American Institute of Mathematical Sciences

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March  2019, 14(1): 53-77. doi: 10.3934/nhm.2019004

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

Yao-Li Chuang 1,2, , Tom Chou 1,3, and  Maria R. D'Orsogna 1,2,,

1. 

Dept. of Biomathematics, UCLA, Los Angeles, CA 90095-1766, USA
2. 

Dept. of Mathematics, CSUN, Los Angeles, CA 91330-8313, USA
3. 

Dept. of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

* Corresponding author: Maria R. D'Orsogna

Received  May 2018 Published  January 2019

Fund Project: This work was made possible by support from grants ARO W1911NF-14-1-0472, ARO W1911NF-16-1-0165 (MRD), and NSF DMS-1516675 (TC).

Successfully integrating newcomers into native communities has become a key issue for policy makers, as the growing number of migrants has brought cultural diversity, new skills, but also, societal tensions to receiving countries. We develop an agent-based network model to study interacting "hosts" and "guests" and to identify the conditions under which cooperative/integrated or uncooperative/segregated societies arise. Players are assumed to seek socioeconomic prosperity through game theoretic rules that shift network links, and cultural acceptance through opinion dynamics. We find that the main predictor of integration under given initial conditions is the timescale associated with cultural adjustment relative to social link remodeling, for both guests and hosts. Fast cultural adjustment results in cooperation and the establishment of host-guest connections that are sustained over long times. Conversely, fast social link remodeling leads to the irreversible formation of isolated enclaves, as migrants and natives optimize their socioeconomic gains through in-group connections. We discuss how migrant population sizes and increasing socioeconomic rewards for host-guest interactions, through governmental incentives or by admitting migrants with highly desirable skills, may affect the overall immigrant experience.

Keywords: Sociological model, network dynamics, game theory, opinion dynamics, agent-based model.
Mathematics Subject Classification: Primary: 90B15, 91D30; Secondary: 05C40, 05C57.
Citation: Yao-Li Chuang, Tom Chou, Maria R. D'Orsogna. A network model of immigration: Enclave formation vs. cultural integration. Networks and Heterogeneous Media, 2019, 14 (1) : 53-77. doi: 10.3934/nhm.2019004
References:
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P. Barron, K. Kaiser and M. Pradhan, Local Conflict in Indonesia: Measuring Incidence and Identifying Patterns, World Bank Policy Research Paper 3384, 2004.

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J. W. Berry, Living successfully in two cultures, International Journal of Intercultural Relations, 29 (2005), 697-712. 

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J. W. Berry, U. Kim, T. Minde and D. Mok, Comparative studies of acculturative stress, The International Migration Review, 21 (1987), 491-511. 

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P. Boyle, K. Halfacree and V. Robinson, Exploring contemporary migration, 2nd edition, Pearson Education Limited, London and New York, 2013.

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E. M. Chaney, Foreword: The world economy and contemporary migration, The International Migration Review, 13 (1979), 204-212.  doi: 10.2307/2545027.

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Y.-S. Chiang, Cooperation could evolve in complex networks when activated conditionally on network characteristics, Journal of Artificial Societies and Social Simulation, 16 (2013), 6. doi: 10.18564/jasss.2148.

[10]

Y.-L. Chuang, M. R. D'Orsogna and T. Chou, A bistable belief dynamics model for radicalization within sectarian conflict, Quarterly of Applied Mathematics, 75 (2017), 19-37.  doi: 10.1090/qam/1446.

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M. D. Cohen, R. L. Riolo and R. Axelrod, The role of social structure in the maintenance of cooperative regimes, Rationality and Society, 13 (2001), 5-32.  doi: 10.1177/104346301013001001.

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A. P. Damm, Determinants of recent immigrants'location choices: Quasi-experimental evidence, Journal of Population Economics, 22 (2009), 145-174. 

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[28]

D. J. Haw and J. Hogan, A dynamical systems model of unorganized segregation, The Journal of Mathematical Sociology, 42 (2018), 113-127.  doi: 10.1080/0022250X.2018.1427091.

[29]

A. D. Henry, P. Pralat and C.-Q. Zhang, Emergence of segregation in evolving social networks, PNAS, 108 (2011), 8605-8610.  doi: 10.1073/pnas.1014486108.

[30] P. Ireland, Becoming Europe: Immigration Integration And The Welfare State, University of Pittsburgh Press, Pittsburgh, PA, 2004. 
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M. A. Javarone, A. E. Atzeni and S. Galam, Emergence of cooperation in the Prisoner's Dilemma driven by conformity, in Applications of Evolutionary Computation. EvoApplications 2015. Lecture Notes in Computer Science (eds. A. Mora and G. Squillero), vol. 9028, Springer, Cham, 2015, 155–163. doi: 10.1007/978-3-319-16549-3_13.

[32]

W. Kandel and J. Cromartie, New patterns of Hispanic settlement in rural America, Technical Report 99, United States Department of Agriculture, 2004.

[33]

T. B. Klos, Decentralized interaction and co-adaptation in the repeated prisoners dilemma, Computational and Mathematical Organization Theory, 5 (1999), 147165.

[34]

R. Koopmans, Trade-offs between equality and difference: Immigrant integration, multiculturalism and the welfare state in cross-national perspective, Journal of Ethnic and Migration Studies, 36 (2010), 1-26.  doi: 10.1080/13691830903250881.

[35]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, in Communications in Difference Equations (eds. S. Elaydi, G. Ladas, J. Popenda and J. Rakowski), Amsterdam: Gordon and Breach, 2000, 227–236.

[36]

D. T. Lichter, D. Parisi and M. C. Taquino, Emerging patterns of Hispanic residential segregation: Lessons from rural and small–town America, Rural Sociology, 81 (2016), 483-518.  doi: 10.1111/ruso.12108.

[37]

D. T. Lichter, D. Parisi, M. C. Taquino and S. M. Grice, Residential segregation in new Hispanic destinations: Cities, suburbs, and rural communities compared, Social Science Research, 39 (2010), 215-230.  doi: 10.1016/j.ssresearch.2009.08.006.

[38]

A. Németh and K. Takács, The evolution of altruism in spatially structured populations, Journal of Artificial Societies and Social Simulations, 10 (2007), 1-13. 

[39]

A. Nowak, J. Szamrej and B. Latané, From private attitude to public opinion: A dynamic theory of social impact, Psychological Review, 97 (1990), 362-376.  doi: 10.1037/0033-295X.97.3.362.

[40]

N. Priest, Y. Paradies, A. Ferdinand, L. Rouhani and M. Kelaher, Patterns of intergroup contact in public spaces: Micro-ecology of segregation in Australian communities, Societies, 4 (2014), 30-44.  doi: 10.3390/soc4010030.

[41]

R. Riolo, The Effects of Tag-Mediated Selection of Partners in Evolving Populations Playing the Iterated Prisoner's Dilemma, Technical report, Santa Fe Institute, Santa Fe, NM, 1997.

[42]

R. L. Riolo, M. D. Cohen and R. Axelrod, Evolution of cooperation without reciprocity, Nature, 414 (2001), 441-443.  doi: 10.1038/35106555.

[43]

F. C. Santos, J. M. Pacheco and T. Lenaerts, Cooperation prevails when individuals adjust their social ties, PLoS Computational Biology, 2 (2006), e140. doi: 10.1371/journal.pcbi.0020140.

[44]

M. Semyonov and A. Tyree, Community segregation and the costs of ethnic subordination, Social Forces, 59 (1981), 649-666. 

[45]

E. S. Stewart, UK dispersal policy and onward migration: Mapping the current state of knowledge, Journal of Refugee Studies, 25 (2012), 25-49.  doi: 10.1093/jrs/fer039.

[46]

A. Szolnoki and M. Perc, Competition of tolerant strategies in the spatial public goods game, New Journal of Physics, 18 (2016), 083021. doi: 10.1088/1367-2630/18/8/083021.

[47]

UNHCR, Global trends: Forced displacement in 2017, Technical report, The UN Refugee Agency, The United Nations, 2018, http://www.unhcr.org/5b27be547.pdf.

[48]

Q. Wang, H. Wang, Z. Zhang, Y. Li, Y. Liu and M. Perc, Heterogeneous investments promote cooperation in evolutionary public goods games, Physica A, 502 (2018), 570-575. 

[49]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. 

[50]

G. Weisbuch, G. Deffuant, F. Amblard and J.-P. Nadal, Meet, discuss, and segregate, Complexity, 7 (2002), 55-63.  doi: 10.1002/cplx.10031.

[51]

J. S. White, R. Hamad, X. Li, S. Basu, H. Ohlsson, J. Sundquist and K. Sundquist, Long-term effects of neighbourhood deprivation on diabetes risk: Quasi-experimental evidence from a refugee dispersal policy in Sweden, The Lancet Diabetes and Endocrinology, 4 (2016), 517-524.  doi: 10.1016/S2213-8587(16)30009-2.

show all references

References:
[1]

R. Axelrod, The dissemination of culture: A model with local convergence and global polarization, The Journal of Conflict Resolution, 41 (1997), 203-226.  doi: 10.1177/0022002797041002001.

[2]

P. Barron, K. Kaiser and M. Pradhan, Local Conflict in Indonesia: Measuring Incidence and Identifying Patterns, World Bank Policy Research Paper 3384, 2004.

[3]

J. W. Berry, Acculturation and adaptation in a new society, International Migration Quarterly Review, 30 (1992), 69-85.  doi: 10.1111/j.1468-2435.1992.tb00776.x.

[4]

J. W. Berry, Living successfully in two cultures, International Journal of Intercultural Relations, 29 (2005), 697-712. 

[5]

J. W. Berry, U. Kim, T. Minde and D. Mok, Comparative studies of acculturative stress, The International Migration Review, 21 (1987), 491-511. 

[6]

P. Boyle, K. Halfacree and V. Robinson, Exploring contemporary migration, 2nd edition, Pearson Education Limited, London and New York, 2013.

[7] S. Castles and M. J. Miller, The Age of Migration: International Population Movements in the Modern World, The Guilford Press, New York, 2003. 
[8]

E. M. Chaney, Foreword: The world economy and contemporary migration, The International Migration Review, 13 (1979), 204-212.  doi: 10.2307/2545027.

[9]

Y.-S. Chiang, Cooperation could evolve in complex networks when activated conditionally on network characteristics, Journal of Artificial Societies and Social Simulation, 16 (2013), 6. doi: 10.18564/jasss.2148.

[10]

Y.-L. Chuang, M. R. D'Orsogna and T. Chou, A bistable belief dynamics model for radicalization within sectarian conflict, Quarterly of Applied Mathematics, 75 (2017), 19-37.  doi: 10.1090/qam/1446.

[11]

M. D. Cohen, R. L. Riolo and R. Axelrod, The role of social structure in the maintenance of cooperative regimes, Rationality and Society, 13 (2001), 5-32.  doi: 10.1177/104346301013001001.

[12]

M. H. Crawford and B. C. Campbell (eds.), Causes and Consequences of Human Migration, Cambridge University Press, Cambridge, UK, 2012. doi: 10.1017/CBO9781139003308.

[13]

A. P. Damm, Determinants of recent immigrants'location choices: Quasi-experimental evidence, Journal of Population Economics, 22 (2009), 145-174. 

[14]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98.  doi: 10.1142/S0219525900000078.

[15]

M. H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121. 

[16]

K. Fehl, D. J. van der Post and D. Semmann, Co-evolution of behaviour and social network structure promotes human cooperation, Ecology Letters, 14 (2011), 546-551.  doi: 10.1111/j.1461-0248.2011.01615.x.

[17]

K. Felijakowski and R. Kosinski, Bounded confidence model in complex networks, International Journal of Modern Physics C, 24 (2013), 1350049, 12pp. doi: 10.1142/S0129183113500496.

[18]

K. Felijakowski and R. Kosinski, Opinion formation and self-organization in a social network in an intelligent agent system, ACTA Physica Polonica B, 45 (2014), 2123-2134.  doi: 10.5506/APhysPolB.45.2123.

[19]

M. Fossett, Ethnic preferences, social distance dynamics, and residential segregation: Theoretical explorations using simulation analysis, The Journal of Mathematical Sociology, 30 (2006), 185-273.  doi: 10.1080/00222500500544052.

[20]

N. E. Friedkin, Choice shift and group polarization, American Sociological Review, 64 (1999), 856-875. 

[21]

A. Gabel, P. L. Krapivsky and S. Redner, Highly dispersed networks by enhanced redirection, Physical Review E, 88 (2013), 050802(R). doi: 10.1103/PhysRevE.88.050802.

[22]

S. Galam, Heterogeneous beliefs, segregation, and extremism in the making of public opinions, Physical Review E, 71 (2005), 046123. doi: 10.1103/PhysRevE.71.046123.

[23]

S. Galam, Stubbornness as an unfortunate key to win a public debate: An illustration from sociophysics, Society, 15 (2016), 117-130.  doi: 10.1007/s11299-015-0175-y.

[24]

S. Galam and M. A. Javarone, Modeling radicalization phenomena in heterogeneous populations, PLOS One, 11 (2016), e0155407. doi: 10.1371/journal.pone.0155407.

[25]

B. Golub and M. O. Jackson, Naíve learning in social networks and the wisdom of crowds, American Economic Journal: Microeconomics, 2 (2010), 112-149.  doi: 10.1257/mic.2.1.112.

[26]

D. Hales, Cooperation without memory or space: Tags, groups and the Prisoner's Dilemma, in Multi-Agent-Based Simulation. (eds. S. Moss and P. Davidsson), Springer, Berlin/Heidelberg, 2000, 157–166. doi: 10.1007/3-540-44561-7_12.

[27]

R. A. Hammond and R. Axelrod, The evolution of ethnocentrism, Journal of Conflict Resolution, 50 (2006), 926-936.  doi: 10.1177/0022002706293470.

[28]

D. J. Haw and J. Hogan, A dynamical systems model of unorganized segregation, The Journal of Mathematical Sociology, 42 (2018), 113-127.  doi: 10.1080/0022250X.2018.1427091.

[29]

A. D. Henry, P. Pralat and C.-Q. Zhang, Emergence of segregation in evolving social networks, PNAS, 108 (2011), 8605-8610.  doi: 10.1073/pnas.1014486108.

[30] P. Ireland, Becoming Europe: Immigration Integration And The Welfare State, University of Pittsburgh Press, Pittsburgh, PA, 2004. 
[31]

M. A. Javarone, A. E. Atzeni and S. Galam, Emergence of cooperation in the Prisoner's Dilemma driven by conformity, in Applications of Evolutionary Computation. EvoApplications 2015. Lecture Notes in Computer Science (eds. A. Mora and G. Squillero), vol. 9028, Springer, Cham, 2015, 155–163. doi: 10.1007/978-3-319-16549-3_13.

[32]

W. Kandel and J. Cromartie, New patterns of Hispanic settlement in rural America, Technical Report 99, United States Department of Agriculture, 2004.

[33]

T. B. Klos, Decentralized interaction and co-adaptation in the repeated prisoners dilemma, Computational and Mathematical Organization Theory, 5 (1999), 147165.

[34]

R. Koopmans, Trade-offs between equality and difference: Immigrant integration, multiculturalism and the welfare state in cross-national perspective, Journal of Ethnic and Migration Studies, 36 (2010), 1-26.  doi: 10.1080/13691830903250881.

[35]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, in Communications in Difference Equations (eds. S. Elaydi, G. Ladas, J. Popenda and J. Rakowski), Amsterdam: Gordon and Breach, 2000, 227–236.

[36]

D. T. Lichter, D. Parisi and M. C. Taquino, Emerging patterns of Hispanic residential segregation: Lessons from rural and small–town America, Rural Sociology, 81 (2016), 483-518.  doi: 10.1111/ruso.12108.

[37]

D. T. Lichter, D. Parisi, M. C. Taquino and S. M. Grice, Residential segregation in new Hispanic destinations: Cities, suburbs, and rural communities compared, Social Science Research, 39 (2010), 215-230.  doi: 10.1016/j.ssresearch.2009.08.006.

[38]

A. Németh and K. Takács, The evolution of altruism in spatially structured populations, Journal of Artificial Societies and Social Simulations, 10 (2007), 1-13. 

[39]

A. Nowak, J. Szamrej and B. Latané, From private attitude to public opinion: A dynamic theory of social impact, Psychological Review, 97 (1990), 362-376.  doi: 10.1037/0033-295X.97.3.362.

[40]

N. Priest, Y. Paradies, A. Ferdinand, L. Rouhani and M. Kelaher, Patterns of intergroup contact in public spaces: Micro-ecology of segregation in Australian communities, Societies, 4 (2014), 30-44.  doi: 10.3390/soc4010030.

[41]

R. Riolo, The Effects of Tag-Mediated Selection of Partners in Evolving Populations Playing the Iterated Prisoner's Dilemma, Technical report, Santa Fe Institute, Santa Fe, NM, 1997.

[42]

R. L. Riolo, M. D. Cohen and R. Axelrod, Evolution of cooperation without reciprocity, Nature, 414 (2001), 441-443.  doi: 10.1038/35106555.

[43]

F. C. Santos, J. M. Pacheco and T. Lenaerts, Cooperation prevails when individuals adjust their social ties, PLoS Computational Biology, 2 (2006), e140. doi: 10.1371/journal.pcbi.0020140.

[44]

M. Semyonov and A. Tyree, Community segregation and the costs of ethnic subordination, Social Forces, 59 (1981), 649-666. 

[45]

E. S. Stewart, UK dispersal policy and onward migration: Mapping the current state of knowledge, Journal of Refugee Studies, 25 (2012), 25-49.  doi: 10.1093/jrs/fer039.

[46]

A. Szolnoki and M. Perc, Competition of tolerant strategies in the spatial public goods game, New Journal of Physics, 18 (2016), 083021. doi: 10.1088/1367-2630/18/8/083021.

[47]

UNHCR, Global trends: Forced displacement in 2017, Technical report, The UN Refugee Agency, The United Nations, 2018, http://www.unhcr.org/5b27be547.pdf.

[48]

Q. Wang, H. Wang, Z. Zhang, Y. Li, Y. Liu and M. Perc, Heterogeneous investments promote cooperation in evolutionary public goods games, Physica A, 502 (2018), 570-575. 

[49]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. 

[50]

G. Weisbuch, G. Deffuant, F. Amblard and J.-P. Nadal, Meet, discuss, and segregate, Complexity, 7 (2002), 55-63.  doi: 10.1002/cplx.10031.

[51]

J. S. White, R. Hamad, X. Li, S. Basu, H. Ohlsson, J. Sundquist and K. Sundquist, Long-term effects of neighbourhood deprivation on diabetes risk: Quasi-experimental evidence from a refugee dispersal policy in Sweden, The Lancet Diabetes and Endocrinology, 4 (2016), 517-524.  doi: 10.1016/S2213-8587(16)30009-2.

Figure 2.  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
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Figure 1.  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
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Figure 3.  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 $
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Figure 4.  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
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Figure 5.  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
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Figure 6.  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
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Figure 7.  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 $\kappa$
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Figure 8.  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
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Figure 9.  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 $\kappa$
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Table 1.  List of variables and parameters of the model
SymbolDescriptiondefault values
$x_i$attitude-1 to 1
$A_{\rm in}$maximal utility through in-group connection $10$
$A_{\rm out}$maximal utility through out-group connection $1$ to $100$
$\sigma$sensitivity to attitude difference $1$
$\kappa$attitude adjustment timescale $100$ to $1000$
$\alpha$cost of adding connections $3$
$N$total population $2000$
$N_{\rm g}$guest population $20$ to $200$
$N_{\rm h}$host population $N - N_{\rm g}$
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SymbolDescriptiondefault values
$x_i$attitude-1 to 1
$A_{\rm in}$maximal utility through in-group connection $10$
$A_{\rm out}$maximal utility through out-group connection $1$ to $100$
$\sigma$sensitivity to attitude difference $1$
$\kappa$attitude adjustment timescale $100$ to $1000$
$\alpha$cost of adding connections $3$
$N$total population $2000$
$N_{\rm g}$guest population $20$ to $200$
$N_{\rm h}$host population $N - N_{\rm g}$
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