In the compromise model of continuous opinions proposed by Deffuant et al, the states of two agents in a network can start to converge if they are neighbors and if their opinions are sufficiently close to each other, below a given threshold of tolerance ǫ. In directed networks, if agent i is a neighbor of agent j, j need not be a neighbor of i. In Watts-Strogatz networks we performed simulations to find the averaged number of final opinions < F > and their distribution as a function of ǫ and of the network structural disorder. In directed networks < F > exhibits a rich structure, being larger than in undirected networks for higher values of ǫ, and smaller for lower values of ǫ.

Understanding the emergence of strong controversial issues in modern societies is a key issue in opinion studies. A commonly diffused idea is the fact that the increasing of homophily in social networks, due to the modern ICT, can be a driving force for opinion polariation. In this paper we address the problem with a modelling approach following three basic steps. We first introduce a network morphogenesis model to reconstruct network structures where homophily can be tuned with a parameter. We show that as homophily increases the emergence of marked topological community structures in the networks raises. Secondly, we perform an opinion dynamics process on homophily dependent networks and we show that, contrary to the common idea, homophily helps consensus formation. Finally, we introduce a tunable external media pressure and we show that, actually, the combination of homophily and media makes the media effect less effective and leads to strongly polarized opinion clusters.

We study the Axelrod's cultural adaptation model using the concept of cluster size entropy Sc that gives information on the variability of the cultural cluster size present in the system. Using networks of different topologies, from regular to random, we find that the critical point of the wellknown nonequilibrium monocultural-multicultural (order-disorder) transition of the Axelrod model is unambiguously given by the maximum of the Sc(q) distributions. The width of the cluster entropy distributions can be used to qualitatively determine whether the transition is first-or second-order. By scaling the cluster entropy distributions we were able to obtain a relationship between the critical cultural trait qc and the number F of cultural features in regular networks. We also analyze the effect of the mass media (external field) on social systems within the Axelrod model in a square network. We find a new partially ordered phase whose largest cultural cluster is not aligned with the external field, in contrast with a recent suggestion that this type of phase cannot be formed in regular networks. We draw a new q − B phase diagram for the Axelrod model in regular networks.

We propose a thermodynamic version of the Axelrod model of social influence. In one-dimensional (1D) lattices, the thermodynamic model becomes a coupled Potts model with a bonding interaction that increases with the site matching traits. We analytically calculate thermodynamic and critical properties for a 1D system and show that an order-disorder phase transition only occurs at T = 0 independent of the number of cultural traits q and features F . The 1D thermodynamic Axelrod model belongs to the same universality class of the Ising and Potts models, notwithstanding the increase of the internal dimension of the local degree of freedom and the state-dependent bonding interaction. We suggest a unifying proposal to compare exponents across different discrete 1D models. The comparison with our Hamiltonian description reveals that in the thermodynamic limit the original out-of-equilibrium 1D Axelrod model with noise behaves like an ordinary thermodynamic 1D interacting particle system.