Triangle Estimation Using Tripartite Independent Set Queries

We present a new randomized algorithm for estimating the number of triangles in massive graphs revealed as a stream of edges in arbitrary order. It exploits the fact that graphs arising from various domains often have small vertex covers, which enables us to reduce the space usage and sample complexity of triangle counting algorithms. The algorithm runs in four passes over the edge set and uses constant processing time per edge. We obtain precise bounds on the complexity and the approximation guarantee of our algorithm. For graphs where even the minimum vertex cover is prohibitively large, we extend the approach to achieve a trade-off between the number of passes and the space usage. Experiments on real-world graphs validate our theoretical analysis and show that the new algorithm yields more accurate estimates than state-ofthe-art approaches using the same amount of space.

2010

In this article, we study the problem of approximate local triangle counting in large graphs. Namely, given a large graph G = (V, E) we want to estimate as accurately as possible the number of triangles incident to every node v ∈ V in the graph. We consider the question both for undirected and directed graphs. The problem of computing the global number of triangles in a graph has been considered before, but to our knowledge this is the first contribution that addresses the problem of approximate local triangle counting with a focus on the efficiency issues arising in massive graphs and that also considers the directed case. The distribution of the local number of triangles and the related local clustering coefficient can be used in many interesting applications. For example, we show that the measures we compute can help detect the presence of spamming activity in largescale Web graphs, as well as to provide useful features for content quality assessment in social networks.

Lecture Notes in Computer Science, 2013

The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. We study the amount of memory required by a randomized algorithm to solve this problem. In case the algorithm is allowed one pass over the stream, we present a best possible lower bound of Ω(m) for graphs G with m edges on n vertices. If a constant number of passes is allowed, we show a lower bound of Ω(m/T), T the number of triangles. We match, in some sense, this lower bound with a 2-pass O(m/T 1/3)-memory algorithm that solves the problem of distinguishing graphs with no triangles from graphs with at least T triangles. We present a new graph parameter ρ(G)-the triangle density, and conjecture that the space complexity of the triangles problem is Ω(m/ρ(G)). We match this by a second algorithm that solves the distinguishing problem using O(m/ρ(G))-memory.

2005

We present three streaming algorithms that (ε, δ)–approximate the number of triangles in graphs. Similar to the previous algorithms [3], the space usage of presented algorithms are inversely proportional to the number of triangles while, for some families of graphs, the space usage is improved. We also prove a lower bound, based on the number of triangles, which indicates that our first algorithm behaves almost optimally on graphs with constant degrees.

Proceeding of the 14th ACM …, 2008

In this paper we study the problem of local triangle counting in large graphs. Namely, given a large graph G=(V; E) we want to estimate as accurately as possible the number of triangles incident to every node υ∈ V in the graph. The problem of computing the global number of triangles in a graph has been considered before, but to our knowledge this is the first paper that addresses the problem of local triangle counting with a focus on the efficiency issues arising in massive graphs. The distribution of the local number of ...

Proceedings of the VLDB Endowment

A bipartite network is a network with two disjoint vertex sets and its edges only exist between vertices from different sets. It has received much interest since it can be used to model the relationship between two different sets of objects in many applications (e.g., the relationship between users and items in E-commerce). In this paper, we study the problem of efficient bi-triangle counting for a large bipartite network, where a bi-triangle is a cycle with three vertices from one vertex set and three vertices from another vertex set. Counting bi-triangles has found many real applications such as computing the transitivity coefficient and clustering coefficient for bipartite networks. To enable efficient bi-triangle counting, we first develop a baseline algorithm relying on the observation that each bi-triangle can be considered as the join of three wedges. Then, we propose a more sophisticated algorithm which regards a bi-triangle as the join of two super-wedges, where a wedge is ...

Proceedings of the 20th international conference on World wide web, 2011

The clustering coefficient of a node in a social network is a fundamental measure that quantifies how tightly-knit the community is around the node. Its computation can be reduced to counting the number of triangles incident on the particular node in the network. In case the graph is too big to fit into memory, this is a non-trivial task, and previous researchers showed how to estimate the clustering coefficient in this scenario. A different avenue of research is to to perform the computation in parallel, spreading it across many machines. In recent years MapReduce has emerged as a de facto programming paradigm for parallel computation on massive data sets. The main focus of this work is to give MapReduce algorithms for counting triangles which we use to compute clustering coefficients. Our contributions are twofold. First, we describe a sequential triangle counting algorithm and show how to adapt it to the MapReduce setting. This algorithm achieves a factor of 10-100 speed up over the naive approach. Second, we present a new algorithm designed specifically for the MapReduce framework. A key feature of this approach is that it allows for a smooth tradeoff between the memory available on each individual machine and the total memory available to the algorithm, while keeping the total work done constant. Moreover, this algorithm can use any triangle counting algorithm as a black box and distribute the computation across many machines. We validate our algorithms on real world datasets comprising of millions of nodes and over a billion edges. Our results show both algorithms effectively deal with skew in the degree distribution and lead to dramatic speed ups over the naive implementation.

2020

Triangle enumeration is a fundamental problem in large-scale graph analysis. For instance, triangles are used to solve practical problems like community detection and spam filtering. On the other hand, there is a large amount of data stored on database management systems (DBMSs), which can be modeled and analyzed as graphs. Alternatively, graph data can be quickly loaded into a DBMS. Our paper shows how to adapt and optimize a randomized distributed triangle enumeration algorithm with SQL queries, which is a significantly different approach from programming graph algorithms in traditional languages such as Python or C++. We choose a parallel columnar DBMS given its fast query processing, but our solution should work for a row DBMS as well. Our randomized solution provides a balanced workload for parallel query processing, being robust to the existence of skewed degree vertices. We experimentally prove our solution ensures a balanced data distribution, and hence workload, among machi...

Discrete Mathematics, 2011

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G − I, there is a vertex w ∈ I such that {u, v, w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NPcomplete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P = NP.

Proceedings of the twenty-fifth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems - PODS '06, 2006

We present two space bounded random sampling algorithms that compute an approximation of the number of triangles in an undirected graph given as a stream of edges. Our first algorithm does not make any assumptions on the order of edges in the stream. It uses space that is inversely related to the ratio between the number of triangles and the number of triples with at least one edge in the induced subgraph, and constant expected update time per edge. Our second algorithm is designed for incidence streams (all edges incident to the same vertex appear consecutively). It uses space that is inversely related to the ratio between the number of triangles and length 2 paths in the graph and expected update time O(log |V | · (1 + s · |V |/|E|)), where s is the space requirement of the algorithm. These results significantly improve over previous work . Since the space complexity depends only on the structure of the input graph and not on the number of nodes, our algorithms scale very well with increasing graph size and so they provide a basic tool to analyze the structure of large graphs. They have many applications, for example, in the discovery of Web communities, the computa- * .