February  2021, 14(1): 45-76. doi: 10.3934/krm.2020048

Opinion formation systems via deterministic particles approximation

1. 

DISIM, Università degli Studi dell'Aquila, via Vetoio 1 (Coppito), 67100 L'Aquila (AQ), Italy

2. 

EPFL SB, Station 8, CH-1015 Lausanne, Switzerland

* Corresponding author: Simone Fagioli

Received  April 2020 Revised  August 2020 Published  February 2021 Early access  September 2020

We propose an ODE-based derivation for a generalized class of opinion formation models either for single and multiple species (followers, leaders, trolls). The approach is purely deterministic and the evolution of the single opinion is determined by the competition between two mechanisms: the opinion diffusion and the compromise process. Such deterministic approach allows to recover in the limit an aggregation/(nonlinear)diffusion system of PDEs for the macroscopic opinion densities.

 

Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Citation: Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic and Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048
References:
[1]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20140138. doi: 10.1098/rsta.2014.0138.

[2]

G. Albi, L. Pareschi and M. Zanella, On the Optimal Control of Opinion Dynamics on Evolving Networks, vol. 494, 58–67, Springer, Cham, 2016.

[3]

G. AlbiL. Pareschi and M. Zanella, Opinion dynamics over complex networks: Kinetic modelling and numerical methods, Kinetic and Related Models, 10 (2017), 1-32.  doi: 10.3934/krm.2017001.

[4]

G. Albi, P. Pareschi, G. Toscani and M. Zanella, Recent advances in opinion modeling: Control and social influence, 49–98, Birkhäuser-Springer, 2017.

[5]

G. AlettiG. Naldi and G. Toscani, First-order continuous models of opinion formation, SIAM J. Appl. Math., 67 (2007), 837-853.  doi: 10.1137/060658679.

[6]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[7]

N. Ansini and S. Fagioli, Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation, Communications in Mathematical Sciences, 18 (2020), 459-486.  doi: 10.4310/CMS.2020.v18.n2.a8.

[8]

N. Bellomo, G. Ajmone Marsan and A. Tosin, Complex Systems and Society. Modeling and Simulation. SpringerBriefs in Mathematics, Springer, 2013. doi: 10.1007/978-1-4614-7242-1.

[9]

E. Ben-Naim, Opinion dynamics: Rise and fall of political parties, Europhysics Letters, 69 (2005), 671. doi: 10.1209/epl/i2004-10421-1.

[10]

S. Biswas and P. Sen, Critical noise can make the minority candidate win: The u.s. presidential election cases, Phys. Rev. E, 96 (2017), 032303. doi: 10.1103/PhysRevE.96.032303.

[11]

D. Borra and T. Lorenzi, A hybrid model for opinion formation, Zeitschrift für angewandte Mathematik und Physik, 64 (2013), 419–437. doi: 10.1007/s00033-012-0259-z.

[12]

L. Boudin and F. Salvarani, Opinion dynamics: Kinetic modelling with mass media, application to the scottish independence referendum, Physica A: Statistical Mechanics and its Applications, 444 (2016), 448 – 457. doi: 10.1016/j.physa.2015.10.014.

[13]

G. R. Boynton, The reach of politics via twitter? Can that be real?, Open Journal of Political Science, 3 (2013), 91-97.  doi: 10.4236/ojps.2013.33013.

[14]

J. A. Carrillo and G. Toscani, Wasserstein metric and large–time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics, (In Honour of the Salvatore Rionero 70th Birthday), 234–244. World Sci. Publ., Hackensack, NJ, 2004.

[15]

C. CastellanoS. Fortunato and V. Loreto, Statistical physics of social dynamics, Review of Modern Physics, 81 (2009), 591-646.  doi: 10.1103/RevModPhys.81.591.

[16] S. CresciM. N. La Polla and M. Tesconi, Il fenomeno dei Fake Follower in Twitter, 151–191, Pisa University Press, Pisa, 2017. 
[17]

E. De Cristofaro, A. Friedman, G. Jourjon, M. A. Kaafar and M. Z. Shafiq, Paying for likes?: Understanding facebook like fraud using honeypots., 2014 ACM 14th Internet Measurement Conference (IMC), 129–136. doi: 10.1145/2663716.2663729.

[18]

G. Deffuant, F. Amblard, G. Weisbuch and T. Faure, How can extremism prevail? A study based on the relative agreement interaction model, Journal of Artificial Societies and Social Simulation, 5.

[19]

M. Di FrancescoS. Fagioli and E. Radici, Deterministic particle approximation for nonlocal transport equations with nonlinear mobility, Journal of Differential Equations, 266 (2019), 2830-2868.  doi: 10.1016/j.jde.2018.08.047.

[20]

M. Di FrancescoS. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Bollettino dell'Unione Matematica Italiana, 10 (2017), 487-501.  doi: 10.1007/s40574-017-0132-2.

[21]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for rational mechanics and analysis, 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.

[22]

M. Di Francesco and G. Stivaletta, Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux, Discrete & Continuous Dynamical Systems - A, 40 (2020), 233-266.  doi: 10.3934/dcds.2020010.

[23]

B. DüringP. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708.  doi: 10.1098/rspa.2009.0239.

[24]

B. During and M.-T. Wolfram, Opinion dynamics: Inhomogeneous boltzmann-type equations modelling opinion leadership and political segregation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20150345. doi: 10.1098/rspa.2015.0345.

[25]

S. Fagioli and E. Radici, Solutions to aggregationdiffusion equations with nonlinear mobility constructed via a deterministic particle approximation, Math. Mod. and Meth. in App. Sci., 28 (2018), 1801-1829.  doi: 10.1142/S0218202518400067.

[26]

G. FurioliA. PulvirentiE. Terraneo and G. Toscani, Wright-fisher-type equations for opinion formation, large time behavior and weighted logarithmic-sobolev inequalities, Annales de l'Institut Henri Poincaré C, Analyse non linaire, 36 (2019), 2065-2082.  doi: 10.1016/j.anihpc.2019.07.005.

[27]

S. Galam, Sociophysics: A Physicists Modeling of Psycho-Political Phenomena, (Understanding Complex Systems), Springer, 2012. doi: 10.1007/978-1-4614-2032-3.

[28]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM J. Numer. Anal., 43 (2006), 2590-2606.  doi: 10.1137/040608672.

[29]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence, models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5.

[30]

A. KleinH. Ahlf and V. Sharma, Social activity and structural centrality in online social networks, Telematics and Informatics, 32 (2015), 321-332.  doi: 10.1016/j.tele.2014.09.008.

[31]

A. D. I. KramerJ. E. Guillory and J. T. Hancock, Experimental evidence of massive scale emotional contagion through social networks, Proceedings of the National Academy of Sciences, 11 (2014), 8788-8789.  doi: 10.1073/pnas.1320040111.

[32]

H. Lavenant and B. Maury, Opinion propagation on social networks: a mathematical standpoint, ESAIM: Proceedings and Surveys, 67 (2020), 285-335. doi: 10.1051/proc/202067016.

[33] P. F. LazarsfeldB. Berelson and H. Gaudet, The People's Choice: How the Voter Makes Up His Mind in a Presidential Campaign, Columbia University Press, 1968.  doi: 10.7312/laza93930.
[34]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Archive for Rational Mechanics and Analysis, 172 (2004), 407-428.  doi: 10.1007/s00205-004-0307-8.

[35]

D. Matthes and B. Söllner, Convergent Lagrangian discretization for drift-diffusion with nonlocal aggregation, vol. 16,313–351, Springer INdAM series, 2017.

[36]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[37]

G. Naldi, L. Pareschi and G. Toscani, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3.

[38] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods., Oxford University Press, 2013. 
[39]

L. Pareschi and G. Toscani, Wealth distribution and collective knowledge: A boltzmann approach, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130396. doi: 10.1098/rsta.2013.0396.

[40]

L. PareschiG. ToscaniA. Tosin and M. Zanella, Hydrodynamic models of preference formation in multi-agent societies, Journal of Nonlinear Science, 29 (2019), 2761-2796.  doi: 10.1007/s00332-019-09558-z.

[41]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (2003), 395-431. 

[42]

G. Russo, Deterministic diffusion of particles, Comm. on Pure and Applied Mathematics, 43 (1990), 697-733.  doi: 10.1002/cpa.3160430602.

[43]

F. Santambrogio, Optimal Transport for Applied Mathematicians, vol. 87 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-20828-2.

[44]

F. Slanina and H. Lavička, Analytical results for the sznajd model of opinion formation., Eur.Phys. J. B, 35 (2003), 279-288.  doi: 10.1140/epjb/e2003-00278-0.

[45]

S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276.  doi: 10.1038/35065725.

[46]

K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community., Int. J. Mod. Phys. C, 11 (2000), 1157-1165.  doi: 10.1142/S0129183100000936.

[47]

G. Toscani, Kinetic models of opinion formation, Comm. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.

[48]

G. ToscaniC. Brugna and S. Demichelis, Kinetic models for the trading of goods, Journal of Statistical Physics, 151 (2013), 549-566.  doi: 10.1007/s10955-012-0653-0.

[49]

G. Toscani, A. Tosin and M. Zanella, Opinion modeling on social media and marketing aspects, Phys. Rev. E, 98 (2018), 022315. doi: 10.1103/PhysRevE.98.022315.

[50]

C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.

[51]

S. Yardi, D. Romero and G. Schoenebeck, Detecting spam in a twitter network, First Monday, 15.

show all references

References:
[1]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20140138. doi: 10.1098/rsta.2014.0138.

[2]

G. Albi, L. Pareschi and M. Zanella, On the Optimal Control of Opinion Dynamics on Evolving Networks, vol. 494, 58–67, Springer, Cham, 2016.

[3]

G. AlbiL. Pareschi and M. Zanella, Opinion dynamics over complex networks: Kinetic modelling and numerical methods, Kinetic and Related Models, 10 (2017), 1-32.  doi: 10.3934/krm.2017001.

[4]

G. Albi, P. Pareschi, G. Toscani and M. Zanella, Recent advances in opinion modeling: Control and social influence, 49–98, Birkhäuser-Springer, 2017.

[5]

G. AlettiG. Naldi and G. Toscani, First-order continuous models of opinion formation, SIAM J. Appl. Math., 67 (2007), 837-853.  doi: 10.1137/060658679.

[6]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[7]

N. Ansini and S. Fagioli, Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation, Communications in Mathematical Sciences, 18 (2020), 459-486.  doi: 10.4310/CMS.2020.v18.n2.a8.

[8]

N. Bellomo, G. Ajmone Marsan and A. Tosin, Complex Systems and Society. Modeling and Simulation. SpringerBriefs in Mathematics, Springer, 2013. doi: 10.1007/978-1-4614-7242-1.

[9]

E. Ben-Naim, Opinion dynamics: Rise and fall of political parties, Europhysics Letters, 69 (2005), 671. doi: 10.1209/epl/i2004-10421-1.

[10]

S. Biswas and P. Sen, Critical noise can make the minority candidate win: The u.s. presidential election cases, Phys. Rev. E, 96 (2017), 032303. doi: 10.1103/PhysRevE.96.032303.

[11]

D. Borra and T. Lorenzi, A hybrid model for opinion formation, Zeitschrift für angewandte Mathematik und Physik, 64 (2013), 419–437. doi: 10.1007/s00033-012-0259-z.

[12]

L. Boudin and F. Salvarani, Opinion dynamics: Kinetic modelling with mass media, application to the scottish independence referendum, Physica A: Statistical Mechanics and its Applications, 444 (2016), 448 – 457. doi: 10.1016/j.physa.2015.10.014.

[13]

G. R. Boynton, The reach of politics via twitter? Can that be real?, Open Journal of Political Science, 3 (2013), 91-97.  doi: 10.4236/ojps.2013.33013.

[14]

J. A. Carrillo and G. Toscani, Wasserstein metric and large–time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics, (In Honour of the Salvatore Rionero 70th Birthday), 234–244. World Sci. Publ., Hackensack, NJ, 2004.

[15]

C. CastellanoS. Fortunato and V. Loreto, Statistical physics of social dynamics, Review of Modern Physics, 81 (2009), 591-646.  doi: 10.1103/RevModPhys.81.591.

[16] S. CresciM. N. La Polla and M. Tesconi, Il fenomeno dei Fake Follower in Twitter, 151–191, Pisa University Press, Pisa, 2017. 
[17]

E. De Cristofaro, A. Friedman, G. Jourjon, M. A. Kaafar and M. Z. Shafiq, Paying for likes?: Understanding facebook like fraud using honeypots., 2014 ACM 14th Internet Measurement Conference (IMC), 129–136. doi: 10.1145/2663716.2663729.

[18]

G. Deffuant, F. Amblard, G. Weisbuch and T. Faure, How can extremism prevail? A study based on the relative agreement interaction model, Journal of Artificial Societies and Social Simulation, 5.

[19]

M. Di FrancescoS. Fagioli and E. Radici, Deterministic particle approximation for nonlocal transport equations with nonlinear mobility, Journal of Differential Equations, 266 (2019), 2830-2868.  doi: 10.1016/j.jde.2018.08.047.

[20]

M. Di FrancescoS. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Bollettino dell'Unione Matematica Italiana, 10 (2017), 487-501.  doi: 10.1007/s40574-017-0132-2.

[21]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for rational mechanics and analysis, 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.

[22]

M. Di Francesco and G. Stivaletta, Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux, Discrete & Continuous Dynamical Systems - A, 40 (2020), 233-266.  doi: 10.3934/dcds.2020010.

[23]

B. DüringP. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708.  doi: 10.1098/rspa.2009.0239.

[24]

B. During and M.-T. Wolfram, Opinion dynamics: Inhomogeneous boltzmann-type equations modelling opinion leadership and political segregation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20150345. doi: 10.1098/rspa.2015.0345.

[25]

S. Fagioli and E. Radici, Solutions to aggregationdiffusion equations with nonlinear mobility constructed via a deterministic particle approximation, Math. Mod. and Meth. in App. Sci., 28 (2018), 1801-1829.  doi: 10.1142/S0218202518400067.

[26]

G. FurioliA. PulvirentiE. Terraneo and G. Toscani, Wright-fisher-type equations for opinion formation, large time behavior and weighted logarithmic-sobolev inequalities, Annales de l'Institut Henri Poincaré C, Analyse non linaire, 36 (2019), 2065-2082.  doi: 10.1016/j.anihpc.2019.07.005.

[27]

S. Galam, Sociophysics: A Physicists Modeling of Psycho-Political Phenomena, (Understanding Complex Systems), Springer, 2012. doi: 10.1007/978-1-4614-2032-3.

[28]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM J. Numer. Anal., 43 (2006), 2590-2606.  doi: 10.1137/040608672.

[29]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence, models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5.

[30]

A. KleinH. Ahlf and V. Sharma, Social activity and structural centrality in online social networks, Telematics and Informatics, 32 (2015), 321-332.  doi: 10.1016/j.tele.2014.09.008.

[31]

A. D. I. KramerJ. E. Guillory and J. T. Hancock, Experimental evidence of massive scale emotional contagion through social networks, Proceedings of the National Academy of Sciences, 11 (2014), 8788-8789.  doi: 10.1073/pnas.1320040111.

[32]

H. Lavenant and B. Maury, Opinion propagation on social networks: a mathematical standpoint, ESAIM: Proceedings and Surveys, 67 (2020), 285-335. doi: 10.1051/proc/202067016.

[33] P. F. LazarsfeldB. Berelson and H. Gaudet, The People's Choice: How the Voter Makes Up His Mind in a Presidential Campaign, Columbia University Press, 1968.  doi: 10.7312/laza93930.
[34]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Archive for Rational Mechanics and Analysis, 172 (2004), 407-428.  doi: 10.1007/s00205-004-0307-8.

[35]

D. Matthes and B. Söllner, Convergent Lagrangian discretization for drift-diffusion with nonlocal aggregation, vol. 16,313–351, Springer INdAM series, 2017.

[36]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[37]

G. Naldi, L. Pareschi and G. Toscani, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3.

[38] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods., Oxford University Press, 2013. 
[39]

L. Pareschi and G. Toscani, Wealth distribution and collective knowledge: A boltzmann approach, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130396. doi: 10.1098/rsta.2013.0396.

[40]

L. PareschiG. ToscaniA. Tosin and M. Zanella, Hydrodynamic models of preference formation in multi-agent societies, Journal of Nonlinear Science, 29 (2019), 2761-2796.  doi: 10.1007/s00332-019-09558-z.

[41]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (2003), 395-431. 

[42]

G. Russo, Deterministic diffusion of particles, Comm. on Pure and Applied Mathematics, 43 (1990), 697-733.  doi: 10.1002/cpa.3160430602.

[43]

F. Santambrogio, Optimal Transport for Applied Mathematicians, vol. 87 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-20828-2.

[44]

F. Slanina and H. Lavička, Analytical results for the sznajd model of opinion formation., Eur.Phys. J. B, 35 (2003), 279-288.  doi: 10.1140/epjb/e2003-00278-0.

[45]

S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276.  doi: 10.1038/35065725.

[46]

K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community., Int. J. Mod. Phys. C, 11 (2000), 1157-1165.  doi: 10.1142/S0129183100000936.

[47]

G. Toscani, Kinetic models of opinion formation, Comm. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.

[48]

G. ToscaniC. Brugna and S. Demichelis, Kinetic models for the trading of goods, Journal of Statistical Physics, 151 (2013), 549-566.  doi: 10.1007/s10955-012-0653-0.

[49]

G. Toscani, A. Tosin and M. Zanella, Opinion modeling on social media and marketing aspects, Phys. Rev. E, 98 (2018), 022315. doi: 10.1103/PhysRevE.98.022315.

[50]

C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.

[51]

S. Yardi, D. Romero and G. Schoenebeck, Detecting spam in a twitter network, First Monday, 15.

Figure 1.  Convergence for different initial data to the stationary state (47). The left column shows the evolution in time for the reconstructed density, where the initial data are (48), (49) and (50) respectively, while the right column shows the opinions evolution in time. The (magenta) stars-line represent the mean opinion $ m_1(t) $ in each case. Note that, in the second simulation, $ m_1 $ is not conserved in time but still converges to zero
Figure 2.  Convergence for different initial data to the stationary state (51), where the initial data are (48), (49) and (50) respectively. Note that also in this case the mean opinion $ m_1 $ (star magenta line in the right column) converges to zero asymptotically
Figure 3.  Comparison of different stationary states (46) for different values of $ \alpha $. On the left the diffusion coefficient $ \lambda^2 = 0.5 $, on the right $ \lambda^2 = 0.03 $
Figure 4.  Convergence for the initial data (48), (49) and (50) to the stationary state (53), with $ \alpha = 1 $. Note that the nonlinearity in the diffusion produces stationary solutions with supports that are smaller than the ones we have seen in the linear diffusion case
Figure 5.  Stationary states in (53) for different values of $ \alpha $
Figure 6.  Opinion dynamics in presence of equally-strong leaders. Top-left initial data for followers(black), left leaders (red) and right leaders (blue). Top-right and bottom-left opinions evolutions for all the groups in short and long term-respectively. Bottom-right opinions evolutions for the followers
Figure 7.  Opinion dynamics in presence of one strong group of leaders and one weak group of leaders. Top-left initial data for followers(black), left leaders (red) and right leaders (blue). Top-right and bottom-left opinions evolutions for all the groups in short and long term-respectively. Bottom-right opinions evolutions for the followers
Figure 8.  Evolution of system (12). The left column concerns the case of two equally strong groups of leaders, the right column instead describes the situation where the right leader is weaker. Trolls are plotted in green and are associated to the right leaders. Top: initial data. Centre: opinions dynamic for all species. Bottom: comparison between the followers paths with or without trolls
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