User:Douglas R. White/Sandbox

Sandbox inserts
This is where I do experiments (e.g., math formulas) before posting elsewhere. Please excuse the iterative history of this page. (This sandbox is for Lecture 3, Complexity summer school), http://en.wikipedia.org/wiki/Help:Formula

Realistic dynamical modeling, continued
The next two decades are critical for determining the survivability of city systems. This is not a simple problem of resources or potentially reversible environmental degradation. Cities now hold the majority of the world’s population (Bett. et al. PNAS 2007). A large part of urban growth is due to displacement of impoverished rural populations and to the displacement of refugees from war-torn region. Rural impoverishment and displacement to crowded urban areas are often compensated by having children as a safety net for mutual aid and needs for labor. At any one point in time there is little correlation between population density and levels of violence, but when tracked over time (Turchin & Korotayev), as populations with limited resources grow beyond capacity, regional sociopolitical violence is an observed outcome, with slow but quickening historical cycling within large politically enclosed regions (Bett.). In a world cross-cultural sample, correlations appear between increased internal conflict within polities and averaged by levels of consolidation of political hierarchy (Turchin & Korotayev). Our analysis, using the same sample of 186 societies ethnographically described at different points in time, shows significantly higher average levels of internal warfare for the 20th century (drw 07) than previous centuries. Some observers have noted that governments in the 20th century were the biggest killers of their own populations. The massive growth of urban populations and proportion urbanization occurred in the last 19th and 20th centuries. After 1962 world population growth slowed but not percent urbanization, which is slowly climbing an S-curve as it leaves a largely impoverished minority in rural areas. Mega-cities will undoubtedly continue to grow. With global warming on an unprecedented scale, questions of sustainability of city systems take on new urgency as … This poses problems not only of individual city management (Bett.), but …

Individual city populations varying over time… studied by Chandler 1987… Mike Batty 2006… Denise Pumain… highly dynamic. Is this simply urban and regional competition? (Modelski and Thompson 1996)

We begin the study of regional city systems with results from the study of urban growth. The growth of cities is constrained by availability of resources and rates of their consumption. The general growth equation (Bettencourt, Lobo, Helbing, Kühnert, & West 2006 assumes a quantity R of resources used per individual, on average, per unit time, while a resource quantity E is required to add one person to the population. An allocation of resources is expressed as $$Y=RN+E(dN/dt)$$, that is, sustenance and replacement, where $$dN/dt$$ is the growth rate. Then

$$\frac{dN(t)}{dt}=(\frac{Y_0}{E})N^\beta(t)-(\frac{R}{E})N(t). \longrightarrow (1)$$

The solution to this equation is given by

$$N(t)=\bigg[\frac{Y_0}{R}+(N^{1-\beta}(0)-\frac{Y_0}{R})exp\Big[-\frac{R}{E}{(1-\beta})t\Big]\bigg]^\dfrac{1}{1-\beta}. \to (2) $$

When $$\beta<1$$, population growth ceases at large times as a finite carrying capacity is reached, which is characteristic of biological species. In this case, “cities and, more generally, social organizations that are driven by economies of scale are destined to eventually stop growing”

Exercise: Create a distribution with (2) and grow it by (1), with $$\beta>1$$. Use Excel to fit to a power-law (Pareto), the R programs to fit to a q-exponential (Pareto II). Does the growth curve follow a power law, as claimed? (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=961842) You can also fit the power-law (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=961842) and q-exponential growth formulas. (See q-exponential, q-analog).

Ref B. 2007 Growth, innovation, scaling, and the pace of life in cities. Luís M. A. Bettencourt, José Lobo, Dirk Helbing, Christian Kühnert, and Geoffrey B. West PNAS 104(17):7301-7306.

BACK TO Lecture 3

User:Douglas R. White/MLE for q-exponential degree distributions
Notes that deal with statistical issues arising from the generative feedback network or "Social-circles" network model.

User:Douglas R. White/Social-circles network model
This is a trial editing of a wikipedia page of that name.