There are few studies that look closely at how the topology of the Internet evolves over time; most focus on snapshots taken at a particular point in time. In this paper, we investigate the evolution of the topology of the Autonomous Systems graph of the Internet, examining how eight commonly-used topological measures change from January 2002 to January 2010. We find that the distributions of most of the measures remain unchanged, except for average path length and clustering coefficient. The average path length has slowly and steadily increased since 2005 and the average clustering coefficient has steadily declined. We hypothesize that these changes are due to changes in peering policies as the Internet evolves. We also investigate a surprising feature, namely that the maximum degree has changed little, an aspect that cannot be captured without modeling link deletion. Our results suggest that evaluating models of the Internet graph by comparing steady-state generated topologies to snapshots of the real data is reasonable for many measures. However, accurately matching time-variant properties is more difficult, as we demonstrate by evaluating ten well-known models against the 2010 data.
In addition to studying business relationships, Dhamdhere el al @cite_11 reported on changes in average degree and average path length over time. Their results on path length agree with ours, although their study included only data up to 2007, so the trends are less clear. Because the degree distribution does not change, it is likely that the shift they see in average degree is a result of a steadily increasing sample size. Another study @cite_26 used spectral analysis to investigate clustering on the AS graph, and study coreness and changing path diversity. This analysis, however, covers short time spans (at most two and a half years), and only considers data before 2004.
@cite_26 @cite_11
1653839352 2112976607 2809138487 1519020025
The degree distribution is one of the most fundamental graph properties of interest for real-world graphs. It has been widely observed in numerous domains that graphs typically have a tailed or scale-free degree distribution. While the average degree is usually quite small, the variance is quite high and there are vertices with degrees at all scales. We focus on the problem of approximating the degree distribution of a large streaming graph, with small storage. We design an algorithm headtail, whose main novelty is a new estimator of infrequent degrees using truncated geometric random variables. We give a mathematical analysis of headtail and show that it has excellent behavior in practice. We can process streams will millions of edges with storage less than 1 and get extremely accurate approximations for all scales in the degree distribution. We also introduce a new notion of Relative Hausdorff distance between tailed histograms. Existing notions of distances between distributions are not suitable, since they ignore infrequent degrees in the tail. The Relative Hausdorff distance measures deviations at all scales, and is a more suitable distance for comparing degree distributions. By tracking this new measure, we are able to give strong empirical evidence of the convergence of headtail. We consider a family of random graphs with a given expected degree sequence. Each edge is chosen independently with probability proportional to the product of the expected degrees of its endpoints. We examine the distribution of the sizes volumes of the connected components which turns out depending primarily on the average degree d and the second-order average degree d . Here d denotes the weighted average of squares of the expected degrees. For example, we prove that the giant component exists if the expected average degree d is at least 1, and there is no giant component if the expected second-order average degree d is at most 1. Examples are given to illustrate that both bounds are best possible. College students rely on increasingly data-rich environments when making learning-relevant decisions about the courses they take and their expected time commitments. However, we know little about how their exposure to such data may influence student course choice, effort regulation, and performance. We conducted a large-scale field experiment in which all the undergraduates at a large, selective university were randomized to an encouragement to use a course-planning web application that integrates information from official transcripts from the past fifteen years with detailed end-of-course evaluation surveys. We found that use of the platform lowered students' GPA by 0.28 standard deviations on average. In a subsequent field experiment, we varied access to information about course grades and time commitment on the platform and found that access to grade information in particular lowered students' overall GPA. Our exploratory analysis suggests these effects are not due to changes in the portfolio of courses that students choose, but rather by changes to their behavior within courses. Department of Physics and Astronomy,University of New Mexico, Albuquerque NM 87131(aaron,moore)@cs.unm.edu(Dated: February 2, 2008)A great deal of eﬀort has been spent measuring topological features of the Internet. However, itwas recently argued that sampling based on taking paths or traceroutes through the network froma small number of sources introduces a fundamental bias in the observed degree distribution. Weexamine this bias analytically and experimentally. For Erd˝os-R´enyirandom graphs with mean degreec, we show analytically that traceroute sampling gives an observed degree distribution P(k) ∼ k