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Recently, graph convolutional networks (GCN) have made substantial progress in semi-supervised learning (SSL). However, established GCN-based methods have two major limitations. First, GCN-based methods are restricted by the oversmoothing issue that limits their ability to extract knowledge from distant but informative nodes. Second, most available GCN-based methods exploit only the feature information of unlabeled nodes, and the pseudo-labels of unlabeled nodes, which contain important information about the data distribution, are not fully utilized. To address these issues, we propose a novel end-to-end ensemble framework, which is named mixed-order graph convolutional networks (MOGCN). MOGCN consists of two modules. (1) It constructs multiple simple GCN learners with multi-order adjacency matrices, which can directly capture the high-order connectivity among the nodes to alleviate the problem of oversmoothing. (2) To efficiently combine the results from multiple GCN learners, MOGCN employs a novel ensemble module, in which the pseudo-labels of unlabeled nodes from various GCN learners are used to augment the diversity among the learners. We conduct experiments on three public benchmark datasets to evaluate the performance of MOGCN on semi-supervised node classification tasks. The experimental results demonstrate that MOGCN consistently outperforms state-of-the-art methods.

In matrix-object data, an object (or a sample) is described by more than one feature vector (record) and all of those feature vectors are responsible for the observed classification of the object. A task for matrix-object data is to cluster it into a set of groups by analyzing and utilizing the information of feature vectors. Matrix-object data are widespread in many real applications. Previous studies typically address data sets that an object is generally represented by a feature vector, which may be violated in many real-world tasks. In this paper, we propose a k-multi-numeric-values-representatives (abbr. k-Mnv-Rep) algorithm to cluster numeric matrix-object data. In this algorithm, a new dissimilarity measure between two numeric matrix-objects is defined and a new heuristic method of updating cluster centers is given. Furthermore, we also propose a k-multi-values-representatives (abbr. k-Mv-Rep) algorithm to cluster hybrid matrix-object data. The two proposed algorithms break the limitations of the previous studies, and can be applied to address matrix-object data sets that exist widely in many real-world tasks. The benefits and effectiveness of the two algorithms are shown by some experiments on real and synthetic data sets.

Collaborative filtering is a fundamental technique in recommender systems, for which memory-based and matrix-factorization-based collaborative filtering are the two types of widely used methods. However, the performance of these two types of methods is limited in the case of sparse data, particularly with extremely sparse data. To improve the effectiveness of the methods in a sparse scenario, this paper proposes a multi-factor similarity measure that captures linear and nonlinear correlations between users resulting from extreme behavior. Subsequently, a fusion method that simultaneously considers the multi-factor similarity and the global rating information in a probability matrix factorization framework is proposed. In our framework, users’ local relations are integrated into the global ratings optimization process, so that prediction accuracy and robustness are improved in sparse data, particularly in extremely sparse data. To verify the performance of the proposed methods, we conduct experiments on four public datasets. The experimental results show that the fusion method is superior to the typical matrix factorization models used in collaborative filtering and significantly improves both the prediction results and robustness in sparse data.

Multi-attribute group decision making (MAGDM) is a common activity for multi-variable complicated decision making situations by integrating collective wisdom. Aiming at fusing granular computing with three-way decisions (3WD) to study scheme synthesis and analysis of solution space, multi-granularity three-way decisions (MG-3WD) provide multi-dimension problem solving methods for MAGDM problems. By using MG-3WD frameworks, this paper intends to study viable strategies of processing consensus and conflicting opinions provided by different decision makers in the interval-valued hesitant fuzzy (IVHF) MAGDM problem. More specifically, after reviewing the relevant literature, four kinds of IVHF multigranulation decision-theoretic rough sets (MG-DTRSs) over two universes are proposed according to different risk appetites of experts firstly. Then, we explore some fundamental propositions of newly proposed models. Afterwards, solutions to MAGDM problems in the context of mergers and acquisitions (M&A) target selections by using the presented IVHF MG-DTRSs over two universes are constructed. At last, a M&A target selection case study, along with a sensitivity analysis and a comparative analysis, is applied to illustrate the established decision making approaches.

The notion of hesitant fuzzy linguistic term sets (HFLTSs), which enables experts to utilize a few possible linguistic terms to evaluate varieties of common qualitative information, plays a significant role in handling situations in cases where these experts are hesitant in offering linguistic expressions. For addressing the challenges of information analysis and information fusion in hesitant fuzzy linguistic (HFL) group decision making, in accordance with the multi-granularity three-way decisions paradigm, the primary purpose of this study is to develop the notion of multigranulation decision-theoretic rough sets (MG-DTRSs) into the HFL background within the two-universe framework. Having revisited the relevant literature, we first propose a hybrid model named adjustable HFL MG-DTRSs over two universes by introducing an adjustable parameter for the expected risk appetite of experts, in which both optimistic and pessimistic versions of HFL MG-DTRSs over two universes are special cases of the adjustable version. Second, some of the fundamental properties of the proposed model are discussed. Then, on the basis of the presented hybrid model, a group decision making approach within the HFL context is further constructed. Finally, a practical example, a comparative analysis, and a validity test concerning person-job fit problems are explored to reveal the rationality and practicability of the constructed decision making rule.

Detecting protein complexes from available protein–protein interaction (PPI) networks is an important task, and several related algorithms have been proposed. These algorithms usually consider a single topological metric and ignore the rich topological characteristics and inherent organization information of protein complexes. However, the effective use of such information is crucial to protein complex detection. To overcome this deficiency, this study presents a heuristic clustering algorithm to identify protein complexes by fully exploiting the topological information of PPI networks. By considering the clustering coefficient and the node degree, a new nodal metric is proposed to quantify the importance of each node within a local subgraph. An iterative paradigm is used to incrementally identify seed proteins and expand each seed to a cluster. First, among the unclustered nodes, the node with the highest nodal metric is selected as a new seed. Then, the seed is expanded to a cluster by adding candidate nodes recursively from its neighbors according to both the density of the cluster and the connection between a candidate node and the cluster. The experimental results demonstrate that the proposed algorithm outperforms other competing algorithms in terms of F-measure and accuracy.

Efficiently describing and discovering communities in a network is an important research concept for graph clustering. In the paper, we present a community description model that evaluates the local importance of a node in a community and its importance concentration in all communities to reflect its representability to the community. Based on the description model, we propose a new evaluation criterion and an iterative search algorithm for community detection (ISCD). The new algorithm can quickly discover communities in a large-scale network, due to the average linear-time complexity with the number of edges. Furthermore, we provide an initial method of input parameters including the number of communities and the initial partition before algorithm implementation, which can enhance the local-search quality of the iterative algorithm. The proposed algorithm with the initial method is called ISCD+. Finally, we compare the effectiveness and efficiency of the ISCD+ algorithm with six representative algorithms on several real network data sets. The experimental results illustrate that the proposed algorithm is suitable to address large-scale networks.

Human granulation intelligence means that people can observe and analyze the same problem from various granulation points of view, which generally acknowledge an essential feature of human intelligence. Each granulation view can generate a granular structure through dividing a cognitive target into some meaningful information granules. This means that a large number of granular structures can be generated from the cognitive target. However, people can group these granular structures and select some representative ones for problem solving. This leads to an interesting research topic: how to efficiently and effectively group a family of granular structures. To address this issue, we first introduce a granular structure distance to measure the difference between two granular structures within a unified knowledge representation. Then, we propose a framework for grouping granular structures, called GGS algorithm, which is used to efficiently partition them. Moreover, two indices denoted as DIS and APD are also designed for evaluating the performance of a grouping result of granular structures. Finally, experiments carried out for nine data sets show that the GGS algorithm comes as a sound solution from perspectives of its convergence, effectiveness and scalability. In this way we have proposed and experimented with the general framework for discovering the structure inherent in granular structures, which can be afterwards used to simulate intelligent behavior of human’s abilities of granular structure selection.

Clustering analysis is a fundamental technique in machine learning, which is also widely used in information granulation. Multiple clustering systems granulate a data set into multiple granular structures. Therefore, clustering ensemble can serve as an important branch of multigranulation information fusion. Many approaches have been proposed to solve the clustering ensemble problem. This paper focuses on the direct approaches which involve two steps: finding cluster correspondence and utilizing a fusion strategy to produce a final result. The existing direct approaches mainly discuss the process of finding cluster correspondence, while the fusing process is simply done by voting. In this paper, we mainly focus on the fusing process and propose a Dempster-Shafer evidence theory-based clustering ensemble algorithm. The advantage of the algorithm is that the information of an object's surrounding cluster structure is taken into consideration by using its neighbors to describe it. First, we find neighbors of each object and generate its label probability outputs in every base partition. Second, these label probability outputs are integrated based on DS theory. Theoretically, our method is superior to other voting methods. Besides, several experiments show that the proposed algorithm is statistically better than seven other clustering ensemble methods.

Rough set theory is one of important tools of soft computing, and rough approximations are the essential elements in rough set models. However, the existing fuzzy rough set model for set-valued data, which is directly constructed based on a kind of similarity relation, fail to explicitly define fuzzy rough approximations. To solve this issue, in this paper, we propose two types of fuzzy rough approximations, and define two corresponding relative positive region reducts. Furthermore, two discernibility matrices and two discernibility functions are introduced to acquire these new proposed reducts, and the relationships among the new reducts and the existing reducts are also be provided. Theoretical analyses demonstrate that the new types of reducts have less redundancy and are more diverse (no lower number of reducts) than those obtained by means of the existing matrices, and experimental results illustrate the new reducts found by our methods outperform those obtained by existing method.

Multigranulation rough set (MGRS) theory provides two kinds of qualitative combination rules that are generated by optimistic and pessimistic multigranulation fusion functions. They are used to aggregate multiple granular structures from a set theoretic standpoint. However, the two combination rules seem to lack robustness because one is too relaxed and the other too restrictive to solve some practical problems. Dempster's combination rule in the evidence theory has been employed to aggregate information coming from multiple sources. However, it fails to deal with conflict evidence. To overcome these limitations, we focus on the combination of granular structures with both reliability and conflict from multiple sources, which has been a challenging task in the field of granular computing. We first address the connection between multigranulation rough set theory and the evidence theory. Then, a two-grade fusion approach involved in the evidence theory and multigranulation rough set theory is proposed, which is based on a well-defined distance function among granulation structures. Finally, an illustrative example is given to show the effectiveness of the proposed fusion method. The results of this study will be useful for pooling the uncertain data from different sources and significant for establishing a new direction of granular computing.

This paper proposes a new method to trend analysis of categorical data streams. A data stream is partitioned into a sequence of time windows and the records in each window are assumed to carry a number of concepts represented as clusters. A data labeling algorithm is proposed to identify the concepts or clusters of a window from the concepts of the preceding window. The expression of a concept is presented and the distance between two concepts in two consecutive windows is defined to analyze the change of concepts in consecutive windows. Finally, a trend analysis algorithm is proposed to compute the trend of concept change in a data stream over the sequence of consecutive time windows. The methods for measuring the significance of an attribute that causes the concept change and the outlier degrees of objects are presented to reveal the causes of concept change. Experiments on real data sets are presented to demonstrate the benefits of the trend analysis method.

Multigranulation rough sets (MGRS) is one of desirable directions in rough set theory, in which lower/upper approximations are approximated by granular structures induced by multiple binary relations. It provides a new perspective for decision making analysis based on rough set theory. In decision making analysis, people often adopt the decision strategy “Seeking common ground while eliminating differences” (SCED). This strategy implies that one reserves common decisions while deleting inconsistent decisions. From this point of view, the objective of this study is to develop a new multigranulation rough set based decision model based on SCED strategy, called pessimistic multigranulation rough sets. We study this model from three aspects, which are lower/upper approximation and their properties, decision rules and attribute reduction, in this paper.

The fast global k-means (FGKM) clustering algorithm is one of the most effective approaches for resolving the local convergence of the k-means clustering algorithm. Numerical experiments show that it can effectively determine a global or near global minimizer of the cost function. However, the FGKM algorithm needs a large amount of computational time or storage space when handling large data sets. To overcome this deficiency, a more efficient FGKM algorithm, namely FGKM+A, is developed in this paper. In the development, we first apply local geometrical information to describe approximately the set of objects represented by a candidate cluster center. On the basis of the approximate description, we then propose an acceleration mechanism for the production of new cluster centers. As a result of the acceleration, the FGKM+A algorithm not only yields the same clustering results as that of the FGKM algorithm but also requires less computational time and fewer distance calculations than the FGKM algorithm and its existing modifications. The efficiency of the FGKM+A algorithm is further confirmed by experimental studies on several UCI data sets.

The classical multigranulation rough set (MGRS) theory offers a formal theoretical framework for solving the complex problem under multigranulation environment. However, it is noticeable that MGRS theory cannot be applied in multi-source information systems with a covering environment in the real world. To address this issue, we firstly present in this paper three types of covering based multigranulation rough sets, in which set approximations are defined by different covering approximation operators. Then, by using two different approximation strategies, i.e., seeking common reserving difference and seeking common rejecting difference, two kinds of covering based multigranulation rough set are presented, namely, a covering based optimistic multigranulation rough set and a covering based pessimistic multigranulation rough sets. Finally, we develop some properties and several uncertainty measures of the covering based multigranulation rough sets. These results will enrich the MGRS theory and enlarge its application scope.

The roughness of a rough set arises from the existence of its boundary region. In such a boundary region, each object has a non-zero rough membership degree. When an object's rough membership degree is regarded as its fuzzy membership degree, a rough set can induce a fuzzy set. This relationship motivates us to assert that there may exist some inherent relations between the roughness of a rough set and the fuzziness of the fuzzy set induced from the rough set. This assertion leads us to the question: Can the existing fuzzy entropies be used to evaluate the roughness of a rough set? To answer this question, we first analyze how the boundary region varies when the partition of the universe becomes coarser, and then exploit this analysis in the introduction of a more appropriate definition on the roughness of a rough set. To determine whether a fuzzy entropy can be used to evaluate the roughness of a rough set or not, we develop three methods for estimating the ability of a fuzzy entropy to measure the roughness. The experiments show that these methods are very effective and can be applied to select a fuzzy entropy as a measure of the roughness of a rough set.

To discover knowledge from hybrid data using rough sets, researchers have developed several fuzzy rough set models and a neighborhood rough set model. These models have been applied to many hybrid data processing applications for a particular purpose, thus neglecting the issue of selecting an appropriate model. To address this issue, this paper mainly concerns the relationships among these rough set models. Investigating fuzzy and neighborhood hybrid granules reveals an important relationship between these two granules. Analyzing the relationships among rough approximations of these models shows that Hu's fuzzy rough approximations are special cases of neighborhood and Wang's fuzzy rough approximations, respectively. Furthermore, one-to-one correspondence relationships exist between Wang's fuzzy and neighborhood rough approximations. This study also finds that Wang's fuzzy and neighborhood rough approximations are cut sets of Dubois' fuzzy rough approximations and Radzikowska and Kerre's fuzzy rough approximations, respectively.

The original rough set model was developed by Pawlak, which is mainly concerned with the approximation of sets described by a single binary relation on the universe. In the view of granular computing, the classical rough set theory is established through a single granulation. This paper extends Pawlak's rough set model to a multi-granulation rough set model (MGRS), where the set approximations are defined by using multi equivalence relations on the universe. A number of important properties of MGRS are obtained. It is shown that some of the properties of Pawlak's rough set theory are special instances of those of MGRS. Moreover, several important measures, such as accuracy [email protected], quality of [email protected] and precision of [email protected], are presented, which are re-interpreted in terms of a classic measure based on sets, the Marczewski-Steinhaus metric and the inclusion degree measure. A concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in MGRS as well. Finally, we discuss how to extract decision rules using MGRS. Unlike the decision rules (''AND'' rules) from Pawlak's rough set model, the form of decision rules in MGRS is ''OR''. Several pivotal algorithms are also designed, which are helpful for applying this theory to practical issues. The multi-granulation rough set model provides an effective approach for problem solving in the context of multi granulations.

Set-valued ordered information systems can be classified into two categories: disjunctive and conjunctive systems. Through introducing two new dominance relations to set-valued information systems, we first introduce the conjunctive/disjunctive set-valued ordered information systems, and develop an approach to queuing problems for objects in presence of multiple attributes and criteria. Then, we present a dominance-based rough set approach for these two types of set-valued ordered information systems, which is mainly based on substitution of the indiscernibility relation by a dominance relation. Through the lower/upper approximation of a decision, some certain/possible decision rules from a so-called set-valued ordered decision table can be extracted. Finally, we present attribute reduction (also called criteria reduction in ordered information systems) approaches to these two types of ordered information systems and ordered decision tables, which can be used to simplify a set-valued ordered information system and find decision rules directly from a set-valued ordered decision table. These criteria reduction approaches can eliminate those criteria that are not essential from the viewpoint of the ordering of objects or decision rules.

In rough set theory, accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification. Although these measures are effective, they have some limitations when the lower/upper approximation of a set under one knowledge is equal to that under another knowledge. To overcome these limitations, we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table. An axiomatic definition of knowledge granulation for an information system is given, under which these three measures are modified. Theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table, respectively, and have a much simpler and more comprehensive form than the existing ones.

As two classical measures, approximation accuracy and consistency degree can be employed to evaluate the decision performance of a decision table. However, these two measures cannot give elaborate depictions of the certainty and consistency of a decision table when their values are equal to zero. To overcome this shortcoming, we first classify decision tables in rough set theory into three types according to their consistency and introduce three new measures for evaluating the decision performance of a decision-rule set extracted from a decision table. We then analyze how each of these three measures depends on the condition granulation and decision granulation of each of the three types of decision tables. Experimental analyses on three practical data sets show that the three new measures appear to be well suited for evaluating the decision performance of a decision-rule set and are much better than the two classical measures.