On the universality of the scaling of fluctuations in traffic on complex networks

We study the scaling of fluctuations with the mean of traffic in complex networks using a model where the arrival and departure of "packets" follow exponential distributions, and the processing capability of nodes is either unlimited or finite. The model presents a wide variety of exponents between 1/2 and 1 for this scaling, revealing their dependence on the few parameters considered, and questioning the existence of universality classes. We also report the experimental scaling of the fluctuations in the Internet for the Abilene backbone network. We found scaling exponents between 0.71 and 0.86 that do not fit with the exponent 1/2 reported in the literature.Comment: 4 pages, 4 figure

The Dynamics of Internet Traffic: Self-Similarity, Self-Organization, and Complex Phenomena

The Internet is the most complex system ever created in human history. Therefore, its dynamics and traffic unsurprisingly take on a rich variety of complex dynamics, self-organization, and other phenomena that have been researched for years. This paper is a review of the complex dynamics of Internet traffic. Departing from normal treatises, we will take a view from both the network engineering and physics perspectives showing the strengths and weaknesses as well as insights of both. In addition, many less covered phenomena such as traffic oscillations, large-scale effects of worm traffic, and comparisons of the Internet and biological models will be covered.Comment: 63 pages, 7 figures, 7 tables, submitted to Advances in Complex System

Anomalous electrical and frictionless flow conductance in complex networks

We study transport properties such as electrical and frictionless flow conductance on scale-free and Erdos-Renyi networks. We consider the conductance G between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G, with a power-law tail distribution \Phi_{SF}(G) \sim G^{g_G}, where g_G = 2\lambda - 1, where \lambda is the decay exponent for the scale-free network degree distribution. We confirm our predictions by simulations of scale-free networks solving the Kirchhoff equations for the conductance between a pair of nodes. The power-law tail in \Phi_{SF}(G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erdos-Renyi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical 'transport backbone' picture we suggest that the conductances of scale-free and Erdos-Renyi networks can be approximated by ck_Ak_B/(k_A+k_B) for any pair of nodes A and B with degrees k_A and k_B. Thus, a single quantity c, which depends on the average degree of the network, characterizes transport on both scale-free and Erdos-Renyi networks. We determine that c tends to 1 for increasing , and it is larger for scale-free networks. We compare the electrical results with a model for frictionless transport, where conductance is defined as the number of link-independent paths between A and B, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. Finally, we use a recent data set for the AS (autonomous system) level of the Internet and confirm that our results are valid in this real-world example.Comment: 8 pages, 11 figure

Transport of multiple users in complex networks

We study the transport properties of model networks such as scale-free and Erd\H{o}s-R\'{e}nyi networks as well as a real network. We consider the conductance $G$ between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of $G$, with a power-law tail distribution $\Phi_{\rm SF}(G)\sim G^{-g_G}$, where $g_G=2\lambda -1$, and $\lambda$ is the decay exponent for the scale-free network degree distribution. We confirm our predictions by large scale simulations. The power-law tail in $\Phi_{\rm SF}(G)$ leads to large values of $G$, thereby significantly improving the transport in scale-free networks, compared to Erd\H{o}s-R\'{e}nyi networks where the tail of the conductivity distribution decays exponentially. We develop a simple physical picture of the transport to account for the results. We study another model for transport, the \emph{max-flow} model, where conductance is defined as the number of link-independent paths between the two nodes, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. We then extend our study to the case of multiple sources, where the transport is define between two \emph{groups} of nodes. We find a fundamental difference between the two forms of flow when considering the quality of the transport with respect to the number of sources, and find an optimal number of sources, or users, for the max-flow case. A qualitative (and partially quantitative) explanation is also given

Traffic matrix estimation on a large IP backbone: a comparison on real data

This paper considers the problem of estimating the point-to-point traffic matrix in an operational IP backbone. Contrary to previous studies, that have used a partial traffic matrix or demands estimated from aggregated Netflow traces, we use a unique data set of complete traffic matrices from a global IP network measured over five-minute intervals. This allows us to do an accurate data analysis on the time-scale of typical link-load measurements and enables us to make a balanced evaluation of different traffic matrix estimation techniques. We describe the data collection infrastructure, present spatial and temporal demand distributions, investigate the stability of fan-out factors, and analyze the mean-variance relationships between demands. We perform a critical evaluation of existing and novel methods for traffic matrix estimation, including recursive fanout estimation, worst-case bounds, regularized estimation techniques, and methods that rely on mean-variance relationships. We discuss the weaknesses and strengths of the various methods, and highlight differences in the results for the European and American subnetworks

A critical look at power law modelling of the Internet

This paper takes a critical look at the usefulness of power law models of the Internet. The twin focuses of the paper are Internet traffic and topology generation. The aim of the paper is twofold. Firstly it summarises the state of the art in power law modelling particularly giving attention to existing open research questions. Secondly it provides insight into the failings of such models and where progress needs to be made for power law research to feed through to actual improvements in network performance.Comment: To appear Computer Communication

Fluctuation-driven capacity distribution in complex networks

Maximizing robustness and minimizing cost are common objectives in the design of infrastructure networks. However, most infrastructure networks evolve and operate in a highly decentralized fashion, which may significantly impact the allocation of resources across the system. Here, we investigate this question by focusing on the relation between capacity and load in different types of real-world communication and transportation networks. We find strong empirical evidence that the actual capacity of the network elements tends to be similar to the maximum available capacity, if the cost is not strongly constraining. As more weight is given to the cost, however, the capacity approaches the load nonlinearly. In particular, all systems analyzed show larger unoccupied portions of the capacities on network elements subjected to smaller loads, which is in sharp contrast with the assumptions involved in (linear) models proposed in previous theoretical studies. We describe the observed behavior of the capacity-load relation as a function of the relative importance of the cost by using a model that optimizes capacities to cope with network traffic fluctuations. These results suggest that infrastructure systems have evolved under pressure to minimize local failures, but not necessarily global failures that can be caused by the spread of local damage through cascading processes

Global Modeling and Prediction of Computer Network Traffic

We develop a probabilistic framework for global modeling of the traffic over a computer network. This model integrates existing single-link (-flow) traffic models with the routing over the network to capture the global traffic behavior. It arises from a limit approximation of the traffic fluctuations as the time--scale and the number of users sharing the network grow. The resulting probability model is comprised of a Gaussian and/or a stable, infinite variance components. They can be succinctly described and handled by certain 'space-time' random fields. The model is validated against simulated and real data. It is then applied to predict traffic fluctuations over unobserved links from a limited set of observed links. Further, applications to anomaly detection and network management are briefly discussed