Many natural and man-made systems are structured in the form of graphs. Typical examples include communication networks,biological systems, social networks and transportation infrastructures. The increasing popularity of these graph data has initiated a fertile research area in information extraction and data mining that benefits various application fields such as sociology,marketing, biomedicine and counterterrorism. However,as more real world graph data have been made publicly available,the privacy preservation, which already has a rich body of research on transactional data [1-4], becomes an important concern associated with graph analysis. In this paper, we focus on protecting sensitive identities of individuals in a graph from background knowledge attacks. That is, if certain local knowledge can uniquely identify some vertices in a graph and is known by an adversary, the privacy of these entities can be breached, even if the data has been perturbed before publication.
Some recent studies [5, 6] show that the simple technique of anonymizing graphs by removing the identities/labels of vertices before publishing the actual graph does not always guarantee privacy. For example, the work in study of Zhou and Pei identifies neighborhood attacks that an adversary has knowledge about neighbors of a target vertex and the relationship among the neighbors. Another study  discusses a specific knowledge attack, assuming an adversary has prior knowledge of the degree of a target vertex. They argue that, if the degree of a vertex is unique in the degree sequence of all vertices, this entity can be easily re-identified, even without its original label.
A common character of the studies mentioned above is that they all concentrate on simple graphs (i.e., undirected, unweighted
and loopless graphs) and avoid weighted networks that are often perceived as being harder to analyze than their unweighted counterparts are. However, as has long been appreciated, many networks are intrinsically weighted, their edges having differing strengths. There may be stronger or weaker social ties between individuals in a social network. There may be longer or shorter distances between stations in a transportation network. There may be more or less bandwidth or data flow between routers/clients in a communication network. There may be more or less flux along particular reaction pathways in a metabolic network.There may be more or less energy or carbon flow between predator-prey pairs in a food web.
In this paper, we discuss weighted-related properties that may lead to potential background knowledge attacks in a graph.Two important properties are introduced here: 1) volume, which is sum of weights for a node; and 2) histogram, which represents the neighborhood weight distribution of a node. From both theoretical and practical views, a weighted graph provides more unique structural information than a simple graph that increases the risk of identity disclosure. For example, a realworld graph, called NetSci, which has 1; 589 nodes and will be introduced in the experimental section, consists of 1% nodes with unique degree values but more than 6% with unique volume values. We will show the details in the experimental part.Furthermore, many preservation algorithms for simple graphs may not be extended or adapted to their weighted version. For instance, Fig. 1 states two weighted graphs, which are 3-degree anonymous and 4-degree anonymous respectively, according to the definition in . However the weight values provide some extra unique information for a vertex, e.g., the volume of the vertex a is unique in Fig. 1a. In the real world, such information may be known as background knowledge and used by adversaries for re-identification.
While the privacy preservation has the highest priority in our discussion, we also consider another aspect, which is to maintain the so-called community in graphs. In societies, a community can be a variety of group organizations, such as families,friendship circles or colleagues. In online social networks, a community can be a virtual interest group. In protein-protein interaction networks, a community can be a group of proteins having the same specific function within a cell. As one of the major features of graphs representing real systems is community structure , detecting such communities is of great importance in various disciplines, such as sociology, biology and computer science. This is usually known as community detection in the graph clustering area, and has been a vibrant research field for the last few years [10-14]. Therefore, it is either necessary or practicable to ensure a privacy preservation approach to maintain the community structure of a published dataset.
- We discuss the identity disclosure problems in weighted graphs with certain weight properties as background knowledge.Two weight characteristics are considered: 1) volume:the sum of weights for a vertex; and 2) histogram: the neighborhood weight distribution of a vertex, and we show empirically how high the disclosure risk is with these weight attacks to breach real-world graphs.
- We formalize a general model for weighted graph anonymization,which is to modify edges and weights in a graph to prevent weight-related attacking.
- We provide a complete solution for the weight anonymization problem introduced in this paper, and show theoretically and empirically how the proposed methods perform on both data privacy and utility.
- We consider the change of graph spectrum as information loss incurred in graph perturbation, and use the algebraic connectivity as a quantitative metric.
- We also theoretically justify the impact of weight modification on the graph spectrum.
- We explore the impact of the proposed approaches on community detection that is a concrete and popular application in the field of graph mining and analysis.
The remainder of this paper is organized as follows. Section II brief overviews the literature on the graph anonymization problem. In Section III, we formally define a general model for weighted graph anonymization and provide two concrete cases of weight-related attacks. We also discuss the use of a graph spectrum as the information loss during anonymization in this part. Section IV focuses on methods against both weight attacks. We present the experimental results in Section V. In Section VI, we explore the issue of community detection with graph anonymization. We conclude this paper in Section VII.
In recent years, a large number of techniques, such as data swapping , microaggregation , and k-anonymization [4,16, 17], have been proposed for the identity disclosure problem in relational databases. Most of these fundamental studies are group-based that means the approaches will partition data according to a certain metric and generalize/suppress data tuples in each group. An extensive study can be found in .
Hay et al.  point out the risk that simply removing the identifiers (or label) of the nodes does not always guarantee privacy.They study a spectrum of adversary external information and its power to re-identify individuals in a social network. Two types of adversary knowledge are formalized in detail: 1) vertex refinement queries that reveal the structure of a graph around a vertex. For a node v, such information includes its label, degree,the list of its neighbors’ degree, and so on. 2) subgraph knowledge queries, which investigate the uniqueness of a subgraph around the target node. The above two directions are also extended by the following studies [7, 8] that we will describe in the following part. The paper also provides a solution to anonymize the social network data based on random perturbation.
The work in  describes a family of attacks on a social network whose labels for vertices are replaced with meaningless unique identifiers. The idea behind these attacks is to create or find unique subgraphs embedded in an arbitrary network. Then,adversaries can learn whether edges exist between specific targeted pairs of nodes.
The work in  identifies an essential type of privacy attack,called neighborhood attacks. That is, if an adversary has some knowledge about the neighbors of a target node and the relationship among them, it is possible to reidentify the node from a social network. The authors modelled the k-neighborhood anonymization problem systematically and proposed an anonymization approach based on the neighborhood component coding technique, but admitted that the algorithm would be a computational serious challenge as the neighborhood size increased. It is believed that this observation is closely related to the subgraph knowledge queries discussed in .
Liu and Terzi  study a specific graph-anonymity model called k-degree anonymity that prevents the re-identification of individuals by adversaries with a priori knowledge of the degrees of certain nodes.
They provide a two-step framework for the k-degree anonymity problem: degree anonymization and graph reconstruction.The first step is solved by a linear-time dynamic programming algorithm, and the second step is solved by a set of graph construction algorithms that are related to the realizability of degree sequences. Experiments on a large spectrum of synthetic and real network datasets demonstrate that their algorithms are efficient and can effectively preserve the graph utility, while satisfying k-degree anonymity.
Zheleva and Getoor  consider the problem of protecting sensitive relationships among the individuals in the anonymized social networks. This is closely related to the link-prediction problem that has been widely studied in the link mining community.The work in  studies how anonymization algorithms based on randomly adding and removing edges change certain graph properties. More specifically, they focus on the change caused in the spectrum of the network.
Liu et al.  take weights as consideration for privacy preservation in social networks. They study situations, such as in a business transaction network, in which weights are attached to network edges that are considered confidential. Then, they provide two perturbation strategies for this application. In particular,their methods yield an approximate length of the shortest path, while maintaining the shortest path between selected pairs of nodes, but also maximize privacy preservation of the original weights. The research in  extends the above work by formulating an abstract model based on linear programming. However,the objective of their work focuses on maintaining a certain linear property of a social network by reassigning edge weights.
The work in  proposes a historically important algorithm in the field of community detection. The main idea behind is to first calculate the centrality for all edges and then invoke an iterative process that in each iteration removes the edge with the largest centrality and recomputes centralities on the running graph. A large variety of methods have been since developed based on the Girvan-Newman algorithm. Tyler et al.  presented a modified version of the algorithm that aims to improve the speed of the calculation. The work in  provides a modification of the Girvan-Newman algorithm in which vertices,rather than edges, are removed based on a vertex-based centrality metric. In addition, as the Girvan-Newman algorithm is unable to detect overlapping communities, both studies in and  explore the modified algorithms in which vertices can be split between communities.
Another popular type of community detection methods is based on the so-called modularity  that is a stopping criterion for the Girvan-Newman algorithm. Since the problem of modularity optimization has been proved NP-complete ,several heuristic algorithms are proposed to find approximations of the modularity maximum in a reasonable time, which are based on different techniques, including greedy technique, simulated annealing , extremal optimization , and spectral optimization . An extensive study of community detection can be found in  for further reading.
In this section, we first provide some preliminaries and notation used throughout the paper. Then, a general model is proposed to define the
An undirected and weighted graph
Definition 3.1. (weight bag) For each vertex vi ∈ V, we define a weight bag for
A weight bag can be a multiset, since different edges can have the same weight in real world cases. Then the weight set can be described as the complete set of weight bags in
In this paper, we allow weights to be integers, rational numbers or real numbers but non-negative. Given a graph
In general, a large variety of weight properties derived from the weight set can be used to identify a vertex. Let f be a function mapping
The information loss is a critical measure to quantify the utility of a perturbed graph, and we will introduce efficient metrics in the next section. Most previous work [9, 21, 22] uses a twostep strategy that splits the problem into two sub-problems,called property anonymization and graph reconstruction, to solve graph perturbation problems. The idea behind this is to reform the structure of a graph according to either its randomized weight matrix or anonymized property sequence. For each step, the methods are devoted to minimizing the information loss incurred by perturbation or reconstruction. Such a strategy is quite attractive for large graph anonymity due to its reasonable complexity in real applications. Therefore, all methods proposed in this paper will follow this track.
From a general view, each privacy preserving approach has its limitations due to the wide concepts of data privacy and utility,which means it only works efficiently on protecting data from a certain family of attacks and maintaining pre-specified data utility. There is no exception here. The task to determine an efficient measure becomes even more complex in graph analysis due to its topological structure.
Differentiating from some other problems, such as
1) Anonymizing Cost:The privacy metric for anonymity is indistinguishing level k in general. Besides the security requirements,we also expect the perturbation to have a minor effect on the original data. As our basic operations for perturbation is to add, delete or reallocate edges/weights in graphs, the method for weight anonymization is naturally required to minimize the changes of weights. Some previous work in  quantifies anonymizing cost, using the number of edges changed before and after graph perturbation. We first consider extending this metric as anonymizing cost in the WGA problem. Given a sequence of certain weight property w = [w1 . . . , wn], the anonymizing cost incurred by property anonymization is mathematically formalized as follows,
where m is the number of anonymizing groups, gi and wi* represent the number of objects and the anonymizing object in each group. Although it is quite simple and intuitive to use CA as the anonymizing cost incurred in the perturbation, the underlying relationship between such anonymization cost and the topological structure of a graph is still unclear. However, our extensive experimental results with real-world data have shown its efficiency,which implies underlying relations with real graph properties.
2) Spectrum-based Information Loss: Our next task is to define an efficient metric for information loss. In this paper, we consider a spectrum-based metric derived from the spectral graph theory. Recall that, the Laplacian matrix of a weighted graph G(V, E) is defined as L = D ? A, where D and A are the degree matrix and the adjacency matrix respectively. The set of eigenvalues of L, λ = (λ1, λ2, . . . , λn), is called the spectrum of G. If the graph only contains one component, we have 0 = λ1 ≤λ2 ≤ . . . ≤ λn. Many studies [33-35] point out that the spectrum plays a central role in understanding of graphs, since it is closely related to most invariants of a graph, such as mean distance,diameter, connectivity, expanding properties, maximum cut, isoperimetric number and randomness of the graph. Particularly,the second smallest eigenvalue λ2, known as algebraic connectivity, has been discovered in its application to several difficult problems in graph theory. In this paper, we use the bias of λ2 to measure the information loss. Let λ2 be the second smallest eigenvalue of the anonymized graph G'. Then, the information loss C is defined as
Therefore, an anonymizing algorithm aims to find a perturbation to minimize Equation 2, while guaranteeing the privacy of entities in the graph.
Now, we discuss the relation between
The proof of THEOREM 3.4 is omitted here due to the space limitation. THEOREM 3.4 shows the change of weights can approximately estimate the bias of λ2, this leads to an efficient way to develop anonymization algorithms. It is ideal to quantify the relation between λ2 and the modification of weights with a determined function λ2 =
Equation 1 only provides an efficient measure to achieve the minimal information loss in designing an anonymization algorithm.However, the bias of the spectrum can be influenced by reconstructing a graph as well. The work in  provides theoretical analysis of reconstructing a weighted graph from its spectrum, which maintains the eigenvalues perfectly. However,the conditions for reconstructability of weighted graphs are only sufficient and it can be costly to implement. Therefore, we provide efficient methods for graph construction with the objective of minimizing the information loss C in a later section.
There exist a variety of weight properties in a weighted graph. Based on the general model for weight anonymity, we will discuss two types of weight-related attacks and apply the general model on these anonymization problems.
1) Volume Attack: We first consider a weight property, called volume, which describes the sum of a weight bag. That is, the function f = Σ, and for a vertex vi, Pi =Σj=1di wij. Notice that, the value of the volume is sometimes used as degree in research on weighted graphs, but we use it separately for a clearer statement.Using si to specifically represent the volume of vi, we can form a sequence S = [s1, . . . , sn] (s1 ≥ s2 ≥ . . . ≥ sn), where n is the number of vertices. It is easy to find that, if there exists a unique si in S and the value is known by an adversary, the entity represented by vi will be identified from the published graph.We term this privacy breaching process a volume attack. Correspondingly,we can define k-volume anonymity for a graph as follows.
For example, the graph (a) in Fig. 1 is 1-volume anonymous,since
2) Histogram Attack: Another weight-related attack refers to the histogram of a weight bag. In statistics, a histogram is a graphical display of tabular frequencies. Let Bi be the set of histogram bins for the vertex vi and each bin consists of weights falling in the bin range. We use Bi = (bi1, bi2, . . . , bim) to represent the set of bin frequencies for vi, where bij is the number of elements in the corresponding bin and m is the number of bins.Here, we assume the partitioning of histogram bins is the same for all vertices. Then, we can define k-histogram anonymous for a graph, as follows.
A special case of histogram attack is that weights are all integers and the bin width in every histogram is 1. Then, the similarity of histograms can be transferred to the similarity of weight bags. We say two weight bags are equal if and only if they have the same elements and the same multiplicity for each element. The graph (a) in Fig. 1 is 1-histogram anonymous as
3) Discussion: It is worth discussing the relationships among various weight anonymity problems, since this may provide alternatives to design anonymization algorithms. The degree anonymity is also considered as a special case of weight anonymity.We have the following proposition.
The proof is obvious with the equality property of weight bags. This proposition shows that an approach for histogram anonymity achieves degree and volume anonymity at the same or a higher security level. We have to point out that Proposition 3.7 is just a sufficient condition. That is, degree and volume anonymity cannot guarantee the same level of histogram anonymity.
In this section, we consider methods to protect a graph from histogram attack. Based on our general model and the definition of k-histogram anonymity, the histogram anonymization (HA)problem is formally defined as follows.
The solution to Problem 4.1 follows the two-step strategy discussed in the previous section. It first tries to find the optimal anonymity
The first challenge in histogram anonymization is to perform appropriate data preparation for the calculation of information loss, since weight bags in a weight set may come in different sizes. Considering each weight bag as a point in a unified multidimensional space, the preparation procedure is required to maintain these points as densely as possible. Mathematically,for a vertex vi, its weight bag wi = [wi1, . . . . , widi] can be seen as a vector in the di-dimension space, where di is the degree of vt. If let
Let us say Uq = [wq1, . . . . , wq(m?1), 0] and
Lemma 4.2 is a necessary condition for an anonymization algorithm to achieve the lowest cost. It implies that, to guarantee the lowest information loss, we have to assign 0s to the end of weight bags whose sizes are smaller than m. For instance, in
Algorithm 1. Histogram anonymization algorithm
Fig. 1b, if we connect the vertices a and d with weight 1, we have the weight set as
Another issue in the HA problem is to determine the anonymous object for data generalization in each anonymous group.Generally, these objects are expected to be chosen from the original dataset. However, this constraint is too strict for histogram anonymization, as the unique weight operation is allowed only so far. Here, we relax the constraint to allow the method building the ‘largest’ vector as the anonymous object. That is,for a group
The complexity of this problem has not been assessed, as far as we know. However, the work in  discusses a similar problem with the u* being the mean vector of a group that has been proven NP-hard. Although it is unclear whether the HA problem can be inducted from this existing problem, we can prove its NP-hardness using a similar induction process, omitted here due to space limitation. Therefore, we describe an efficient heuristic algorithm as a solution in Algorithm 1.
The computational complexity of Algorithm 1 is O(n2log□).Here, we form a symmetric n × n distance matrix in which each entry represents the Euclidean distance between two weight bags in
Graph construction is the second stage of the two-step strategy that aims to construct a graph with an anonymized weight set. The graph construction based on a specified degree or volume sequence has been extensively studied in previous work[38-40]. Although the
The WGC algorithm takes the specified weight set W as inputs and returns either a successfully constructed graph or“Fail”, meaning
The computational complexity of Algorithm 2 is
Notice that, Step 7 makes a trade-off between efficiency and accuracy in Algorithm 2, as there may exist a group of vs and this may result in various information losses by connecting vr to different vs. We provide a sort-then-switch procedure on the edge set of a constructed graph to optimize the information loss incurred in graph construction (the implementation of this procedure is omitted here). The procedure takes a constructed graph
Algorithm 2 Weighted graph construction algorithm
with the sort-then-switch procedure.
In this section, we evaluate the performance of the proposed graph anonymization algorithms. The experiments are conducted on a 2.16 GHz Intel Core 2 Duo Mac with 4 GB of 667 MHz DDR2 SDRAM running the Macintosh OS X 10.5.8 operating system. All algorithms are implemented using Matlab 7.0.
We use both synthetic and real-world datasets. For experiments with synthetic data, we generate a random weighted graph
We also use two real-world graph datasets, named BkFrat and NetSci, respectively. All these graphs are weighted and undirected. The BkFrat graph  concerns interactions among students living in a fraternity at a West Virginia college. The graph contains 58 nodes with all integer weights in the range of[0, 51]. The NetSci graph  contains a coauthorship network of scientists working on network theory and experiments.
The version given here consists of 1; 589 scientists and assigns real weights as described in . All the testing data have been simply generalized by removing all real labels for their vertices. All the real-world graph datasets are available at http://www-personal.umich.edu/~mejn/netdata/.
Our first experiment is to show how possible weight attacks may occur on realworld graphs. We consider both volume attack and histogram attack, and provide results of degree attack as a comparison. Table 1 shows our results. The parameter α is a threshold to assess a breach. That is, while the number of vertices sharing the same value of a weight property is no larger than α, these vertices are considered to be disclosed. It clearly shows that weight attacks are a real issue for graph data publishing.All testing datasets have relatively high risk of entity disclosure.For BkFrat data, the success rate of volume attack withα = 1 is as high as 93%, and even as high as 98% for histogram attack, implying that most of its vertices can be uniquely identified.In addition, the disclosure risk grows quite fast as αincreases. For example, the success rate of volume attack on the NetSci dataset increases nearly 10&
Weight attacks on real-world data
Moreover, the results show that weight attacks have much higher success rate to breach a dataset than degree attack. This means such attacks are more practical in real-world scenarios.In addition, both weight attacks maintain similar success rates,while the impact of degree attack is decreased significantly, as the data size increased. The result for NetSci data shows there are 11:01% vertices with high disclosure risk, as
In this section, we assess the qualitative performance of information loss incurred by applying histogram anonymization.As a comparison, we also implement a greedy anonymization algorithm for volume attack, termed GreedyVA, modified from degree anonymization  by replacing the degree sequence with volume sequence.
The graphs in Fig. 2 describe the relations between anonymization cost
1) Impacts on Graph Spectrum: We evaluate and compare the information loss for the complete histogram anonymization.Fig. 3 summarizes the impacts on the spectrum λ2 on the Y-axis,varying k for the testing datasets. Two construction methods are compared here: WGC (HA-WGC) and WGC with the sort-thenswitch procedure (HA-WGCS). From the plots, we can observe the sort-then-switch procedure can significantly improve the utility performance of histogram anonymization. For example,in RandGraph, such a procedure reduces the bias of information loss from 1.4 to 0.6 for k = 250. The gap is insignificant for NetSci compared to other data, and it is not obvious to decide on the major reason.
2) Impacts on Real Graph Characteristics. As mentioned above, the spectrum of a graph has close relations with many real graph characteristics. In this section, we evaluate the information loss based on real graph characteristics of the test data with the algorithms of HA-WGC and HA-WGCS. We focus on two of the most robust metrics of network topology. The first is the global clustering coefficient that is a measure of the degree to which nodes in a graph tend to cluster together. A generalized clustering coefficient is formally defined in  as
where ωΔ is the total value of triangles and ω3 is the total value
of triplets. In addition, we define the value of a triplet as the geometric mean of the weights of ties. The second one is the weighted average path length,
Figs. 4 and 5 show the relative changes of the real graph properties
So far, we have extensively explored the problem of privacy preservation on weighted graphs against weight-related attacks.The theoretical and experimental results show that our algorithms perform effectively on not only protecting graphs from identity disclosure, but also maintaining elementary graph properties,such as spectrum, clustering coefficient, and average path length. Such utility preservation can guarantee the data quality for statistical analysis applications. However, it is poor in answering the following question: how the proposed methods will affect data quality for data mining analysis.
In this section, we discuss how to extend our approaches to ensure the quality of graph clustering with published data. This is known as community detection in the related field. Instead of providing a comprehensive solution for privacy preserving graph mining, our intention is just to show that the proposed methods can be used to solve concrete data mining problems with only slight modification.
According to the basic preliminaries in Section III-A, we can introduce notation for community description. A natural approach for graph clustering is based on the concept of graph cut. In graph theory, a cut of a graph
Then, we can initially introduce a popular definition of community based on edge connectivity, as follows:
(1) H ⊆ C,
(2) sC ≥ sH' for each subgraph H' ⊆ G with H ⊆ H',
(3) H' ⊆ C for each subgraph H' ⊆ G with H ⊆ H'and sC = sH'.
The above definition interprets a community as the largest subgraph of maximal edge connectivity among all subgraphs of
Appropriate metrics need to be defined in advance to quantify the information loss of graph anonymization on community detection. We first consider an intuitive metric, termed
Another important metric for community detection is modularity,known as a global quality function to identify good partitions. One of the most popular concepts of modularity is proposed by Newman . It is based on the idea that a random graph is not expected to have a cluster structure, so the possible existence of clusters is revealed by the comparison between the density of edges in a subgraph and the density one would expect to have in the subgraph, if the vertices of the graph were attached, regardless of community structure. This expected edge density depends on the chosen null model, i.e. a copy of the original graph keeping some of its structural properties but without community structure. Mathematically, the modularity
pproximates the ratio of edges that one would expect to have inside the community from chance alone. Let
In this part, we provide a community preservation procedure aiming to minimize the change of the community set
In this section, we evaluate the impact of graph anonymization on community detection in terms of the proposed metrics.The experimental setup and testing data are the same as that in Section V.
1) Intra-group Error ZH: We first assess the intra-group error defined in Equation 3 for graph anonymization. We compare two approaches: the basic HA-WGCS algorithm and the HA-WGCS algorithm with the community preserving procedure(HA-CP).
Fig. 6 summarizes the intra-group error
preservation during the perturbation process. In addition, it reveals that this method can still maintain a relatively low information loss, even for a large
2) Modularity Bias ZM:The second experiment explore the relationship between the modularity bias defined in Equation 4 and k. Fig. 7 shows the relative changes of ZM with HA-CP and HA-WGCS by varying k. The modularity bias increases as k increases that follows the similar trend of ZH. However, the gradients are not as steep as that of ZH, especially when k is not large. This implies that the impact of perturbation on modularity is insignificant as the intra-group error, as ZM is a global measurement.In addition, HA-CP performs much better than HAWGCS in all cases, as expected. Moreover, the BkFrat data have the smallest difference, especially when k is not large (< 10).The reason for this is still unclear and we suppose it is related to certain structural properties of the dataset itself.
It is clear that the community preservation scheme can significantly improve the quality of community detection. However,in all experiments, we can see that HA-WGCS also has reasonable performance on community preservation, when the privacy level is not too high. This implies that in the real-world scenarios, HAWGCS is also practicable.
In this paper, we discussed a class of important background knowledge attacks in a weighted graph. We provided a general model for the weight anonymization problem to defend against weight-related attacks. As a proof of concept, we considered the k-volume anonymity and the k-histogram anonymity as two cases that could occur in the real-world privacy graph data publication.We provide a complete solution to achieve both volume anonymity and histogram anonymity using the graph spectrum as an effective metric for information loss. We also analyzed the complexity of the models, and experimentally validate our analysis using both synthetic and real-world weighted graphs. Finally, we extend our work to preserve the quality of community detection that is a popular application in the graph mining field.
Many issues of this work need that to be addressed clearly merit further research. As a NP-hard problem, it is worth developing approximation algorithms for the histogram anonymization problem. Also, if we allow random weight modification(increment and decrement), the impact on graph spectrum has to be reconsidered. In addition, this paper only evaluated the clustering coefficient and average path length as real-graph properties,and the affect on other topological structures is still unclear.