# Eikonal Blog

## Concepts

Graph theory:

• nodes and edges
• degree = number of edges for a given node
• isolated nodes
• connected nodes
• hub = well connected node
• Scale-free networks = average number of nodes stays constant
• preferential attachment:
• the fraction of nodes with k edges: $p(k) \sim k^{-\gamma}$
• a long tail distribution
• Degree of distribution

## Papers

• “Power laws, Pareto distributions and Zipf’s law” by M.E.J. Newman (arXiv) – http://arxiv.org/abs/cond-mat/0412004
• When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf’s law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, solar flares, moon craters, wars and people’s personal fortunes all appear to follow power laws. The origin of power-law behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them.
• more papers by M.E.J. Newman at arXiv – arxiv.org/find/cond-mat/1/au:+Newman_M/0/1/0/all/0/1