Denser than the densest subgraph

Tsourakakis, Charalampos E.; Bonchi, Francesco; Gionis, Aristides; Gullo, Francesco; Tsiarli, Maria A.

doi:10.1145/2487575.2487645

Cited by 249 publications

(44 citation statements)

References 30 publications

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“…This algorithm can be viewed as a local search algorithm, which consists of expanding and shrinking phases, and the combination of which is a key ingredient that enables the algorithm to search for a better solution. Our experimental results show that this algorithm outputs a higher quality solution than the current best algorithms do . Moreover, the solutions it obtained are almost optimum for all of the small real‐world graphs we tested.…”

Section: Introductionmentioning

confidence: 80%

“…The exact algorithm uses mixed integer quadratic programming (MIQP) and so it takes exponential time in the worst case, but recent developments in state‐of‐the‐art MIQP solvers allow us to compute the optimum solutions for small real‐world graphs (graphs with up to 1,000 vertices) in a reasonable amount of time. Regarding the computational complexity of the densest subgraph problem, the authors of conjecture that it is NP‐hard to obtain the optimum solution when using the QC metric; we conjecture that it is also NP‐hard when using our new metric. In Section 3.3, we give evidence that supports this conjecture by showing the NP‐hardness of optimizing the densest subgraph problem using our new density metric with

β = 2

.…”

Section: Introductionmentioning

confidence: 82%

“…When we solve the densest subgraph problem, there are many definitions that could be used for the density metric

f (S)

, but there is no consensus on which metric we should use. The average degree metric

f (S) = e [S] / | S |

has been widely used in the graph theory community , but in practice, it is known that the densest subgraph obtained by using this metric is typically a loosely‐connected large subgraph (rather than a tightly‐connected small subgraph) . This phenomenon comes from the fact that the average degree metric tends to give a higher value for a large graph than a small graph.…”

Section: Introductionmentioning

confidence: 99%

“…Since the average degree metric has some drawbacks, various other density metrics have been proposed. For example, suggests using a quasi‐clique (QC) metric

f (S) = e [S] - α true(\begin{matrix} | S | \\ 2 \end{matrix} true)

with the parameter

α

, Tsourakakis uses a triangle density metric

f (S) = t [S] / | S |

, where

t [S]

represents the number of triangles in

G [S]

, and the authors of use yet another density metric.…”

Section: Introductionmentioning

confidence: 99%

“…Our second contribution is an exact algorithm to obtain an optimum solution for the densest subgraph with respect to a density metric of the form

f (S) = e [S] / h (| S |)

f (S) = e [S] - h (| S |)

, where

h (x)

is a convex increasing function with respect to

x

. This form includes our discounted average degree metric

f (S) = e [S] / | S |^{β}

and the QC metric

f (S) = e [S] - α true(\begin{matrix} | S | \\ 2 \end{matrix} true)

. The exact algorithm uses mixed integer quadratic programming (MIQP) and so it takes exponential time in the worst case, but recent developments in state‐of‐the‐art MIQP solvers allow us to compute the optimum solutions for small real‐world graphs (graphs with up to 1,000 vertices) in a reasonable amount of time.…”

Section: Introductionmentioning

confidence: 99%

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

Yanagisawa

Hara

2017

Networks

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Detecting the densest subgraph is one of the most important problems in graph mining and has a variety of applications. Although there are many possible metrics for subgraph density, there is no consensus on which density metric we should use. In this article, we suggest a new density metric, the discounted average degree, which has some desirable properties of the subgraph's density. We also show how to obtain an optimum densest subgraph for small graphs with respect to several density metrics, including our new density metric, by using mixed integer programming. Finally, we develop a new heuristic algorithm to quickly obtain a good approximate solution for large graphs. Our computational experiments on realworld graphs showed that our new heuristic algorithm outperformed other heuristics in terms of the quality of the solutions.

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Section: Introductionmentioning

confidence: 80%

β = 2

.…”

Section: Introductionmentioning

confidence: 82%

“…When we solve the densest subgraph problem, there are many definitions that could be used for the density metric

f (S)

, but there is no consensus on which metric we should use. The average degree metric

f (S) = e [S] / | S |

Section: Introductionmentioning

confidence: 99%

“…Since the average degree metric has some drawbacks, various other density metrics have been proposed. For example, suggests using a quasi‐clique (QC) metric

f (S) = e [S] - α true(\begin{matrix} | S | \\ 2 \end{matrix} true)

with the parameter

α

, Tsourakakis uses a triangle density metric

f (S) = t [S] / | S |

, where

t [S]

represents the number of triangles in

G [S]

, and the authors of use yet another density metric.…”

Section: Introductionmentioning

confidence: 99%

“…Our second contribution is an exact algorithm to obtain an optimum solution for the densest subgraph with respect to a density metric of the form

f (S) = e [S] / h (| S |)

f (S) = e [S] - h (| S |)

, where

h (x)

is a convex increasing function with respect to

x

. This form includes our discounted average degree metric

f (S) = e [S] / | S |^{β}

and the QC metric

f (S) = e [S] - α true(\begin{matrix} | S | \\ 2 \end{matrix} true)

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Yanagisawa

Hara

2017

Networks

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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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