Common Graph Research Articles

The Monophonic-triangular Number of a Graph

A path \(x_1, x_2, \dots, x_n\) in a connected graph \( G \) that has no edge \( x_i x_j \) \((j \geq i+3)\) is called a monophonic-triangular path or mt-path. A non-empty subset \( M \) of \( V(G) \) is a monophonic-triangular set or mt-set of \( G \) if every member in \( V(G) \) exists in a mt-path joining some pair of members in \( M \). The monophonic-triangular number or mt-number is the lowest cardinality of an mt-set of \( G \) and it is symbolized by \( mt(G) \). The general properties satisfied by mt-sets are discussed. Also, we establish \( mt \)-number boundaries and discover similar results for a few common graphs. Graphs \( G \) of order \( p \) with \( mt(G) = p \), \( p – 1 \), or \( p – 2 \) are characterized.

Bathtub curve and Allan-Plaȋt chart: two problematic graphs in reliability engineering

Abstract In the reliability literature, there are completely different representations for two of the most common graphs: the graph that expresses the failure rate over the life of a product (bathtub curve) and the one that graphically identifies the Weibull triparametric model (linear regression of the Weibull curve using the Allan-Plaît graph). Even if the identification of the Weibull model is no longer done graphically, but with the help of dedicated calculation programs, the mathematical bases of the linearization of the F(t) fault distribution function by logarithm remain equally important. The present paper aims to establish which of the graphic variants are the correct ones and to identify the causes that led to the incorrect graphic representations. Based on an extensive theoretical documentation (analysis of numerous works in the field of reliability, published over the last decades, in the country and abroad) it is concluded that in foreign literature these graphs are always correct and that they were taken correctly in Romanian works from the 70’s to the 80’s of the last century, but starting with the 90’s there are fundamentally different representations, especially in the field of reliability engineering. Finally, the causes that led to these non-compliant graphical representations are identified. As these misrepresentations seem to persist, this paper is intended to be a signal of this issue.

Self-Guided Partial Graph Propagation for Incomplete Multiview Clustering.

In this work, we study a more realistic challenging scenario in multiview clustering (MVC), referred to as incomplete MVC (IMVC) where some instances in certain views are missing. The key to IMVC is how to adequately exploit complementary and consistency information under the incompleteness of data. However, most existing methods address the incompleteness problem at the instance level and they require sufficient information to perform data recovery. In this work, we develop a new approach to facilitate IMVC based on the graph propagation perspective. Specifically, a partial graph is used to describe the similarity of samples for incomplete views, such that the issue of missing instances can be translated into the missing entries of the partial graph. In this way, a common graph can be adaptively learned to self-guide the propagation process by exploiting the consistency information, and the propagated graph of each view is in turn used to refine the common self-guided graph in an iterative manner. Thus, the associated missing entries can be inferred through graph propagation by exploiting the consistency information across all views. On the other hand, existing approaches focus on the consistency structure only, and the complementary information has not been sufficiently exploited due to the data incompleteness issue. By contrast, under the proposed graph propagation framework, an exclusive regularization term can be naturally adopted to exploit the complementary information in our method. Extensive experiments demonstrate the effectiveness of the proposed method in comparison with state-of-the-art methods. The source code of our method is available at the https://github.com/CLiu272/TNNLS-PGP.

High-Accuracy Analytical Model for Heterogeneous Cloud Systems with Limited Availability of Physical Machine Resources Based on Markov Chain

The article presents the results of a study on modeling cloud systems. In this research, the authors developed both analytical and simulation models. System analysis was conducted at the level of virtual machine support, corresponding to Infrastructure as a Service (IaaS). The models assumed that virtual machines of different sizes are offered as part of IaaS, reflecting the heterogeneous nature of modern systems. Additionally, it was assumed that due to limitations in access to physical server resources, only a portion of these resources could be used to create virtual machines. The model is based on Markov chain analysis for state-dependent systems. The system was divided into an external structure, represented by a collection of physical machines, and an internal structure, represented by a single physical machine. The authors developed a novel approach to determine the equivalent traffic, approximating the real traffic appearing at the input of a single physical machine under the assumptions of request distribution. As a result, it was possible to determine the actual request loss probability in the entire system. The results obtained from both models (simulation and analytical) were summarized in common graphs. The studies were related to the actual parameters of commercially offered physical and virtual machines. The conducted research confirmed the high accuracy of the analytical model and its independence from the number of different instances of virtual machines and the number of physical machines. Thus, the model can be used to dimension cloud systems.

Integration of UML Class Diagrams Based on Semantics and Structure

As a high-level reuse, software design reuse has received more and more attention because of its important impact on the subsequent development stages. Usually, the design models are typically represented as some graphs or diagrams, in which Unified Modeling Language (UML) class diagram is so widely used in software design that it has become the de facto standard. There has been some research on the reuse of UML class diagrams so far, mainly focusing on matching and retrieval. However, it is worth noting that there are many similar class diagrams modeling the same object or some related class diagrams modeling different aspects of the same object in the reuse repository. As a matter of fact, the primary step to achieve a high-quality reuse is to have high-quality software artifacts, so the well-designed UML class diagrams become a necessary resource for software design reuse. Therefore, it is necessary to integrate these class diagrams so that they have stronger modeling ability, and eliminating redundancy is another benefit of the integration. Up to now, there has been little discussion about the integration of class diagrams, so we propose an integration approach based on semantics and structure in this paper. The equivalent elements that can identify the semantically merged parts are defined, and the possible conflict items are listed from both semantic and structural aspects. The integration procedure composed of three stages is proposed, in which an approach combining semantic common class diagrams (SCCDs) and structural common graph sequence (SCGS) is combined to determine the merged parts, the integration issue of heterogeneous class diagrams is considered from the proposed abstract models, and the conflict resolution for each conflict item is described by examples. The experimental results show the effectiveness of our proposed integration approach.

Single-Image 3-D Reconstruction: Rethinking Point Cloud Deformation.

Single-image 3-D reconstruction has long been a challenging problem. Recent deep learning approaches have been introduced to this 3-D area, but the ability to generate point clouds still remains limited due to inefficient and expensive 3-D representations, the dependency between the output and the number of model parameters, or the lack of a suitable computing operation. In this article, we present a novel deep-learning-based method to reconstruct a point cloud of an object from a single still image. The proposed method can be decomposed into two steps: feature fusion and deformation. The first step extracts both global and point-specific shape features from a 2-D object image, and then injects them into a randomly generated point cloud. In the second step, which is deformation, we introduce a new layer termed as GraphX that considers the interrelationship between points like common graph convolutions but operates on unordered sets. The framework can be applicable to realistic image data with background as we optionally learn a mask branch to segment objects from input images. To complement the quality of point clouds, we further propose an objective function to control the point uniformity. In addition, we introduce different variants of GraphX that cover from best performance to best memory budget. Moreover, the proposed model can generate an arbitrary-sized point cloud, which is the first deep method to do so. Extensive experiments demonstrate that we outperform the existing models and set a new height for different performance metrics in single-image 3-D reconstruction.

Consensus Affinity Graph Learning via Structure Graph Fusion and Block Diagonal Representation for Multiview Clustering

Learning a robust affinity graph is fundamental to graph-based clustering methods. However, some existing affinity graph learning methods have encountered the following problems. First, the constructed affinity graphs cannot capture the intrinsic structure of data well. Second, when fusing all view-specific affinity graphs, most of them obtain a fusion graph by simply taking the average of multiple views, or directly learning a common graph from multiple views, without considering the discriminative property among diverse views. Third, the fusion graph does not maintain an explicit cluster structure. To alleviate these problems, the adaptive neighbor graph learning approach and the data self-expression approach are first integrated into a structure graph fusion framework to obtain a view-specific structure affinity graph to capture the local and global structures of data. Then, all the structural affinity graphs are weighted dynamically into a consensus affinity graph, which not only effectively incorporates the complementary affinity structure of important views but also has the capability of preserving the consensus affinity structure that is shared by all views. Finally, a k–block diagonal regularizer is introduced for the consensus affinity graph to encourage it to have an explicit cluster structure. An efficient optimization algorithm is developed to tackle the resultant optimization problem. Extensive experiments on benchmark datasets validate the superiority of the proposed method.

GLDL: Graph Label Distribution Learning

Label Distribution Learning (LDL), as a more general learning setting than generic single-label and multi-label learning, has been commonly used in computer vision and many other applications. To date, existing LDL approaches are designed and applied to data without considering the interdependence between instances. In this paper, we propose a Graph Label Distribution Learning (GLDL) framework, which explicitly models three types of relationships: instance-instance, label-label, and instance-label, to learn the label distribution for networked data. A label-label network is learned to capture label-to-label correlation, through which GLDL can accurately learn label distributions for nodes. Dual graph convolution network (GCN) Co-training with heterogeneous message passing ensures two GCNs, one focusing on instance-instance relationship and the other one targeting label-label correlation, are jointly trained such that instance-instance relationship can help induce label-label correlation and vice versa. Our theoretical study derives the error bound of GLDL. For verification, four benchmark datasets with label distributions for nodes are created using common graph benchmarks. The experiments show that considering dependency helps learn better label distributions for networked data, compared to state-of-the-art LDL baseline. In addition, GLDL not only outperforms simple GCN and graph attention networks (GAT) using distribution loss but is also superior to its variant considering label-label relationship as a static network. GLDL and its benchmarks are the first research endeavors to address LDL for graphs. Code and benchmark data are released for public access.

DCLP: Neural Architecture Predictor with Curriculum Contrastive Learning

Neural predictors have shown great potential in the evaluation process of neural architecture search (NAS). However, current predictor-based approaches overlook the fact that training a predictor necessitates a considerable number of trained neural networks as the labeled training set, which is costly to obtain. Therefore, the critical issue in utilizing predictors for NAS is to train a high-performance predictor using as few trained neural networks as possible. Although some methods attempt to address this problem through unsupervised learning, they often result in inaccurate predictions. We argue that the unsupervised tasks intended for the common graph data are too challenging for neural networks, causing unsupervised training to be susceptible to performance crashes in NAS. To address this issue, we propose a CurricuLum-guided Contrastive Learning framework for neural Predictor (DCLP). Our method simplifies the contrastive task by designing a novel curriculum to enhance the stability of unlabeled training data distribution during contrastive training. Specifically, we propose a scheduler that ranks the training data according to the contrastive difficulty of each data and then inputs them to the contrastive learner in order. This approach concentrates the training data distribution and makes contrastive training more efficient. By using our method, the contrastive learner incrementally learns feature representations via unsupervised data on a smooth learning curve, avoiding performance crashes that may occur with excessively variable training data distributions. We experimentally demonstrate that DCLP has high accuracy and efficiency compared with existing predictors, and shows promising potential to discover superior architectures in various search spaces when combined with search strategies. Our code is available at: https://github.com/Zhengsh123/DCLP.

GraphScale: Scalable Processing on FPGAs for HBM and Large Graphs

Recent advances in graph processing on FPGAs promise to alleviate performance bottlenecks with irregular memory access patterns. Such bottlenecks challenge performance for a growing number of important application areas like machine learning and data analytics. While FPGAs denote a promising solution through flexible memory hierarchies and massive parallelism, we argue that current graph processing accelerators either use the off-chip memory bandwidth inefficiently or do not scale well across memory channels. In this work, we propose GraphScale, a scalable graph processing framework for FPGAs. GraphScale combines multi-channel memory with asynchronous graph processing (i.e., for fast convergence on results) and a compressed graph representation (i.e., for efficient usage of memory bandwidth and reduced memory footprint). GraphScale solves common graph problems like breadth-first search, PageRank, and weakly connected components through modular user-defined functions, a novel two-dimensional partitioning scheme, and a high-performance two-level crossbar design. Additionally, we extend GraphScale to scale to modern high-bandwidth memory (HBM) and reduce partitioning overhead of large graphs with binary packing.

Incomplete Multiview Nonnegative Representation Learning With Graph Completion and Adaptive Neighbors.

Despite incomplete multiview clustering (IMC) being widely studied in the past decade, it is still difficult to model the correlation among multiple views due to the absence of partial views. Most existing works for IMC only mine the correlation among multiple views from available views and ignore the importance of missing views. To address this issue, we propose a novel Incomplete Multiview Nonnegative representation learning model with Graph completion and Adaptive neighbors (IMNGA), which performs common graph learning, missing graph completion, and consensus nonnegative representation learning simultaneously. In IMNGA, the common graph on all views and the incomplete graph of each view are used to reconstruct the completed graph of the corresponding view, where the common graph satisfies the neighbor constraints of incomplete multiview data and consensus representation. IMNGA gets consensus representation by factorizing completed and incomplete graphs, where consensus representation satisfies the common graph constraint. IMNGA shows its effectiveness by outperforming other state-of-the-art methods.

Characterization of Fractional Mixed Domination Number of Paths and Cycles

Let G′ be a simple, connected, and undirected (UD) graph with the vertex set M(G′) and an edge set N(G′). In this article, we define a function f:M∪N⟶0,1 as a fractional mixed dominating function (FMXDF) if it satisfies fRmx=∑yϵRmxfy≥1 for all x∈MG′∪NG′, where Rmx indicates the closed mixed neighbourhood of x, that is the set of all y∈MG′∪NG′ such that y is adjacent to x and y is incident with x and also x itself. Here, pf=∑x∈M∪Nfx is the poundage (or weight) of f. The fractional mixed domination number (FMXDN) is denoted by γfm∗G′ and is designated as the lowest poundage among all FMXDFs of G′. We compute the FMXDN of some common graphs such as paths, cycles, and star graphs, the middle graph of paths and cycles, and shadow graphs. Furthermore, we compute upper bounds for the sum of the two fractional dominating parameters, resulting in the inequality γf1′Τ+γfm∗Τ≤r+p−radΤ−α, where γf1′ and γfm∗ are the fractional edge domination number and FMXDN, respectively. Finally, we compare γfm∗ to other resolvability-related parameters such as metric and fault-tolerant metric dimensions on some families of graphs.

An Efficient Graph Learning System for Emotion Recognition Inspired by the Cognitive Prior Graph of EEG Brain Network.

Benefiting from the high-temporal resolution of electroencephalogram (EEG), EEG-based emotion recognition has become one of the hotspots of affective computing. For EEG-based emotion recognition systems, it is crucial to utilize state-of-the-art learning strategies to automatically learn emotion-related brain cognitive patterns from emotional EEG signals, and the learned stable cognitive patterns effectively ensure the robustness of the emotion recognition system. In this work, to realize the efficient decoding of emotional EEG, we propose a graph learning system Graph Convolutional Network framework with Brain network initial inspiration and Fused attention mechanism (BF-GCN) inspired by the brain cognitive mechanism to automatically learn graph patterns from emotional EEG and improve the performance of EEG emotion recognition. In the proposed BF-GCN, three graph branches, i.e., cognition-inspired functional graph branch, data-driven graph branch, and fused common graph branch, are first elaborately designed to automatically learn emotional cognitive graph patterns from emotional EEG signals. And then, the attention mechanism is adopted to further capture the brain activation graph patterns that are related to emotion cognition to achieve an efficient representation of emotional EEG signals. Essentially, the proposed BF-CGN model is a cognition-inspired graph learning neural network model, which utilizes the spectral graph filtering theory in the automatic learning and extracting of emotional EEG graph patterns. To evaluate the performance of the BF-GCN graph learning system, we conducted subject-dependent and subject-independent experiments on two public datasets, i.e., SEED and SEED-IV. The proposed BF-GCN graph learning system has achieved 97.44% (SEED) and 89.55% (SEED-IV) in subject-dependent experiments, and the results in subject-independent experiments have achieved 92.72% (SEED) and 82.03% (SEED-IV), respectively. The state-of-the-art performance indicates that the proposed BF-GCN graph learning system has a robust performance in EEG-based emotion recognition, which provides a promising direction for affective computing.

Edge geo chromatic number of a graph

The Edge Geodetic Number is also known as the smallest size among all sets of edges that make up Edge Geodetic sets and is represented by the symbol g1(G). Assuming G is a finite connected graph, we can define a set S contained in V(G) as an Edge Geodetic set if, for any pair of vertices u and v in set S, every edge in G lies on the shortest path between u and v. A set C contained in E(G) is referred to as a chromatic index set when it contains all of the proper k-edge colors of G. The chromatic index is determined by the set's lowest cardinality among its edges. The "Chromatic Index Number" is a term used to refer to the collection of these minimum cardinality chromatic index values and is represented by the symbol chi'(G). By combining an Edge Geodetic set with a Chromatic Index set, the Edge Geo Chromatic set of G is created, providing an innovative concept. The Edge Geo Chromatic Number, abbreviated as X, is the least cardinality among all Edge Geo Chromatic sets of G. In essence, an Edge Geodetic set and a Chromatic Index set combine to form an Edge Geo Chromatic graph. We talk about an edge geo chromatic linked graph with n orders. For a number of common graphs, we have computed the Edge Geo Chromatic Number and established its bounds. Furthermore, we have shown that when G represents a graph of diameter d, then chi'_gc(G) lessthan or equal to n - d +2.

Enhancing Privacy in Graph Algorithms: Data-Oblivious Approaches to DFS and Dijkstra's Algorithm

Data obliviousness is characterized by a consistent sequence of operations irrespective of input data and data-independent memory accesses, making it a suitable solution for users utilizing outsourced storage data services who aim to conceal their data access patterns. In ACM SIGSAC 2013, data-oblivious algorithms were introduced for breadth-first search, single-source single-destination (SSSD) algorithms, maximum flow, and minimum spanning tree. In this study, we present novel data-oblivious algorithms designed for depth-first search and single-source shortest path (Dijkstra’s algorithm). Our proposed data-oblivious algorithms demonstrate efficiency comparable to non-data-oblivious counterparts, particularly for graphs containing fewer than 1000 nodes. This research contributes to advancing data privacy in outsourced storage services by providing effective data-oblivious solutions for common graph algorithms.

Stability, Vertex Stability, and Unfrozenness for Special Graph Classes

Frei et al. (J. Comput. Syst. Sci. 123, 103–121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for Θ2P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\Theta _{2}^{\ extrm{P}}}$$\\end{document}, the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\alpha }$$\\end{document} (the independence number), β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\beta }$$\\end{document} (the vertex cover number), ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\omega }$$\\end{document} (the clique number), and χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\chi }$$\\end{document} (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. In order to improve these general time bounds even further, we then focus on trees, forests, bipartite graphs, and co-graphs.

A Multi-Agent System in Education Facility Design

This paper deals with a multi-agent system which supports the designer in solving complex design tasks. The behaviour of design agents is modelled by sets of grammar rules. Each agent uses a graph grammar or a shape grammar and a database of facts concerning the subtask it is responsible for. The course of the design process is determined by the interaction between specialised agents. Space layouts of designs are represented by attributed graphs encoding both topological structures and semantic properties of solutions. The agents work in parallel on the common graph, independently generating layouts of different design components while specified node labels evoke agents using shape grammars. The agents’ cooperation allows them to combine a form-oriented approach with a functional-structural one in the design process, where the agents generate the general 3D form of the object based on design requirements together with the space layout based on the functional aspects of the solution. Based on the given design criteria, the agents search for admissible solutions within the design space that constitutes their operating environment. The proposed approach is illustrated by the example of designing kindergarten facilities.

Learning Bi-Typed Multi-Relational Heterogeneous Graph Via Dual Hierarchical Attention Networks

Bi-typed multi-relational heterogeneous graph (BMHG) is one of the most common graphs in practice, for example, academic networks, e-commerce user behavior graph and enterprise knowledge graph. It is a critical and challenge problem on how to learn the numerical representation for each node to characterize subtle structures. However, most previous studies treat all node relations in BMHG as the same class of relation without distinguishing the different characteristics between the intra-type relations and inter-type relations of the bi-typed nodes, causing the loss of significant structure information. To address this issue, we propose a novel <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</b> ual <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</b> ierarchical <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> ttention <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</b> etworks (DHAN) based on the bi-typed multi-relational heterogeneous graphs to learn comprehensive node representations with the intra-type and inter-type attention-based encoder under a hierarchical mechanism. Specifically, the former encoder aggregates information from the same type of nodes, while the latter aggregates node representations from its different types of neighbors. Moreover, to sufficiently model node multi-relational information in BMHG, we adopt a newly proposed hierarchical mechanism. By doing so, the proposed dual hierarchical attention operations enable our model to fully capture the complex structures of the bi-typed multi-relational heterogeneous graphs. Experimental results on various tasks against the state-of-the-arts sufficiently confirm the capability of DHAN in learning node representations on the BMHGs.

GraphTune: An Efficient Dependency-Aware Substrate to Alleviate Irregularity in Concurrent Graph Processing

With the increasing need for graph analysis, massive Concurrent iterative Graph Processing (CGP) jobs are usually performed on the common large-scale real-world graph. Although several solutions have been proposed, these CGP jobs are not coordinated with the consideration of the inherent dependencies in graph data driven by graph topology. As a result, they suffer from redundant and fragmented accesses of the same underlying graph dispersed over distributed platform, because the same graph is typically irregularly traversed by these jobs along different paths at the same time. In this work, we develop GraphTune , which can be integrated into existing distributed graph processing systems, such as D-Galois, Gemini, PowerGraph, and Chaos, to efficiently perform CGP jobs and enhance system throughput. The key component of GraphTune is a dependency-aware synchronous execution engine in conjunction with several optimization strategies based on the constructed cross-iteration dependency graph of chunks. Specifically, GraphTune transparently regularizes the processing behavior of the CGP jobs in a novel synchronous way and assigns the chunks of graph data to be handled by them based on the topological order of the dependency graph so as to maximize the performance. In this way, it can transform the irregular accesses of the chunks into more regular ones so that as many CGP jobs as possible can fully share the data accesses to the common graph. Meanwhile, it also efficiently synchronizes the communications launched by different CGP jobs based on the dependency graph to minimize the communication cost. We integrate it into four cutting-edge distributed graph processing systems and a popular out-of-core graph processing system to demonstrate the efficiency of GraphTune. Experimental results show that GraphTune improves the throughput of CGP jobs by 3.1∼6.2, 3.8∼8.5, 3.5∼10.8, 4.3∼12.4, and 3.8∼6.9 times over D-Galois, Gemini, PowerGraph, Chaos, and GraphChi, respectively.

A negative survey based privacy preservation method for topology of social networks

With the development of social platforms, the social network has aroused wide attention. Since social networks contain a lot of personal sensitive information, many privacy preservation methods have been designed for social networks to allay concerns about privacy disclosure of people. However, most of the existing methods disturb the social networks too much to ensure the utility of social networks. To this end, we propose a negative survey based privacy preservation method, called NetNS, to preserve the topology privacy of social networks, where a dedicated negative survey model is developed to disturb edges in social networks in order to preserve the topology privacy of them. The theoretical analysis indicates that the developed NetNS is efficient, and can resist two common graph structure attacks including friendship attack and subgraph attack, while empirical studies conducted on three real-world social networks show that compared to six existing privacy preservation algorithms tailored for the topology of social networks, the developed NetNS can provide disturbed social networks with better utility while achieving same privacy preservation level.