Arboricity and subgraph listing algorithms
SIAM Journal on Computing
Planar orientations with low out-degree and compaction of adjacency matrices
Theoretical Computer Science
Randomized algorithms
A graph-constructive approach to solving systems of geometric constraints
ACM Transactions on Graphics (TOG)
A critical point for random graphs with a given degree sequence
Random Graphs 93 Proceedings of the sixth international seminar on Random graphs and probabilistic methods in combinatorics and computer science
On power-law relationships of the Internet topology
Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication
Proceedings of the 9th international World Wide Web conference on Computer networks : the international journal of computer and telecommunications netowrking
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
The Size of the Giant Component of a Random Graph with a Given Degree Sequence
Combinatorics, Probability and Computing
Main-memory triangle computations for very large (sparse (power-law)) graphs
Theoretical Computer Science
Fast Counting of Triangles in Large Real Networks without Counting: Algorithms and Laws
ICDM '08 Proceedings of the 2008 Eighth IEEE International Conference on Data Mining
Graph Twiddling in a MapReduce World
Computing in Science and Engineering
Is a friend a friend?: investigating the structure of friendship networks in virtual worlds
CHI '10 Extended Abstracts on Human Factors in Computing Systems
Subcubic Equivalences between Path, Matrix and Triangle Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Counting triangles and the curse of the last reducer
Proceedings of the 20th international conference on World wide web
Finding, counting and listing all triangles in large graphs, an experimental study
WEA'05 Proceedings of the 4th international conference on Experimental and Efficient Algorithms
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Triangle enumeration is a fundamental graph operation. 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 where 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 l4/3-norm of the degree sequence. 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).