implications

The experimental results demonstrate the efficacy of the gSH framework across a variety of real-world datasets, including social networks and web graphs. By maintaining sample sizes substantially smaller than the full dataset (often around 1% or less of the total edges), the gSH maintains relative errors typically under 1% for most graph statistics. This efficiency suggests that gSH can perform well on large datasets where other sampling methodologies may struggle due to higher computational or storage requirements.

The research further explores the adaptability of the framework to varying graph densities and characteristics, with experimental evidence showing consistent performance across diverse test cases. The adaptability of gSH to different network types showcases its potential for being widely applicable in various big-data contexts where graph structures are prevalent.

Comparative Analysis and Future Directions

In comparison to related work, notably the algorithms for triangle counting and sampling present in the literature, gSH stands out for its balance between accuracy and resource efficiency. While traditional algorithms, such as those based on wedge sampling and reservoir methods, often require significant storage or lead to higher relative errors, gSH provides a structured solution that holds its ground both in accuracy and computational efficiency.

The paper provides a pathway for future developments in graph sampling techniques. Potential extensions could include further reduction in storage requirements or adaptations for dynamic graph structures where nodes and edges evolve over time. Additionally, exploring its application to more complex graph metrics, such as those involving community detection or resilience metrics, remains a promising direction for subsequent research.

In conclusion, the "Graph Sample and Hold" paper presents a robust and efficient framework for graph analytics that aligns well with the needs of current big-data analysis paradigms, setting the stage for more nuanced and adaptable analytical approaches in the future.