NetKI: A kirchhoff index based statistical graph embedding in nearly linear time

Abstract

Recent advancements in learning from graph-structured data have shown promising results on the graph classification task. However, due to their high time complexities, making them scalable on large graphs, with millions of nodes and edges, remains a challenge. In this paper, we propose NetKI, an algorithm to extract sparse representation from a given graph with n nodes and m edges in $O(m{∊}^{-2}{\log }^{4}n)$ time. Our approach follows the notion of Kirchhoff index that encodes the structure of the graph by estimating effective resistance - relying on this approach yields nearly linear time graph representation method that allows scalability on sufficiently large graphs. Through extensive experiments, we show that NetKI provides improved results in terms of running time on large networks and the classification accuracy is within range 2% from the state-of-the-art results.

Introduction

Many real-world systems involve interactions between pairs of entities. Examples of such systems include social networks, financial systems, molecular graph structures, resistor networks, recommender systems, and protein–protein interaction networks. All of these systems and many more can be represented as graphs, that encode real-world objects (i.e., nodes) and the pair-wise interactions (i.e., edges) between them. More recently, graphs have also been used to model grid-structured data, such as images, to make the computations faster [1]. Encoding graphs into low-dimensional embeddings enables one to employ the traditional machine learning and data mining algorithms on graph-structured data. This approach facilitates researchers to solve hard problems on graphs such as graph classification. The graph classification problem refers to the understanding of complex graph structures among different classes, and it has a variety of real-world applications ranging from text classification to molecules toxicity prediction and classifying community structures in a social network [2]. However, graph classification poses several challenges such as permutation invariance, scalability, and, the runtime efficiency of the encoding. Generally, the same graph can be represented by many adjacency matrices due to a lack order on nodes of a graph, therefore, the ideal encoding procedure for graph classification should be invariant under permutations of the nodes. Moreover, it should preserve the atomic structure of the graph based on the local and global positions of pairs of nodes. Unfortunately, existing graph classification approaches often require a pairwise comparison among the graphs or are based solely on a statistical and spectral representations, which is hard to compute [3], [4], [5], [6]. Therefore, appropriate representation methods are required to encode the atomic structure of the graph succinctly that are efficient. To address the aforementioned challenges, in this paper, we propose using a well-known measure called the Kirchhoff index for extracting graph representations. The Kirchhoff index encapsulates the atomic structure of the graph because it incorporates the information regarding the number of paths and quality of paths. It features in many other contexts, such as Markov chains (the average commute time of a Markov chain on the graph), experiment design, and Euclidean distance embeddings [7], [8]. Recently, it has also been employed to quantify the resilience of networks based on noisy data in distributed networked control systems [9]. In fact, it has more equivalent descriptions including the Laplacian spectrum and the effective resistance which makes it suitable for extracting graph representations. Typically, Kirchhoff index is defined in terms of effective resistance using the analogy of graphs in an electrical circuit. A graph can be transformed into an electrical circuit by replacing each edge with a unit resistance. The effective resistance between any two nodes in such a network refers to the equivalent electrical resistance between those two nodes. The sum of the effective resistance between all pair of nodes is referred to as the Kirchhoff index of a graph. We note that a higher value of effective resistance between nodes indicates an inferior or less robust overall connection between the nodes (as measured by the number and quality of paths between them) [8], [10], [11], [7], [12]. Since there is always a path between two nodes in a connected network (that is, any two nodes are always connected, maybe via some other nodes), the effective resistance between any two nodes is well defined. Consequently, the effective resistance of a graph, which is the sum of all pair-wise effective resistances between nodes, is also well defined. To illustrate this, consider the example in Fig. 1.

In (b), there are four paths between nodes 1 and 2 with lengths $1,3,4$, and 5, and the effective resistance between the nodes is $0.72$. In (c), there are three paths between nodes 1 and 2 with lengths $1,3$, and 5, and the effective resistance between is $0.73$. We see only a slight increase in effective resistance (indicating a slightly less robust connection) as the path of length four does not exist anymore. The effect is minimal because the paths of smaller lengths, which are more important, are still there. In (d), there are only two paths between nodes 1 and 2 with lengths 3, and 5. The edge between nodes 1 and 2, hence the path of length 1, is removed. The effective resistance between the nodes increases significantly as the path with the shortest length having the maximum effect is deleted. In (e), there is only one path of length 3 between nodes 1 and 2, which increase the effective resistance from $2.75$ to $3.0$. Thus, we observe that effective resistance between two nodes (adjacent or non-adjacent) measure robustness in terms of the number and quality (length) of paths between nodes. Since Kirchhoff index measures the number as well as the quality of paths, we argue in this paper that this is a suitable measure to generate meaningful and expressive graph representations.

Tapping the recent advancements in solving linear systems in graph Laplacian and for measuring edge centrality in graphs, we propose an efficient method to compute the Kirchhoff index for graph classification task [13], [14]. The major contributions of this study are as follows.

• We propose an efficient Kirchhoff index based graph classification method that encapsulates graph structure in terms of the number and quality of paths between all pairs of nodes.

• We empirically show that proposed method provides improved results in terms of running time on large networks while the classification accuracy is close to the state-of-the-art methods that are, otherwise, impractical for large graphs.

• We are making our implementation publicly available for other researchers to use and improve upon.