Persistent individual bias in a voter model with quenched disorder

Path-integral approach to sparse non-Hermitian random matrices

The theory of large random matrices has proved an invaluable tool for the study of systems with disordered interactions in many quite disparate research areas. Widely applicable results, such as the celebrated elliptic law for dense random matrices, allow one to deduce the statistical properties of the interactions in a complex dynamical system that permit stability. However, such simple and universal results have so far proved difficult to come by in the case of sparse random matrices. Here, we perform an expansion in the inverse connectivity, and thus derive general modified versions of the classic elliptic and semicircle laws, taking into account the sparse correction. This is accomplished using a dynamical approach, which maps the hermitized resolvent of a random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling one to construct Feynman diagrams, which facilitate the perturbative analysis. Additionally, in order to demonstrate the broad utility of the path integral framework, we derive a generic non-Hermitian generalization of the Marchenko-Pastur law, and we also show how one can handle non-negligible higher-order statistics (i.e., non-Gaussian statistics) in dense ensembles.

Q-voter model with independence on signed random graphs: Homogeneous approximations

The q-voter model with independence is generalized to signed random graphs and studied by means of Monte Carlo simulations and theoretically using the mean-field approximation and different forms of the pair approximation. In the signed network with quenched disorder, positive and negative signs associated randomly with the links correspond to reinforcing and antagonistic interactions, promoting, respectively, the same or opposite orientations of two-state spins representing agents' opinions; otherwise, the opinions are called mismatched. With probability 1-p, the agents change their opinions if the opinions of all members of a randomly selected q neighborhood are mismatched, and with probability p, they choose an opinion randomly. The model on networks with finite mean degree 〈k〉 and fixed fraction of the antagonistic interactions r exhibits ferromagnetic transition with varying the independence parameter p, which can be first or second order, depending on q and r, and disappears for large r. Besides, numerical evidence is provided for the occurrence of the spin-glass-like transition for large r. The order and critical lines for the ferromagnetic transition on the p vs r phase diagram obtained in Monte Carlo simulations are reproduced qualitatively by the mean-field approximation. Within the range of applicability of the pair approximation, for the model with 〈k〉 finite but 〈k〉≫q, predictions of the homogeneous pair approximation concerning the ferromagnetic transition show much better quantitative agreement with numerical results for small r but fail for larger r. A more advanced signed homogeneous pair approximation is formulated which distinguishes between classes of active links with a given sign connecting nodes occupied by agents with mismatched opinions; for the model with 〈k〉≫q its predictions agree quantitatively with numerical results in a whole range of r where the ferromagnetic transition occurs.

Eigenvalue spectra and stability of directed complex networks

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction network between components has on the eigenvalue spectrum. We build on previous results, which usually only take into account the mean degree of the network, by allowing for nontrivial network degree heterogeneity. We derive closed-form expressions for the eigenvalue spectrum of the adjacency matrix of a general weighted and directed network. Using these results, which are valid for any large well-connected complex network, we then derive compact formulas for the corrections (due to nonzero network heterogeneity) to well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semicircle law, the Girko circle law, and the elliptic law and any outlier eigenvalues. We also derive a surprisingly neat analytical expression for the eigenvalue density of a directed Barabási-Albert network. We are thus able to deduce that network heterogeneity is mostly a destabilizing influence in complex dynamical systems.

Pair approximation for the q-voter models with quenched disorder on networks

Using two models of opinion dynamics, the q-voter model with independence and the q-voter model with anticonformity, we discuss how the change of disorder from annealed to quenched affects phase transitions on networks. To derive phase diagrams on networks, we develop the pair approximation for the quenched versions of the models. This formalism can be also applied to other quenched dynamics of similar kind. The results indicate that such a change of disorder eliminates all discontinuous phase transitions and broadens ordered phases. We show that although the annealed and quenched types of disorder lead to the same result in the q-voter model with anticonformity at the mean-field level, they do lead to distinct phase diagrams on networks. These phase diagrams shift towards each other as the average node degree of a network increases, and eventually, they coincide in the mean-field limit. In contrast, for the q-voter model with independence, the phase diagrams move towards the same direction regardless of the disorder type, and they do not coincide even in the mean-field limit. To validate our results, we carry out Monte Carlo simulations on random regular graphs and Barabási-Albert networks. Although the pair approximation may incorrectly predict the type of phase transitions for the annealed models, we have not observed such errors for their quenched counterparts.

Consensus, polarization, and coexistence in a continuous opinion dynamics model with quenched disorder

A model of opinion dynamics is introduced in which each individual's opinion is measured on a bounded continuous spectrum. Each opinion is influenced heterogeneously by every other opinion in the population. It is demonstrated that consensus, polarization and a spread of moderate opinions are all possible within this model. Using dynamic mean-field theory, we are able to identify the statistical features of the interactions between individuals that give rise to each of the aforementioned emergent phenomena. The nature of the transitions between each of the observed macroscopic states is also studied. It is demonstrated that heterogeneity of interactions between individuals can lead to polarization, that mostly antagonistic or contrarian interactions can promote consensus at a moderate opinion, and that mostly reinforcing interactions encourage the majority to take an extreme opinion.