Scalable Approximation Algorithm for Graph Summarization

Abstract

Massive sizes of real-world graphs, such as social networks and web graph, impose serious challenges to process and perform analytics on them. These issues can be resolved by working on a small summary of the graph instead . A summary is a compressed version of the graph that removes several details, yet preserves it's essential structure. Generally, some predefined quality measure of the summary is optimized to bound the approximation error incurred by working on the summary instead of the whole graph. All known summarization algorithms are computationally prohibitive and do not scale to large graphs. In this paper we present an efficient randomized algorithm to compute graph summaries with the goal to minimize reconstruction error. We propose a novel weighted sampling scheme to sample vertices for merging that will result in the least reconstruction error. We provide analytical bounds on the running time of the algorithm and prove approximation guarantee for our score computation. Efficiency of our algorithm makes it scalable to very large graphs on which known algorithms cannot be applied. We test our algorithm on several real world graphs to empirically demonstrate the quality of summaries produced and compare to state of the art algorithms. We use the summaries to answer several structural queries about original graph and report their accuracies.Graph summarization and compression is a widely studied problem and has applications in diverse domains. There are broadly two types of graph summaries (represented as supergraphs described above): lossless and lossy. The original

g-Sum: A Graph Summarization Approach for a Single Large Social Network

“…Web-Stanford Network: Nodes represent pages from Stanford University and edges represent hyperlinks between the pages. There are 2,81,903 nodes and 19,92,636 edges in the dataset [24][25][26].…”

Querying and mining complex graphs : uncertainty and multi-relations

“…Aggregation-based graph summary. Notable techniques under this category are pattern mining and community based summarization [27,148], supernode and edgecorrection (thus lossless) [134,156], supernode and reconstruction-error (thus lossy) [16,112,146]. Supernode based aggregation methods [16,112,134,146,156] are most similar to ours and are summarised in Table 6.1.…”

“…We show the complete procedure in Algorithm 1. It initializes a heap h v for every visited node v, and pushes v's each out-neighbor nbr with a geometric random instance, i.e., nbr, X(nbr) into h v (lines [12][13][14][15][16][17][18]. It also maintains a counter c v to keep track of the number of times v has been visited.…”

“…Extract the set of edges E P on path P We ensure that the number of included candidate edges from E + in P 1 does not exceed k during the process (line [11][12][13][14][15][16]. The included candidate edges in G * are reported as our solution.…”

“…For a lossy summarization [16,112,146], only the graph summary G S = (V S , E S ) is created; no additional correction list is stored. The summary G S is a complete graph in this case, including all self-loops, i.e., E S = V S × V S .…”