Strong isoperimetric inequality for the edge graph of a tiling of the plane

1993 ◽  
Vol 61 (6) ◽  
pp. 584-595
Author(s):  
A. Calogero
Keyword(s):  
Isoperimetric Inequality ◽  
Edge Graph ◽  
Strong Isoperimetric Inequality

Efficient Directed Densest Subgraph Discovery

2021 ◽  
Vol 50 (1) ◽  
pp. 33-40
Author(s):  
Chenhao Ma ◽  
Yixiang Fang ◽  
Reynold Cheng ◽  
Laks V.S. Lakshmanan ◽  
Wenjie Zhang ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
State Of The Art ◽  
Exact Algorithms ◽  
Large Datasets ◽  
Dense Subgraph ◽  
Extensive Evaluation ◽  
Wide Range ◽  
Edge Graph ◽  
Community Mining ◽  
Densest Subgraph

Given a directed graph G, the directed densest subgraph (DDS) problem refers to the finding of a subgraph from G, whose density is the highest among all the subgraphs of G. The DDS problem is fundamental to a wide range of applications, such as fraud detection, community mining, and graph compression. However, existing DDS solutions suffer from efficiency and scalability problems: on a threethousand- edge graph, it takes three days for one of the best exact algorithms to complete. In this paper, we develop an efficient and scalable DDS solution. We introduce the notion of [x, y]-core, which is a dense subgraph for G, and show that the densest subgraph can be accurately located through the [x, y]-core with theoretical guarantees. Based on the [x, y]-core, we develop both exact and approximation algorithms. We have performed an extensive evaluation of our approaches on eight real large datasets. The results show that our proposed solutions are up to six orders of magnitude faster than the state-of-the-art.

An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift

2005 ◽  
Vol 340 (5) ◽  
pp. 347-352 ◽  
Author(s):  
François Hamel ◽  
Nikolai Nadirashvili ◽  
Emmanuel Russ
Keyword(s):  
Isoperimetric Inequality ◽  
Principal Eigenvalue
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