Mathematics and the Internet: A Source of Enormous Confusion and Great Potential

Graph theory models the Internet mathematically, and a number of plausible mathematically intersecting network models for the Internet have been developed and studied. Simultaneously, Internet researchers have developed methodology to use real data to validate, or invalidate, proposed Internet models. The authors look at these parallel developments, particularly as they apply to scale-free network models of the preferential attachment type

Understanding Internet topology: principles, models, and validation

Building on a recent effort that combines a first-principles approach to modeling router-level connectivity with a more pragmatic use of statistics and graph theory, we show in this paper that for the Internet, an improved understanding of its physical infrastructure is possible by viewing the physical connectivity as an annotated graph that delivers raw connectivity and bandwidth to the upper layers in the TCP/IP protocol stack, subject to practical constraints (e.g., router technology) and economic considerations (e.g., link costs). More importantly, by relying on data from Abilene, a Tier-1 ISP, and the Rocketfuel project, we provide empirical evidence in support of the proposed approach and its consistency with networking reality. To illustrate its utility, we: 1) show that our approach provides insight into the origin of high variability in measured or inferred router-level maps; 2) demonstrate that it easily accommodates the incorporation of additional objectives of network design (e.g., robustness to router failure); and 3) discuss how it complements ongoing community efforts to reverse-engineer the Internet

A contrasting look at self-organization in the Internet and next-generation communication networks

This article examines contrasting notions of self-organization in the Internet and next-generation communication networks, by reviewing in some detail recent evidence regarding several of the more popular attempts to explain prominent features of Internet structure and behavior as "emergent phenomena." In these examples, what might appear to the nonexpert as "emergent self-organization" in the Internet actually results from well conceived (albeit perhaps ad hoc) design, with explanations that are mathematically rigorous, in agreement with engineering reality, and fully consistent with network measurements. These examples serve as concrete starting points from which networking researchers can assess whether or not explanations involving self-organization are relevant or appropriate in the context of next-generation communication networks, while also highlighting the main differences between approaches to self-organization that are rooted in engineering design vs. those inspired by statistical physics

Systematic Topology Analysis and Generation Using Degree Correlations

We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS- and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.Comment: Final versio

Modeling Data-Plane Power Consumption of Future Internet Architectures

With current efforts to design Future Internet Architectures (FIAs), the evaluation and comparison of different proposals is an interesting research challenge. Previously, metrics such as bandwidth or latency have commonly been used to compare FIAs to IP networks. We suggest the use of power consumption as a metric to compare FIAs. While low power consumption is an important goal in its own right (as lower energy use translates to smaller environmental impact as well as lower operating costs), power consumption can also serve as a proxy for other metrics such as bandwidth and processor load. Lacking power consumption statistics about either commodity FIA routers or widely deployed FIA testbeds, we propose models for power consumption of FIA routers. Based on our models, we simulate scenarios for measuring power consumption of content delivery in different FIAs. Specifically, we address two questions: 1) which of the proposed FIA candidates achieves the lowest energy footprint; and 2) which set of design choices yields a power-efficient network architecture? Although the lack of real-world data makes numerous assumptions necessary for our analysis, we explore the uncertainty of our calculations through sensitivity analysis of input parameters

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

Generating Representative ISP Technologies From First-Principles

Understanding and modeling the factors that underlie the growth and evolution of network topologies are basic questions that impact capacity planning, forecasting, and protocol research. Early topology generation work focused on generating network-wide connectivity maps, either at the AS-level or the router-level, typically with an eye towards reproducing abstract properties of observed topologies. But recently, advocates of an alternative "first-principles" approach question the feasibility of realizing representative topologies with simple generative models that do not explicitly incorporate real-world constraints, such as the relative costs of router configurations, into the model. Our work synthesizes these two lines by designing a topology generation mechanism that incorporates first-principles constraints. Our goal is more modest than that of constructing an Internet-wide topology: we aim to generate representative topologies for single ISPs. However, our methods also go well beyond previous work, as we annotate these topologies with representative capacity and latency information. Taking only demand for network services over a given region as input, we propose a natural cost model for building and interconnecting PoPs and formulate the resulting optimization problem faced by an ISP. We devise hill-climbing heuristics for this problem and demonstrate that the solutions we obtain are quantitatively similar to those in measured router-level ISP topologies, with respect to both topological properties and fault-tolerance