A new method for graph stream summarization based on both the structure and concepts

2.1 Structural summarization

In [12] a method has been proposed for graph structural summarization. In this method, a graph is compressed by partitioning similar nodes into one group and dissimilar nodes into different groups. For a compressed graph, a super-edge is the aggregated edges between a pair of supernodes. In this method, a graph is compressed based on the Minimum Description Length (MDL) idea. Firstly, they developed a greedy algorithm and secondly to reduce the runtime of the algorithm, they proposed a randomized version.

Riondato et al. [3] proposed another method to summarize structural graphs. In this method, the aim is to guarantee the quality of a summary and minimizing the reconstruction errors. Riondato et al. have presented a connection between graph summarization and geometric clustering problems for the first time. Based on this connection, the authors developed a polynomial-time algorithm to generate the best possible summary of the expected size.

Tian et al. [13] proposed three distributed summarization algorithms named DistGreedy, DistRandom and DistLSH to summarize large scale graphs. These algorithms differ in how they select a pair of nodes to merge, which they select greedy, randomly, and using locality sensitive hashing theory, respectively.

Chen et al. [14] proposed a method based on producing randomized summary graphs for identifying frequent patterns. Structural summaries can be beneficial for frequent pattern mining. In fact, instead of mining massive and time-consuming original graphs, summary graphs are mined.

In fact spectral graph clustering can be used for structural summarization. Spectral graph clustering partitions a graph based on eigenvalues and eigenvectors of the graph adjacency matrix [15, 16, 17, 18, 19]. This technique can be beneficial in image segmentation and social network analysis. There are a number of applications that use spectral graph clustering for finding communities in networks [20]. In this applications, initially a large graph is converted into a small one by summarization and then use spectral graph clustering to cluster the resultant small summary graph [21].

Community detection algorithms has many applications and recently, many articles [22, 23, 24, 25, 26] have been published on this subject. Graph summarization can be beneficial for detecting communities in a network.

Of-course, there are some similar methods/models [27, 28], subgraph mining models,which are limited in comparison with summarization methods. These models rather than summarizing, choose one or more subsets from graphs.

2.2 Attribute-based summarization

In [7], a summarization method with two novel operations Summarization by grouping Nodes on Attributes and Pairwise relations (SNAP) and k-SNAP has been proposed. These operations are used for grouping nodes and summarizing attributed graphs. Tian et al. defined attribute and relation compatible grouping. They also improved SNAP operation by proposing k-SNAP operation. In k-SNAP operation, k is the summary size where is determined by the user. The k-SNAP operation improved by Zhang et al. [8] by proposing the CANAL algorithm in 2009. The CANAL algorithm is used to categorize attribute values automatically, and also to provide a criterion to measure the quality of a summary.

In 2008, the OLAP framework has been proposed by Chen et al. [29]. In the OLAP framework, the cubes are created on the graph based on dimensions and measures. In this framework, a graph is summarized based on both selected attributes and input information.

Figure 1

Original graph (left) and its summary (right)

2.3 Hybrid summarization

In [6] a method was proposed for clustering a graph based on both the structure and vertex attributes. In this method, for a given graph a new graph, named the augmented graph, with real and virtual links is constructed. Because of attribute-based similarity of vertices, the virtual links are added to the new graph. In the augmented graph, both real and virtual links are considered to measure the similarity of two nodes. If the number of associated attributes is relatively high, the augmented graph will be massive and finally the runtime of the algorithm is high.

In [11] another method has been proposed to hybrid summarization of a graph. In this method, initially a graph is summarized based on vertex attributes, without take into consideration the graph structure, and then by moving nodes between super-nodes adjust the summary to the graph structure. This method for situations where the structure has an important impact in constructing summary may be inefficient.

In [30] a method has been proposed for attributed graphs which constructs a hybrid summary by considering MDL principle to model the graph summarization problem into a code cost function and utilizing greedy method to compute an optimal summary. In this method, the user’s needs and also the ontology of the graph have not been considered.

2.4 Graph stream summarization

There has been some research work on the subject of graph stream summarization but the contribution of these works in the scope of graph stream summarization is not significant. Major research work done on graph stream are as follows.

In this [31] a novel Graph Stream Sketch (GSS) has been proposed to summarize graph streams with linear space cost (O(|E|) and constant update time complexity (O(1)). The aim of Gou et al. has been constructing a summary for query answering with the controllable errors.

In [32] the focus is on calculating the rank of a node in a graph stream with the minimum passes over the stream and the minimum space, of-course up to an adaptive error. Therefore, algorithms and models has presented in this regard.

In [33] Feigenbaum et al. have been interested in the trade-offs between model parameters such as perdata-item processing time, required space, and the required number of passes over the stream. These trade-offs have been considered for solving problems such as Spanner Construction, BFS-Tree Construction, Graph Distance Lower-Bounds.

In [34] a new variation of streaming model with a helper which can provide annotations for data streams have been proposed by Cormode et al. They have discussed that by giving linear sized annotations, the memory for many problems is reduced to constant time.

In [35] Feigenbaum proposed a new streaming model and formulized it. They believed this model is necessary for proposing efficient algorithms to solve problems on massive graphs. They have considered an upper bound for required spaces foe such algorithms. They applied the proposed model on special problems.

In [36] Aggarwal also et al. proposed a method for graph stream clustering by introducing micro-clusters and compressing them with hash functions. The proposed method can be beneficial for special applications. Aggarwal proposed a new method for classification a massive domain graph stream [37]. Aggarwal has proposed a probabilistic approach for constructing a summary that can be stored in main memory. Aggarwal used this method for determining special patterns in a graph stream.

There are other works on graph stream such as the problems of connectivity [35], counting subgraphs e.g. triangles [38, 39], calculating the degree of nodes [40], spanners [41], sparsification [42]. Thus to the best of our knowledge, there is no capable method for summarizing a graph stream which converts a graph to a smaller one by removing unnecessary details and preserving overall properties.

3.1 Algorithm

The proposed method has been summarized in Algorithm 1. In this algorithm, summarizing a graph, comparing two summary graphs and calculating the distance of two summary graphs are not clear and should be illustrated more. In the following subsections, we illustrate the structure of a summary graph, similarity of two supernodes, comparing two summary graphs and finally the proposed method is illustrated by an example.

Algorithm 1: Graph stream summarization

Input: A graph stream;

Output: A reference summary graph;

1: Begin

2: consider the window w0 on the graph stream;

3: Construct the graph of w₀ and summarize it as SG_R.

4: While(true)

5: consider the next window w_c on the graph stream;

6: construct the graph of w_c and summarize it as SG_c.

7: compare two summary graphs and let d=dist(SG_R, SG_c);

8: if(d > threshold) replace SG_R with SG_c;

9: else ignore SG_c.

10: Endwhile;

11: end.

3.2 The summary structure

In the proposed method, an attributed graph is summarized based on both the structure and vertex attributes. Every super-node in the summary graph is a vector of structural and semantical attributes. Structural attributes are the number of vertices in the super-node, the degree of the super-node and the percentage of vertices, which are relevant with nodes of the other super-nodes. Semantical attributes are considered as the percentage of vertices, which have a value on an attribute. In fact, for every value of a vertex attribute this percentage value is calculated. In Section 3.5 we illustrate the summary structure by an example.

3.3 Similarity of two super-nodes

Based on the proposed structure for the summary graph, a super-node is a vector of attributes and the similarity of two super-nodes is calculated based on their vectors. The similarity of two super-nodes is calculated using Equation (1),which also uses Equations (2) through (6). Initially, the distance of two super-nodes is calculated and then by subtracting this value from one, the similarity of two supernodes is obtained.

The distance between two super-nodes is equal to summation of structural and attribute-based distance of two super-nodes, which is presented by Equation (2).

The number of vertices, the degree of super-nodes and the number of vertices which relevant to vertices of other super-nodes (the weight of edges) are considered as structural attributes. These structural attributes determine the structural distance of two super-nodes. Equation (3) describes the structural distance of two super-nodes. As seen in Equation (3), the value of structural distance belongs to [0,1]. For each of the three parts of Equation (3), if the denominator is zero, the value of that part is considered to be zero.

The attribute-based distance between two super-nodes with k attributes and each attribute with k values is calculated using Equation (4).

where

where

where n_p and d_q are the number of vertices in V_p and the degree of V_p, respectively.

3.4 Comparing two summary graphs

To compute the distance of two summary graphs, initially the similarity of each pair of super-nodes of two summary graphs is calculated using Equation (1). The super-node pairs with the most similarity are associated. After associating the super-nodes of two summary graphs, the distance of two summary graphs is calculated. The distance is equal to summation of distances of each pair of matched super-nodes. The approach for calculating the distance of two summary graphs has been described in Algorithm 2.

Algorithm 2: Calculating distance two summary graphs

Input: summary graphs: GS1 and GS2 and the size of summary graph: size;

Output: distance of two summaries;

1: Begin

2: Calculate the distance of every two super-nodes

3: Add every super-node pair with its calculated distance to ascending priority queue q;

4: n=summary graph size;

5: While(n>0)

6: Remove a super-node pair;

7: Match two super-node of the pair;

8: n–;

9: endwhile;

10: set dsit to sum of distances of the matched super-node pairs;

11: end.

3.5 Illustrating the proposed method with an example

To clarify the issue, we consider two summary graphs SG₁ and SG₂ with above-mentioned structure, each with three super-nodes and two attributes. Attributes are gender and education level, gender with values of Male and Female and education level with values of BSc., MSc. and Ph.D. As we see in Figure 3, the summary graph shows the overall and important information of the original graph. For example, super-node V₁ shows a group of 600 people where 20% are in relationship with people of V₂, 80% are in relationship with people of V₃, 65% are female and 35% are male, 30% are bachelor of science, 30% are master of science and 30% are Doctor of Philosophy.

Figure 3

First summary with three super-nodes and two attributes. Attribute values, size of super-nodes and the percentage of vertices, which are in relationship to vertices of the other super-nodes, are shown in the summary.

Figure 4

Second summary with three super-nodes and two attributes. Attribute values, size of super-nodes and the percentage of vertices, which are in relationship to vertices of the other supernodes, shown in the summary.

As already mentioned, initially the similarity of every pair of super-nodes of two summary graphs is calculated. For clarify, in the following we calculate the similarity of two super-nodes V₁ and V1′,step-by-step.

For saving time, we have refused to provide computational steps for other super-nodes pairs and only have entered their final similarity values in Table 1.

Based on Table 1, matched super-nodes are (V2,V2′)(V3,V3′)and(V1,V1′)respectively. The first component of each pair is a super-node of the first summary graph and the second component is its matched super-node of the second summary graph. According to this matching, the distance of two summary graphs SG₁ and SG₂ is calculated as follows:

4.1 Dataset

Amazon co-purchasing network

This data is available in address^[1] and includes information about different products such as the books, music CDs, DVDs and VHS video tapes. There are 548,552 products and for each product, the information such as title, salesrank, list of similar products, category and reviews is available. This data are about Amazon co-purchasing products of 2003 and has been collected in summer 2006 by Jure Leskovec with crawling Amazon website. The information of this products and their graph streams are presented in Table 2. Rows second to fifth show four directed graph streams. Each graph is a graph stream where each edge (x, y) shows product y has frequently co-purchased with product x. We chose Id, ASIN, group and salesrank fields for providing experiments.

4.2 Evaluation

To the best of our knowledge, our proposed method is a novel general-purpose method for graph stream summarization and there is no competitor method for exact evaluation. We believed that comparing the results of our proposed method with the changes of real constructed graphs are more reasonable and reliable than comparing to other competitor methods.

Therefore, for evaluating the proposed method for graph stream summarization, we chose amazon0302 file and set the window size to 1000 edges. We considered the first window over the first 1000 edges of the file and constructed the first graph. For every window, the vertices are those, which are appeared at least as one end of the first 5000 edges. The graph of the first window has been summarized and resulted a summary graph with the size of 7. The summary graph is maintained as a reference. The next windows are also considered over the graph stream and in the following tasks such as summarizing graphs, comparing every coming graph with the reference summary and changing the reference summary(if necessary) are done. In this experiment, window size was fixed, 1000 edges, but usually the first window is considered bigger than the others are. In the following, 5 summary graphs are presented in Figures 5 through 9. In these Figures, dis and toy represent discontinued and toy products. These two categories do not belong to the main categories which are mentioned in the description of the dataset.

Figure 5

The first summary graph of the size of 7

The semantic of each summary graph has been extracted and shown in Table 3. Semantic changes of two consecutive summary graphs are presented in Table 4. Figures 5 through 9 and Tables 3 and 4 are used to discuss the experimental results.

Figure 6

The second summary graph of the size of 7

Figure 7

The third summary graph of the size of 7

Figure 8

The fourth summary graph of the size of 7

Figure 9

The fifth summary graph of the size of 7

4.3 Time complexity

In the proposed method, the dominant time belongs to the summarization algorithm. According to summarization algorithm in [9, 43], initially the similarity of each pair of nodes is calculated and after that the graph is summarized by merging nodes/super-nodes. In the worst case, the summarization algorithm performs at most |V| merge operations to obtain the expected summary. Henceforth, the time complexity of this method is O(|E|×|V|). Time complexity of other processes such as calculating distance between two super-nodes, matching super-nodes of two summary graphs and finally calculating the distance of two summary graphs is less than the runtime of summarization algorithm.