We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

/pdf/the-review-of-economic-studies-209dovqisz.pdf

The review of economic studies

Innovation indicators throughout the innovation process: An extensive literature analysis

Cambridge University Press. Paperback. Book Condition: New. Paperback. 480 pages. Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems. This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Paperback.

A First Course in the Numerical Analysis of Differential Equations: Stiff equations

To slow the spread of the novel coronavirus, many universities shifted to online instruction and now face the question of whether and how to resume in-person instruction. This article uses transcript data from a medium-sized American university to describe three enrollment networks that connect students through classes and in the process create social conditions for the spread of infectious disease: a university-wide network, an undergraduate-only network, and a liberal arts college network. All three networks are 'small worlds' characterized by high clustering, short average path lengths, and multiple independent paths connecting students. Students from different majors cluster together, but gateway courses and distributional requirements create cross-major integration. Connectivity declines when large courses of 100 students or more are removed from the network, as might be the case if some courses are taught online, but moderately sized courses must also be removed before less than half of student-pairs are connected in three steps and less than two-thirds in four steps. In all simulations, most students are connected through multiple independent paths. Hybrid models of instruction can reduce but not eliminate the potential for epidemic spread through the small worlds of course enrollments.

https://sociologicalscience.com/download/vol-7/may/SocSci_v7_222to241.pdf

The Small World Network of College Classes: Implications for Epidemic Spread on a University Campus

Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method

Dynamical processes, such as the diffusion of knowledge, opinions, pathogens, "fake news," innovation, and others, are highly dependent on the structure of the social network in which they occur. However, questions on why most social networks present some particular structural features, namely, high levels of transitivity and degree assortativity, when compared to other types of networks remain open. First, we argue that every one-mode network can be regarded as a projection of a bipartite network, and we show that this is the case using two simple examples solved with the generating functions formalism. Second, using synthetic and empirical data, we reveal how the combination of the degree distribution of both sets of nodes of the bipartite network-together with the presence of cycles of lengths four and six-explain the observed values of transitivity and degree assortativity coefficients in the one-mode projected network. Bipartite networks with top node degrees that display a more right-skewed distribution than the bottom nodes result in highly transitive and degree assortative projections, especially if a large number of small cycles are present in the bipartite structure.

Transitivity and degree assortativity explained: The bipartite structure of social networks.

Bipartite (two-mode) networks are important in the analysis of social and economic systems as they explicitly show conceptual links between different types of entities. However, applications of such networks often work with a projected (one-mode) version of the original bipartite network. The topology of the projected network, and the dynamics that take place on it, are highly dependent on the degree distributions of the two different node types from the original bipartite structure. To date, the interaction between the degree distributions of bipartite networks and their one-mode projections is well understood for only a few cases, or for networks that satisfy a restrictive set of assumptions. Here we show a broader analysis in order to fill the gap left by previous studies. We use the formalism of generating functions to prove that the degree distributions of both node types in the original bipartite network affect the degree distribution in the projected version. To support our analysis, we simulate several types of synthetic bipartite networks using a configuration model where node degrees are assigned from specific probability distributions, ranging from peaked to heavy-tailed distributions. Our findings show that when projecting a bipartite network onto a particular set of nodes, the degree distribution for the resulting one-mode network follows the distribution of the nodes being projected on to, but only so long as the degree distribution for the opposite set of nodes does not have a heavier tail. Furthermore, we show that bipartite degree distributions are not the only feature driving topology formation of projected networks, in contrast to what is commonly described in the literature.

Degree distributions of bipartite networks and their projections.

The total number of patents produced by a country (or the number of patents produced per capita) is often used as an indicator for innovation. Here we present evidence that the distribution of patents amongst applicants within many countries is well-described by power laws with exponents that vary between 1.66 (Japan) and 2.37 (Poland). We suggest that this exponent is a useful new metric for studying innovation. Using simulations based on simple preferential attachment-type rules that generate power laws, we find we can explain some of the variation in exponents between countries, with countries that have larger numbers of patents per applicant generally exhibiting smaller exponents in both the simulated and actual data. Similarly we find that the exponents for most countries are inversely correlated with other indicators of innovation, such as research and development intensity or the ubiquity of export baskets. This suggests that in more advanced economies, which tend to have smaller values of the exponent, a greater proportion of the total number of patents are filed by large companies than in less advanced countries.

https://s3-eu-west-1.amazonaws.com/pfigshare-u-files/1092881/APS13_ONEALE.pdf

Power law distributions of patents as indicators of innovation.

/pdf/power-law-distributions-of-patents-as-indicators-of-nnqgjj0jdk.pdf

Dion R. J. O’Neale

Papers

Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method

Transitivity and degree assortativity explained: The bipartite structure of social networks.

Degree distributions of bipartite networks and their projections.

Power law distributions of patents as indicators of innovation.

Power Law Distributions of Patents as Indicators of Innovation