The degree distribution is one of the most fundamental properties used in the analysis of massive graphs. There is a large literature on graph sampling, where the goal is to estimate properties (especially the degree distribution) of a large graph through a small, random sample. The degree distribution estimation poses a significant challenge, due to its heavy-tailed nature and the large variance in degrees. We design a new algorithm, SADDLES, for this problem, using recent mathematical techniques from the field of sublinear algorithms. The SADDLES algorithm gives provably accurate outputs for all values of the degree distribution. For the analysis, we define two fatness measures of the degree distribution, called the h-index and the z-index. We prove that SADDLES is sublinear in the graph size when these indices are large. A corollary of this result is a provably sublinear algorithm for any degree distribution bounded below by a power law. We deploy our new algorithm on a variety of real datasets and demonstrate its excellent empirical behavior. In all instances, we get extremely accurate approximations for all values in the degree distribution by observing at most 1 of the vertices. This is a major improvement over the state-of-the-art sampling algorithms, which typically sample more than 10 of the vertices to give comparable results. We also observe that the h and z-indices of real graphs are large, validating our theoretical analysis.
Inspired by Feige (36th STOC, 2004), we initiate a study of sublinear randomized algorithms for approximating average parameters of a graph. Specifically, we consider the average degree of a graph and the average distance between pairs of vertices in a graph. Since our focus is on sublinear algorithms, these algorithms access the input graph via queries to an adequate oracle. We consider two types of queries. The first type is standard neighborhood queries (i.e., what is the ith neighbor of vertex v?), whereas the second type are queries regarding the quantities that we need to find the average of (i.e., what is the degree of vertex v? and what is the distance between u and v?, respectively). Loosely speaking, our results indicate a difference between the two problems: For approximating the average degree, the standard neighbor queries suffice and in fact are preferable to degree queries. In contrast, for approximating average distances, the standard neighbor queries are of little help whereas distance queries are crucial. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 Supported by Israel Internet Association (ISOC-IL). This article is dedicated in memory of Shimon Even (1935–2004). Property testing algorithms are "ultra"-efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decision they need to make usually concerns properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this monograph we survey results in property testing, where our emphasis is on common analysis and algorithmic techniques. Among the techniques surveyed are the following: The self-correcting approach, which was mainly applied in the study of property testing of algebraic properties; The enforce-and-test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the dense-graphs model), as well as in other contexts; Szemeredi's Regularity Lemma, which plays a very important role in the analysis of algorithms for testing graph properties (in the dense-graphs model); The approach of Testing by implicit learning, which implies efficient testability of membership in many functions classes; and Algorithmic techniques for testing properties of sparse graphs, which include local search and random walks. We prove the following inequality: for every positive integer n and every collection X 1 ,..., X n of nonnegative independent random variables that each has expectation 1, the probability that their sum remains below n+1 is at least α > 0. Our proof produces a value of α = 1 13 ≅ 0.077, but we conjecture that the inequality also holds with α = 1 e ≅ 0.368.As an example for the use of the new inequality, we consider the problem of estimating the average degree of a graph by querying the degrees of some of its vertices. We show the following threshold behavior: approximation factors above 2 require far less queries than approximation factors below 2. The new inequality is used in order to get tight (up to multiplicative constant factors) relations between the number of queries and the quality of the approximation. We show how the degree approximation algorithm can be used in order to quickly find those edges in a network that belong to many shortest paths. 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. Specifically, we consider algorithms that may query the degree of any vertex of their choice and may ask for any neighbor of any vertex of their choice. The main focus of this work is on the basic problem of counting the number of length-2 paths and more generally on counting the number of... The connectivity of the Internet crucially depends on the relationships between thousands of Autonomous Systems (ASes) that exchange routing information using the Border Gateway Protocol (BGP). These relationships can be modeled as a graph, called the AS-graph, in which the vertices model the ASes, and the edges model the peering arrangements between the ASes. Based on topological studies, it is widely believed that the Internet graph contains a central dense-core: Informally, this is a small set of high-degree, tightly interconnected ASes that participate in a large fraction of end-to-end routes. Finding this dense-core is a very important practical task when analyzing the Internet's topology. In this work we introduce a randomized sublinear algorithm that finds a dense-core of the AS-graph. We mathematically prove the correctness of our algorithm, bound the density of the core it returns, and analyze its running time. We also implemented our algorithm and tested it on real AS-graph data and on real undirected version of WWW network data. Our results show that the core discovered by our algorithm is nearly identical to the cores found by existing algorithms - at a fraction of the running time. We revisit the classic problem of estimating the degree distribution moments of an undirected graph. Consider an undirected graph @math with @math vertices, and define (for @math ) @math . Our aim is to estimate @math within a multiplicative error of @math (for a given approximation parameter @math ) in sublinear time. We consider the sparse graph model that allows access to: uniform random vertices, queries for the degree of any vertex, and queries for a neighbor of any vertex. For the case of @math (the average degree), @math queries suffice for any constant @math (Feige, SICOMP 06 and Goldreich-Ron, RSA 08). Gonen-Ron-Shavitt (SIDMA 11) extended this result to all integral @math , by designing an algorithms that performs @math queries. We design a new, significantly simpler algorithm for this problem. In the worst-case, it exactly matches the bounds of Gonen-Ron-Shavitt, and has a much simpler proof. More importantly, the running time of this algorithm is connected to the degeneracy of @math . This is (essentially) the maximum density of an induced subgraph. For the family of graphs with degeneracy at most @math , it has a query complexity of @math . Thus, for the class of bounded degeneracy graphs (which includes all minor closed families and preferential attachment graphs), we can estimate the average degree in @math queries, and can estimate the variance of the degree distribution in @math queries. This is a major improvement over the previous worst-case bounds. Our key insight is in designing an estimator for @math that has low variance when @math does not have large dense subgraphs.  We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a sublinear-time algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries, and neighbor queries. We show that for any given approximation parameter @math , the algorithm provides an estimate @math such that, with high constant probability, @math , where @math is the number of triangles in the graph @math . The expected query complexity of the algorithm is @math , where @math is the number of vertices in the graph and @math is the number of edges. The expected running time of the algorithm is $( n t^ 1 3 + m^ 3 2 t )...