# Influence Networks and the Faraday Cage

This issue contains two Research Spotlights articles, the first of which applies mathematical modeling and analysis to a topic in the social sciences. “Opinion Dynamics and the Evolution of Social Power in Influence Networks,” by Peng Jia, Anahita MirTabatabaei, Noah E. Friedkin, and Francesco Bullo, describes the effect of a dynamically changing social influence network when forming opinions. Previous models have considered influence of a network on a single binary decision. In contrast, this paper considers an interpersonal network of individuals who are forming a sequence of opinions. The model for influence combines a fixed social influence network with an evolving structure based on the individual appraisal of prior decisions. The paper presents results on how power structures form within such a social network. In particular, two possible equilibria of this network are autocracy of a single individual and democracy of equally shared influence. The paper shows that there is a computable quantity of centrality, which serves as a threshold between the attraction of equilibria. These results are then applied in the case of three real social networks: a professional manufacturing association, a Facebook circle, and a research group of biological sciences faculty. The use of network dynamics applied to social problems is a relatively new and quite rich area. The article is a nice example, illustrating the types of questions that can be addressed within this burgeoning field, and is thus of wide appeal. It will be of particular interest to people with an interest in networks, dynamical systems, and mathematical social science.

The second article addresses a very classical topic, answering a question that comes to the mind of every person who has witnessed during a visit to a science museum the most dramatic of science demonstrations, the electric sparks of the Van de Graaff generator. Namely, how does it work? “Mathematics of the Faraday Cage,” by S. Jonathan Chapman, David P. Hewett, and Lloyd N. Trefethen, explains the how and the why of the Faraday cage effect, in which a wire mesh cube or screen blocks electric fields of long enough wavelength. In addition to the Van de Graaff generator, this is also the effect that allows you to see through the metal mesh to watch what is cooking in your microwave oven without being cooked. Faraday discovered the effect in 1836, and essentially every physics and engineering textbook since then has contained an explanation, but amazingly no detailed mathematical explanation of this effect has previously been given. This paper gives such an analysis, yielding a number of quantitative statements, most notably uncovering a mistake in the strength of the effect given in The Feynman Lectures on Physics. It presents models and numerical and theoretical results for a variant of Laplace’s equation in the plane for a mesh of discrete point charges as well as a resulting continuum model. The paper is enjoyable to read, and by leaving some technical mathematical details given in the appendix, the main points are clear and the mathematics easy to follow. This paper is an excellent read for a curious mathematician.

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Opinion Dynamics and the Evolution of Social Power in Influence Network

SIAM Rev., 57(3), 367–397.

Mathematics of the Faraday Cage

*SIAM Review*, 57(3), 398-417.