Counting Stars and Other Small Subgraphs in Sublinear-Time

Abstract

Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the World-Wide-Web. Several polynomial-time algorithms have been suggested for counting or detecting the number of occurrences of certain network motifs. However, a need for more efficient algorithms arises when the input graph is very large, as is indeed the case in many applications of motif counting.In this paper we design sublinear-time algorithms for approximating the number of copies of certain constant-size subgraphs in a graph G. That is, our algorithms do not read the whole graph, but rather query parts of the graph.

Sublinear-Time Algorithms for Approximating Graph Parameters

“…in expectation, where this bound is essentially optimal [15] (up to a dependence on 1/ and polylogarithmic factors in n). For example, when s = 2 this function behaves as follows.…”

“…Observe that for s = 1 we have that μ 1 = d (and M 1 = 2m where m is the number of edges in the graph), while for s = 2, the variance of the degree distribution is μ 2 − μ 2 1 . Gonen et al [15] gave a sublinear-time algorithm for approximating μ s . Technically, their algorithm approximates the number of stars in a graph (with a given size s), but a simple modification yields an algorithm for moments estimation.…”

“…The Lower Bound and Graphs with Bounded Arboricity. The lower bound constructions showing that the complexity of the aforementioned algorithms for approximating μ s is essentially optimal [15], are based on "locally dense" graphs. In particular, the first (and simpler) lower bound (corresponding to the first term,…”

“…Gonen et al [15] also considered the problem of approximating the number of triangles in a graph G, denoted t = t(G). They showed a linear lower bound when the algorithm may use degree and neighbor queries and m = Θ(n).…”

Approximating the Caro-Wei Bound for Independent Sets in Graph Streams

“…Our work is the first to consider the graph frequency moments (or degree moments) in the data streaming model. They have previously been considered in the property testing literature [21,22,23], where the input graph can only be queried a sublinear number of times. There are important connections between the degree moments and network science and various other disciplines.…”