Bounded confidence models of opinion dynamics in social networks have been actively studied in recent years, in particular, opinion formation and extremism propagation along with other aspects of social dynamics. In this work, after an analysis of limitations of the Deffuant-Weisbuch (DW) bounded confidence, relative agreement model, we propose the mixed model that takes into account two psychological types of individuals. Concord agents (C-agents) are friendly people; they interact in a way that their opinions get closer always. Agents of the other psychological type show partial antagonism in their interaction (PA-agents). Opinion dynamics in heterogeneous social groups, consisting of agents of the two types, was studied on different social networks: Erdos-Renyi random graphs, small-world networks and complete graphs. Limit cases of the mixed model, pure C-and PA-societies, were also studied. We found that group opinion formation is, qualitatively, almost independent of the topology of networks used in this work. Opinion fragmentation, polarization and consensus are observed in the mixed model at different proportions of PA-and C-agents, depending on the value of initial opinion tolerance of agents. As for the opinion formation and arising of "dissidents", the opinion dynamics of the C-agents society was found to be similar to that of the DW model, except for the rate of opinion convergence. Nevertheless, mixed societies showed dynamics and bifurcation patterns notably different to those of the DW model. The influence of biased initial conditions over opinion formation in heterogeneous social groups was also studied versus the initial value of opinion uncertainty, varying the proportion of the PA-to C-agents. Bifurcation diagrams showed impressive evolution of collective opinion, in particular, radical change of left to right consensus or vice versa at an opinion uncertainty value equal to 0.7 in the model with the PA/C mixture of population near 50/50.

Ideology is one of the defining elements of opinion dynamics. In this paper, we report the e ects of the nonlinear interaction of ideological a inity with the psychological reaction of agents in the frame of a multiparametric mathematical model of opinion dynamics. Computer simulations of artificial networked societies composed of agents of two psychological types were used for studying opinion formation; the simulations showed a phenomenon of preferential self-organization into groups of ideological a inity at the first stages of opinion evolution. The separation into ideologically akin opinion groups (ideological a inity) was more notable in societies composed mostly of concord agents; a larger opinion polarization was associated with the increase of agents' initial average opinion uncertainty. We also observed a sensibility of opinion dynamics to the initial conditions of opinion and uncertainty, indicating potential instabilities. A measure of convergence was introduced to facilitate the analysis of transitions between the opinion states of networked societies and to detect social instability events. We found that the average of opinion uncertainty distribution reaches a steady state with values lower than the initial average value, sometimes nearing zero, which points at socially apathetic agents. Our analyses showed that the model can be utilized for further investigation on opinion dynamics and can be extended to other social phenomena.

A long-standing conjecture states that the crossing number of the Cartesian product of cycles C m _C n is (m&2) n, for every m, n satisfying n m 3. A crossing is proper if it occurs between edges in different principal cycles. In this paper drawings of C m _C n with the principal n-cycles pairwise disjoint or the principal m-cycles pairwise disjoint are analyzed, and it is proved that every such drawing has at least (m&2) n proper crossings. As an application of this result, we prove that the crossing number of C m _C n is at least (m&2) nÂ2, for all integers m, n such that n m 4. This is the best general lower bound known for the crossing number of C m _C n .

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