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.
Due to the complexity of nodes' counting, numerous research works focus on estimating the cardinality and mean edges degree in a graph. The authors in @cite_19 @cite_12 state a method for determining the total number of nodes in a graph using data flooding and random walks. They present an algorithm for node estimation in large-scale wireless sensor networks using random walks that travel through the network based on a predefined probability distribution.
@cite_19 @cite_12
2090914728 2075709248 2062833795 2000985696
Graph datasets with billions of edges, such as social and Web graphs, are prevalent. To be feasible, computation on such large graphs should scale linearly with graph size. All-distances sketches (ADSs) are emerging as a powerful tool for scalable computation of some basic properties of individual nodes or the whole graph. ADSs were first proposed two decades ago (Cohen 1994) and more recent algorithms include ANF (Palmer, Gibbons, and Faloutsos 2002) and hyperANF (Boldi, Rosa, and Vigna 2011). A sketch of logarithmic size is computed for each node in the graph and the computation in total requires only a near linear number of edge relaxations. From the ADS of a node, we can estimate neighborhood cardinalities (the number of nodes within some query distance) and closeness centrality. More generally we can estimate the distance distribution, effective diameter, similarities, and other parameters of the full graph. We make several contributions which facilitate a more effective use of ADSs for scalable analysis of massive graphs. We provide, for the first time, a unified exposition of ADS algorithms and applications. We present the Historic Inverse Probability (HIP) estimators which are applied to the ADS of a node to estimate a large natural class of queries including neighborhood cardinalities and closeness centralities. We show that our HIP estimators have at most half the variance of previous neighborhood cardinality estimators and that this is essentially optimal. Moreover, HIP obtains a polynomial improvement over state of the art for more general domain queries and the estimators are simple, flexible, unbiased, and elegant. The ADS generalizes Min-Hash sketches, used for approximating cardinality (distinct count) on data streams. We obtain lower bounds on Min-Hash cardinality estimation using classic estimation theory. We illustrate the power of HIP, both in terms of ease of application and estimation quality, by comparing it to the HyperLogLog algorithm ( 2007), demonstrating a significant improvement over this state-of-the-art practical algorithm. We also study the quality of ADS estimation of distance ranges, generalizing the near-linear time factor-2 approximation of the diameter. Networks are characterized by nodes and edges. While there has been a spate of recent work on estimating the number of nodes in a network, the edge-estimation question appears to be largely unaddressed. In this work we consider the problem of estimating the average degree of a large network using efficient random sampling, where the number of nodes is not known to the algorithm. We propose a new estimator for this problem that relies on access to node samples under a prescribed distribution. Next, we show how to efficiently realize this ideal estimator in a random walk setting. Our estimator has a natural and simple implementation using random walks; we bound its performance in terms of the mixing time of the underlying graph. We then show that our estimators are both provably and practically better than many natural estimators for the problem. Our work contrasts with existing theoretical work on estimating average degree, which assume that a uniform random sample of nodes is available and the number of nodes is known. Sampling from large graphs is an area of great interest, especially since the emergence of huge structures such as Online Social Networks (OSNs) and the World Wide Web (WWW). These networks, when viewed as graphs, often contain hundreds of millions of vertices and billions of edges. The large scale properties of a network can be summarized in terms of parameters of the underlying graph, such as the total number of vertices, edges and triangles. The large size of these networks makes it computationally expensive to obtain such structural properties of the underlying graph by exhaustive search. If we can estimate these properties by taking small but representative samples from the network, then size is no longer such a problem. In this paper we present a general framework to estimate network properties using random walks. These methods work under the assumption we are able to obtain local characteristics of a vertex during each step of the random walk, for example the number and labels of the neighboring vertices of a specific vertex These assumptions are relatively reasonable in practice, but may add some additional query cost to each step of the random walk. We also present some practical methods to estimate the total number of edges links m, number of vertices nodes n and number of connected triads of vertices (triangles) t in graphs with degree distributions which follow a power-law and higher number of triangles higher than expected in random graphs. We use these graphs since they tend to better correspond to the structure of large online networks, and in fact some of the data used are taken from such a network. Additionally we present experimental estimates for n, m, t we obtained using our methods on real or manufactured networks. In order to make the methods practical, the total number of steps made by the walk was limited to at most the size n of the network. In fact the results appear to converge for a lower number of steps, indicating that our proposed methods are feasible in practice. In this paper, we study the impact of edge weights on distances in sparse random graphs. We interpret these weights as delays and take them as independent and identically distributed exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some standard regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of @math when the size of the graph @math tends to infinity. We also derive the exact value for the prefactor. These results allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average.