A network model of immigration: Enclave formation vs. cultural integration

Simulated network dynamics leading to (a) complete segregation, and (b) integration between guest (red) and host (blue) populations. Shading of node colors represents the degree of hostility $|x_i^t|$ of node $i$ towards those of its opposite group, according to the color scheme shown in Fig. 1. Initial conditions are randomly connected guest and host nodes with attitudes $x_{i, {\rm guest}}^{0} = -1$ and $x_{i, {\rm host}}^{0} = 1$. Other parameters are $N_{\rm h} = 900, N_{\rm g} = 100$, $\alpha = 3$, $A_{\rm in} = A_{\rm out} = 10$, $\sigma = 1$. The two panels differ only for $\kappa$, the attitude adjustment timescale, with $\kappa = 1000$ in panel (a) and $\kappa = 100$ in panel (b). (a) For slowly changing attitudes ($\kappa = 1000$), hostile attitudes persist over time, eventually leading to segregated clusters. (b) For fast changing attitudes ($\kappa = 100$), guests initially become more cooperative, as shown by the lighter red colors. Over time, a more connected host--guest cluster arises with hosts eventually adopting more cooperative attitudes as well

Model diagram. Each node $i$ is characterized by a variable attitude $-1 \le x_i^t \le 1$ at time $t$. Negative values, depicted in red, indicate guest nodes; positive values represent hosts, colored in blue. The magnitude $\vert x_i^t \vert$ represents the degree of hostility of node $i$ towards members of the other group. Each node is shaded accordingly. All nodes $j, k$ linked to the central node $i$ represent the green-shaded social circle $\Omega_i^t$ of node $i$ at time $t$. The utility $U_i^t$ of node $i$ depends on its attitude relative to that of its $m^t_i $ connections in $\Omega_i^t$ and on $m^t_i$. Nodes maximize their utility by adjusting their attitudes $x_i^t$ and by establishing or severing connections, reshaping the network over time

Dynamics of the average utility per node $ \langle U_i^t \rangle_{\rm guest} $ in panels (a) and (c), and of the average attitudes $ \langle x^t_{i} \rangle_{\rm guest}, \langle x^t_{i} \rangle_{\rm host} $ in panels (b) and (d) for $ N_{\rm g} = 200 $ (a, b) and $ N_{\rm g} = 20 $ (c, d) guests in a total population of $ N = 2000 $ nodes. Parameters are $ \alpha = 3 $, $ A_{\rm in} = A_{\rm out} = 10 $, and $ \sigma = 1 $, and $ \kappa = 100 $ (faster) and $ \kappa = 1000 $ (slower) attitude adjustment. Initial attitudes are $ x_{i, {\text{host}}}^0 = 1 $ and $ x_{i, {\rm {guest}}}^0 = -1 $, with random connections between nodes so that on average each node is connected to $ m_i^0 = 10 $ others at $ t = 0 $, representing full insertion of guests into the community. Network remodeling (solid-red curve) and attitude adjustment (blue-dashed and green-dotted curves) are considered separately; their interplay is illustrated in full model simulations (purple-dot-dashed and magenta-double-dotted-dashed). Utility is increased in all cases, but attitude adjustment is more efficient at the onset due to the initially set cross-group connections. Network remodeling allows for higher utilities at longer times. For the full model, fast adjustment ($ \kappa = 100 $) leads to well integrated societies for $ N_{\rm g} = 200 $ as $ t \to \infty $, given that $ \langle x^t_{i} \rangle_{\rm host} \to 0^{+} $ and $ \langle x^t_{i} \rangle_{\rm guest} \to 0^{-} $; for $ N_{\rm g} = 20 $ hosts and guests segregate, with guests adopting collaborative attitudes, $ \langle x^t_{i} \rangle_{\rm host} \to 0.93 $ and $ \langle x^t_{i} \rangle_{\rm guest} \to 0^{-} $. Under slow adjustment ($ \kappa = 1000 $) hosts and guests will remain hostile and segregated with $ \langle x^t_{i} \rangle_{\rm host} \to 0.95 $, $ \langle x^t_{i} \rangle_{\rm guest} \to -0.34 $ for $ N_{\rm g} = 200 $ and $ \langle x^t_{i} \rangle_{\rm host} \to 0.99, \langle x^t_{i} \rangle_{\rm guest} \to 0^- $ for $ N_{\rm g} = 20 $

Dynamics of the integration index $I^t_{\rm int}$ in panels (a) and (c) and of the out-group reward fraction $v^t_{\rm out}$ in panels (b) and (d). Parameters and initial conditions are the same as in Fig. 3. (a, b) Large migrant population $N_{\rm g} = 200$. Here, $I_{\rm int}^t \to 0$ and $v_{\rm out}^t \to 0$ at long times when only network remodeling is allowed, and nodes seek links with conspecifics. If only attitude adjustment is allowed, $I_{\rm int}^t$ remains fixed due to the quenched network connectivity, while $v_{\rm out}^t$ increases as guests and hosts adopt more cooperative attitudes. For the full model, slow attitude changes ($\kappa = 1000$) lead to segregation and $I_{\rm int}^t \to 0$, $v_{\rm int}^t \to 0$ as $t \to \infty$. Fast attitude changes ($\kappa = 100$) lead to non-zero values of $I_{\rm int}^t$ and $v_{\rm out}^t$, indicating a more cooperative society. (c, d) Small migrant population $N_{\rm g} = 20$. Results are similar to the previous case except for the full model where $I_{\rm int}^t \to 0$, $v_{\rm out}^t \to 0$ as $t \to \infty$ for both $\kappa = 1000$ and $\kappa = 100$. For low values of $N_{\rm g}$ segregation arises under both fast and slow attitude changes

Dynamics of the integration index $I^t_{\rm out}$ in panel (a) and of the out-group reward fraction $v^t_{\rm out}$ in panel (b) for initially cooperative hosts. Parameters are the same as for the full model in Fig. 3, with initially cooperative hosts and uncooperative guests at $x_{i, {\rm host}}^0 = 0^+$ and $x_{i, {\rm guest}}^0 = -1$. (a) $I_{\rm int}^t$ decreases at the onset, eventually rising towards integration, where $I_{\rm int}^t \to 1$ as $t \to \infty$. The initial decrease is more pronounced for slow attitude adjustment ($\kappa = 1000$) and for larger guest populations ($N_{\rm g} = 200$) as described in the text. (b) $v_{\rm out}^t$ increases over long times as attitude adjustment allows for more cooperation between guests and hosts. Under slow attitude adjustment ($\kappa = 1000$) and large guest populations ($N_{\rm g} = 200$), $v_{\rm out}^t$ decreases at the onset, with players seeking in-group connections. As guests and hosts become more cooperative $v_{\rm out}^t$ increases

Dynamics of the integration index $I^t_{\rm out}$ in panel (a) and of the out-group reward fraction $v^t_{\rm out}$ in panel (b) under different initial random connectivities. Parameters are the same as in Fig 3 with initial hostile attitudes $x_{i, {\rm host}}^0 = 1$ and $x_{i, {\rm guest}}^0 = -1$. In the blue-solid curve $I_{\rm int}^0 = 0.91$; in the green-dashed curve $I_{\rm int}^0 = 0.37$; in the red-dotted curve $I_{\rm int}^0 = 0.06$. (a) For all three cases, $I_{\rm int}^t$ decreases from the initial values, but only the initially poorly connected case of $I_{\rm int}^0 = 0.06$ leads to full segregation, indicated by $I_{\rm int}^t \to 0$ as $t \to \infty$. For the other two cases, $I_{\rm int}^t \to 1$. (b) For all three cases $v_{\rm out}^t$ increases at the onset due to attitude adjustment, and later decreases due to network remodeling. Only $I_{\rm int}^0 = 0.06$ leads to long-time $v_{\rm out}^t \to 0$: as guest-host connections are severed, no socioeconomic utility can be shared. For the other two cases, $v_{\rm out}^t$ increases at long times, suggesting increasing rewards through cross-group connections

Integration index at steady state. In panel (a) $\langle I^*_{\rm int} \rangle$ is averaged over 20 realizations and plotted as a function of $A_{\rm out} / A_{\rm in}$ with $\kappa = \infty$. The bar indicates the variance. In panel (b) single representations $I^*_{\rm int}$ are shown as a function of $\kappa$ with $A_{\rm out} / A_{\rm in} = 2$. Other parameters are set at $\alpha = 3$ and $\sigma = 1$, with $N_{\rm h} = 1800$ and $N_{\rm g} = 200$. In both panels red solid circles represent initially unconnected, hostile hosts and guests, $x_{i, {\rm host}}^0 = 1$, $x_{i, {\rm guest}}^0 = -1$; blue triangles correspond to fully cooperative initial conditions $x_{i, {\rm host}}^0 = x_{i, {\rm guest}}^0 = 0$. When the ratio $A_{\rm out} / A_{\rm in}$ increases, the long-time state of the network changes from segregation to uniform mixture, and finally to reversed segregation. The transition for the default initial conditions occurs at larger $A_{\rm out} / A_{\rm in}$ ratios, compared to the cooperative initial conditions, as the former require higher compensation from out-group connections to overlook the hostile attitudes between guests and hosts. In panel (b) each data point corresponds to one realization. Increasing attitude adjustment time scale $\kappa$ leads to increased likelihood of segregation. A bimodal regime emerges for intermediate $\kappa$

Time $\tau_{\rm seg}$ to reach $\langle I_{\rm int}^*\rangle = 0.1$, where 90$\%$ of guest nodes are segregated as a function of (a) the sensitivity to the reward function $\sigma$, (b) the relative guest population $N_{\rm g}/N$ and (c) the total population $N$ assuming $N_{\rm g} = 0.1 N$. Other parameters are set to $\alpha = 3$, $A_{\rm in} = A_{\rm out} = 10$, $\kappa = 600$ in all panels. In panel (a) $N_{\rm g} = 200$, $N = 2000$; in panel (b) $\sigma = 1$ and $N = 2000$; in panel (c) $\sigma = 1$. In all three cases, guests and hosts are initially unconnected and hostile to each other, $x_{i, {\rm host}}^0 = 1$ and $x_{i, {\rm guest}}^0 = -1$. Each data point and its error bar represent the mean and the variance over $20$ simulations. In panel (a) increasing $\sigma$ allows for more tolerance to attitude differences, increasing the time to segregation. In panel (b) the higher guest population ratio leads to faster segregation as guests are more likely to establish in-group connections, forming guest only enclaves. In panel (c) the time to segregation increases with the overall population, for a constant $10\%$ guest population

Integration index at steady state. $\langle I^*_{\rm int} \rangle$ is averaged over 10 realizations and plotted as a function of $\kappa$ and $N_{\rm g} / N$ with $\alpha = 3$ in panel (a), and as a function of $\kappa$ and $\alpha$ with $N_{\rm g} / N = 0.1$ in panel (b). Other parameters are set at $A_{\rm in} = 10$, $A_{\rm out} = 20$, $\sigma = 1$, and $N = 2000$. In both panels guests and hosts are initially unconnected, with hostile attitudes, $x_{i, {\rm host}}^0 = 1$, $x_{i, {\rm guest}}^0 = -1$. In panel (a), for smaller $N_{\rm g} / N$, the transition from segregation to integration (or reverse segregation) occurs at larger $\kappa$. In panel (b) increasing $\alpha$ causes the transition point to shift towards larger $\kappa$