A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees

Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erd˝os-R´enyi model, such as dependent edges and non-binomial degrees. In this paper, we use a characterization due to Erd˝os and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence.Statistic

Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs

Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent beta, and high clustering that can be controlled via the temperature T. We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to T = 0. We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation. Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input. Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases

Parallel Algorithms for Generating Random Networks with Given Degree Sequences

Random networks are widely used for modeling and analyzing complex processes. Many mathematical models have been proposed to capture diverse real-world networks. One of the most important aspects of these models is degree distribution. Chung--Lu (CL) model is a random network model, which can produce networks with any given arbitrary degree distribution. The complex systems we deal with nowadays are growing larger and more diverse than ever. Generating random networks with any given degree distribution consisting of billions of nodes and edges or more has become a necessity, which requires efficient and parallel algorithms. We present an MPI-based distributed memory parallel algorithm for generating massive random networks using CL model, which takes $O(\frac{m+n}{P}+P)$ time with high probability and $O(n)$ space per processor, where $n$, $m$, and $P$ are the number of nodes, edges and processors, respectively. The time efficiency is achieved by using a novel load-balancing algorithm. Our algorithms scale very well to a large number of processors and can generate massive power--law networks with one billion nodes and $250$ billion edges in one minute using $1024$ processors.Comment: Accepted in NPC 201

Enumerating Maximal Bicliques from a Large Graph using MapReduce

We consider the enumeration of maximal bipartite cliques (bicliques) from a large graph, a task central to many practical data mining problems in social network analysis and bioinformatics. We present novel parallel algorithms for the MapReduce platform, and an experimental evaluation using Hadoop MapReduce. Our algorithm is based on clustering the input graph into smaller sized subgraphs, followed by processing different subgraphs in parallel. Our algorithm uses two ideas that enable it to scale to large graphs: (1) the redundancy in work between different subgraph explorations is minimized through a careful pruning of the search space, and (2) the load on different reducers is balanced through the use of an appropriate total order among the vertices. Our evaluation shows that the algorithm scales to large graphs with millions of edges and tens of mil- lions of maximal bicliques. To our knowledge, this is the first work on maximal biclique enumeration for graphs of this scale.Comment: A preliminary version of the paper was accepted at the Proceedings of the 3rd IEEE International Congress on Big Data 201

Parallel Graph Decompositions Using Random Shifts

We show an improved parallel algorithm for decomposing an undirected unweighted graph into small diameter pieces with a small fraction of the edges in between. These decompositions form critical subroutines in a number of graph algorithms. Our algorithm builds upon the shifted shortest path approach introduced in [Blelloch, Gupta, Koutis, Miller, Peng, Tangwongsan, SPAA 2011]. By combining various stages of the previous algorithm, we obtain a significantly simpler algorithm with the same asymptotic guarantees as the best sequential algorithm