Estimating and Sampling Graphs with Multidimensional Random Walks (1002.1751v2)

Abstract: Estimating characteristics of large graphs via sampling is a vital part of the study of complex networks. Current sampling methods such as (independent) random vertex and random walks are useful but have drawbacks. Random vertex sampling may require too many resources (time, bandwidth, or money). Random walks, which normally require fewer resources per sample, can suffer from large estimation errors in the presence of disconnected or loosely connected graphs. In this work we propose a new $m$-dimensional random walk that uses $m$ dependent random walkers. We show that the proposed sampling method, which we call Frontier sampling, exhibits all of the nice sampling properties of a regular random walk. At the same time, our simulations over large real world graphs show that, in the presence of disconnected or loosely connected components, Frontier sampling exhibits lower estimation errors than regular random walks. We also show that Frontier sampling is more suitable than random vertex sampling to sample the tail of the degree distribution of the graph.

• The paper introduces Frontier Sampling, a method using dependent multidimensional random walks to reduce sampling errors in sparse and disconnected graphs.
• It leverages joint steady-state distributions to achieve faster convergence and more accurate estimation of graph properties, notably improving degree distribution metrics.
• The approach is scalable and versatile, making it highly applicable for real-world large-scale network analyses and complex social network studies.

Analytical Overview of "Estimating and Sampling Graphs with Multidimensional Random Walks"

This paper presents an innovative approach to sampling and estimating the properties of graphs utilizing multidimensional random walks, specifically a technique termed Frontier Sampling (FS). The authors identify key deficiencies in existing methods such as random vertex sampling and random edge sampling, particularly emphasizing challenges encountered in effectively sampling complex, large-scale networks. The fundamental aim of the described method is to mitigate errors found in sparsely connected components of graphs and enhance the accuracy of characteristic estimations, especially in disconnected scenarios common in real-world networks.

Context and Methodology

In network science, understanding graph properties through sampling is crucial because querying entire graphs is resource-intensive. Contemporary methods like independent random walks struggle with disconnected graph components, leading to significant estimation errors. In addressing this, the authors propose FS—a multidimensional random walk where m dependent walkers traverse the graph simultaneously. FS maintains the desirable properties of traditional random walks but offers improved estimation accuracy by reducing bias and the effects of disconnected components.

FS notably performs random walks over the m-th Cartesian power of a graph, and each walker in FS contributes to a shared edge frontier, in contrast to traditional independent usage. By coordinating multiple dependent walkers, FS systematically leverages initial random vertex samples to distribute the walkers in a way that aligns with the underlying graph structure, converging faster to the steady state and uniform edge sampling.

Key Contributions and Findings

1. Enhanced Estimation Accuracy: Frontier Sampling demonstrates marked improvements in estimation accuracy over both single and multiple independent random walkers. This advantage is most pronounced when dealing with loosely connected graphs, prevalent in structured graph-based scenarios such as social networks. Simulation results corroborate these findings with FS exhibiting smaller Mean Squared Errors (MSEs) in degree distribution and assortative mixing coefficient estimations.
2. Theoretical Insights: The paper establishes theoretical foundations showing FS reduces the transient time of random walks. The authors prove the joint steady-state distribution of FS remains closer to a uniform distribution compared to multiple independent random walkers. By rigorously deriving probabilities and distributions, they reinforce the statistical robustness of FS for discrete graph estimation tasks.
3. Versatile Application and Scalability: A key advantage is FS's adaptability in real-world scenarios where graph structures are irregular, passing through disconnected components more effectively than traditional methods. Moreover, the distributed nature of FS enhances its scalability, extending its applicability in varied parallel and large-scale network analytics operations without intensive coordination.

Practical Implications and Future Directions

The proposed FS method stands as a powerful tool in the analysis of modern complex networks, with implications extending across fields such as social network analysis, Internet topology mapping, and large-scale information retrieval—all confronting problems of sparsity or disconnected graph components. The authors’ systematic exploration of sampling tails of degree distributions further indicates FS's potential in planning network interventions or improving network robustness analyses.

Future work could focus on optimizing the computational efficiency of FS in streaming graph data scenarios or exploring the integration of FS with non-Markovian dynamics in dynamic or evolving networks. Another exploration could be enhancing the adaptability of FS in heterogeneous networks with mixed vertex connectivity traits, leveraging machine learning to inform walker decision processes.

In conclusion, by introducing FS and demonstrating its superiority in handling challenging network properties, the authors advance the capabilities of graph sampling techniques, providing a robust framework that bridges significant gaps in graph estimation methodologies. This work not only challenges existing paradigms but also sets a foundation for subsequent innovative advancements in graph sampling technologies.