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The authors have declared that no competing interests exist.

Conceived and designed the experiments: MB MD PN WR. Performed the experiments: MB MD. Analyzed the data: MB MD PN WR. Contributed reagents/materials/analysis tools: MB MD PN WR. Wrote the paper: MB MD PN WR. Advisory: PN.

This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations. Such developments can range from chaotic to fully ordered. We demonstrate that in a dynamic space of interactive human behaviour cities produce a wealth of gravitational attractors whose size and shape depend on the resistance of space emerging inter alia from transport friction costs. This finding offers original insights into the complex evolution of spatial systems and appears to be consistent with the principles of central place theory known from the spatial sciences and geography. Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

The chief goal of this paper is to demonstrate that social-spatial interactions may generate a wealth of phase transitions varying from chaotic patterns to organised structures, even in geographic spaces endowed with robust spatial-order features. Attempts to identify spatial order through hexagonal networks embracing settlement units of various sizes have been made in the spatial sciences for a long period, witness the pioneering work of Reynaud [

Our study will present the equations of motion (see

a) Chaos and order in a hexagon with cities of equal masses, with the following parameters:

In the simulation process, the transport cost parameter _{c} (critical), h, m and k:
_{c} = 0.5.

The agent starts from the extreme upper (NW) square and proceeds through successive, neighbouring squares arranged in layers. In this case, it is attracted by six towns. The field from which it started is marked by the colour of the town that attracted it. The procedure stops when the agent has reached the square in the lower right corner (SE).

How should

It is worthwhile reiterating that the main point of this paper is to demonstrate that agents behave in a seemingly unpredictable manner when the transport cost (friction) is low. This means that a city can successfully attract very distant agents in low-friction systems. It is practically impossible to guess which city is to attract the agent without performing detailed numerical calculations, because even a slight displacement of the initial position of the agent can change its final destination. This is generally known as deterministic chaos, and often referred to as the “butterfly effect”.

Since the pioneering works of [_{i} and _{i} are the population and the distance from the

The Newtonian dynamics generated from

The equations of motion are given below:
_{A} = 1 without loss of generality. The friction coefficient in the behavioural parameter in our model is related to the principle of least effort of agents in dynamic space. For each starting point

In this algorithm, ‘steps’ are time steps of the numerical integration of the equation of motion. At every time step the agent changes his position, and in each new position energy (potential, kinetic and total) is calculated. The closer to the hexagon point (city), the lower the agent’s potential energy. Potential energy (red line) is calculated using

In

Fit: 3.2 exp(−0.24^{2}). Data for the red line were obtained from the numerical solution of the equation of motion

The model in _{d} is the dissipated energy, _{p} is the initial energy,

As shown above, a significant factor affecting the ordering of spatial interactions is transport costs. At minimum or zero costs, there is a practically unlimited freedom of movement in space (‘the flying carpet’ phenomenon with zero friction costs—see Rietveld [

a) CPT-derived pattern of towns in the Noord Oost Polder. Source: own preparation; b) chaotic pattern at

Next, we present an empirical illustration from the US.

a) Referral diagram of the Fresno medical care region [

So far we have used arbitrary units, and therefore it is of considerable importance to rewrite _{c}, scales as:

We notice that the transition to chaos can be triggered either via increasing

The concept and identification of attraction basins—or attractors, to use the modern terminology—have been known since Newton’s times. However, it was only the discoveries by Mandelbrot, Hausdorff, Sierpinski and other influential physicists and mathematicians that have made this concept a universal research framework, especially within the framework of social physics and spatial sciences. In this paper the close relationship between gravitational impacts and spatial behaviour in socio-economic systems has been analysed. One of the factors that determines them is the resistance of space. This is a result of cost-minimizing spatial motions of interactive agents in geographical space. If the resistance is weak, spatial behaviour patterns assume a chaotic form. And vice versa, stronger resistance causes the system to stabilise by producing gravitational attractors, i.e. as a result of a phase transition. Then, another property is also revealed: when gravitational forces stabilise or disappear, the elements of the system interact via network linkages (see Nijkamp and Reggiani [

We enclose a fractal built on a square for

(TIFF)