The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. 
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. 
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. 
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

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Statistical mechanics of complex networks

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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The Structure and Function of Complex Networks

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Using Multivariate Statistics

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.

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Power-Law Distributions in Empirical Data

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Demonstrates that Ethernet LAN traffic is statistically self-similar, that none of the commonly used traffic models is able to capture this fractal-like behavior, that such behavior has serious implications for the design, control, and analysis of high-speed, cell-based networks, and that aggregating streams of such traffic typically intensifies the self-similarity ("burstiness") instead of smoothing it. These conclusions are supported by a rigorous statistical analysis of hundreds of millions of high quality Ethernet traffic measurements collected between 1989 and 1992, coupled with a discussion of the underlying mathematical and statistical properties of self-similarity and their relationship with actual network behavior. The authors also present traffic models based on self-similar stochastic processes that provide simple, accurate, and realistic descriptions of traffic scenarios expected during B-ISDN deployment. >

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On the self-similar nature of Ethernet traffic (extended version)

A number of empirical studies of traffic measurements from a variety of working packet networks have demonstrated that actual network traffic is self-similar or long-range dependent in nature-in sharp contrast to commonly made traffic modeling assumptions. We provide a plausible physical explanation for the occurrence of self-similarity in local-area network (LAN) traffic. Our explanation is based on convergence results for processes that exhibit high variability and is supported by detailed statistical analyzes of real-time traffic measurements from Ethernet LANs at the level of individual sources. This paper is an extended version of Willinger et al. (1995). We develop here the mathematical results concerning the superposition of strictly alternating ON/OFF sources. Our key mathematical result states that the superposition of many ON/OFF sources (also known as packet-trains) with strictly alternating ON- and OFF-periods and whose ON-periods or OFF-periods exhibit the Noah effect produces aggregate network traffic that exhibits the Joseph effect. There is, moreover, a simple relation between the parameters describing the intensities of the Noah effect (high variability) and the Joseph effect (self-similarity). An extensive statistical analysis of high time-resolution Ethernet LAN traffic traces confirms that the data at the level of individual sources or source-destination pairs are consistent with the Noah effect. We also discuss implications of this simple physical explanation for the presence of self-similar traffic patterns in modern high-speed network traffic.

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Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level

Self-Similar Network Traffic: An Overview (K. Park & W. Willinger). Wavelets for the Analysis, Estimation, and Synthesis of Scaling Data (P. Abry, et al.). Simulations with Heavy-Tailed Workloads (M. Crovella & L. Lipsky). Queueing Behavior Under Fractional Brownian Traffic (I. Norros). Heavy Load Queueing Analysis with LRD On/Off Sources (F. Brichet, et al.). The Single Server Queue: Heavy Tails and Heavy Traffic (O. Boxma & J. Cohen). Fluid Queues, On/Off Processes, and Teletraffic Modeling with Highly Variable and Correlated Inputs (S. Resnick & G. Samorodnitsky). Bounds on the Buffer Occupancy Probability with Self-Similar Input Traffic (N. Likhanov). Buffer Asymptotics for M/G/ Input Processes (A. Makowski & M. Parulekar). Asymptotic Analysis of Queues with Subexponential Arrival Processes (P. Jelenkovi). Traffic and Queueing from an Unbounded Set of Independent Memoryless On/Off Sources (P. Jacquet). Long-Range Dependence and Queueing Effects for VBR Video (D. Heyman & T. Lakshman). Analysis of Transient Loss Performance Impact of Long-Range Dependence in Network Traffic (G.-L. Li & V. Li). The Protocol Stack and Its Modulating Effect on Self-Similar Traffic (K. Park, et al.). Characteristics of TCP Connection Arrivals (A. Feldmann). Engineering for Quality of Service (J. Roberts). Network Design and Control Using On/Off and Multilevel Source Traffic Models with Heavy-Tailed Distributions (N. Duffield & W. Whitt). Congestion Control for Self-Similar Network Traffic (T. Tuan & K. Park). Quality of Service Provisioning for Long-Range-Dependent Real-Time Traffic (A. Adas & A. Mukherjhee). Toward an Improved Understanding of Network Traffic Dynamics (R. Riedi & W. Willinger). Future Directions and Open Problems in Performance Evaluation and Control of Self-Similar Network Traffic (K. Park). Index.

Self-Similar Network Traffic and Performance Evaluation

We analyze 20 large sets of actual variable-bit-rate (VBR) video data, generated by a variety of different codecs and representing a wide range of different scenes. Performing extensive statistical and graphical tests, our main conclusion is that long-range dependence is an inherent feature of VBR video traffic, i.e., a feature that is independent of scene (e.g., video phone, video conference, motion picture video) and codec. In particular, we show that the long-range dependence property allows us to clearly distinguish between our measured data and traffic generated by VBR source models currently used in the literature. These findings give rise to novel and challenging problems in traffic engineering for high-speed networks and open up new areas of research in queueing and performance analysis involving long-range dependent traffic models. A small number of analytic queueing results already exist, and we discuss their implications for network design and network control strategies in the presence of long-range dependent traffic. >

Long-range dependence in variable-bit-rate video traffic

Various methods for estimating the self-similarity parameter and/or the intensity of long-range dependence in a time series are available. Some are more reliable than others. To discover the ones t...

Walter Willinger

Papers

On the self-similar nature of Ethernet traffic (extended version)

Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level

Self-Similar Network Traffic and Performance Evaluation

Long-range dependence in variable-bit-rate video traffic

Estimators for long-range dependence: an empirical study