Many ``real-world'' networks are clearly defined while most ``social'' networks are to some extent subjective. Indeed, the accuracy of empirically-determined social networks is a question of some concern because individuals may have distinct perceptions of what constitutes a social link. One unambiguous type of connection is sexual contact. Here we analyze data on the sexual behavior of a random sample of individuals, and find that the cumulative distributions of the number of sexual partners during the twelve months prior to the survey decays as a power law with similar exponents $\alpha \approx 2.4$ for females and males. The scale-free nature of the web of human sexual contacts suggests that strategic interventions aimed at preventing the spread of sexually-transmitted diseases may be the most efficient approach.

The Web of Human Sexual Contacts

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Computers And Intractability A Guide To The Theory Of Np Completeness

We propose a new exact method for shortest-path distance queries on large-scale networks. Our method precomputes distance labels for vertices by performing a breadth-first search from every vertex. Seemingly too obvious and too inefficient at first glance, the key ingredient introduced here is pruning during breadth-first searches. While we can still answer the correct distance for any pair of vertices from the labels, it surprisingly reduces the search space and sizes of labels. Moreover, we show that we can perform 32 or 64 breadth-first searches simultaneously exploiting bitwise operations. We experimentally demonstrate that the combination of these two techniques is efficient and robust on various kinds of large-scale real-world networks. In particular, our method can handle social networks and web graphs with hundreds of millions of edges, which are two orders of magnitude larger than the limits of previous exact methods, with comparable query time to those of previous methods.

Fast Exact Shortest-Path Distance Queries on Large Networks by Pruned Landmark Labeling

Fast exact shortest-path distance queries on large networks by pruned landmark labeling

We consider the point-to-point (approximate) shortest-path query problem, which is the following generalization of the classical single-source (SSSP) and all-pairs shortest-path (APSP) problems: we are first presented with a network (graph). A so-called preprocessing algorithm may compute certain information (a data structure or index) to prepare for the next phase. After this preprocessing step, applications may ask shortest-path or distance queries, which should be answered as fast as possible.Due to its many applications in areas such as transportation, networking, and social science, this problem has been considered by researchers from various communities (sometimes under different names): algorithm engineers construct fast route planning methods; database and information systems researchers investigate materialization tradeoffs, query processing on spatial networks, and reachability queries; and theoretical computer scientists analyze distance oracles and sparse spanners. Related problems are considered for compact routing and distance labeling schemes in networking and distributed computing and for metric embeddings in geometry as well.In this survey, we review selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time. We survey methods for general graphs as well as specialized methods for restricted graph classes, in particular for those classes with arguable practical significance such as planar graphs and complex networks.

/pdf/shortest-path-queries-in-static-networks-t3dlcej7at.pdf

Shortest-path queries in static networks

Shortest distance query is a fundamental operation in large-scale networks. Many existing methods in the literature take a landmark embedding approach, which selects a set of graph nodes as landmarks and computes the shortest distances from each landmark to all nodes as an embedding. To answer a shortest distance query, the precomputed distances from the landmarks to the two query nodes are used to compute an approximate shortest distance based on the triangle inequality. In this paper, we analyze the factors that affect the accuracy of distance estimation in landmark embedding. In particular, we find that a globally selected, query-independent landmark set may introduce a large relative error, especially for nearby query nodes. To address this issue, we propose a query-dependent local landmark scheme, which identifies a local landmark close to both query nodes and provides more accurate distance estimation than the traditional global landmark approach. We propose efficient local landmark indexing and retrieval techniques, which achieve low offline indexing complexity and online query complexity. Two optimization techniques on graph compression and graph online search are also proposed, with the goal of further reducing index size and improving query accuracy. Furthermore, the challenge of immense graphs whose index may not fit in the memory leads us to store the embedding in relational database, so that a query of the local landmark scheme can be expressed with relational operators. Effective indexing and query optimization mechanisms are designed in this context. Our experimental results on large-scale social networks and road networks demonstrate that the local landmark scheme reduces the shortest distance estimation error significantly when compared with global landmark embedding and the state-of-the-art sketch-based embedding.

/pdf/approximate-shortest-distance-computing-a-query-dependent-4ss03utxzk.pdf

Approximate Shortest Distance Computing: A Query-Dependent Local Landmark Scheme

Subgraph matching finds a set I of all occurrences of a pattern graph in a target graph. It has a wide range of applications while suffers an expensive computation. This efficiency issue has been studied extensively. All existing approaches, however, turn a blind eye to the output crisis, that is, when the system has to materialize I as a preprocessing/intermediate/final result or an index, the cost of the export of I dominates the overall cost, which could be prohibitive even for a small pattern graph.This paper studies subgraph matching via two problems. 1) Is there an ideal compression of I? 2) Will the compression of I reversely boost the computation of I? For the problem 1), we propose a technique called VCBC to compress I to code (I) which serves effectively the same as I. For problem 2), we propose a subgraph matching computation framework CBF which computes code(I)instead of I to bring down the output cost. CBF further reduces the overall cost by reducing the intermediate results. Extensive experiments show that the compression ratio of VCBC can be up to 105 which also significantly lowers the output cost of CBF. Extensive experiments show the superior performance of CBF over existing approaches.

/pdf/subgraph-matching-on-compression-and-computation-lqatqjv9d4.pdf

Subgraph matching: on compression and computation

Shortest distance queries are essential not only in graph analysis and graph mining tasks but also in database applications, when a large graph needs to be dealt with. Such shortest distance queries are frequently issued by end-users or requested as a subroutine in real applications. For intensive queries on large graphs, it is impractical to compute shortest distances on-line from scratch, and impractical to materialize all-pairs shortest distances. In the literature, 2-hop distance labeling is proposed to index the all-pairs shortest distances. It assigns distance labels to vertices in a large graph in a pre-computing step off-line and then answers shortest distance queries on-line by making use of such distance labels, which avoids exhaustively traversing the large graph when answering queries. However, the existing algorithms to generate 2-hop distance labels are not scalable to large graphs. Finding an optimal 2-hop distance labeling is NP-hard, and heuristic algorithms may generate large size distance labels while still needing to pre-compute all-pairs shortest paths. In this paper, we propose a multi-hop distance labeling approach, which generates a subset of the 2-hop distance labels as index off-line. We can compute the multi-hop distance labels efficiently by avoiding pre-computing all-pairs shortest paths. In addition, our multi-hop distance labeling is small in size to be stored. To answer a shortest distance query between two vertices, we first generate the query-specific small set of 2-hop distance labels for the two vertices based on our multi-hop distance labels stored and compute the shortest distance between the two vertices based on the 2-hop distance labels generated on-line. We conducted extensive performance studies on large real graphs and confirmed the efficiency of our multi-hop distance labeling scheme.

/pdf/the-exact-distance-to-destination-in-undirected-world-1rm3rr9uu2.pdf

The exact distance to destination in undirected world

Distance labeling approaches are widely adopted to speed up the online performance of shortest distance queries. The construction of the distance labeling, however, can be exhaustive especially on big graphs. For a major category of large graphs, small-world networks, the state-of-the-art approach is Pruned Landmark Labeling (PLL). PLL prunes distance labels based on a node order and directly constructs the pruned labels by performing breadth-first searches in the node order. The pruning technique, as well as the index construction, has a strong sequential nature which hinders PLL from being parallelized. It becomes an urgent issue on massive small-world networks whose index can hardly be constructed by a single thread within a reasonable time. This paper scales distance labeling on small-world networks by proposing a Parallel Shortest-distance Labeling (PSL) scheme and further reducing the index size by exploiting graph and label properties. PSL insightfully converts the PLL's node-order dependency to a shortest-distance dependence, which leads to a propagation-based parallel labeling in D rounds where D denotes the diameter of the graph. Extensive experimental results verify our efficiency on billion-scale graphs and near-linear speedup in a multi-core environment.

/pdf/scaling-distance-labeling-on-small-world-networks-290ktkxjda.pdf

Scaling Distance Labeling on Small-World Networks

It is quite common for networks emerging nowadays to have labels or textual contents on the nodes. On such networks, we study the problem of top-k nearest keyword (k-NK) search. In a network G modeled as an undirected graph, each node is attached with zero or more keywords, and each edge is assigned with a weight measuring its length. Given a query node q in G and a keyword λ, a k-NK query seeks k nodes which contain λ and are nearest to q. k-NK is not only useful as a stand-alone query but also as a building block for tackling complex graph pattern matching problems.The key to an accurate k-NK result is a precise shortest distance estimation in a graph. Based on the latest distance oracle technique, we build a shortest path tree for a distance oracle and use the tree distance as a more accurate estimation. With such representation, the original k-NK query on a graph can be reduced to answering the query on a set of trees and then assembling the results obtained from the trees. We propose two efficient algorithms to report the exact k-NK result on a tree. One is query time optimized for a scenario when a small number of result nodes are of interest to users. The other handles k-NK queries for an arbitrarily large k efficiently. In obtaining a k-NK result on a graph from that on trees, a global storage technique is proposed to further reduce the index size and the query time. Extensive experimental results conform with our theoretical findings, and demonstrate the effectiveness and efficiency of our k-NK algorithms on large real graphs.

/pdf/top-k-nearest-keyword-search-on-large-graphs-s4wzy6njzi.pdf

Miao Qiao

Papers

Approximate Shortest Distance Computing: A Query-Dependent Local Landmark Scheme

Subgraph matching: on compression and computation

The exact distance to destination in undirected world

Scaling Distance Labeling on Small-World Networks

Top-K nearest keyword search on large graphs