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For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/ poly(n) close to those of a uniformly random walk in ℓ1 distance. We first give an algorithm for local access to random walks on a given undirected d-regular graph with eO( 1 1−λ √ n) runtime per query, where λ is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n, d) are expanders with high probability, this gives an eO(√ n) algorithm for a graph drawn from G(n, d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than Ω(√ n/ log(n)) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree nϵ graphs for any ϵ ∈ (0, 1]) and Cartesian product.
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Scalable Graph Synthesis with Adj and 1 — Adj
Graph synthesis is a long-standing research problem. Many deep neural networks that learn about latent characteristics of graphs and generate fake graphs have been proposed. However, in many cases their scalability is too high to be used to synthesize large graphs. Recently, one work proposed an interesting scalable idea to learn and generate random walks that can be merged into a graph. Due to its difficulty, however, the random walk-based graph synthesis failed to show state-of-the-art performance in many cases. We present an improved random walk-based method by using negative random walks. In our experiments with 6 datasets and 8 baseline methods, our method shows the best performance in almost all cases. We achieve both high scalability and generation quality.
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- Award ID(s):
- 1822094
- PAR ID:
- 10276745
- Date Published:
- Journal Name:
- Proceedings of the 2021 SIAM International Conference on Data Mining (SDM)
- Page Range / eLocation ID:
- 307-315
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611976700.35
