March  2017, 10(1): 1-32. doi: 10.3934/krm.2017001

Opinion dynamics over complex networks: Kinetic modelling and numerical methods

1. 

TU München, Faculty of Mathematics, Boltzmannstra$\mathcal{B}$ e 3, D-85748, Garching (München), Germany

2. 

University of Ferrara, Department of Mathematics and Computer Science, Via N. Machiavelli 35,44121, Ferrara, Italy

Received  March 2016 Revised  May 2016 Published  November 2016

In this paper we consider the modeling of opinion dynamics over time dependent large scale networks. A kinetic description of the agents' distribution over the evolving network is considered which combines an opinion update based on binary interactions between agents with a dynamic creation and removal process of new connections. The number of connections of each agent influences the spreading of opinions in the network but also the way connections are created is influenced by the agents' opinion. The evolution of the network of connections is studied by showing that its asymptotic behavior is consistent both with Poisson distributions and truncated power-laws. In order to study the large time behavior of the opinion dynamics a mean field description is derived which allows to compute exact stationary solutions in some simplified situations. Numerical methods which are capable to describe correctly the large time behavior of the system are also introduced and discussed. Finally, several numerical examples showing the influence of the agents' number of connections in the opinion dynamics are reported.

Citation: Giacomo Albi, Lorenzo Pareschi, Mattia Zanella. Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinetic & Related Models, 2017, 10 (1) : 1-32. doi: 10.3934/krm.2017001
References:
[1]

D. Acemoglu and O. Asuman, Opinion dynamics and learning in social networks, Dynamic Games and Applications, 1 (2011), 3-49.  doi: 10.1007/s13235-010-0004-1.  Google Scholar

[2]

R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, 74 (2002), 1-47.  doi: 10.1103/RevModPhys.74.47.  Google Scholar

[3]

G. Albi and L. Pareschi, Binary interaction algorithm for the simulation of flocking and swarming dynamics, SIAM Journal on Multiscale Modeling and Simulations, 11 (2013), 1-29.  doi: 10.1137/120868748.  Google Scholar

[4]

G. AlbiM. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Communications in Mathematical Sciences, 13 (2015), 1407-1429.  doi: 10.4310/CMS.2015.v13.n6.a3.  Google Scholar

[5]

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

[6]

G. Albi, L. Pareschi and M. Zanella, On the optimal control of opinion dynamics on evolving networks, IFIP TC7 2015 Proceedings, to appear. doi: 10.1155/2015/850124.  Google Scholar

[7]

L.A.N. AmaralA. ScalaM. Bathélemy and H. E. Stanley, Classes of small-world networks, Proceedings of the National Academy of Sciences of the United States of America, 97 (2000), 11149-11152.  doi: 10.1073/pnas.200327197.  Google Scholar

[8]

A. Aydogdo, M. Caponigro, S. McQuade, B. Piccoli, N. Pouradier Duteil, F. Rossi and E. Trélat, Interaction network, state space and control in social dynamics, In Active Particle Volume 1, Theory, Methods, and Applications, Eds. N. Bellomo, P. Degond, E. Tadmor. To appear. Google Scholar

[9]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.  doi: 10.1126/science.286.5439.509.  Google Scholar

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A.-L. BarabásiR. Albert and H. Jeong, Mean-field theory for scale-free random networks, Physica A: Statistical Mechanics and its Applications, 272 (1999), 173-187.   Google Scholar

[11]

L. Boudin, R. Monaco and F. Salvarani, Kinetic model for multidimensional opinion formation, Physical Review E, 81 (2010), 036109. doi: 10.1103/PhysRevE.81.036109.  Google Scholar

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C. Brugna and G. Toscani, Kinetic models of opinion formation in the presence of personal conviction, Physical Review E, 92 052818,2015. doi: 10.1103/PhysRevE.92.052818.  Google Scholar

[13]

C. BuetS. Cordier and V. Dos Santos, A conservative and entropy scheme for a simplified model of granular media, Transport Theory and Statistical Physics, 33 (2004), 125-155.  doi: 10.1081/TT-120037804.  Google Scholar

[14]

C. Buet and S. Dellacherie, On the Chang and Cooper numerical scheme applied to a linear Fokker-Planck equation, Communications in Mathematical Sciences, 8 (2010), 1079-1090.  doi: 10.4310/CMS.2010.v8.n4.a15.  Google Scholar

[15]

J.S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck equation, Journal of Computational Physics, 6 (1970), 1-16.  doi: 10.1016/0021-9991(70)90001-X.  Google Scholar

[16]

A. DasS. Gollapudi and K. Munagala, Modeling opinion dynamics in social networks, Proceedings of the 7th ACM International Conference on Web Search and Data Mining, (2014), 403-412.  doi: 10.1145/2556195.2559896.  Google Scholar

[17]

M. Dolfin and M. Lachowicz, Modeling opinion dynamics: how the network enhances consensus, Networks & Heterogeneous Media, 10 (2015), 877-896.  doi: 10.3934/nhm.2015.10.877.  Google Scholar

[18]

B. DüringP.A. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society of London A, 465 (2009), 3687-3708.  doi: 10.1098/rspa.2009.0239.  Google Scholar

[19]

B. Düring and M. -T. Wolfram, Opinion dynamics: Inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation Proceedings of the Royal Society of London A, 471 (2015), 20150345. doi: 10.1098/rspa.2015.0345.  Google Scholar

[20]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence, models, analysis and simulation, Journal of Artifcial Societies and Social Simulation, 5(3): 2,2002. Google Scholar

[21]

E.W. LarsenC.D. LevermoreG.C. Pomraning and J.G. Sanderson, Discretization methods for one-dimensional Fokker-Planck operators, Journal of Computational Physics, 61 (1985), 359-390.  doi: 10.1016/0021-9991(85)90070-1.  Google Scholar

[22]

M. E. J. Newman, The structure and function on complex networks, SIAM Review, 45 (2003), 167-256.  doi: 10.1137/S003614450342480.  Google Scholar

[23]

M. Mohammadi and A. Borzí, Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations, Journal of Numerical Mathematics, 23 (2015), 271-288.  doi: 10.1515/jnma-2015-0018.  Google Scholar

[24]

L. Pareschi and G. Russo, An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM: Proceedings, EDP Sciences, 10 (2001), 35-75.  doi: 10.1051/proc:2001004.  Google Scholar

[25]

L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. Google Scholar

[26]

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

[27]

L. Pareschi, P. Vellucci and M. Zanella, Kinetic Models of Collective Decision-making in the Presence of Equality Bias, Physica A: Statistical Mechanics and its Applications, 2016. doi: 10.1016/j.physa.2016.10.003.  Google Scholar

[28]

S. Patterson and B. Bamieh, Interaction-driven opinion dynamics in online social networks, S. Patterson and B. Bamieh, Interaction-driven opinion dynamics in online social networks, Proceedings of the First Workshop on Social Media Analytics, (2010), 98-110.  doi: 10.1145/1964858.1964872.  Google Scholar

[29]

W. Quattrociocchi, G. Caldarelli and A. Scala, Opinion dynamics on interacting networks: Media competition and social influence Scientific Reports, 4 (2014), p4938. doi: 10.1038/srep04938.  Google Scholar

[30]

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

[31]

K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 1197-1165.  doi: 10.1142/S0129183100000936.  Google Scholar

[32]

G. Toscani, Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.  Google Scholar

[33]

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

[34]

Y.-B. XieT. Zhou and B.-H. Wang, Scale-free networks without growth, Physica A, 387 (2008), 1683-1688.  doi: 10.1016/j.physa.2007.11.005.  Google Scholar

show all references

References:
[1]

D. Acemoglu and O. Asuman, Opinion dynamics and learning in social networks, Dynamic Games and Applications, 1 (2011), 3-49.  doi: 10.1007/s13235-010-0004-1.  Google Scholar

[2]

R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, 74 (2002), 1-47.  doi: 10.1103/RevModPhys.74.47.  Google Scholar

[3]

G. Albi and L. Pareschi, Binary interaction algorithm for the simulation of flocking and swarming dynamics, SIAM Journal on Multiscale Modeling and Simulations, 11 (2013), 1-29.  doi: 10.1137/120868748.  Google Scholar

[4]

G. AlbiM. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Communications in Mathematical Sciences, 13 (2015), 1407-1429.  doi: 10.4310/CMS.2015.v13.n6.a3.  Google Scholar

[5]

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

[6]

G. Albi, L. Pareschi and M. Zanella, On the optimal control of opinion dynamics on evolving networks, IFIP TC7 2015 Proceedings, to appear. doi: 10.1155/2015/850124.  Google Scholar

[7]

L.A.N. AmaralA. ScalaM. Bathélemy and H. E. Stanley, Classes of small-world networks, Proceedings of the National Academy of Sciences of the United States of America, 97 (2000), 11149-11152.  doi: 10.1073/pnas.200327197.  Google Scholar

[8]

A. Aydogdo, M. Caponigro, S. McQuade, B. Piccoli, N. Pouradier Duteil, F. Rossi and E. Trélat, Interaction network, state space and control in social dynamics, In Active Particle Volume 1, Theory, Methods, and Applications, Eds. N. Bellomo, P. Degond, E. Tadmor. To appear. Google Scholar

[9]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.  doi: 10.1126/science.286.5439.509.  Google Scholar

[10]

A.-L. BarabásiR. Albert and H. Jeong, Mean-field theory for scale-free random networks, Physica A: Statistical Mechanics and its Applications, 272 (1999), 173-187.   Google Scholar

[11]

L. Boudin, R. Monaco and F. Salvarani, Kinetic model for multidimensional opinion formation, Physical Review E, 81 (2010), 036109. doi: 10.1103/PhysRevE.81.036109.  Google Scholar

[12]

C. Brugna and G. Toscani, Kinetic models of opinion formation in the presence of personal conviction, Physical Review E, 92 052818,2015. doi: 10.1103/PhysRevE.92.052818.  Google Scholar

[13]

C. BuetS. Cordier and V. Dos Santos, A conservative and entropy scheme for a simplified model of granular media, Transport Theory and Statistical Physics, 33 (2004), 125-155.  doi: 10.1081/TT-120037804.  Google Scholar

[14]

C. Buet and S. Dellacherie, On the Chang and Cooper numerical scheme applied to a linear Fokker-Planck equation, Communications in Mathematical Sciences, 8 (2010), 1079-1090.  doi: 10.4310/CMS.2010.v8.n4.a15.  Google Scholar

[15]

J.S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck equation, Journal of Computational Physics, 6 (1970), 1-16.  doi: 10.1016/0021-9991(70)90001-X.  Google Scholar

[16]

A. DasS. Gollapudi and K. Munagala, Modeling opinion dynamics in social networks, Proceedings of the 7th ACM International Conference on Web Search and Data Mining, (2014), 403-412.  doi: 10.1145/2556195.2559896.  Google Scholar

[17]

M. Dolfin and M. Lachowicz, Modeling opinion dynamics: how the network enhances consensus, Networks & Heterogeneous Media, 10 (2015), 877-896.  doi: 10.3934/nhm.2015.10.877.  Google Scholar

[18]

B. DüringP.A. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society of London A, 465 (2009), 3687-3708.  doi: 10.1098/rspa.2009.0239.  Google Scholar

[19]

B. Düring and M. -T. Wolfram, Opinion dynamics: Inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation Proceedings of the Royal Society of London A, 471 (2015), 20150345. doi: 10.1098/rspa.2015.0345.  Google Scholar

[20]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence, models, analysis and simulation, Journal of Artifcial Societies and Social Simulation, 5(3): 2,2002. Google Scholar

[21]

E.W. LarsenC.D. LevermoreG.C. Pomraning and J.G. Sanderson, Discretization methods for one-dimensional Fokker-Planck operators, Journal of Computational Physics, 61 (1985), 359-390.  doi: 10.1016/0021-9991(85)90070-1.  Google Scholar

[22]

M. E. J. Newman, The structure and function on complex networks, SIAM Review, 45 (2003), 167-256.  doi: 10.1137/S003614450342480.  Google Scholar

[23]

M. Mohammadi and A. Borzí, Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations, Journal of Numerical Mathematics, 23 (2015), 271-288.  doi: 10.1515/jnma-2015-0018.  Google Scholar

[24]

L. Pareschi and G. Russo, An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM: Proceedings, EDP Sciences, 10 (2001), 35-75.  doi: 10.1051/proc:2001004.  Google Scholar

[25]

L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. Google Scholar

[26]

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

[27]

L. Pareschi, P. Vellucci and M. Zanella, Kinetic Models of Collective Decision-making in the Presence of Equality Bias, Physica A: Statistical Mechanics and its Applications, 2016. doi: 10.1016/j.physa.2016.10.003.  Google Scholar

[28]

S. Patterson and B. Bamieh, Interaction-driven opinion dynamics in online social networks, S. Patterson and B. Bamieh, Interaction-driven opinion dynamics in online social networks, Proceedings of the First Workshop on Social Media Analytics, (2010), 98-110.  doi: 10.1145/1964858.1964872.  Google Scholar

[29]

W. Quattrociocchi, G. Caldarelli and A. Scala, Opinion dynamics on interacting networks: Media competition and social influence Scientific Reports, 4 (2014), p4938. doi: 10.1038/srep04938.  Google Scholar

[30]

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

[31]

K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 1197-1165.  doi: 10.1142/S0129183100000936.  Google Scholar

[32]

G. Toscani, Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.  Google Scholar

[33]

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

[34]

Y.-B. XieT. Zhou and B.-H. Wang, Scale-free networks without growth, Physica A, 387 (2008), 1683-1688.  doi: 10.1016/j.physa.2007.11.005.  Google Scholar

Figure 1.  Stationary states of (21) with relaxation coefficients ${V_r}={V_a}=1$, mean density of connectivity $\gamma=30$, ${{c}_{\max }}=1500$ and several values of the attraction parameters $\alpha$, and having fixed $\beta = 0$. Left: convergence toward the Poisson distribution for big values of $\alpha$. Right: convergence toward a power-law distribution in the limit $\alpha\rightarrow 0$, we indicated with $p_{\infty}^{(-k)}, k=1,2,3$ the $\alpha-$dependent stationary solutions for $\alpha=10^{-1},10^{-2},10^{-3}$, respectively.
Figure 2.  Stationary solutions of type $f_\infty(w,c) = g_\infty(w)p_\infty(c)$, where $ g_\infty(w)$ is given by (51) with $\kappa =1$, $m_w = 0$, $\sigma^2 = 0.05$ and $p_\infty(c)$ defined by (25), with ${V_r}={V_a}=1$, $\gamma=30$ and $\alpha = 10$ on the left and $\alpha = 0.1$ on the right.
Figure 3.  Stationary solutions captured via Monte Carlo simulations, with $N_s=2\times10^4$ samples. Parameters of the model are chosen as follows $\sigma^2 = 0.05$, ${V_r}={V_a}=1$, $\beta = 0$, $\alpha = 10$ on the right hand side and $\alpha =0.1$ on the left hand side.
Figure 4.  Test $\#1$. One-dimensional setting: on the left, convergence of (58) to the stationary solution (51), of the Fokker-Planck equation, for decreasing values of the parameter $\varepsilon$, $g^N_0$ represents the initial distribution. On the right, convergence of the Monte-Carlo (55) to the reference solution (24) for increasing values of the the number of samples $N_s$.
Figure 5.  Test $\#1.$ One-dimensional setting: on the left, the solution of the Chang-Cooper type scheme with the flux (70) is indicated with $g^N_T$ and compared with the stationary solution (51), also the initial data $g^N_0$ (72) is reported. On the right we report the decay of the $L^1$ relative error (73) for different choices of the quadrature rule, mid-point rule (67) and Milne's rule (respectively of $2^{nd}$ and $4^{th}$ order).
Figure 6.  Test $\#2.$ From left to right and from the top to the bottom: evolution of the density $f(w,c,t)$ at different time steps. The plot $(a)$ represents the initial data $f_0(w,c)$ (74) and plot $(d)$ the stationary solution. On the plane $(z,c)$ we depict with a blu line the marginal distribution $p(c,t)$ of the solution at time $t$, with red line we represent the reference marginal distribution of the stationary solution.
Figure 7.  Test $\#2.$ Decay of the $L^1$ relative error with respect to the stationary solution (51). On the left, fixed characteristic rates $V=\{10^3,10^4,10^5\}$, on the right, variable characteristic rates defined as in (76) with $U =\{10^3,10^4,10^5\}$. In both cases for increasing values of the characteristic rate $V$ and $U$ the stationary state is reached faster.
Figure 8.  Test $\#2.$ Evolution at time $t = 1$ of the initial data $f_0(w, c)$ (74) as isoline plot. On the left in the case of constant characteristic rate on the right variable characteristic rates defined as in (76). The right plot shows that for lower opinion's density the evolution along the connection is faster and slower where the opinions are more concentrated.
Figure 9.  Test $\#3.$ From left to right and from the first row to the second row, evolution of the initial data (78) in time frame $[0,T]$ with $T = 2$. The evolution shows how a small portion of density with high connectivity can bias the majority of the population towards their position. (Note: The density is scaled according to the marginal distribution $\rho(c,t)$ in order to better show its evolution, the actual marginal density $\rho(c,t)$ is depicted in the background, scaled by a factor 10).
Figure 10.  Test $\#3.$ On the left-hand side final and initial state of the marginal distribution $g(w,t)$ of the opinion, the green line represents the evolution of the average opinion $\bar{m}(t)$, the red and blue dashed lines represent respectively the opinions $\bar{w}_L=0.75$ and $\bar{w}_F=-0.5$, which are the two leading opinions of the initial data (78).
Figure 11.  Test $\#4.$ Evolution of the Fokker-Planck model (41) where the interaction are described by (80) with $\Delta=0.25$, in the time frame $[0,T]$ with $T = 100$. The evolution shows the emergence of three main opinion clusters, which are not affected by the connectivity variable. (Note: In order to better show its evolution, we represent the solution as $\log(f(w,c,t)+\epsilon)$, with $\epsilon = 0.001$.)
Figure 12.  Test $\#4.$ Evolution of the solution of the Fokker-Planck model (41), where the interaction are described by (79) with $\Delta(c)=d_0c/{c_{\max }}$ and $d_0=1.01$, in the time frame $[0,T]$ with $T = 100$. The choice of $\Delta(c)$ reflects in the heterogeneous emergence of clusters with respect to the connectivity level: for higher level of connectivity consensus is reached, instead for lower levels of connectivity multiple opinion clusters are present. (Note: In order to better show its evolution, we represent the solution as $\log(f(w,c,t)+\epsilon)$, with $\epsilon = 0.001$.
Table 1.  Parameters in the various test cases
Test $\sigma^2$ $\sigma^2_F$ $\sigma^2_L$ ${{c}_{\max }}$ ${V_r}$ ${V_a}$ $\gamma_0$ $\alpha$ $\beta$
#1 $5\times10^{-2}$ $6\times10^{-2}$ $-$ $250$ $1$ 1 $30$ $1\times10^{-1} $ $0$
#2 $5\times10^{-2}$ $6\times10^{-2}$ $-$ $250$ $-$ $-$ $30$ $1\times10^{-1} $ $0$
#3 $5\times10^{-3}$ $4\times10^{-2}$ $2.5\times10^{-2}$ 250 $1$ $1$ 30 $1\times10^{-4}$ $0$
#4 $1\times10^{-3}$ $-$ $-$ 250 $1$ $1$ 30 $1\times10^{-1}$ $0$
Test $\sigma^2$ $\sigma^2_F$ $\sigma^2_L$ ${{c}_{\max }}$ ${V_r}$ ${V_a}$ $\gamma_0$ $\alpha$ $\beta$
#1 $5\times10^{-2}$ $6\times10^{-2}$ $-$ $250$ $1$ 1 $30$ $1\times10^{-1} $ $0$
#2 $5\times10^{-2}$ $6\times10^{-2}$ $-$ $250$ $-$ $-$ $30$ $1\times10^{-1} $ $0$
#3 $5\times10^{-3}$ $4\times10^{-2}$ $2.5\times10^{-2}$ 250 $1$ $1$ 30 $1\times10^{-4}$ $0$
#4 $1\times10^{-3}$ $-$ $-$ 250 $1$ $1$ 30 $1\times10^{-1}$ $0$
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