There is a large, popular, and growing literature on "scale-free" networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scale-free graphs and few rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and that the most celebrated claims regarding the Internet and biology are verifiably false. In this paper, we introduce a structural metric that allows us to differentiate between all simple, connected graphs having an identical degree sequence, which is of particular interest when that sequence satisfies a power law relationship. We demonstrate that the proposed structural metric yields considerable insight into the claimed properties of SF graphs and provides one possible measure of the extent to which a graph is scale-free. This structural view can be related to previously studied graph properties such as the various notions of self-similarity, likelihood, betweenness and assortativity. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the current literature, and offers a rigorous and quantitative alternative, while suggesting the potential for a rich and interesting theory. This paper is aimed at readers familiar with the basics of Internet technology and comfortable with a theorem-proof style of exposition, but who may be unfamiliar with the existing literature on scale-free networks.

The search for unifying properties of complex networks is popular, challenging, and important. For modeling approaches that focus on robustness and fragility as unifying concepts, the Internet is an especially attractive case study, mainly because its applications are ubiquitous and pervasive, and widely available expositions exist at every level of detail. Nevertheless, alternative approaches to modeling the Internet often make extremely different assumptions and derive opposite conclusions about fundamental properties of one and the same system. Fortunately, a detailed understanding of Internet technology combined with a unique ability to measure the network means that these differences can be understood thoroughly and resolved unambiguously. This article aims to make recent results of this process accessible beyond Internet specialists to the broader scientific community and to clarify several sources of basic methodological differences that are relevant beyond either the Internet or the two specific approaches focused on here (i.e., scale-free networks and highly optimized tolerance networks).complex network ͉ HOT ͉ Internet topology ͉ network design ͉ scale-free network

A detailed understanding of the many facets of the Internet's topological structure is critical for evaluating the performance of networking protocols, for assessing the effectiveness of proposed techniques to protect the network from nefarious intrusions and attacks, or for developing improved designs for resource provisioning. Previous studies of topology have focused on interpreting measurements or on phenomenological descriptions and evaluation of graph-theoretic properties of topology generators. We propose a complementary approach of combining a more subtle use of statistics and graph theory with a first-principles theory of router-level topology that reflects practical constraints and tradeoffs. While there is an inevitable tradeoff between model complexity and fidelity, a challenge is to distill from the seemingly endless list of potentially relevant technological and economic issues the features that are most essential to a solid understanding of the intrinsic fundamentals of network topology. We claim that very simple models that incorporate hard technological constraints on router and link bandwidth and connectivity, together with abstract models of user demand and network performance, can successfully address this challenge and further resolve much of the confusion and controversy that has surrounded topology generation and evaluation.

We propose a definition of infrastructure resilience that is tied to the operation (or function) of an infrastructure as a system of interacting components and that can be objectively evaluated using quantitative models. Specifically, for any particular system, we use quantitative models of system operation to represent the decisions of an infrastructure operator who guides the behavior of the system as a whole, even in the presence of disruptions. Modeling infrastructure operation in this way makes it possible to systematically evaluate the consequences associated with the loss of infrastructure components, and leads to a precise notion of "operational resilience" that facilitates model verification, validation, and reproducible results. Using a simple example of a notional infrastructure, we demonstrate how to use these models for (1) assessing the operational resilience of an infrastructure system, (2) identifying critical vulnerabilities that threaten its continued function, and (3) advising policymakers on investments to improve resilience.