Abstract-Analysis of large networks is a critical component of many of today's application environments, including online social networks, protein interactions in biological networks, and Internet traffic analysis. The arrival of massive network graphs with hundreds of millions of nodes, e.g. social graphs, presents a unique challenge to graph analysis applications. Most of these applications rely on computing distances between node pairs, which for large graphs can take minutes to compute using traditional algorithms such as breadth-first-search (BFS). In this paper, we study ways to enable scalable graph process ing for today's massive networks. We explore the design space of graph coordinate systems, a new approach that accurately approximates node distances in constant time by embedding graphs into coordinate spaces. We show that a hyperbolic embedding produces relatively low distortion error, and propose Rigel, a hyperbolic graph coordinate system that lends itself to efficient parallelization across a compute cluster. Rigel produces significantly more accurate results than prior systems, and is naturally parallelizable across compute clusters, allowing it to provide accurate results for graphs up to 43 million nodes. Finally, we show that Rigel's functionality can be easily extended to locate (near-) shortest paths between node pairs. After a one time preprocessing cost, Rigel answers node-distance queries in 10's of microseconds, and also produces shortest path results up to 18 times faster than prior shortest-path systems with similar levels of accuracy. Index Terms-Online Social Network, Graph Coordinate Sys tem. Fast and scalable analysis of large networks is a critical component of many of today's application environments, in cluding online social networks, biological protein interaction networks, and analysis of the Internet router backbone. For example, a social game network might search for "central" users to help deploy new games, while a social auction site [1] wants to tell a buyer if a specific item is being auctioned by someone in her social circles. Ideally, such queries should be answered quickly, regardless of the size of the graph, or even if graphs themselves are changing over time. Unfortunately, these goals are simply unattainable for to day's online social networks. This is because numerous graph analysis problems such as centrality computation, node sepa ration, and community detection all rely on the simple node distance (length of shortest path) primitive, which scales badly with graph (or network) size. For graphs generated from social networks such as Facebook (700 million nodes), Linkedin (l00 million) and Twitter (200 million), computing the shortest path distance between a single pair of nodes can take a minute or 77 more using traditional algorithms such as breadth-first-search (BFS) [2]. Similarly, variants such as Dijkstra and Floyd Warshall also fail to scale to these network sizes. Without an efficient alternative for node distance compu tation, recent work has focused on exploring efficient ap proximation algorithms [2], [3], [4]. Our prior work [4], described the idea of graph coordinate systems, which embeds graph nodes into points on a coordinate system. The resulting coordinates can be used to quickly approximate node distance queries on the original graph. Our initial system, Orion, was a centralized system that approximated node distances by mapping nodes to the Euclidean coordinate system [4]. It has several limitations in practice. First, Orion's initial graph embedding process is centralized and computationally expensive, which presents a significant performance bottleneck for larger graphs. Second, Orion's results produce error rates between 15% and 20%, which limits the types of applications it can serve. Finally, it is unable to produce actual paths connecting node pairs, which is often necessary for a number of graph applications. In this work, we seek to extend work on graph coordinate systems by developing a practical system that provides signifi cant improvement in accuracy, scalability, and extended func tionality. We systematically explore decisions in the design of a graph coordinate system, and make two key observations. First, we propose to extend our work on graph coordinate systems, by embedding large graphs in a hyperbolic space for lower distance distortion errors. Our embedding algorithm naturally parallelizes the costly embedding process across multiple servers, allowing our system to quickly embed multi million node graphs. Second, we propose a novel way to use graph coordinates to efficiently locate shortest paths between node pairs. The result of our work is Rigel, a hyperbolic graph coordinate system that supports queries for both node distance and shortest paths on today's large social graphs. After a one-time, easily parallelizable, preprocessing phase, Rigel can resolve queries in tens of microseconds, even for massive social graphs up to 43 million nodes. Our paper describes four key contributions. • In Sections III and IV, we describe the detailed design of Rigel, and show how we can minimize embedding time by effectively parallelizing the most computationally expensive parts of the graph embedding process. • We evaluate a distributed prlW�tt��17tfSilQrkJ6lYJl
The number of landmarks should be equal to, or even higher than the dimension of the embedding space REF .
Efficient shortest paths on massive social graphs
7th International Conference on Collaborative Computing: Networking, Applications and Worksharing (CollaborateCom)
7th International Conference on Collaborative Computing: Networking, Applications and Worksharing (CollaborateCom)
Computer Science