# American Institute of Mathematical Sciences

April  2022, 15(2): 187-212. doi: 10.3934/krm.2021051

## Kinetic equations for processes on co-evolving networks

 Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstr. 11, D 91058 Erlangen, Germany

Received  May 2021 Revised  December 2021 Published  April 2022 Early access  January 2022

Fund Project: The author acknowledges partial financial support by European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 777826 (NoMADS) and the German Science Foundation (DFG) through CRC TR 154 "Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks", Subproject C06

The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between them are updated in time. In our derivation we follow the basic paradigm of statistical mechanics: We start from paradigmatic microscopic models and derive a Liouville-type equation in a high-dimensional space including not only the node states in the network (corresponding to positions in mechanics), but also the edge weights between them. We then derive a natural (finite) marginal hierarchy and pass to an infinite limit.

We will discuss the closure problem for this hierarchy and see that a simple mean-field solution can only arise if the weight distributions between nodes of equal states are concentrated. In a more interesting general case we propose a suitable closure at the level of a two-particle distribution (including the weight between them) and discuss some properties of the arising kinetic equations. Moreover, we highlight some structure-preserving properties of this closure and discuss its analysis in a minimal model. We discuss the application of our theory to some agent-based models in literature and discuss some open mathematical issues.

Citation: Martin Burger. Kinetic equations for processes on co-evolving networks. Kinetic and Related Models, 2022, 15 (2) : 187-212. doi: 10.3934/krm.2021051
##### References:
 [1] G. Albi, M. Burger, J. Haskovec, P. Markowich, M. Schlottbom, N. Bellomo, P. Degond and E. Tadmor, eds., Continuum modeling of biological network formation, Active Particles, Birkhäuser, Cham, 1 (2017), 1-48. [2] G. Albi, L. Pareschi and M. Zanella, Opinion dynamics over complex networks: Kinetic modeling and numerical methods, Kinet. Relat. Models, 10 (2017), 1-32.  doi: 10.3934/krm.2017001. [3] W. Arendt, Resolvent positive operators, Proc. London Math. Soc., 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321. [4] N. Ayi and N. Pouradier-Duteil, Mean-field and graph limits for collective dynamics models with time-varying weights, J. Differential Equations, 299 (2021), 65-110.  doi: 10.1016/j.jde.2021.07.010. [5] F. Baumann, P. Lorenz-Spreen, I. M. Sokolov and M. Starnini, Modeling echo chambers and polarization dynamics in social networks, Phys. Rev. Lett., 124 (2020), 048301, 6 pp. doi: 10.1103/PhysRevLett.124.048301. [6] A. M. Belaza, K. Hoefman, J. Ryckebusch, A. Bramson, M. van den Heuvel and K. Schoors, Statistical physics of balance theory, PLoS One, 12 (2017), e0183696. [7] A. Benatti, H. F. de Arruda, F. N. Silva, C. H. Comin and L. da Fontoura Costa, Opinion diversity and social bubbles in adaptive Sznajd networks, J. Stat. Mech. Theory Exp., 2 (2020), 023407, 16 pp. doi: 10.1088/1742-5468/ab6de3. [8] L. Berlyand, R. Creese, P. E. Jabin and M. Potomkin, Continuum approximations to systems of correlated interacting particles, J. Stat. Phys., 174 (2019), 808-829.  doi: 10.1007/s10955-018-2205-8. [9] L. Berlyand, P. E. Jabin and M. Potomkin, Complexity reduction in many particle systems with random initial data, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 446-474.  doi: 10.1137/140969786. [10] L. Boltzmann, Weitere studien über das wärmegleichgewicht unter gasmolekülen, Sitzungsberichte Akademie der Wissenschaften, 66 (1872), 275-370. [11] G. Boschi, C. Cammarota and R. Kühn, Opinion dynamics with emergent collective memory: A society shaped by its own past, Phys. A, 558 (2020), 124909, 19 pp. doi: 10.1016/j.physa.2020.124909. [12] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Series in Applied Mathematics, 4, Gauthier-Villars, Paris, 2000. [13] M. Burger, Transport metrics for Vlasov hierarchies, In J. D. Benamou, V. Ehrlacher, D. Matthes, eds., Applications of Optimal Transportation in the Natural Sciences, Oberwolfach Proceedings, Mathematisches Forschungsinstitut Oberwolfach, 7 (2017), 392-395. [14] M. Burger, Network-structured kinetic models of social interactions, Vietnam J. Math., 49 (2021), 937-956.  doi: 10.1007/s10013-021-00505-8. [15] M. Carney and B. Davies, Agent-based modeling, scientific reproducibility, and taphonomy: A successful model implementation case study, J. Computer Applications in Archaeology, 3 (2020), 182-196. [16] A. Carro, R. Toral and M. San Miguel, The role of noise and initial conditions in the asymptotic solution of a bounded confidence continuous-opinion model, J. Stat. Phys., 151 (2013), 131-149.  doi: 10.1007/s10955-012-0635-2. [17] C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York, 1969. [18] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [19] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [20] U. Chitra and C. Musco, Understanding filter bubbles and polarization in social networks, preprint, arXiv: 1906.08772, 2019. [21] R. Conte and M. Paolucci, On agent-based modeling and computational social science, Frontiers in Psychology, 5 (2014), 668. [22] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. [23] M. Degroot, Reaching a consensus, J. American Statistical Association, 69 (1974), 118-121. [24] R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123. [25] E. Donkin, P. Dennis, A. Ustalakov, J. Warren and A. Clare, Replicating complex agent based models, a formidable task, Environmental Modelling and Software, 92 (2017), 142-151. [26] M. Fraia and A. Tosin, The Boltzmann legacy revisited: Kinetic models of social interactions, Mat. Cult. Soc. Riv. Unione Mat. Ital., 5 (2020), 93-109. [27] N. E. Friedkin, A formal theory of social power, The Journal of Mathematical Sociology, 12 (1986), 103-126. [28] N. E. Friedkin and E. Johnsen, Social influence and opinions, J. Math. Sociology, 15 (1990), 193-206. [29] N. E. Friedkin and E. Johnsen, Social influence networks and opinion change Models of opinion formation, Advances in Group Processes, 16 (1999), 1-29. [30] R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. doi: 10.1137/1.9781611971477. [31] F. Golse, On the dynamics of large particle systems in the mean field limit, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, Cham, 3 (2016), 1–144. doi: 10.1007/978-3-319-26883-5_1. [32] R. Golse, C. Mouhot and V. Ricci, Empirical measures and Vlasov hierarchies, Kinet. Relat. Models, 6 (2013), 919-943.  doi: 10.3934/krm.2013.6.919. [33] Y. Gu, Y. Sun and J. Gao, The Co-evolution model for social network evolving and opinion migration, In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (2017), 175–184. [34] R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002), 3. [35] L. Hörmander, The Analysis of Linear Partial Differential Operators. I., 2${nd}$ edition, Distribution Theory and Fourier Analysis. Springer, Berlin, 1990. doi: 10.1007/978-3-642-61497-2. [36] D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Physical Review Letters, 111 (2013), 138701. [37] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359. [38] J. G. Kirkwood, Statistical mechanics of fluid mixtures, J. Chem. Phys., 3 (1935), 300-313.  doi: 10.1063/1.1749657. [39] J. Kohne, N. Gallagher, Z. M. Kirgil, R. Paolillo, L. Padmos and F. Karimi, The role of network structure and initial group norm distributions in norm conflict, In Computational Conflict Research, Springer, Cham (2020), 113–140. [40] M. Konig, C. S. Hsieh and X. Liu, A structural model for the coevolution of networks and behavior, Review of Economics and Statistics, (2020), accepted. [41] R. Lambiotte, Majority rule on heterogeneous networks, J. Phys. A, 41 (2008), 224021, 6 pp. doi: 10.1088/1751-8113/41/22/224021. [42] J. Maas and A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, J. Stat. Phys., 181 (2020), 2257-2303.  doi: 10.1007/s10955-020-02663-4. [43] H. P. Maia, S. C. Ferreira and M. L. Martins, Adaptive network approach for emergence of societal bubbles, Physica A: Statistical Mechanics and its Applications, 572 (2021), 125588. [44] J. C. Maxwell, On the dynamical theory of gases, Philosophical Transactions of the Royal Society of London, 157 (1867), 49-88. [45] S. McQuade, B. Piccoli and N. Pouradier Duteil, Social dynamics models with time-varying influence, Math. Models Methods Appl. Sci., 29 (2019), 681-716.  doi: 10.1142/S0218202519400037. [46] B. Min and M. San Miguel, Fragmentation transitions in a coevolving nonlinear voter model, Scientific Reports, 7 (2017), 1-9. [47] S. Mischler, C. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8. [48] G. Naldi, L. Pareschi and G. Toscani, eds., Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Springer, New York, 2010. doi: 10.1007/978-0-8176-4946-3. [49] L. Neuhäuser, M. T. chaub, A. Mellor and R. Lambiotte, Opinion dynamics with multi-body interactions, in: S.Lasaulce, P.Mertikopoulos, A.Orda, eds., Network Games, Control and Optimization, Springer, Cham, 2021. [50] H. Neunzert and J. Wick, Theoretische und numerische Ergebnisse zur nichtlinearen Vlasov-Gleichung, Numerische Lösung Nichtlinearer Partieller Differential-und Integrodifferentialgleichungen, Springer, Berlin, Heidelberg, 267 (1972), 159–185. [51] A. Nigam, K. Shin, A. Bahulkar, B. Hooi, D. Hachen, B. K. Szymanski, C. Faloutsolos and N. V. Chawla, ONE-M: modeling the co-evolution of opinions and network connections, In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, Springer, Cham, (2018), 122–140. [52] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243. [53] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford 2014. [54] T. Paul, M. Pulvirenti and S. Simonella, On the size of chaos in the mean field dynamics, Arch. Ration. Mech. Anal., 231 (2019), 285-317.  doi: 10.1007/s00205-018-1280-y. [55] T. M. Pham, A. C. Alexander, J. Korbel, R. Hanel and S. Thurner, Balanced and fragmented phases in societies with homophily and social balance, Scientific Reports, 11 (2021), 17188. [56] T. M. Pham, I. Kondor, R. Hanel and S. Thurner, The effect of social balance on social fragmentation, Journal of the Royal Society Interface, 17 (2020), 172. [57] G. Poole and T. Boullion, A survey on M-matrices, SIAM Review, 16 (1974), 419-427.  doi: 10.1137/1016079. [58] N. Pouradier-Duteil, Mean-field limit of collective dynamics with time-varying weights, preprint, arXiv: 2103.06527, 2021. [59] T. Raducha and M. San Miguel, Emergence of complex structures from nonlinear interactions and noise in coevolving networks, Scientific Reports, 10 (2020), 1-14. [60] C. Ravazzi and A. Proskurnikov, Dynamical Social Networks, Encyclopedia of Systems and Control, Springer, London, 2020. [61] M. Saeedian, M. San Miguel and R. Toral, Absorbing-state transition in a coevolution model with node and link states in an adaptive network: Network fragmentation transition at criticality, New Journal of Physics, 22 (2020), 113001. [62] A. Singer, Maximum entropy formulation of the Kirkwood superposition approximation, The Journal of Chemical Physics, 121 (2004), 3657-3666. [63] T. A. Snijders, C. E. Steglich and M. Schweinberger, Modeling the co-evolution of networks and behavior, Longitudinal Models in the Behavioral and Related Sciences, 31 (2007), 41-71. [64] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, Berlin, Heidelberg, New York 1991. doi: 10.1007/978-3-642-84371-6. [65] H. Spohn and H. Neunzert, On the Vlasov hierarchy, Math. Methods Appl. Sci., 3 (1981), 445-455.  doi: 10.1002/mma.1670030131. [66] K. Sugishita, M. A. Porter, M. Beguerisse-Díaz and N. Masuda, Opinion dynamics in tie-decay networks, Phys. Rev., 3 (2021), 023249. [67] S. Thurner, Virtual social science, Computational Social Science and Complex Systems, 203 (2019), 1. [68] S. Thurner, R. Hanel and P. Klimek, Introduction to the Theory of Complex Systems, Oxford University Press, 2018. [69] E. M. Tur and J. M. Azagra-Caro, The coevolution of endogenous knowledge networks and knowledge creation, Journal of Economic Behavior and Organization, 145 (2018), 424-434. [70] F. Vazquez, V. M.Eguiluz and M. San Miguel, Generic absorbing transition in coevolution dynamics, Physical Review Letters, 100 (2008), 108702. [71] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-305.  doi: 10.1016/S1874-5792(02)80004-0. [72] D. H. Zanette, Beyond networks: Opinion formation in triplet-based populations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 3311-3319.  doi: 10.1098/rsta.2009.0066.

show all references

##### References:
 [1] G. Albi, M. Burger, J. Haskovec, P. Markowich, M. Schlottbom, N. Bellomo, P. Degond and E. Tadmor, eds., Continuum modeling of biological network formation, Active Particles, Birkhäuser, Cham, 1 (2017), 1-48. [2] G. Albi, L. Pareschi and M. Zanella, Opinion dynamics over complex networks: Kinetic modeling and numerical methods, Kinet. Relat. Models, 10 (2017), 1-32.  doi: 10.3934/krm.2017001. [3] W. Arendt, Resolvent positive operators, Proc. London Math. Soc., 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321. [4] N. Ayi and N. Pouradier-Duteil, Mean-field and graph limits for collective dynamics models with time-varying weights, J. Differential Equations, 299 (2021), 65-110.  doi: 10.1016/j.jde.2021.07.010. [5] F. Baumann, P. Lorenz-Spreen, I. M. Sokolov and M. Starnini, Modeling echo chambers and polarization dynamics in social networks, Phys. Rev. Lett., 124 (2020), 048301, 6 pp. doi: 10.1103/PhysRevLett.124.048301. [6] A. M. Belaza, K. Hoefman, J. Ryckebusch, A. Bramson, M. van den Heuvel and K. Schoors, Statistical physics of balance theory, PLoS One, 12 (2017), e0183696. [7] A. Benatti, H. F. de Arruda, F. N. Silva, C. H. Comin and L. da Fontoura Costa, Opinion diversity and social bubbles in adaptive Sznajd networks, J. Stat. Mech. Theory Exp., 2 (2020), 023407, 16 pp. doi: 10.1088/1742-5468/ab6de3. [8] L. Berlyand, R. Creese, P. E. Jabin and M. Potomkin, Continuum approximations to systems of correlated interacting particles, J. Stat. Phys., 174 (2019), 808-829.  doi: 10.1007/s10955-018-2205-8. [9] L. Berlyand, P. E. Jabin and M. Potomkin, Complexity reduction in many particle systems with random initial data, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 446-474.  doi: 10.1137/140969786. [10] L. Boltzmann, Weitere studien über das wärmegleichgewicht unter gasmolekülen, Sitzungsberichte Akademie der Wissenschaften, 66 (1872), 275-370. [11] G. Boschi, C. Cammarota and R. Kühn, Opinion dynamics with emergent collective memory: A society shaped by its own past, Phys. A, 558 (2020), 124909, 19 pp. doi: 10.1016/j.physa.2020.124909. [12] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Series in Applied Mathematics, 4, Gauthier-Villars, Paris, 2000. [13] M. Burger, Transport metrics for Vlasov hierarchies, In J. D. Benamou, V. Ehrlacher, D. Matthes, eds., Applications of Optimal Transportation in the Natural Sciences, Oberwolfach Proceedings, Mathematisches Forschungsinstitut Oberwolfach, 7 (2017), 392-395. [14] M. Burger, Network-structured kinetic models of social interactions, Vietnam J. Math., 49 (2021), 937-956.  doi: 10.1007/s10013-021-00505-8. [15] M. Carney and B. Davies, Agent-based modeling, scientific reproducibility, and taphonomy: A successful model implementation case study, J. Computer Applications in Archaeology, 3 (2020), 182-196. [16] A. Carro, R. Toral and M. San Miguel, The role of noise and initial conditions in the asymptotic solution of a bounded confidence continuous-opinion model, J. Stat. Phys., 151 (2013), 131-149.  doi: 10.1007/s10955-012-0635-2. [17] C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York, 1969. [18] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [19] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [20] U. Chitra and C. Musco, Understanding filter bubbles and polarization in social networks, preprint, arXiv: 1906.08772, 2019. [21] R. Conte and M. Paolucci, On agent-based modeling and computational social science, Frontiers in Psychology, 5 (2014), 668. [22] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. [23] M. Degroot, Reaching a consensus, J. American Statistical Association, 69 (1974), 118-121. [24] R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123. [25] E. Donkin, P. Dennis, A. Ustalakov, J. Warren and A. Clare, Replicating complex agent based models, a formidable task, Environmental Modelling and Software, 92 (2017), 142-151. [26] M. Fraia and A. Tosin, The Boltzmann legacy revisited: Kinetic models of social interactions, Mat. Cult. Soc. Riv. Unione Mat. Ital., 5 (2020), 93-109. [27] N. E. Friedkin, A formal theory of social power, The Journal of Mathematical Sociology, 12 (1986), 103-126. [28] N. E. Friedkin and E. Johnsen, Social influence and opinions, J. Math. Sociology, 15 (1990), 193-206. [29] N. E. Friedkin and E. Johnsen, Social influence networks and opinion change Models of opinion formation, Advances in Group Processes, 16 (1999), 1-29. [30] R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. doi: 10.1137/1.9781611971477. [31] F. Golse, On the dynamics of large particle systems in the mean field limit, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, Cham, 3 (2016), 1–144. doi: 10.1007/978-3-319-26883-5_1. [32] R. Golse, C. Mouhot and V. Ricci, Empirical measures and Vlasov hierarchies, Kinet. Relat. Models, 6 (2013), 919-943.  doi: 10.3934/krm.2013.6.919. [33] Y. Gu, Y. Sun and J. Gao, The Co-evolution model for social network evolving and opinion migration, In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (2017), 175–184. [34] R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002), 3. [35] L. Hörmander, The Analysis of Linear Partial Differential Operators. I., 2${nd}$ edition, Distribution Theory and Fourier Analysis. Springer, Berlin, 1990. doi: 10.1007/978-3-642-61497-2. [36] D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Physical Review Letters, 111 (2013), 138701. [37] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17.  doi: 10.1137/S0036141096303359. [38] J. G. Kirkwood, Statistical mechanics of fluid mixtures, J. Chem. Phys., 3 (1935), 300-313.  doi: 10.1063/1.1749657. [39] J. Kohne, N. Gallagher, Z. M. Kirgil, R. Paolillo, L. Padmos and F. Karimi, The role of network structure and initial group norm distributions in norm conflict, In Computational Conflict Research, Springer, Cham (2020), 113–140. [40] M. Konig, C. S. Hsieh and X. Liu, A structural model for the coevolution of networks and behavior, Review of Economics and Statistics, (2020), accepted. [41] R. Lambiotte, Majority rule on heterogeneous networks, J. Phys. A, 41 (2008), 224021, 6 pp. doi: 10.1088/1751-8113/41/22/224021. [42] J. Maas and A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, J. Stat. Phys., 181 (2020), 2257-2303.  doi: 10.1007/s10955-020-02663-4. [43] H. P. Maia, S. C. Ferreira and M. L. Martins, Adaptive network approach for emergence of societal bubbles, Physica A: Statistical Mechanics and its Applications, 572 (2021), 125588. [44] J. C. Maxwell, On the dynamical theory of gases, Philosophical Transactions of the Royal Society of London, 157 (1867), 49-88. [45] S. McQuade, B. Piccoli and N. Pouradier Duteil, Social dynamics models with time-varying influence, Math. Models Methods Appl. Sci., 29 (2019), 681-716.  doi: 10.1142/S0218202519400037. [46] B. Min and M. San Miguel, Fragmentation transitions in a coevolving nonlinear voter model, Scientific Reports, 7 (2017), 1-9. [47] S. Mischler, C. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8. [48] G. Naldi, L. Pareschi and G. Toscani, eds., Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Springer, New York, 2010. doi: 10.1007/978-0-8176-4946-3. [49] L. Neuhäuser, M. T. chaub, A. Mellor and R. Lambiotte, Opinion dynamics with multi-body interactions, in: S.Lasaulce, P.Mertikopoulos, A.Orda, eds., Network Games, Control and Optimization, Springer, Cham, 2021. [50] H. Neunzert and J. Wick, Theoretische und numerische Ergebnisse zur nichtlinearen Vlasov-Gleichung, Numerische Lösung Nichtlinearer Partieller Differential-und Integrodifferentialgleichungen, Springer, Berlin, Heidelberg, 267 (1972), 159–185. [51] A. Nigam, K. Shin, A. Bahulkar, B. Hooi, D. Hachen, B. K. Szymanski, C. Faloutsolos and N. V. 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Thurner, The effect of social balance on social fragmentation, Journal of the Royal Society Interface, 17 (2020), 172. [57] G. Poole and T. Boullion, A survey on M-matrices, SIAM Review, 16 (1974), 419-427.  doi: 10.1137/1016079. [58] N. Pouradier-Duteil, Mean-field limit of collective dynamics with time-varying weights, preprint, arXiv: 2103.06527, 2021. [59] T. Raducha and M. San Miguel, Emergence of complex structures from nonlinear interactions and noise in coevolving networks, Scientific Reports, 10 (2020), 1-14. [60] C. Ravazzi and A. Proskurnikov, Dynamical Social Networks, Encyclopedia of Systems and Control, Springer, London, 2020. [61] M. Saeedian, M. San Miguel and R. Toral, Absorbing-state transition in a coevolution model with node and link states in an adaptive network: Network fragmentation transition at criticality, New Journal of Physics, 22 (2020), 113001. [62] A. 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