Journal of Statistical Mechanics: Theory and Experiment
Volume 2007
August 2007
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R Lambiotte and M Ausloos J. Stat. Mech. (2007) P08026 doi:10.1088/1742-5468/2007/08/P08026
Coexistence of opposite opinions in a network with communities
R Lambiotte and M Ausloos
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| Figure 1. Sketch of a time step, where one node (surrounded in black) and two of its neighbours (surrounded in grey) are selected. The majority rule implies that the blue node becomes red. |
| Figure 2. Typical realizations of coupled random networks for small values of ν. The network is composed of N = 100 nodes and pin = 0.1. The system obviously consists of two communities that are less and less discernible for larger values of ν. We have assigned different shapes (circles and rectangles) to nodes belonging to different communities. The graphs were plotted thanks to the visone graphical tools [18]. |
Figure 3. Computer simulations of MR on coupled random networks with
N = 104 nodes and
pin = 0.01. The simulations are stopped after 103 steps/node and the results averaged over 100 realizations of the random process. The vertical dashed lines point to the theoretical transition value obtained from equation (28) and to the critical value
q = 3/5 (d). The solid lines correspond to the theoretical predictions (19) and (21).
The simulations are either started with a symmetric initial condition
a1 = 1, a2 = 1 or with an asymmetric initial condition a1 = 1, a2 = 0. (a) Bifurcation diagram of |a1–a2| as
a function of ν, for simulations starting from asymmetric initial conditions. The system ceases to be asymmetric above
νc ≈ 0.15. (b) and (c)
Bifurcation diagram of a1 as a function of ν, starting the simulations from asymmetric or symmetric initial conditions for
q = 0 (b)
and q = 0.2 (c). The systems behave qualitatively in the same way when
q = 0 and q = 0.2, except that the symmetric state is frozen when
q = 0 (all the nodes have and keep the same opinion, a1 = 1)
while a1 fluctuates around its asymptotic value when  . (d) Bifurcation diagram of a1 as a function of q, starting the simulations from asymmetric or symmetric initial conditions for
ν = 0.05. In
that case, the system may undergo two transitions: one from the asymmetric to the symmetric state at q ≈ 0.485 and one from the symmetric to the disordered state at
q = 3/5. |
Figure 4. Phase diagram of MR on CRN. Three phases may take place: (i) a disordered phase when
q > 3/5; (ii) a symmetric phase when q < 3/5; (iii) an asymmetric phase when q < 3/5 and when  . A system in the asymmetric state, but close to the transition line, e.g. the green triangle, may lose its stability due to an increase of the number of inter-community links (along
ν) or to an increase of the internal fluctuations (along
q). The asymmetric state differs from the symmetric and from the disordered states by the fact that it exhibits correlations between the shape of the nodes (i.e. the community to which they belong) and their colour (i.e. their opinion). |
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