A novel parallel distance metric-based approach for diversified ranking on large graphs

Highlights

• A generalized distance metric based on a subadditive set function over the symmetry difference of neighbors of pairs of nodes is introduced to capture the pairwise dis-similarity over pairs of nodes.

• Diversified ranking on graphs (DRG) is formulated as a Max-Sum k-dispersion problem with metrical edge weights.

• A centralized linear time 2-approximation algorithm GA is developed to significantly solve the problem of DRG.

• A highly parallelizable algorithm is developed for DRG, which can be easily implemented in MapReduce style parallel computation models using GA as a basic reducer.

Abstract

Diversified ranking on graphs ($\mathrm{DRG}$) is an important and challenging issue in researching graph data mining. Traditionally, this problem is modeled by a submodular optimization objective, and solved by applying a cardinality constrained monotone submodular maximization. However, the existing submodular objectives do not directly capture the dis-similarity over pairs of nodes, while most of algorithms cannot easily take full advantage of the power of a distributed cluster computing platform, such as Spark, to significantly promote the efficiency of algorithms. To overcome the deficiencies of existing approaches, in this paper, a generalized distance metric based on a subadditive set function over the symmetry difference of neighbors of pairs of nodes is introduced to capture the pairwise dis-similarity over pairs of nodes. In our approach,$\mathrm{DRG}$ is formulated as a Max-Sum k-dispersion problem with metrical edge weights, which is NP-hard, in association with the proposed distance metric, a centralized linear time 2-approximation algorithm GA is then developed to significantly solve the problem of$\mathrm{DRG}$. Moreover, we develop a highly parallelizable algorithm for$\mathrm{DRG}$, which can be easily implemented in MapReduce style parallel computation models using GA as a basic reducer. Finally, extensive experiments are conducted on real network datasets to verify the effectiveness and efficiency of our proposed approaches.

Introduction

In recent years, with the support of web 2.0 technology, various social network applications, such as Facebook and WeChat etc., have been rapidly developed, and result in massive accumulated data. In order to effectively process and analyze these social network data, many researchers have developed different data analysis method and technology platform [[1], [2]].

Graph data is an important type of social network data. In general, graph data is composed of nodes and edges without total ordering information over all nodes. Thus, graph ranking, which aims to bring nodes into a total order, becomes one of the fundamental tasks for graph data analysis [3]. PageRank [4] is a popular measure of the importance of nodes in a graph. The basic idea of PageRank is that a node in a graph has a higher rank if it is linked by more high-ranking nodes. Generally, PageRank measures the global importance of nodes in a graph. Individual relevance is not well reflected by this measure. To overcome this shortcoming, personalized PageRank (PPR) and its variations [[4], [5]], are proposed to find nodes in a graph that are most relevant to a query or user. However, as discussed in these works [[6], [7]], the top-k ranking list found by PPR may have highly similar nodes due to the fact of that PPR only focuses on the query relevance of nodes. This reduces ranking effectiveness, especially in the case of that the query intent of users is uncertain and ambiguous.

Diversity has been widely recognized and studied as a way of addressing uncertainty and ambiguity in information retrieval applications [8]. To provide more desirable graph ranking results, graph ranking algorithms have to find a tradeoff solution between query relevance and diversification. Consequently, a considerable amount of work on diversified ranking on graphs ($\mathrm{DRG}$) has been undertaken to improve the diversity of top-k ranking results [[9], [10], [11], [12], [13], [14], [15], [16]]. In this paper, we focus on improving the diversity of ranking results based purely on the topological structure of the graph without accounting for information about the link and node.

Li et al. [10] formulated$\mathrm{DRG}$ as a bi-criteria optimization problem where the relevance measure is the sum of Personalized PageRank scores, and diversity is measured by the expansion ratio over the ranking results. Küçüktunç et al. [11] presented expanded relevance, which combines both relevance and diversity into a single function as an optimization objective. Expanded relevance is actually a weighted expansion ratio using personalized PageRank scores as the weights of nodes. Both the expansion ratio and expanded relevance resort to a submodular optimization objective and apply the classic cardinality constrained monotone submodular maximization to solve their problems.

However, two issues must be taken into consideration for improving the effectiveness and efficiency of$\mathrm{DRG}$. First, the key challenge of$\mathrm{DRG}$ is to design well-defined diversity measures. Both expansion ratio and expanded relevance attempt to work on the assumption: two nodes are dissimilar if they do not share common neighbors in a graph, and a set of nodes that are dissimilar to one another implies a set of nodes with large expansion measures. However, both the expansion ratio and expanded relevance do not directly capture dissimilarity over pairs of nodes. In other words, a set of nodes with high expansion ratio or expanded relevance does not always imply that the nodes in the set are dissimilar to one another. The following Example 1 illustrates this observation.

Example 1

Let us firstly review the definition of expansion ratio introduced in [10]. Let$G=〈V,E〉$ be a graph.$S\subseteq V$ is a subset of nodes. The expansion set of$S$ is denoted by$N\left(S\right)$ and defined as$N\left(S\right)=S\cup \left\{v\in V-S|\exists u\in S,\left(u,v\right)\in E\right\}$. Then, the expansion ratio of$S$ is denoted by$\mathrm{er}\left(S\right)=\left|N\left(S\right)\right|∕\left|V\right|$. Thereby, we have the expansion ratio for$\left\{1,2\right\}$ and$\left\{7,8\right\}$ in Fig. 1 respectively:

$$\begin{array}{cc}\hfill \mathrm{er}\left(\left\{1,2\right\}\right)& =|\left\{1,2\right\}\cup \left\{3,4,5,6\right\}|∕\left|\left\{1,2,3,4,5,6,7,8\right\}\right|=0.75\hfill \\ \hfill \mathrm{er}\left(\left\{7,8\right\}\right)& =|\left\{7,8\right\}\cup \left\{4,6\right\}|∕\left|\left\{1,2,3,4,5,6,7,8\right\}\right|=0.5\hfill \end{array}$$

Here$\left\{1,2\right\}$ has higher expansion ratio than$\left\{7,8\right\}$, but 1 and 2 are obviously more similar with each other than that of 7 and 8 since they have more common neighbors (note that, in this paper, we assume as in paper [[10], [11]] that diversified measure is based purely on the topological structure of the graph without accounting for information about the link and node). This illustrates that the expansion ratio does not always capture well the dissimilarity over pairs of nodes. Since the expanded relevance can be considered as a weighted version of expansion ratio, a similar problem also exists for the expanded relevance.  $■$

The second issue is that maximizing a cardinality constrained monotone sub-modular function is NP-hard. A$\left(1-1∕e\right)$-approximation greedy algorithm can be used to solve the problem [17]. The greedy algorithm starts with an empty set and adds a node maximizing marginal gain to the solution set in each iteration. For a graph with$m$ nodes, the greedy algorithm needs$O\left(mk\right)$ evaluations of marginal gains. However, when dealing with massive graphs, the classic greedy algorithm becomes expensive and infeasible. Although submodularity can be exploited to implement accelerated greedy algorithms, such as CELF [18] or lazy-greedy [19], as the scale of a graph increases, even for the small sizes of$k$, the accelerated greedy algorithm is still inefficient. Moreover, the approaches based on submodular optimization are essentially implemented in a sequential procedure, and are not suitable for parallel processing. This results in that these algorithms cannot easily take full advantage of the power of a parallel graph processing platform, such as Spark GraphX [20], to significantly promote the efficiency of algorithms.

Therefore, a natural question to ask is whether it is possible to present a distance measure to capture directly pairwise dissimilarity between nodes and formulate the diversity of ranking results based on this measure. Or, even better, is it possible to develop efficient algorithms derived from the diversity model that are suitable for parallel implementations?

To this end, in this paper, a distance measure is firstly introduced to capture pairwise dis-similarity over pairs of nodes. Based on the defined distance metric,$\mathrm{DRG}$ is formulated as a Max-Sum$k$-dispersion problem (MSk D) [21]. A centralized linear time algorithm and a highly parallelizable MapReduce algorithm are proposed respectively to solve$\mathrm{DRG}$.

More specifically, the key contributions in this paper can be summarized as follows:

• A generalized distance metric is introduced to measure various dissimilarities over pairs of nodes. The distance measure is defined by a set function$f$ over the symmetry difference of neighbors of pairs of nodes. Moreover, if$f$ is a subadditive set function [22] over the set of nodes, the distance measure can be proven as a metric that can be exploited to develop efficient algorithms. Furthermore, since many set functions are subadditive, such as the cardinality of a set, or the sum of weights of nodes, the proposed distance metric is a generalized measurement that can be used to capture various dissimilarities over pairs of nodes by setting different subadditive set functions.

• $\mathit{DRG}$ is formulated as a Max-Sum$\mathit{k}$-dispersion problem. Based on the defined distance metric,$\mathrm{DRG}$ can be defined as a weighted complete graph$C\left(Q\right)$, where$Q$ is the set of query-dependent nodes for a Personalized PageRank. Let the edge weight between two nodes be a distance metric between their corresponding nodes in an original graph. Using these means,$\mathrm{DRG}$ is formulated as a MSk D on$C\left(Q\right)$ that is to seek a size-$k$ subset of nodes having an induced subgraph with a maximum sum of edge weight.

• A highly parallelizable MapReduce approach with approximation guarantee is presented to solve$\mathit{DRG}$ on large graphs. Since MSk D is NP-hard [21], a centralized linear time approximation algorithm GA is firstly proposed to solve$\mathrm{DRG}$ with the benefit of using the metrical distance. Using GA as a basic reducer, we further develop a highly parallelizable approach to solve$\mathrm{DRG}$, such approach can be easily implemented in MapReduce style parallel computation models. Meanwhile, this parallel approach is able to obtain approximation guarantees for$\mathrm{DRG}$. To the best of our knowledge this is the first method that solves$\mathrm{DRG}$ in a MapReduce manner with approximation guarantees.

Extensive experiments are conducted on several representative real-world network datasets in comparison with existing approaches under relevance and various diversified measures. As a result, the experimental results significantly demonstrate the effectiveness and efficiency of our proposed algorithms.

The remainder of this paper is organized as follows. In Section 2, we review the related literature. In Section 3, we discuss the pairwise distance metric and present the formulation of our problem. The diversified algorithms are introduced in Section 4. In Section 5, we report the experimental results. Finally, we conclude the paper in Section 6.