On the streaming complexity of computing local clustering coefficients

Abstract

Due to a large number of applications, the problem of estimating the number of triangles in graphs revealed as a stream of edges, and the closely related problem of estimating the graph's clustering coefficient, have received considerable attention in the last decade. Both efficient algorithms and impossibility results have shed light on the computational complexity of the problem. Motivated by applications in Web mining, Becchetti et al. presented new algorithms for the estimation of the local number of triangles, i.e., the number of triangles incident to individual vertices. The algorithms are shown, both theoretically and experimentally, to efficiently handle the problem. However, at least two passes over the data are needed and thus the algorithms are not suitable for real streaming scenarios.In the present work, we consider the problem of estimating the clustering coefficient of individual vertices in a graph over n vertices revealed as a stream of m edges. As a first result we show that any one pass randomized streaming algorithm that can distinguish a graph with no triangles from a graph having a vertex of degree d with clustering coefficient > 1/2 must use Ω(m/d) bits of space in expectation.Our second result is a new randomized one pass algorithm estimating the local clustering coefficient of each vertex with degree at least d. The space requirement of our algorithm is within a logarithmic factor of the lower bound, thus our approach is close to optimal. We also extend the algorithm to local triangle counting and report experimental results on its performance on real-life graphs.

Graph stream algorithms

“…Algorithms have also been developed for the multi-pass model including a two-pass algorithm usingÕ(m/t 1/3 ) space [17] and an O(log n)-pass semi-streaming algorithm [13]. Kutzkov and Pagh [49] and Jha et al [37] also designed algorithms for estimating the clustering and transitivity coefficients directly.…”

New Bounds for the CLIQUE-GAP Problem Using Graph Decomposition Theory

“…Counting the number of triangles is usually an essential part of obtaining important statistics such as the clustering coefficient and transitivity coefficient [5,21] of a social network. Starting with the work of Bar-Yossef et al [4], triangle counting in the streaming model has received sustained attention by researchers [6,12,19,25].…”

Triangle counting in streamed graphs via small vertex covers

“…Building upon results from linear algebra, researchers proposed techniques for approximate triangle counting not relying on sampling [2,29] but they appear not to be practical. The closely related problem of estimating the clustering coefficient was considered in [6,24]. Motivated by the problem of detection of emerging web communities by analyzing the Web graph [23], Buriol et al [6] studied the problem of counting bipartite cliques of small size.…”