Intriguing effects of underlying star topology in Schelling's model with blocks

We explore the intriguing effects of the underlying star topological structure in the framework of Schelling's segregation model with blocks. The significant consequences exerted by the star topology are both theoretically analyzed and numerically simulated with and without introducing a fraction of altruistic agents, respectively. The collective utility of the model with egoists alone can be optimized and the optimum stationary state is achieved with the underlying star topology of blocks. More surprisingly, once a proportion of altruists is introduced, the average utility gradually decreases as the fraction of altruists increases. This presents a sharp contrast to the results in Schelling's model with a lattice topology of blocks. Furthermore, an adding-link mechanism is introduced to bridge the gap between the lattice and the star topologies and extend our analysis to more general scenarios. A scaling law of the average utility function is found for the star topology of blocks.

Figure 1

Cartoon illustration of a “city” with star spatial topology. It can be viewed as an abstraction of a monocentric city with satellite blocks. A total number of $N$ agents live in the city. The blue (peripheral gray) nodes represent the peripheral blocks and the red (central dark) node represents the central block, the so-called hub (analogous to a central business district in the real world), which has the most edges. The solid lines represent the links of all blocks neighboring the central hub. The dashed lines represent possible links among two peripheral blocks (see Sec. 4).

Figure 2

(a) Evolution of the agents' density ${\rho }_q$ and (b) the utility $u_q$ of the $q\mathrm{th}$ block of the star spatial topology with $H=200$ and $Q=50$. Note that a fraction of altruists $p=0$ is taken in the simulation, which means that no altruists are introduced. The red solid (bold dark) line represents the evolution of (a) the hub's density ${\rho }_{\mathrm{hub}}$ and (b) the utility $u_{\mathrm{hub}}$. The horizontal blue (gray) dotted line in (a) at ${\rho }_q=0.5$ is to guide the eye.

Figure 3

Similar to Fig. 2 but with the fraction of altruists $p=0.5$.

Figure 4

Similar to Fig. 2 but with the fraction of altruists $p=1$.

Figure 5

Average utility function $\langle u\rangle$ as a function of the fraction of altruists $p$ with $H=200$ and $Q=100$ (open squares), 200 (open triangles), and 400 (open circles).

Figure 6

(a) Average utility $\langle u\rangle$ as a function of the adding-link probability $\mathrm{\Pi }$ (note the linear-log scale) for different numbers of blocks $Q=100$ (open squares), 200 (open triangles), and 400 (open circles), with a fraction of altruists $p=0$. Three vertical dashed lines correspond to the critical probability ${\mathrm{\Pi }}_c$ (from right to left) $Q=100$, 200, and 400. (b) Collapse of $\langle u\rangle$ after rescaling with respect to ${\mathrm{\Pi }}^{*}=Q(\mathrm{\Pi }-{\mathrm{\Pi }}_c)$ (in linear scale), using the same data as in (a). The dashed line labels the rescaled critical probability ${\mathrm{\Pi }}_c^{*}$.