Triangle Counting Algorithm Research Articles

A distributed streaming framework for edge–cloud triangle counting in graph streams

The triangle counting problem in graph streams has been extensively studied in social network analysis, recommendation systems, user portraits and other fields. However, cloud computing based streaming algorithms cause high bandwidth occupation and long transmission latency due to limited bandwidth of the cloud. Recently, edge computing is promising to overcome the issue of transmitting large-scale data for cloud computing. However, directly applying edge computing in streaming triangle counting will reduce the accuracy of the triangle count estimation, due to the limitation of local computing at the edge network. We term the cooperations between edge computing and cloud computing for streaming triangle counting as edge–cloud triangle counting in graph streams. In this paper, we first propose a streaming framework for edge–cloud triangle counting in graph streams. Then, we propose a streaming triangle counting algorithm called Trie-based Edge Compression (TbEC) by using the binary trie at the edge network that enables lossless compression and efficient transmission to the cloud. In addition, to extend our algorithms for triangle counting in multigraphs, we present a dual deduplication strategy collaboratively using the trie-based data structure and a Bloom Filter. Our experiments with real-world datasets show that TbEC is (a) Accurate: yielding up to 3.35×more accurate smaller estimation error than the state-of-the-art distributed streaming algorithm, (b) Fast: yielding up to 10.59×faster than the state-of-the-art distributed streaming algorithm, (c) Scalable: scaling linearly with the number of edges in the input graph stream.

A Block-Based Triangle Counting Algorithm on Heterogeneous Environments

Triangle counting is a fundamental building block in graph algorithms. In this article, we propose a block-based triangle counting algorithm to reduce data movement during both sequential and parallel execution. Our block-based formulation makes the algorithm naturally suitable for heterogeneous architectures. The problem of partitioning the adjacency matrix of a graph is well-studied. Our task decomposition goes one step further: it partitions the set of triangles in the graph. By streaming these small tasks to compute resources, we can solve problems that do not fit on a device. We demonstrate the effectiveness of our approach by providing an implementation on a compute node with multiple sockets, cores and GPUs. The current state-of-the-art in triangle enumeration processes the Friendster graph in 2.1 seconds, not including data copy time between CPU and GPU. Using that metric, our approach is 20 percent faster. When copy times are included, our algorithm takes 3.2 seconds. This is 5.6 times faster than the fastest published CPU-only time.

A Note on Detection of Communities in Social Networks

The modern Science of Social Networks has brought significant advances to our understanding of the Structure, dynamics and evolution of the Network. One of the important features of graphs representing the Social Networks is community structure. The communities can be considered as fairly independent components of the social graph that helps identify groups of users with similar interests, locations, friends, or occupations. The community structure is closely tied to triangles and their count forms the basis of community detection algorithms. The present work takes into consideration, a triangle instead of the edge as the basic indicator of a strong relation in the social graph. A simple triangle counting algorithm is then used to evaluate different metrics that are employed to detect strong communities. The methodology is applied to Zachary Social network and discussed. The results bring out the usefulness of counting triangles in a network to detect strong communities in a Social Network.

Memory-Efficient and Accurate Sampling for Counting Local Triangles in Graph Streams

How can we estimate local triangle counts accurately in a graph stream without storing the whole graph? How to handle duplicated edges in local triangle counting for graph stream? Local triangle counting, which computes the number of triangles attached to each node in a graph, is a very important problem with wide applications in social network analysis, anomaly detection, web mining, and the like. In this article, we propose algorithms for local triangle counting in a graph stream based on edge sampling: M ascot for a simple graph, and M ulti BM ascot and M ulti WM ascot for a multigraph. To develop M ascot , we first present two naive local triangle counting algorithms in a graph stream, called M ascot -C and M ascot -A. M ascot -C is based on constant edge sampling, and M ascot -A improves its accuracy by utilizing more memory spaces. M ascot achieves both accuracy and memory-efficiency of the two algorithms by unconditional triangle counting for a new edge, regardless of whether it is sampled or not. Extending the idea to a multigraph, we develop two algorithms M ulti BM ascot and M ulti WM ascot . M ulti BM ascot enables local triangle counting on the corresponding simple graph of a streamed multigraph without explicit graph conversion; M ulti WM ascot considers repeated occurrences of an edge as its weight and counts each triangle as the product of its three edge weights. In contrast to the existing algorithm that requires prior knowledge on the target graph and appropriately set parameters, our proposed algorithms require only one parameter of edge sampling probability. Through extensive experiments, we show that for the same number of edges sampled, M ascot provides the best accuracy compared to the existing algorithm as well as M ascot -C and M ascot -A. We also demonstrate that M ulti BM ascot on a multigraph is comparable to M ascot -C on the counterpart simple graph, and M ulti WM ascot becomes more accurate for higher degree nodes. Thanks to M ascot , we also discover interesting anomalous patterns in real graphs, including core-peripheries in the web, a bimodal call pattern in a phone call history, and intensive collaboration in DBLP.

Counting Triangles in Large Graphs by Random Sampling

The problem of counting triangles in graphs has been well studied in the literature. However, all existing algorithms, exact or approximate, spend at least linear time in the size of the graph (except a recent theoretical result), which can be prohibitive on today's large graphs. Nevertheless, we observe that the ideas in many existing triangle counting algorithms can be coupled with random sampling to yield potentially sublinear-time algorithms that return an approximation of the triangle count without looking at the whole graph. This paper makes these random sampling algorithms more explicit, and presents an experimental and analytical comparison of different approaches, identifying the best performers among a number of candidates.

Triangle Counting in Dynamic Graph Streams

Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in an arbitrary order or as incidence lists. However, with a few exceptions, the algorithms have considered insert-only streams. We present a new algorithm estimating the number of triangles in dynamic graph streams where edges can be both inserted and deleted. We show that our algorithm achieves better time and space complexity than previous solutions for various graph classes, for example sparse graphs with a relatively small number of triangles. Also, for graphs with constant transitivity coefficient, a common situation in real graphs, this is the first algorithm achieving constant processing time per edge. The result is achieved by a novel approach combining sampling of vertex triples and sparsification of the input graph. In the course of the analysis of the algorithm we present a lower bound on the number of pairwise independent 2-paths in general graphs which might be of independent interest. At the end of the paper we discuss lower bounds on the space complexity of triangle counting algorithms that make no assumptions on the structure of the graph.

Counting triangles in real-world networks using projections

Triangle counting is an important problem in graph mining. Two frequently used metrics in complex network analysis that require the count of triangles are the clustering coefficients and the transitivity ratio of the graph. Triangles have been used successfully in several real-world applications, such as detection of spamming activity, uncovering the hidden thematic structure of the web and link recommendation in online social networks. Furthermore, the count of triangles is a frequently used network statistic in exponential random graph models. However, counting the number of triangles in a graph is computationally expensive. In this paper, we propose the EigenTriangle and EigenTriangleLocal algorithms to estimate the number of triangles in a graph. The efficiency of our algorithms is based on the special spectral properties of real-world networks, which allow us to approximate accurately the number of triangles. We verify the efficacy of our method experimentally in almost 160 experiments using several Web Graphs, social, co-authorship, information, and Internet networks where we obtain significant speedups with respect to a straightforward triangle counting algorithm. Furthermore, we propose an algorithm based on Fast SVD which allows us to apply the core idea of the EigenTriangle algorithm on graphs which do not fit in the main memory. The main idea is a simple node-sampling process according to which node i is selected with probability $${\frac{d_i}{2m}}$$ where d i is the degree of node i and m is the total number of edges in the graph. Our theoretical contributions also include a theorem that gives a closed formula for the number of triangles in Kronecker graphs, a model of networks which mimics several properties of real-world networks.