# Complexity And Spatial Networks

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Then, we ran the simulation for several values of T where for each T we found the value of N. We also determined the time by graphical integration of the function N 1 vs. 1). 6 Macro Analysis of the Model: The Rank Size Curve and the Number of Cities The outputs of the model are presented in Figs. 8; Fig. 4 presents the relation N vs. 7) is indeed verified. The relation t vs. T 1=3 is presented in Fig. 5 where a linear relationship between Fig. 4 Results of the model: the number of cities N vs.

5 Application of the Model to Israeli Cities In this section, we show how we use the model to interpret the system of cities in Israel. Our goal is to use the model in order to determine the city size distribution or more precisely the rank-size function for Israel’s real system of cities. Since we know the function NðtÞ is quadratic, we had to find the function KðTÞ which fits this result. For that, the following system of equations needs to be solved: dt 1 ¼ ; dT N ð3:5aÞ dN 1 ¼ ; and dT K ð3:5bÞ N ¼ N0 þ N 1 ð t À t 1 Þ 2 : ð3:5cÞ 40 L.

28) is particularly important for our discussion of cities below: the challenge then is to identify the {Xi} and the {xi} in that case. 31), as: dF ¼ ÀSdT þ Si Xi dxi : ð2:33Þ F can be specified as a function of T and V and then all other properties can be deduced. The free energy (Pippard 1957, p. 56) is a measure of the work that can be done a decrease in F by a system in an isothermal reversible change. Given the second law, it is the maximum amount of work that can be done by a system. 31), the principle of maximizing entropy, which we will invoke below, is equivalent other terms being kept constant to minimizing free energy.