text: This paper introduces a novel algorithm for cardinality, i.e., the number of nodes, estimation in large scale anonymous graphs using statistical inference methods. Applications of this work include estimating the number of sensor devices, online social users, active protein cells, etc. In anonymous graphs, each node possesses little or non-existing information on the network topology. In particular, this paper assumes that each node only knows its unique identifier. The aim is to estimate the cardinality of the graph and the neighbours of each node by querying a small portion of them. While the former allows the design of more efficient coding schemes for the network, the second provides a reliable way for routing packets. As a reference for comparison, this work considers the Best Linear Unbiased Estimators (BLUE). For dense graphs and specific running times, the proposed algorithm produces a cardinality estimate proportional to the BLUE. Furthermore, for an arbitrary number of iterations, the estimate converges to the BLUE as the number of queried nodes tends to the total number of nodes in the network. Simulation results confirm the theoretical results by revealing that, for a moderate running time, asking a small group of nodes is sufficient to perform an estimation of 95 of the whole network.
  @cite_14: We consider estimation of network cardinality by distributed anonymous strategies relying on statistical inference methods. In particular, we focus on the relative Mean Square Error (MSE) of Maximum Likelihood (ML) estimators based on either the maximum or the average of M-dimensional vectors randomly generated at each node. In the case of continuous probability distributions, we show that the relative MSE achieved by the max-based strategy decreases as 1 M, independently of the used distribution, while that of the average-based estimator scales approximately as 2 M. We then introduce a novel strategy based on the average of M-dimensional vectors locally generated from Bernoulli random variables. In this case, the ML estimator, which is the Least Common Multiple (LCM) of the denominators of the irreducible fractions corresponding to the M elements of the average vector, leads to an MSE which goes exponentially to zero as M increases. We then discuss the effects of finite-precision arithmetics in practical dynamic implementations. Numerical experiments reveal that the MSE of the strategy based on Bernoulli trials is two order of magnitude smaller than that based on continuous random variables, at a price of one order of magnitude slower convergence time.
  @cite_11: Computing the transitive closure in directed graphs is a fundamental graph problem. We consider the more restricted problem of computing the number of nodes reachable from every node and the size of the transitive closure. The fastest known transitive closure algorithms run inO(min mn,n2.38 ) time, wherenis the number of nodes andmthe number of edges in the graph. We present anO(m) time randomized (Monte Carlo) algorithm that estimates, with small relative error, the sizes of all reachability sets and the transitive closure. Another ramification of our estimation scheme is a O(m) time algorithm for estimating sizes of neighborhoods in directed graphs with nonnegative edge lengths. Our size-estimation algorithms are much faster than performing the respective explicit computations.
summary: While the authors in @cite_14 propose a model for distributed cardinality estimation in anonymous networks using statistical inference methods, the authors in @cite_11 present a scheme for estimating the number of reachable neighbours for a given node and a size of the transitive closure. They present an @math time complexity algorithm based on Monte Carlo that estimates, with a small error, the sizes of all reachability sets and the transitive closure of a graph.