The k-core of a graph is the largest induced subgraph with minimum degree k. The k-core decomposition is to find the core number of each vertex in a graph, which is the largest value of k that the vertex belongs to a k-core. k-core decomposition has applications in many areas including network analysis, computational biology and graph visualization. The primary reason for it being widely used is the availability of an O(n + m) algorithm. The algorithm was proposed by Batagelj and Zaversnik and is considered the state-of-the-art algorithm for k-core decomposition. However, the algorithm is not suitable for parallelization and to the best of our knowledge there is no algorithm proposed for k-core decomposition on multicore processors. Also, the algorithm has not been experimentally analyzed for large graphs. Since the working set size of the algorithm is large, and the access pattern is highly random, it can be inefficient for large graphs. In this paper, we present an experimental analysis of the algorithm of Batagelj and Zaversnik and propose a new algorithm, ParK, that significantly reduces the working set size and minimizes the random accesses. We provide an experimental analysis of the algorithm using graphs with up to 65 million vertices and 1.8 billion edges. We compare the ParK algorithm with state-of-the-art algorithm and show that it is up to 6 times faster. We also provide a parallel methodology and show that the algorithm is amenable to parallelization on multicore architectures. We ran our experiments on a 4 socket Nehalem-EX processor which has 8 cores per socket and show that the algorithm scales up to 21 times using 32 cores.

ParK: An efficient algorithm for k-core decomposition on multicore processors

k-core decomposition is a social network analytic that can be applied to centrality analysis, visualization, and community detection. There is a simple and well-known linear time sequential algorithm to perform k-core decomposition, and for this reason, k-core decomposition is widely used as a network analytic. In this work, we present a new shared-memory parallel algorithm called PKC for k-core decomposition on multicore platforms. This approach improves on the state-of-the-art implementations for k-core decomposition algorithms by reducing synchronization overhead and creating a smaller graph to process high-degree vertices. We show that PKC consistently outperforms implementations of other methods on a 32-core multicore server and on a collection of large sparse graphs. We achieve a 2.81× speedup (geometric mean of speedups for 17 large graphs) over our implementation of the ParK k-core decomposition algorithm.

Parallel k-Core Decomposition on Multicore Platforms

The $k$ -core, a kind of structure of graphs, is a maximal connected subgraph with the minimum degree greater than or equal to $k$ , and has been used in many fields. The maximum $k$ such that a $k$ -core contains $u$ is the $K$ value of $u$ . Especially, for an edge-weighted graph, the degree of a vertex is the sum of weights of all its incident edges. The core decomposition problem on static graphs and the core maintenance problem on dynamic graphs have been studied in unweighted graphs. We improve the core decomposition algorithm to suit edge-weighted graphs, but it costs too much to update $K$ values of all vertices after the change of large graphs by using it directly. Then we find a small subgraph $H$ which contains all vertices whose $K$ values will change after the change of graphs. By operating on $H$ , the cost will be greatly reduced. Next, we design core maintenance algorithms for edge-weighted graphs in both insertion and deletion cases, which is the major work in this paper. In those core maintenance algorithms, a hierarchical process is added, which help us determine the new $K$ values of vertices in $H$ from the small ones to high. Finally, we conduct extensive experiments on real-world graphs to show the effectiveness and the efficiency that our algorithms have. The results show that our algorithms have the best performance.

Incremental Algorithms of the Core Maintenance Problem on Edge-Weighted Graphs

Wikipedia has seen a huge expansion of content since its inception. Pages within this online
encyclopedia are organised by assigning them to one or more categories, where Wikipedia
maintains a manually constructed taxonomy graph that encodes the semantic relationship
between these categories. An alternative, called the category co-occurrence graph, can be
produced automatically by linking together categories that have pages in common. Properties
of the latter graph and its relationship to the former is the concern of this thesis.
The analytic framework, called t-component, is introduced to formalise the graphs and
discover category clusters connecting relevant categories together. The m-core, a cohesive
subgroup concept as a clustering model, is used to construct a subgraph depending on the
number of shared pages between the categories exceeding a given threshold t. The significant
of the clustering result of the m-core is validated using a permutation test. This is compared
to the k-core, another clustering model.
TheWikipedia category co-occurrence graphs are scale-free with a few category hubs and
the majority of clusters are size 2. All observed properties for the distribution of the largest
clusters of the category graphs obey power-laws with decay exponent averages around 1.
As the threshold t of the number of shared pages is increased, eventually a critical threshold
is reached when the largest cluster shrinks significantly in size. This phenomena is only
exhibited for the m-core but not the k-core. Lastly, the clustering in the category graph
is shown to be consistent with the distance between categories in the taxonomy graph.

Analysis of category co-occurrence in Wikipedia networks

Effective and efficient core computation in signed networks

The k-core of a graph is the maximal subgraph with minimum vertex degree k. When the core with the largest k is found, it can be removed and the next core can be found. In each of these cores (or shells), connected components have been enumerated in an existing graph clustering method. Here we propose a method to find graph clusters that are not just k-cores but also made of r-cliques. Just like the existing k-core clustering approach, our approach is also based on vertex and edge removals that can be made completely using local information (within the neighborhood). Our local and recursive algorithms are based on new graph theoretical insights that relate concepts including neighborhood, clique percolation, and k-core. We demonstrate our method with a gene network from PubMed data and compare our results with that from the existing k-core approach and use MeSH and TF-IDF ranking to show the differences in subject enrichment our approach can make.

Local k-core clustering for gene networks

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Local k-core clustering for gene networks