Discounted average degree density metric and new algorithms for the densest subgraph problem

Yanagisawa, Hiroki; Hara, Satoshi

doi:10.1002/net.21764

Cited by 10 publications

(4 citation statements)

References 20 publications

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“…Generalizing in a different direction, Kawase and Miyauchi [18] introduced the 𝑓 -densest subgraph problem, which seeks a set 𝑆 ⊆ 𝑉 maximizing |𝐸 𝑆 |/𝑓 (|𝑆 |) for a convex or concave function 𝑓 . This includes the special case 𝑓 (𝑆) = |𝑆 | 𝛼 for 𝛼 > 0, which generalizes the earlier notion of discounted average degree considered by Yanagisawa and Hara [45]. When 𝑓 is concave, maximizing the objective will always produce output sets that are larger than or equal to optimizers for the densest subgraph problem.…”

Section: Comparison With Existing Objectivesmentioning

confidence: 76%

The Generalized Mean Densest Subgraph Problem

Veldt

Benson

Kleinberg

2021

Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery &Amp; Data Mining

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Finding dense subgraphs of a large graph is a standard problem in graph mining that has been studied extensively both for its theoretical richness and its many practical applications. In this paper we introduce a new family of dense subgraph objectives, parameterized by a single parameter 𝑝, based on computing generalized means of degree sequences of a subgraph. Our objective captures both the standard densest subgraph problem and the maximum 𝑘-core as special cases, and provides a way to interpolate between and extrapolate beyond these two objectives when searching for other notions of dense subgraphs. In terms of algorithmic contributions, we first show that our objective can be minimized in polynomial time for all 𝑝 ≥ 1 using repeated submodular minimization. A major contribution of our work is analyzing the performance of different types of peeling algorithms for dense subgraphs both in theory and practice. We prove that the standard peeling algorithm can perform arbitrarily poorly on our generalized objective, but we then design a more sophisticated peeling method which for 𝑝 ≥ 1 has an approximation guarantee that is always at least 1/2 and converges to 1 as 𝑝 → ∞. In practice, we show that this algorithm obtains extremely good approximations to the optimal solution, scales to large graphs, and highlights a range of different meaningful notions of density on graphs coming from numerous domains. Furthermore, it is typically able to approximate the densest subgraph problem better than the standard peeling algorithm, by better accounting for how the removal of one node affects other nodes in its neighborhood.

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Section: Comparison With Existing Objectivesmentioning

confidence: 76%

The Generalized Mean Densest Subgraph Problem

Veldt

Benson

Kleinberg

2021

Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery &Amp; Data Mining

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“…The authors in [19] claim that quasi‐clique metric is better than average‐degree, as it was shown that quasi‐clique produces subgraphs that are tightly connected and smaller. In the same vein as [19], authors in [20] proposed another density metric called discounted average degree as

f (S) = \frac{| E (S) |}{| S |^{β}}

, where

β

is a parameter that can be chosen to affect the size of the desired subgraph. They also give four desirable properties of a density metric and show that their discounted average degree metric performs well on satisfying those four properties.…”

Section: Definition Of the Problemmentioning

confidence: 99%

In search of dense subgraphs: How good is greedy peeling?

2021

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The problem of finding the densest subgraph in a given graph has several real‐world applications, particularly in areas like social network analysis, protein, and gene networks. Depending on the application, finding dense subgraphs can be used to determine regions of high importance, similar characteristics, or enhanced interaction. The densest subgraph extraction problem is fundamentally a non‐linear optimization problem. Nevertheless, it can be solved in polynomial time by an exact algorithm based on iteratively solving a series of max‐flow subproblems. Despite its polynomial‐time complexity, the computing time required by exact algorithms on very large graphs could be prohibitive. Thus, to approach graphs with millions of vertices and edges, one has to resort to heuristic algorithms. We provide an efficient implementation of a greedy heuristic from the literature that is extremely fast and has some nice theoretical properties. We also introduce a new heuristic algorithm that is built on top of the greedy and the exact methods. An extensive computational study is presented to evaluate the performance of various algorithms on a benchmark composed of 86 instances taken from the literature and real world. This analysis shows that the proposed heuristic algorithm is very effective on a large number of test instances, often providing either the optimal solution or a near‐optimal solution within short computing times.

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“…Variants allow for size restrictions (Andersen and Chellapilla 2009) or local subgraphs (Andersen 2010). Other measures include edge surplus (Tsourakakis et al 2013), triangle and k-clique density (Tsourakakis 2015), discounted average degree (Yanagisawa and Hara 2018), and minimum internal degree, which defines k-cores (Shin, Eliassi-Rad, and Faloutsos 2016). Related measures underlie k-plexes (Seidman and Foster 1978) and k-trusses (Cohen 2008).…”

Section: Related Workmentioning

confidence: 99%

TellTail: Fast Scoring and Detection of Dense Subgraphs

Hooi

Shin

Lamba

et al. 2020

AAAI

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Suppose you visit an e-commerce site, and see that 50 users each reviewed almost all of the same 500 products several times each: would you get suspicious? Similarly, given a Twitter follow graph, how can we design principled measures for identifying surprisingly dense subgraphs? Dense subgraphs often indicate interesting structure, such as network attacks in network traffic graphs. However, most existing dense subgraph measures either do not model normal variation, or model it using an Erdős-Renyi assumption - but this assumption has been discredited decades ago. What is the right assumption then? We propose a novel application of extreme value theory to the dense subgraph problem, which allows us to propose measures and algorithms which evaluate the surprisingness of a subgraph probabilistically, without requiring restrictive assumptions (e.g. Erdős-Renyi). We then improve the practicality of our approach by incorporating empirical observations about dense subgraph patterns in real graphs, and by proposing a fast pruning-based search algorithm. Our approach (a) provides theoretical guarantees of consistency, (b) scales quasi-linearly, and (c) outperforms baselines in synthetic and ground truth settings.

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Discounted average degree density metric and new algorithms for the densest subgraph problem

Cited by 10 publications

References 20 publications

The Generalized Mean Densest Subgraph Problem

The Generalized Mean Densest Subgraph Problem

In search of dense subgraphs: How good is greedy peeling?

TellTail: Fast Scoring and Detection of Dense Subgraphs

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