向最优社团检测:从树到一般加权网络

Q3 Mathematics
Thang N. Dinh, M. Thai
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引用次数: 23

摘要

许多网络,包括互联网、社会网络和生物关系,都被自然地划分为节点紧密连接的社区,称为社区结构。自从Newman提出使用模块化作为衡量社区结构优劣的标准以来,人们提出了许多有效的方法来最大化模块化,但没有最优性保证。在这项工作中,我们研究了精确的和理论上接近最优的算法来最大化模块化。在第一部分中,我们研究了树图问题的复杂性和近似性。令人惊讶的是,这个问题在树上仍然是np完全的。然后,我们为均匀加权树提供了一个多项式时间算法,并为任意权重的树提供了一个伪多项式时间算法和PTAS。在第二部分中,我们给出了一般图中问题的紧线性规划表达式族。这些公式利用了图的连通性结构,大大减少了约束的数量,从而大大提高了求解线性规划和整数规划的运行时间。因此,数千个顶点的网络可以在几分钟内解决,而目前用数学规划解决的最大实例只有不到250个顶点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Toward Optimal Community Detection: From Trees to General Weighted Networks
Abstract Many networks, including the Internet, social networks, and biological relations, are found to be naturally divided into communities of densely connected nodes, known as community structure. Since Newman’s suggestion of using modularity as a measure to qualify the goodness of community structures, many efficient methods to maximize modularity have been proposed but without optimality guarantees. In this work we study exact and theoretically near-optimal algorithms for maximizing modularity. In the first part, we investigate the complexity and approximability of the problem on tree graphs. Somewhat surprisingly, the problem is still NP-complete on trees. We then provide a polynomial time algorithm for uniform-weighted trees and a pseudopolynomial time algorithm and a PTAS for trees with arbitrary weights. In the second part, we present a family of compact linear programming formulations for the problem in general graphs. These formulations exploit the graph connectivity structure and reduce substantially the number of constraints, thus, they vastly improve the running time for solving linear programming and integer programming. As a result, networks of thousands of vertices can be solved in minutes, whereas the current largest instance solved with mathematical programming has fewer than 250 vertices.
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来源期刊
Internet Mathematics
Internet Mathematics Mathematics-Applied Mathematics
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