Atsushi Miyauchi

Z-Score-Based Modularity for Community Detection in Networks

Identifying community structure in networks is an issue of particular interest in network science. The modularity introduced by Newman and Girvan is the most popular quality function for community detection in networks. In this study, we identify a problem in the concept of modularity and suggest a solution to overcome this problem. Specifically, we obtain a new quality function for community detection. We refer to the function as Z-modularity because it measures the Z-score of a given partition with respect to the fraction of the number of edges within communities. Our theoretical analysis shows that Z-modularity mitigates the resolution limit of the original modularity in certain cases. Computational experiments using both artificial networks and well-known real-world networks demonstrate the validity and reliability of the proposed quality function.

What Is a Network Community?: A Novel Quality Function and Detection Algorithms

In this study, we introduce a novel quality function for a network community, which we refer to as the communitude. The communitude has a strong statistical background. Specifically, it measures the Z-score of a subset of vertices S with respect to the fraction of the number of edges within the subgraph induced by S. Due to the null model of a random graph used in the definition, our quality function focuses not only on the inside of the subgraph but also on the cut edges, unlike some quality functions for extracting dense subgraphs. To evaluate the detection ability of our quality function, we address the communitude maximization problem and its variants for realistic scenarios. For the problems, we propose a two-phase heuristic algorithm together with some modified versions. In the first phase, it repeatedly removes the vertex with the smallest degree, and then obtains the subgraph with maximum communitude over the iterations. In the second phase, the algorithm improves the obtained solution using a simple local search heuristic. This algorithm runs in linear time when the number of iterations is fixed to a constant; thus, it is applicable to massive graphs. Computational experiments using both synthetic graphs and real-world networks demonstrate the validity and reliability of the proposed quality function and algorithms.

Fractional programming formulation for the vertex coloring problem

We devise a new formulation for the vertex coloring problem. Different from other formulations, decision variables are associated with pairs of vertices. Consequently, colors will be distinguishable. Although the objective function is fractional, it can be replaced by a piece-wise linear convex function. Numerical experiments show that our formulation has significantly good performance for dense graphs.

Finding a Dense Subgraph with Sparse Cut

Community detection is one of the fundamental tasks in graph mining, which has many real-world applications in diverse domains. In this study, we propose an optimization model for finding a community that is densely connected internally but sparsely connected to the rest of the graph. The model extends the densest subgraph problem, in which we maximize the density while minimizing the average cut size. We first show that our proposed model can be solved efficiently. Then we design two polynomial-time exact algorithms based on linear programming and a maximum flow algorithm, respectively. Moreover, to deal with larger-sized graphs in practice, we present a scalable greedy algorithm that runs in almost linear time with theoretical performance guarantee of the output. In addition, as our model is closely related to a quality function called the modularity density, we show that our algorithms can also be used to find global community structure in a graph. With thorough experiments using well-known real-world graphs, we demonstrate that our algorithms are highly effective in finding a suitable community in a graph. For example, for web-Google, our algorithm finds a solution with more than 99.1% density and less than 3.1% cut size, compared with a solution obtained by a baseline algorithm for the densest subgraph problem.

Additive Approximation Algorithms for Modularity Maximization

The modularity is a quality function in community detection, which was introduced by Newman and Girvan (2004). Community detection in graphs is now often conducted through modularity maximization: given an undirected graph

A note on the complexity of the maximum edge clique partitioning problem with respect to the clique number

G=(V,E)

Maximizing Barber’s bipartite modularity is also hard

, we are asked to find a partition

Redundant constraints in the standard formulation for the clique partitioning problem

\mathcal{C}

Computing an upper bound of modularity

of

Threshold Influence Model for Allocating Advertising Budgets

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