On Approximating the Number of $k$-Cliques in Sublinear Time

Eden, Talya; Ron, Dana; Seshadhri, C.

doi:10.1137/18m1176701

Cited by 36 publications

(66 citation statements)

References 30 publications

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“…This bound on the query complexity is tight (up to factors polynomial in log n and 1/ ) [7]. The result was recently extended to approximating the number of k-cliques [8], for any given k ≥ 3. 2 The arboricity of a graph G, denoted arb(G), is the minimum number of forests into which its edges can be partitioned.…”

Section: Number Of Triangles and Larger Cliquesmentioning

confidence: 85%

Sublinear-Time Algorithms for Approximating Graph Parameters

Ron

2019

Lecture Notes in Computer Science

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Section: Number Of Triangles and Larger Cliquesmentioning

confidence: 85%

Sublinear-Time Algorithms for Approximating Graph Parameters

Ron

2019

Lecture Notes in Computer Science

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“…Other open questions include using polylog(n) BIS queries to estimate the number of cliques in a graph (see [9] for an algorithm using degree, neighbor and edge existence queries) or to sample a uniformly random edge (see [11] for an algorithm using degree, neighbor and edge existence queries). In general, any graph estimation problems may benefit from BIS or IS queries, possibly in combination with standard queries (such as neighbor queries).…”

Section: Open Directionsmentioning

confidence: 99%

Edge Estimation with Independent Set Oracles

Beame

Har-Peled

Ramamoorthy

et al. 2020

ACM Trans. Algorithms

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We study the problem of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n-vertex graph: one that uses only polylog(n) bipartite independent set queries, and another one that uses n 2/3 • polylog(n) independent set queries.

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“…With a view on large data sets, we develop a test that only relies on small samples from the graph and does not examine the graph in its entirety. Here, we follow the examples of many authors who have inferred various graph properties through sampling (e.g., [3,11,12,17,18,19]).…”

Section: Fig 1 Graphs Displaying Clustered and Unclustered Structurementioning

confidence: 99%

“…The graph testing (through sampling) literature is very rich. Many authors have proposed tests for various graph properties other than the existence of clusters (e.g., [3,11,19,20,25]). Tests for clustering have also been studied in the past.…”

Section: Related Workmentioning

confidence: 99%

A Statistical Test of Heterogeneous Subgraph Densities to Assess Clusterability

Miasnikof

Prokhorenkova

Shestopaloff

et al. 2020

Lecture Notes in Computer Science

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Determining if a graph displays a clustered structure prior to subjecting it to any cluster detection technique has recently gained attention in the literature. Attempts to group graph vertices into clusters when a graph does not have a clustered structure is not only a waste of time but will also lead to misleading conclusions. To address this problem, we introduce a novel statistical test, the δ-test, which is based on comparisons of local and global densities. Our goal is to assess whether a given graph meets the necessary conditions to be meaningfully summarized by clusters of vertices. We empirically explore our test's behavior under a number of graph structures. We also compare it to other recently published tests. From a theoretical standpoint, our test is more general, versatile and transparent than recently published competing techniques. It is based on the examination of intuitive quantities, applies equally to weighted and unweighted graphs and allows comparisons across graphs. More importantly, it does not rely on any distributional assumptions, other than the universally accepted definition of a clustered graph. Empirically, our test is shown to be more responsive to graph structure than other competing tests.

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On Approximating the Number of $k$-Cliques in Sublinear Time

Cited by 36 publications

References 30 publications

Sublinear-Time Algorithms for Approximating Graph Parameters

Sublinear-Time Algorithms for Approximating Graph Parameters

Edge Estimation with Independent Set Oracles

A Statistical Test of Heterogeneous Subgraph Densities to Assess Clusterability

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