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1 Detecting Statistically Significant Communities Zengyou He, Hao Liang, Zheng Chen, Can Zhao, Yan Liu Abstract—Community detection is a key data analysis problem across different fields. During the past decades, numerous algorithms have been proposed to address this issue. However, most work on community detection does not address the issue of statistical significance. Although some research efforts have been made towards mining statistically significant communities, deriving an analytical solution of p-value for one community under the configuration model is still a challenging mission that remains unsolved. The configuration model is a widely used random graph model in community detection, in which the degree of each node is preserved in the generated random networks. To partially fulfill this void, we present a tight upper bound on the p-value of a single community under the configuration model, which can be used for quantifying the statistical significance of each community analytically. Meanwhile, we present a local search method to detect statistically significant communities in an iterative manner. Experimental results demonstrate that our method is comparable with the competing methods on detecting statistically significant communities. Index Terms—Community Detection, Random Graphs, Configuration Model, Statistical Significance. F 1 INTRODUCTION ETWORKS are widely used for modeling the structure function that is able to always achieve the best performance N of complex systems in many fields, such as biology, in all possible scenarios. engineering, and social science. Within the networks, some However, most of these objective functions (metrics) do vertices with similar properties will form communities that not address the issue of statistical significance of commu- have more internal connections and less external links. nities. In other words, how to judge whether one com- Community detection is one of the most important tasks munity or a network partition is real or not based on in network science and data mining, which can reveal some rigorous statistical significance testing procedures. the hierarchy and organization of network structures. The Such testing-based approaches provide many advantages detection of meaningful communities is of great importance over other metrics. First of all, the statistical significance in many real applications [1]. In these diverse domains, each of communities can be quantified in terms of the p-value, detected community may provide us with some interesting which is a universally understood measure between 0 and 1 yet unknown information. in different fields. As a result, the threshold for the p-value Due to the importance of the community detection prob- is easy to specify since it corresponds to the significance lem, numerous algorithms from different perspectives have level. In contrast, numerical values generated from other been proposed during the past decades [1], [2]. Although objective functions are generally data-dependent, which is existing community detection methods have very different hard for people to interpret and determine a universal procedures to detect community structures, these methods threshold across all data sets. Furthermore, the p-value is can be roughly classified into different categories accord- typically derived under a certain random graph model in ing to the objective function and the corresponding search a mathematically sound manner, while many other metrics procedure [3]. The objective function is used for evaluating may be just defined in an ad-hoc manner. the quality of candidate communities, which serves as a In many real applications, it is critical to find network guideline for the search procedure and is critical to the suc- communities that are statistically significant. For instance, arXiv:1806.05602v2 [cs.SI] 11 Aug 2019 cess of the entire community detection algorithm. Probably the detection of protein complexes from protein-protein the modularity proposed by Newman and Girvan [4] is the interaction (PPI) networks corresponds to a special com- most popular objective function in the field of community munity detection problem [8]. However, the PPI networks detection. Based on modularity, some popular community derived from current high-throughput experimental tech- detection methods, such as Louvain [5] and FastGreedy [6], niques are very noisy, in which many spurious edges are have been proposed to search the communities in networks. present while some true edges are missing [9], [10]. More im- In addition, many other objective functions have been pro- portantly, some reported protein complexes (communities in posed to evaluate the goodness of communities as well, as PPI networks) need to be further validated by wet-lab exper- summarized in [7]. Overall, there is still no such an objective iments that are costly and time-consuming. Thus, in these types of applications, the quality of identified communities must be strictly controlled through rigourous hypothesis • Z. He is with School of Software, Dalian University of Technology, testing procedures in order to obtain really meaningful out- Dalian,China, and Key Laboratory for Ubiquitous Network and Service put (e.g. [11], [12], [13]). In fact, the issue of errors and noises Software of Liaoning Province, Dalian, China. E-mail: [email protected] in the network data has also be recognized in many other • H. Liang, Z. Chen and Y. Liu are with School of Software, Dalian fields such as social science [10]. Therefore, the detection of University of Technology, Dalian, China. statistically significant communities will be desirable as well • C. Zhao is with Institute of Information Engineering, CAS. in these fields. Furthermore, the p-value of each community 2 can be used as the metric in emerging applications such as used analytically for assessing the statistical significance network community prioritization [14]. of a single community. Then, we present a local search To address the problem of discovering statistically signif- method to detect statistically significant communities in an icant communities, some research efforts have been made iterative manner. Experimental results on both real data sets towards this direction (e.g. [15], [16], [17], [18], [19], [20], and the LFR benchmark data sets show that our method [21], [22]). In these methods, the first critical issue that is comparable with the competing methods in terms of must be solved is to define a metric to effectively assess different evaluation metrics. the statistical significance of communities. This is because The main contributions of this paper can be summarized the significance-based metric will be used as the objective as follows: function for finding statistically significant communities. Therefore, different measures have been presented for as- • We propose a new method for assessing the statis- sessing the statistical significance of communities, which can tical significance of one single community. To our be classified into two categories: the statistical significance knowledge, this is the first piece of work that de- of one network partition (e.g. [15], [16], [17], [18]) and the livers an analytical p-value (or its upper bound) for statistical significance of a single community (e.g. [19], [20], quantifying a community under the configuration [21], [22]). model. Unlike existing methods under the config- Although the metrics that evaluate a network partition uration model that quantify the statistical signifi- can provide a global view on the set of all generated cance through the membership probabilities of single communities, they generally cannot guarantee that every nodes, our method directly evaluates each candidate single community is statistically significant as well. In ad- community. dition, many different partitions of the same network may • We provide a systematic summarization and analysis lead to quite similar significance values, making it difficult on the existing methods for detecting statistically to determine which partition should be reported as the significant communities, which may serve as the final result. Furthermore, such partition-based significance foundation for the further investigation towards this metrics typically focus on the assessment of a set of non- direction. overlapping communities. Therefore, evaluating the statis- • We present a local search method to conduct the tical significance of each single community is more mean- community detection based on the proposed upper ingful in community detection. Meanwhile, these methods bound of p-value. Extensive empirical studies vali- can be also classified by the techniques for deriving the p- date the effectiveness of our method on community values: analytical methods (e.g. [15], [16], [19], [20], [21]) evaluation and detection. or sampling methods (e.g. [17], [18], [22]). The sampling The remaining parts of this paper are structured as method calculates an empirical p-value through generat- follows: Section 2 presents and discusses related work in ing a number of random graphs. Although the sampling a systematic manner. Section 3 introduces our definition method is easy to understand and implement, it is very on the statistical significance of a single community and time consuming since many random networks have to be the corresponding community detection method. Section 4 generated and analyzed. More importantly, the p-values of shows the experimental results and Section 5 concludes this the same community may be inconsistent in different runs paper. and the precision of the p-value
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