Triangles In Graph Research Articles

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.

Directed closure measures for networks with reciprocity

The study of triangles in graphs is a standard tool in network analysis, leading to measures such as the \emph{transitivity}, i.e., the fraction of paths of length $2$ that participate in triangles. Real-world networks are often directed, and it can be difficult to "measure" this network structure meaningfully. We propose a collection of \emph{directed closure values} for measuring triangles in directed graphs in a way that is analogous to transitivity in an undirected graph. Our study of these values reveals much information about directed triadic closure. For instance, we immediately see that reciprocal edges have a high propensity to participate in triangles. We also observe striking similarities between the triadic closure patterns of different web and social networks. We perform mathematical and empirical analysis showing that directed configuration models that preserve reciprocity cannot capture the triadic closure patterns of real networks.

An efficient exact algorithm for triangle listing in large graphs

This paper presents a new efficient exact algorithm for listing triangles in a large graph. While the problem of listing triangles in a graph has been considered before, dealing with large graphs continues to be a challenge. Although previous research has attempted to tackle the challenge, this is the first contribution that addresses this problem on a compressed copy of the input graph. In fact, the proposed solution lists the triangles without decompressing the graph. This yields interesting improvements in both storage requirement of the graphs and their time processing.

Using shortcut edges to maximize the number of triangles in graphs

In this paper, we consider the following problem: given an undirected graph G=(V,E) and an integer k, find I⊆V2 with |I|≤k in such a way that G′=(V,E∪I) has the maximum number of triangles (a cycle of length 3). We first prove that this problem is NP-hard and then give an approximation algorithm for it.

Why Do Simple Algorithms for Triangle Enumeration Work in the Real World?

_Listing all triangles is a fundamental graph operation. Triangles can have important interpretations in real-world graphs, especially social and other interaction networks. Despite the lack of provably efficient (linear, or slightly super linear) worst-case algorithms for this problem, practitioners run simple, efficient heuristics to find all triangles in graphs with millions of vertices. How are these heuristics exploiting the structure of these special graphs to provide major speedups in running time_? _We study one of the most prevalent algorithms used by practitioners. A trivial algorithm enumerates all paths of length 2, and checks if each such path is incident to a triangle. A good heuristic is to enumerate only those paths of length 2 in which the middle vertex has the lowest degree. It is easily implemented and is empirically known to give remarkable speedups over the trivial algorithm_. _We study the behavior of this algorithm over graphs with heavy-tailed degree distributions, a defining feature of real-world graphs. The erased configuration model (ECM) efficiently generates a graph with asymptotically (almost) any desired degree sequence. We show that the expected running time of this algorithm over the distribution of graphs created by the ECM is controlled by the ℓ_4/3-norm of the degree sequence. Norms of the degree sequence are a measure of the heaviness of the tail, and it is precisely this feature that allows non trivial speedups of simple triangle enumeration algorithms. As a corollary of our main theorem, we prove expected linear-time performance for degree sequences following a power law with exponent α ≥ 7/3, and non trivial speedup whenever_ α ∈ (2, 3).

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

Abstract The number of triangles is a computationally expensive graph statistic frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model), and in important real-world applications such as spam detection, uncovering the hidden thematic structures in the Web, and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms that run fast, use little space, provide accurate estimates of the number of triangles, and preferably are parallelizable. In this paper we present an efficient triangle-counting approximation algorithm that can be adapted to the semistreaming model [Feigenbaum et al. 05]. Its key idea is to combine the sampling algorithm of [Tsourakakis et al. 09, Tsourakakis et al. 11] and the partitioning of the set of vertices into high- and low-degree subsets as in [Alon et al. 97], treating each set appropriately. From a mathematical perspective, we present a simplified pr...

Colorful triangle counting and a MapReduce implementation

In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, let G be a graph with n vertices, t triangles and let Δ be the maximum number of triangles an edge of G is contained in. Our randomized algorithm colors the vertices of G with N=1/p colors uniformly at random, counts monochromatic triangles, i.e., triangles whose vertices have the same color, and scales that count appropriately. We show that if p⩾max(Δlognt,lognt) then for any constant ϵ>0 our unbiased estimate T is concentrated around its expectation, i.e., Pr[|T−E[T]|⩾ϵE[T]]=o(1). Finally, our algorithm is amenable to being parallelized. We present a simple MapReduce implementation of our algorithm.

Complexity issues for the sandwich homogeneous set problem

Graph sandwich problems were introduced by Golumbic et al. (1994) in [12] for DNA physical mapping problems and can be described as follows. Given a property Π of graphs and two disjoint sets of edges E 1 , E 2 with E 1 ⊆ E 2 on a vertex set V , the problem is to find a graph G on V with edge set E s having property Π and such that E 1 ⊆ E s ⊆ E 2 . In this paper, we exhibit a quasi-linear reduction between the problem of finding an independent set of size k ≥ 2 in a graph and the problem of finding a sandwich homogeneous set of the same size k . Using this reduction, we prove that a number of natural (decision and counting) problems related to sandwich homogeneous sets are hard in general. We then exploit a little further the reduction and show that finding efficient algorithms to compute small sandwich homogeneous sets would imply substantial improvement for computing triangles in graphs.

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.

On the Minimal Density of Triangles in Graphs

For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving thatwhere$t\df \lfloor 1/(1-\rho)\rfloor$is the integer such that$\rho\in\bigl[ 1-\frac 1t,1-\frac 1{t+1}\bigr]$.

Pentagons vs. triangles

We study the maximum number of triangles in graphs with no cycle of length 5 and analogously, the maximum number of edges in 3-uniform hypergraphs with no cycle of length 5.

Clustering of spectra and fractals of regular graphs

We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labeled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara–Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind.

Packing and covering triangles in graphs

It is shown that if G is a graph such that the maximum size of a set of pairwise edge-disjoint triangles is v( G), then there is a set C of edges of G of size at most (3 − ε) v( G) such that E( T) ∩ C ≠ ∅ for every triangle T of G, where ε > 3 23 . This is the first nontrivial bound known for a long-standing conjecture of Tuza.

On the Maximum Number of Triangles in Wheel-Free Graphs

Gallai [1] raised the question of determiningt(n), the maximum number of triangles in graphs ofnvertices with acyclic neighborhoods. Here we disprove his conjecture (t(n) ~ n2/8) by exhibiting graphs having n2/7.5 triangles. We improve the upper bound [11] of (n2−n)/6 tot(n) ≤;n2/7.02 +O(n). For regular graphs, we further decrease this bound ton2/7.75 +O(n).

Triangles in a complete chromatic graph

AbstractSuppose that in a complete graph on N points, each edge is given arbitrarily either the color red or the color blue, but the total number of blue edges is fixed at T. We find the minimum number of monochromatic triangles in the graph as a function of N and T. The maximum number of monochromatic triangles presents a more difficult problem. Here we propose a reasonable conjecture supported by examples.

On the maximal number of independent triangles in graphs

On the maximal number of independent triangles in graphs