How complexity emerges in urban systems: Theory of urban morphology

Human beings develop the land and transform land use patterns, constructing artificial structures. Among them, the city is a representative system and its morphology has attracted much attention. While most existing studies have been devoted to individual dynamics and focused on the proximity of specific areas of a city, we here pay attention to the city as a complex system, where interactions between individuals give rise to emergent properties. Specifically, analyzing the big data on every building in Seoul City, we specify the relevant interactions among constituents and probe the emergence of complex land use patterns. In particular, based on the empirical observations, we illustrate that interactions between land uses are frustrated, which serves as a basic postulate of the theory of urban morphology. We examine this conjecture with the help of a layered Ising-type model and disclose that the actual land use pattern emerges at the criticality of the system in the presence of heterogeneously distributed fields. It is also remarked that our model, allowing quantitative predictions, can easily be applied to other cities around the world.

Figure 1

Lattice structure of the urbanized area in Seoul City and its fractal dimension. (a) A coarse-grained lattice with lattice constant $a=250$ m. (b) Half the coordination number $z/2$ versus lattice constant $a$, which gives the dimension $d=1.82\pm 0.01$ in the range $100<a<1500\mathrm{m}$. Note the strong finite-size effects for $a<100$ m and strong boundary effects for $a>1500$ m. (c) Number $N$ of sites versus lattice constant $a$, disclosing the scaling relation between them. Fitted to the solid line of slope $-1.77$, it leads to the fractal dimension $d=1.77\pm 0.01$. (d) Correlation dimension $d_{\mathrm{corr}}$ versus lattice constant $a$, displaying a pattern similar to that in (b). In the range $100<a<1000$ m, we obtain $d=1.78\pm 0.01$.

Figure 2

Rank distributions (red squares) of total gross floor areas for (a) residential, (b) workplace, and (c) commercial buildings. The lattice constant is taken to be $a=138$ m. Power-law behaviors are observed in (a) and (c), down to the 30th percentile indicated by vertical blue lines (6093 among the total of 20 309 boxes in this coarse-grained scale). The black solid lines describe the power-law distributions with exponents (a) 0.36 and (c) 0.49.

Figure 3

Three-category Ising-type description of Seoul City under the median criterion. (a) Sites in the $r$, $w$, and $c$ states are depicted. It is observed that sites near the boundary of Seoul City are mostly in these states. (b) Sites in the $rw$, $wc$, and $cr$ states are displayed. Note that downtown areas are populated mainly by the $wc$-state sites; in effect buildings of the residential category are expelled from the downtown, giving rise to an urban doughnut pattern. (c) Sites in the $rwc$ state as well as in the 0 state. In contrast to $r$-, $w$-, and $c$-state sites, $rwc$-state sites are located away from the boundary. (d) Fraction $N_{\vec{\sigma }}/N$ of sites in state $\vec{\sigma }$ are presented versus the lattice constant $a$. Notably, the fractions of $c$- and $rw$-state sites are smaller than those of other state sites.

Figure 4

Model description. (a) A schematic diagram of the interlayer coupling, displaying frustration due to the antiferromagnetic interaction between the residential layer and the workplace layer. (b) External fields applied in the city and boundary conditions imposed at the boundary.

Figure 5

Order parameters (a) $m_q$ and (c) $m_x$ and their fluctuations (b) ${\chi }_q$ and (d) ${\chi }_x$. The dotted and dashed vertical lines in (b) and (d) indicate the values of $J$ at which ${\chi }_q$ reaches the peak: $J=0.24$ for $q=6$ and $J=0.30$ for $q=2,3$. It is evident that $m_r$ and $m_w$ (similarly, ${\chi }_r$ and ${\chi }_w$) exhibit the same behavior while ${\chi }_c$ peaks at a smaller value of $J$ than ${\chi }_r$ and ${\chi }_w$. The lattice constant and the interlayer coupling constant are taken to be $a=138$ m and $K=0.2$, respectively.

Figure 6

Accuracy factor $Q$ versus intralayer coupling constant $J$ for various values of the interlayer coupling constant $K$. $Q$ reaches the peak in the vicinity of the $Z_6$ symmetry breaking, revealing the criticality of the city. The lattice constant is given by $a=138$ m.

Figure 7

Urban morphology of Seoul City, extracted from (a) the Seoul building data set and (b)–(d) Monte Carlo simulation results. The number of ensembles used in the averaging process is (b) 1 (snapshot, short-time limit), (c) 20 (intermediate time), and (d) 1000 (long-time limit). Fluctuations are mainly observed in the short-time limit while a rather regular pattern is extracted in the long-time limit. Both results are not quite appropriate due to the quasiequilibrium nature of the system.