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

More "normal" than normal: scaling distributions and complex systems

One feature of many naturally occurring or engineered complex systems is tremendous variability in event sizes. To account for it, the behavior of these systems is often described using power law relationships or scaling distributions, which tend to be viewed as "exotic" because of their unusual properties (e.g., infinite moments). An alternate view is based on mathematical, statistical, and data-analytic arguments and suggests that scaling distributions should be viewed as "more normal than normal". In support of this latter view that has been advocated by Mandelbrot for the last 40 years, we review in this paper some relevant results from probability theory and illustrate a powerful statistical approach for deciding whether the variability associated with observed event sizes is consistent with an underlying Gaussian-type (finite variance) or scaling-type (infinite variance) distribution. We contrast this approach with traditional model fitting techniques and discuss its implications for future modeling of complex systems

Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)

Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet Mathematics (2005

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

Psychosocial care for persons affected by emergencies and major incidents: a Delphi study to determine the needs of professional first responders for education, training and support

Background The role of ambulance clinicians in providing psychosocial care in major incidents and emergencies is recognised in recent Department of Health guidance. The study described in this paper identified NHS professional first responders’ needs for education about survivors’ psychosocial responses, training in psychosocial skills, and continuing support. Method Ambulance staff participated in an online Delphi questionnaire, comprising 74 items (Round 1) on 7-point Likert scales. Second-round and third-round participants each received feedback based on the previous round, and responded to modified versions of the original items and to new items for clarification. Results One hundred and two participants took part in Round 1; 47 statements (64%) achieved consensus. In Round 2, 72 people from Round 1 participated; 15 out of 39 statements (38%) achieved consensus. In Round 3, 49 people from Round 2 participated; 15 out of 27 statements (59%) achieved consensus. Overall, there was consensus in the following areas: ‘psychosocial needs of patients’ (consensus in 34/37 items); ‘possible sources of stress in your work’ (8/9); ‘impacts of distress in your work’ (7/10); ‘meeting your own emotional needs’ (4/5); ‘support within your organisation’ (2/5); ‘needs for training in psychosocial skills for patients’ (15/15); ‘my needs for psychosocial training and support’ (5/6). Conclusions Ambulance clinicians recognise their own education needs and the importance of their being offered psychosocial training and support. The authors recommend that, in order to meet patients’ psychosocial needs effectively, ambulance clinicians are provided with education and training in a number of skills and their own psychosocial support should be enhanced

Artificial Artificial Intelligence: Measuring Influence of AI 'Assessments' on Moral Decision-Making

Given AI's growing role in modeling and improving decision-making, how and when to present users with feedback is an urgent topic to address. We empirically examined the effect of feedback from false AI on moral decision-making about donor kidney allocation. We found some evidence that judgments about whether a patient should receive a kidney can be influenced by feedback about participants' own decision-making perceived to be given by AI, even if the feedback is entirely random. We also discovered different effects between assessments presented as being from human experts and assessments presented as being from AI