# A Network Model of Immigration and Coexistence

In the summer of 2015, more than one million refugees from the Middle East, Central Asia, and Africa arrived in Europe via dangerous routes across the Mediterranean Sea and the Balkans [7]. German Chancellor Angela Merkel welcomed the newly arrived with an enthusiastic “Wir schaffen das”—“We can do this”—embodying the collective spirit of optimism that pervaded Europe at the time. The vast majority of migrants were fleeing civil wars, brutal dictatorships, or religious persecution; others were seeking better economic opportunities. Preferred destinations among the more prosperous nations included Germany, Sweden, and the U.K., whereas European law and geography placed most of the burden of processing asylum claims on border nations such as Italy, Greece, and Hungary, which were not prepared to cope with such unprecedented numbers of new arrivals.

Measures including the forced return of illegal migrants to Turkey in exchange for economic concessions attempted to stem the flow. Hungary closed its borders, and Italy eventually closed its ports. European lawmakers were unable to devise a clear burden-sharing system among member states; at the same time, refugees and smugglers quickly found and exploited new migrant routes as existing ones saw increased patrolling and border controls. Eventually, the perception of an unmanageable crisis touched the entire continent. Discontent among the general public grew, as did discussions on safety, integration, European identity, secularism, resource availability, and the role of non-governmental organizations. As a result, the issue of migration has dominated elections across Europe over the past few years, and nationalist parties have enjoyed large gains in many countries.

It is within this larger sociopolitical context that many migrants have settled into European cities, each with their own personal story of adaptation, hurdles, discoveries, kindness, and hostility from strangers. Outcomes have thus far been mixed; refugees have successfully integrated in many communities from Italy to Sweden, but in some cases there have been challenges and mistrust. A common observation is that newcomers who do not adapt well—either by circumstance, aversion from natives, lack of resources and/or motivation, etc.—tend to self-segregate and create insular communities [5]. While these enclaves provide immigrants with advantages and a sense of belonging, they may also prevent them from fully integrating into the larger society.

**Figure 1.**Each node \(i\) is characterized by a variable attitude \(-1 \le x_i^t \le 1\) at time \(t\). Negative (red) values indicate guests and positive (blue) values represent hosts. The magnitude \(\vert x_i^t \vert\) represents node \(i\)'s degree of hostility towards members of the other group. All nodes \(j,k\) that are linked to 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. Nodes maximize their utility by adjusting their attitudes \(x_i^t\) and establishing or severing connections. Figure courtesy of Yao-li Chuang [2].

Attitudes \(x^t_i\) vary between \(-1 \le x^t_i \le 0\) for guests and \(0 \le x^t_i \le 1\) for hosts; the magnitude \(\mid x^t_i \mid\) indicates the degree of hostility towards the other group. Thus, \(x^t_i \to 0^{\pm}\) characterizes most receptive guests or most hospitable hosts, while \(x^t_i = \pm 1\) represents the highest level of xenophobia (see Figure 1). The utility \(U_i^t\) is given by a pairwise reward—to which each node \(j\) linked to \(i\) contributes—and by a cost function for maintaining \(m_i^t\) connections, such that

\[U_i^t = \sum_{j \in \Omega^t_i} A_{ij} \exp \Big( - \frac{( x^t_i - x^t_j)^2}{2 \sigma}\Big) - \exp \Big( \frac{m_i^t}{\alpha}\Big). \]

Here, \(\Omega_i^t\) is the set of nodes linked to \(i\) at time \(t\), so that \(m_i^t\) is given by its cardinality: \(m_i^t = \left\vert \Omega^t_i \right\vert\). The pairwise reward depends on the attitude difference \(\vert x^t_i - x^t_j \vert\) between nodes \(i\) and \(j\); a diminishing attitude difference correlates with an increasingly high reward. Therefore, if both \(i\) and \(j\) are hosts or immigrants, the reward is maximized for \(x_i^t = x_j^t\), leading to consensus within the group. But if \(i\) and \(j\) are from different groups, the reward is optimized only if both nodes adopt more cooperative attitudes: \(x_i^t \to 0^-\) and \(x_j^t \to 0^+\). The parameter \(\sigma\) controls the reward’s sensitivity to attitude differences, the amplitude \(A_{ij}\) specifies the maximum possible reward, and the scaling coefficient \(\alpha\) governs the cost of maintaining active links. Other models have considered residential segregation between two ethnic groups, with nodes seeking “friendly” neighbors with whom to connect. The most famous of these is the seminal Schelling model of segregation [3, 4, 6]. Our utility function \(U_i^t\) adds socioeconomic status as a decision-making factor in the link establishment process.

The dynamics unfold so that connectivities are modified at each time step to maximize utility. Attitudes are changed by imitation, so that

\[x_i^{t+1} = \Bigg\{ \begin{array}{ll} \min \Big( 0,x_i^t + \frac{x_j^t-x_i^t}{\kappa} \Big) \:\: \rm{for \:\: guests}, \\ \max \Big( 0, x_i^t + \frac{x_j^t-x_i^t}{\kappa} \Big) \:\:\rm{for \:\: hosts}, \end {array} \]

where \(\kappa\) governs attitude adjustment. Specifically, the timescale for guest cultural adjustment \(\tau_{\rm g}\) is given by \(\kappa\) and scaled by the probability of a guest being paired with a host \(N_{\rm h}/N\), so that \(\tau_{\rm g} \sim \kappa N/N_{\rm h}.\) Similarly, the host cultural adjustment timescale \(\tau_{\rm h} \sim \kappa N/N_{\rm g}\). Since \(N_{\rm h} \gg N_{\rm g}\), also \(\tau_{\rm h} \gg \tau_{\rm g}\); adjustment times for hosts are longer than for guests. These cultural adjustment timescales are compared with the unitary timescale for social link remodeling. Finally, initial conditions represent the way in which guests are originally settled in the community. One extreme case involves a perfectly executed welcoming program that provides refugees with sufficient social ties to hosts, and where all nodes are randomly connected — regardless of attitudes and utilities. The other extreme case is that of guests who arrive in a completely foreign environment with nonexistent initial resources. Hosts are naturally connected to one another in their own state of equilibrium, and guests are introduced without any links to hosts or each other.

Figure 2 depicts two representative steady-state outcomes. In Figure 2a, hosts and guests segregate and maintain highly hostile attitudes. Any initial cross-group utilities yield low rewards that do not increase over time, so that all ties between hosts and guests are eventually severed. Enclaves emerge when the two separate communities adopt uniform but differing attitudes \(x_i\). In Figure 2b, all nodes develop more cooperative attitudes that increase cross-group rewards, so that hosts and guests remain mixed. Eventually, \(x_i^t \to 0\) on all nodes. For both scenarios, \(|x_i^t - x_j^t| \:\:\to 0\) at steady state, but to which configuration society converges depends on parameter choices and initial conditions.

**Figure 2.**Simulated dynamics leading to complete segregation (2a) and integration (2b) between guest (red) and host (blue) populations. Initial conditions are randomly connected guest and host nodes with attitudes \(x_{i,\: {\rm guest}}^{0} = -1\) and \(x_{i, \:{\rm host}}^{0} = 1\). Panels 2a and 2b differ only for \(\kappa\), the attitude adjustment timescale, with \(\kappa = 1000\) in 2a, where segregated clusters emerge, and \(\kappa = 100\) in 2b, where a connected host-guest cluster arises over time. Figure courtesy of Yao-li Chuang [2].

We find that the main predictor of integration versus segregation is the magnitude of the \(\tau_{\rm g}, \tau_{\rm h}\) timescales relative to the unitary network remodeling time. In the case of slow cultural adjustment, immigrant and host communities tend to segregate as accumulation of socioeconomic wealth occurs more efficiently through insular, in-group connections. Conversely, fast cultural adjustment enables the establishment and sustenance of cross-cultural bridges, allowing different groups to reach consensus and maintain active cooperation. This is shown in Figures 2a and 2b, where the only difference is the \(\kappa\) parameter that drives \(\tau_{\rm g}, \tau_{\rm h}\). We also find that a high guest-to-host ratio \(N_{\rm {g}}/N_{\rm{h}}\) increases the likelihood of in-group connections and reduces communication between immigrant and host populations.

One possible approach to avoid segregation is the promotion of cross-group interactions via government incentives, or if newcomers carry or acquire desired skill sets, for example. Note that cultural adjustment does not necessarily mean that either side must abandon their identity; rather, we find that different groups must adopt tolerant attitudes towards one another, engaging in rapport building and acceptance to bridge differences and promote integration [1]. This is the long-term challenge for the future.

*Maria R. D'Orsogna presented this research during a minisymposium presentation at the 2019 SIAM Conference on Applications of Dynamical Systems, which took place last May in Snowbird, Utah.*

**Acknowledgments:** This work was made possible by support from grant W1911NF-16-1-0165.

**References**

[1] Berry, J.W. (2005). Living successfully in two cultures. *Inter. J. Intercult. Relations, 29*, 697-712.

[2] Chuang, Y.-L., Chou, T., & D’Orsogna, M.R. (2019). A network model of immigration: Enclave formation vs. cultural integration. *Net. Hetero. Media, 14*(1), 53-77.

[3] Gauvin, L., & Nadal, J.-P. (2015). Modeling and understanding social segregation. In H. Kaper & C. Rousseau (Eds.), *Mathematics of planet Earth: Mathematicians reflect on how to discover, organize, and protect our planet* (pp. 155-156). Philadelphia, PA: Society for Industrial and Applied Mathematics.

[4] Hatna, E., & Benenson, I. (2012). The Schelling model of ethnic residential dynamics: Beyond the integrated-segregated dichotomy of patterns. *J. Artific. Societ. Soc. Sim., 15*(1), 6.

[5] Koopmans, R. (2010). Trade-offs between equality and difference: Immigrant integration, multiculturalism and the welfare state in cross-national perspective. *J. Ethnic and Migration Stud., 36*, 1-26.

[6] Schelling, T.C. (1971). Dynamic models of segregation. *J. Math. Sociol., 1*(2), 143-186.

[7] United Nations High Commissioner for Refugees. (2017) *Global trends: Forced displacement in 2016*. Geneva, Switzerland: The United Nations. Retrieved from http://www.unhcr.org/5943e8a34.pdf.