Mapping the Internet generally consists in sampling the network from a limited set of sources by using "traceroute"-like probes. This methodology, akin to the merging of different spanning trees to a set of destinations, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. Here we explore these biases and provide a statistical analysis of their origin. We derive a mean-field analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular we find that the edge and vertex detection probability is depending on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with scale-free topology. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in different network models. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. The numerical study also allows the identification of intervals of the exploration parameters that optimize the fraction of nodes and edges discovered in the sampled graph. This finding might hint the steps toward more efficient mapping strategies.
Work by @cite_21 has shown that power-law like distributions can be obtained for subgraphs of Erd "os-R 'enyi random graphs when the subgraph is the result of a traceroute exploration with relatively few sources and destinations. They discuss the origin of these biases and the effect of the distance between source and target in the mapping process.
@cite_21
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Considerable attention has been focused on the properties of graphs derived from Internet measurements. Router-level topologies collected via traceroute-like methods have led some to conclude that the router graph of the Internet is well modeled as a power-law random graph. In such a graph, the degree distribution of nodes follows a distribution with a power-law tail. We argue that the evidence to date for this conclusion is at best insufficient We show that when graphs are sampled using traceroute-like methods, the resulting degree distribution can differ sharply from that of the underlying graph. For example, given a sparse Erdos-Renyi random graph, the subgraph formed by a collection of shortest paths from a small set of random sources to a larger set of random destinations can exhibit a degree distribution remarkably like a power-law. We explore the reasons for how this effect arises, and show that in such a setting, edges are sampled in a highly biased manner. This insight allows us to formulate tests for determining when sampling bias is present. When we apply these tests to a number of well-known datasets, we find strong evidence for sampling bias. A power law degree distribution is established for a graph evolution model based on the graph class of k-trees. This k-tree-based graph process can be viewed as an idealized model that captures some characteristics of the preferential attachment and copying mechanisms that existing evolving graph processes fail to model due to technical obstacles. The result also serves as a further cautionary note reinforcing the point of view that a power law degree distribution should not be regarded as the only important characteristic of a complex network, as has been previously argued [D. Achlioptas, A. Clauset, D. Kempe, C. Moore, On the bias of traceroute sampling, or power-law degree distribution in regular graphs, in: Proceedings of the 37th ACM Symposium on Theory of Computing, STOC'05, 2005, pp. 694-703; L. Li, D. Alderson, J. Doyle, W. Willinger, Towards a theory of scale-free graphs: Definition, properties, and implications, Internet Mathematics 2 (4) (2005) 431-523; M. Mitzenmacher, The future of power law research, Internet Mathematics, 2 (4) (2005) 525-534]. Understanding the structure of the Internet graph is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining its graph structure is a surprisingly difficult task, as edges cannot be explicitly queried. Instead, empirical studies rely on traceroutes to build what are essentially single-source, all-destinations, shortest-path trees. These trees only sample a fraction of the network's edges, and a recent paper by found empirically that the resuting sample is intrinsically biased. For instance, the observed degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is Poisson.In this paper, we study the bias of traceroute sampling systematically, and, for a very general class of underlying degree distributions, calculate the likely observed distributions explicitly. To do this, we use a continuous-time realization of the process of exposing the BFS tree of a random graph with a given degree distribution, calculate the expected degree distribution of the tree, and show that it is sharply concentrated. As example applications of our machinery, we show how traceroute sampling finds power-law degree distributions in both δ-regular and Poisson-distributed random graphs. Thus, our work puts the observations of on a rigorous footing, and extends them to nearly arbitrary degree distributions. Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.