Degree distribution models are incredibly important tools for analyzing and understanding the structure and formation of social networks, and can help guide the design of efficient graph algorithms. In particular, the Power-law degree distribution has long been used to model the structure of online social networks, and is the basis for algorithms and heuristics in graph applications such as influence maximization and social search. Along with recent measurement results, our interest in this topic was sparked by our own experimental results on social graphs that deviated significantly from those predicted by a Power-law model. In this work, we seek a deeper understanding of these deviations, and propose an alternative model with significant implications on graph algorithms and applications. We start by quantifying this artifact using a variety of real social graphs, and show that their structures cannot be accurately modeled using elementary distributions including the Power-law. Instead, we propose the Pareto-Lognormal (PLN) model, verify its goodness-of-fit using graphical and statistical methods, and present an analytical study of its asymptotical differences with the Power-law. To demonstrate the quantitative benefits of the PLN model, we compare the results of three wide-ranging graph applications on real social graphs against those on synthetic graphs generated using the PLN and Power-law models. We show that synthetic graphs generated using PLN are much better predictors of degree distributions in real graphs, and produce experimental results with errors that are orders-of-magnitude smaller than those produced by the Power-law model.
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. We study data transfer opportunities between wireless devices carried by humans. We observe that the distribution of the intercontact time (the time gap separating two contacts between the same pair of devices) may be well approximated by a power law over the range [10 minutes; 1 day]. This observation is confirmed using eight distinct experimental data sets. It is at odds with the exponential decay implied by the most commonly used mobility models. In this paper, we study how this newly uncovered characteristic of human mobility impacts one class of forwarding algorithms previously proposed. We use a simplified model based on the renewal theory to study how the parameters of the distribution impact the performance in terms of the delivery delay of these algorithms. We make recommendations for the design of well-founded opportunistic forwarding algorithms in the context of human-carried devices Global surveys of genomes measure the usage of essential molecular parts, defined here as protein families, superfamilies or folds, in different organisms. Based on surveys of the first 20 completely sequenced gen- omes, we observe that the occurrence of these parts follows a power-law distribution. That is, the number of distinct parts (F) with a given geno- mic occurrence (V) decays as Fa aVb , with a few parts occurring many times and most occurring infrequently. For a given organism, the distri- butions of families, superfamilies and folds are nearly identical, and this is reflected in the size of the decay exponent b. Moreover, the exponent varies between different organisms, with those of smaller genomes dis- playing a steeper decay (i.e. larger b). Clearly, the power law indicates a preference to duplicate genes that encode for molecular parts which are already common. Here, we present a minimal, but biologically meaning- ful model that accurately describes the observed power law. Although the model performs equally well for all three protein classes, we focus on the occurrence of folds in preference to families and superfamilies. This is because folds are comparatively insensitive to the effects of point mutations that can cause a family member to diverge beyond detectable similarity. In the model, genomes evolve through two basic operations: (i) duplication of existing genes; (ii) net flow of new genes. The flow term is closely related to the exponent b and can accommodate consider- able gene loss; however, we demonstrate that the observed data is repro- duced best with a net inflow, i.e. with more gene gain than loss. Moreover, we show that prokaryotes have much higher rates of gene acquisition than eukaryotes, probably reflecting lateral transfer. A further natural outcome from our model is an estimation of the fold composition of the initial genome, which potentially relates to the common ancestor for modern organisms. Supplementary material pertaining to this work is available from www.partslist.org powerlaw. # 2001 Academic Press