We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Matrix Factorization Techniques for Recommender Systems

What is Twitter

: Current systems for graph computation require a distributed computing cluster to handle very large real-world problems, such as analysis on social networks or the web graph. While distributed computational resources have become more accessible developing distributed graph algorithms still remains challenging, especially to non-experts. In this work, we present GraphChi, a disk-based system for computing efficiently on graphs with billions of edges. By using a well-known method to break large graphs into small parts, and a novel Parallel Sliding Windows algorithm, GraphChi is able to execute several advanced data mining, graph mining and machine learning algorithms on very large graphs, using just a single consumer-level computer. We show, through experiments and theoretical analysis, that GraphChi performs well on both SSDs and rotational hard drives. We build on the basis of Parallel Sliding Windows to propose a new data structure Partitioned Adjacency Lists, which we use to design an online graph database GraphChi-DB.We demonstrate that, on a single PC, GraphChi-DB can process over one hundred thousand graph updates per second, while simultaneously performing computation. GraphChi-DB compares favorably to existing graph databases, particularly on data that is much larger than the available memory. We evaluate our work both experimentally and theoretically. Based on the Parallel Sliding Windows algorithm, we propose new I/O efficient algorithms for solving fundamental graph problems. We also propose a novel algorithm for simulating billions of random walks in parallel on a single computer. By repeating experiments reported for existing distributed systems we show that with only fraction of the resources, GraphChi can solve the same problems in a very reasonable time. Our work makes large-scale graph computation available to anyone with a modern PC.

Large-scale Graph Computation on Just a PC

There has been significant recent interest in parallel frameworks for processing graphs due to their applicability in studying social networks, the Web graph, networks in biology, and unstructured meshes in scientific simulation. Due to the desire to process large graphs, these systems have emphasized the ability to run on distributed memory machines. Today, however, a single multicore server can support more than a terabyte of memory, which can fit graphs with tens or even hundreds of billions of edges. Furthermore, for graph algorithms, shared-memory multicores are generally significantly more efficient on a per core, per dollar, and per joule basis than distributed memory systems, and shared-memory algorithms tend to be simpler than their distributed counterparts.In this paper, we present a lightweight graph processing framework that is specific for shared-memory parallel/multicore machines, which makes graph traversal algorithms easy to write. The framework has two very simple routines, one for mapping over edges and one for mapping over vertices. Our routines can be applied to any subset of the vertices, which makes the framework useful for many graph traversal algorithms that operate on subsets of the vertices. Based on recent ideas used in a very fast algorithm for breadth-first search (BFS), our routines automatically adapt to the density of vertex sets. We implement several algorithms in this framework, including BFS, graph radii estimation, graph connectivity, betweenness centrality, PageRank and single-source shortest paths. Our algorithms expressed using this framework are very simple and concise, and perform almost as well as highly optimized code. Furthermore, they get good speedups on a 40-core machine and are significantly more efficient than previously reported results using graph frameworks on machines with many more cores.

Ligra: a lightweight graph processing framework for shared memory

Graphs are used to model many real objects such as social networks and web graphs. Many real applications in various fields require efficient and effective management of large-scale graph structured data. Although distributed graph engines such as GBase and Pregel handle billion-scale graphs, the user needs to be skilled at managing and tuning a distributed system in a cluster, which is a nontrivial job for the ordinary user. Furthermore, these distributed systems need many machines in a cluster in order to provide reasonable performance. In order to address this problem, a disk-based parallel graph engine called Graph-Chi, has been recently proposed. Although Graph-Chi significantly outperforms all representative (disk-based) distributed graph engines, we observe that Graph-Chi still has serious performance problems for many important types of graph queries due to 1) limited parallelism and 2) separate steps for I/O processing and CPU processing. In this paper, we propose a general, disk-based graph engine called TurboGraph to process billion-scale graphs very efficiently by using modern hardware on a single PC. TurboGraph is the first truly parallel graph engine that exploits 1) full parallelism including multi-core parallelism and FlashSSD IO parallelism and 2) full overlap of CPU processing and I/O processing as much as possible. Specifically, we propose a novel parallel execution model, called pin-and-slide. TurboGraph also provides engine-level operators such as BFS which are implemented under the pin-and-slide model. Extensive experimental results with large real datasets show that TurboGraph consistently and significantly outperforms Graph-Chi by up to four orders of magnitude! Our implementation of TurboGraph is available at ``http://wshan.net/turbograph}" as executable files.

/pdf/turbograph-a-fast-parallel-graph-engine-handling-billion-1awhx24vmi.pdf

TurboGraph: a fast parallel graph engine handling billion-scale graphs in a single PC

As social network services connect people across the world, influence maximization, i.e., finding the most influential nodes (or individuals) in the network, is being actively researched with applications to viral marketing. One crucial challenge in scalable influence maximization processing is evaluating influence, which is #P-hard and thus hard to solve in polynomial time. We propose a scalable influence approximation algorithm, Independent Path Algorithm (IPA) for Independent Cascade (IC) diffusion model. IPA efficiently approximates influence by considering an independent influence path as an influence evaluation unit. IPA are also easily parallelized by simply adding a few lines of OpenMP meta-programming expressions. Also, overhead of maintaining influence paths in memory is relieved by safely throwing away insignificant influence paths. Extensive experiments conducted on large-scale real social networks show that IPA is an order of magnitude faster and uses less memory than the state of the art algorithms. Our experimental results also show that parallel versions of IPA speeds up further as the number of CPU cores increases, and more speed-up is achieved for larger datasets. The algorithms have been implemented in our demo application for influence maximization (available at http://dm.postech.ac.kr/ipa demo), which efficiently finds the most influential nodes in a social network.

Scalable and parallelizable processing of influence maximization for large-scale social networks?

Graph-based approaches to data analysis have become more widespread, which has given need for a query language for graphs. Such a graph query language needs not only SQL-like functionality for querying structured data, but also intrinsic support for typical graph-style applications: reachability analysis, path finding and graph construction.We propose a new query language for the popular Property Graph (PG) data model: the Property Graph Query Language (PGQL). PGQL is based on the paradigm of graph pattern matching, closely follows syntactic structures of SQL, and provides regular path queries with conditions on labels and properties to allow for reachability and path finding queries. Besides intrinsic vertex, edge and path types, PGQL also has the graph as intrinsic type and allows for graph construction and query composition.

PGQL: a property graph query language

With the advent of multicore processors,it has become imperative to write parallel programs if one wishes to exploit the next generation of processors. This paper deals with skyline computation as a case study of parallelizing database operations on multicore architectures. We compare two parallel skyline algorithms: a parallel version of the branch-and-bound algorithm (BBS) and a new parallel algorithm based on skeletal parallel programming. Experimental results show despite its simple design, the new parallel algorithm is comparable to parallel BBS in speed. For sequential skyline computation, the new algorithm far outperforms sequential BBS when the density of skyline tuples is low.

/pdf/parallel-skyline-computation-on-multicore-architectures-43ad8lu0j8.pdf

Parallel Skyline Computation on Multicore Architectures

RDF data are used to model knowledge in various areas such as life sciences, Semantic Web, bioinformatics, and social graphs. The size of real RDF data reaches billions of triples. This calls for a framework for efficiently processing RDF data. The core function of processing RDF data is subgraph pattern matching. There have been two completely different directions for supporting efficient subgraph pattern matching. One direction is to develop specialized RDF query processing engines exploiting the properties of RDF data for the last decade, while the other direction is to develop efficient subgraph isomorphism algorithms for general, labeled graphs for over 30 years. Although both directions have a similar goal (i.e., finding subgraphs in data graphs for a given query graph), they have been independently researched without clear reason. We argue that a subgraph isomorphism algorithm can be easily modified to handle the graph homomorphism, which is the RDF pattern matching semantics, by just removing the injectivity constraint. In this paper, based on the state-of-the-art subgraph isomorphism algorithm, we propose an in-memory solution, TurboHOM++, which is tamed for the RDF processing, and we compare it with the representative RDF processing engines for several RDF benchmarks in a server machine where billions of triples can be loaded in memory. In order to speed up TurboHOM++, we also provide a simple yet effective transformation and a series of optimization techniques. Extensive experiments using several RDF benchmarks show that TurboHOM++consistently and significantly outperforms the representative RDF engines. Specifically, TurboHOM++ outperforms its competitors by up to five orders of magnitude.

/pdf/taming-subgraph-isomorphism-for-rdf-query-processing-2hf1z9agwz.pdf

Jinha Kim

Papers

TurboGraph: a fast parallel graph engine handling billion-scale graphs in a single PC

Scalable and parallelizable processing of influence maximization for large-scale social networks?

PGQL: a property graph query language

Parallel Skyline Computation on Multicore Architectures

Taming subgraph isomorphism for RDF query processing