Coevolving nonlinear voter model with triadic closure

We study a nonlinear coevolving voter model with triadic closure local rewiring. We find three phases with different topological properties and configuration in the steady state: absorbing consensus phase with a single component, absorbing fragmented phase with two components in opposite consensus states, and a dynamically active shattered phase with many isolated nodes. This shattered phase, which does not exist for a coevolving model with global rewiring, has a lifetime that scale exponentially with system size. We characterize the transitions between these phases in terms of the size of the largest cluster, the number of clusters, and the magnetization. Our analysis provides a possible solution to reproduce isolated parts in adaptive networks and high clustering widely observed in social systems

Ordering dynamics in the voter model with aging

The voter model with memory-dependent dynamics is theoretically and numerically studied at the mean-field level. The `internal age', or time an individual spends holding the same state, is added to the set of binary states of the population, such that the probability of changing state (or activation probability $p_i$) depends on this age. A closed set of integro-differential equations describing the time evolution of the fraction of individuals with a given state and age is derived, and from it analytical results are obtained characterizing the behavior of the system close to the absorbing states. In general, different age-dependent activation probabilities have different effects on the dynamics. When the activation probability $p_i$ is an increasing function of the age $i$, the system reaches a steady state with coexistence of opinions. In the case of aging, with $p_i$ being a decreasing function, either the system reaches consensus or it gets trapped in a frozen state, depending on the value of $p_\infty$ (zero or not) and the velocity of $p_i$ approaching $p_\infty$. Moreover, when the system reaches consensus, the time ordering of the system can be exponential ($p_\infty>0$) or power-law like ($p_\infty=0$). Exact conditions for having one or another behavior, together with the equations and explicit expressions for the exponents, are provided

Emergence of complex structures from nonlinear interactions and noise in coevolving networks

We study the joint effect of the non-linearity of interactions and noise on coevolutionary dynamics. We choose the coevolving voter model as a prototype framework for this problem. By numerical simulations and analytical approximations we find three main phases that differ in the absolute magnetization and the size of the largest component: a consensus phase, a coexistence phase, and a dynamical fragmentation phase. More detailed analysis reveals inner differences in these phases, allowing us to divide two of them further. In the consensus phase we can distinguish between a weak or alternating consensus (switching between two opposite consensus states), and a strong consensus, in which the system remains in the same state for the whole realization of the stochastic dynamics. Additionally, weak and strong consensus phases scale differently with the system size. The strong consensus phase exists for superlinear interactions and it is the only consensus phase that survives in the thermodynamic limit. In the coexistence phase we distinguish a fully-mixing phase (both states well mixed in the network) and a structured coexistence phase, where the number of links connecting nodes in different states (active links) drops significantly due to the formation of two homogeneous communities of opposite states connected by a few links. The structured coexistence phase is an example of emergence of community structure from not exclusively topological dynamics, but coevolution. Our numerical observations are supported by an analytical description using a pair approximation approach and an ad-hoc calculation for the transition between the coexistence and dynamical fragmentation phases. Our work shows how simple interaction rules including the joint effect of non-linearity, noise, and coevolution lead to complex structures relevant in the description of social systems

Opinion dynamics in social networks: From models to data

Opinions are an integral part of how we perceive the world and each other. They shape collective action, playing a role in democratic processes, the evolution of norms, and cultural change. For decades, researchers in the social and natural sciences have tried to describe how shifting individual perspectives and social exchange lead to archetypal states of public opinion like consensus and polarization. Here we review some of the many contributions to the field, focusing both on idealized models of opinion dynamics, and attempts at validating them with observational data and controlled sociological experiments. By further closing the gap between models and data, these efforts may help us understand how to face current challenges that require the agreement of large groups of people in complex scenarios, such as economic inequality, climate change, and the ongoing fracture of the sociopolitical landscape.Comment: 22 pages, 3 figure

Statistical Physics Of Opinion Formation: is it a SPOOF?

We present a short review based on the nonlinear $q$-voter model about problems and methods raised within statistical physics of opinion formation (SPOOF). We describe relations between models of opinion formation, developed by physicists, and theoretical models of social response, known in social psychology. We draw attention to issues that are interesting for social psychologists and physicists. We show examples of studies directly inspired by social psychology like: "independence vs. anticonformity" or "personality vs. situation". We summarize the results that have been already obtained and point out what else can be done, also with respect to other models in SPOOF. Finally, we demonstrate several analytical methods useful in SPOOF, such as the concept of effective force and potential, Landau's approach to phase transitions, or mean-field and pair approximations.Comment: 29 pages, 4 figures, new section 6 slightly extended, figures of higher quality, corrected typos, extended references, other minor improvements throughout the tex

Vulnerability of democratic electoral systems

The two most common types of electoral systems (ES) used in electing national legislatures are proportional representation and plurality voting. When they are evaluated, most often the arguments come from social choice theory and political sciences. The former overall uses an axiomatic approach including a list of mathematical criteria a system should fulfill. The latter predominantly focuses on the trade-off between proportionality of apportionment and governability. However, there is no consensus on the best ES, nor on the set of indexes and measures that would be the most important in such assessment. Moreover, the ongoing debate about the fairness of national elections neglects the study of their vulnerabilities. Here we address this research gap with a framework that can measure electoral systems' vulnerability to different means of influence. Using in silico analysis we show that plurality voting systems are less stable than proportional representation. They are also more susceptible to political agitators and media propaganda. A review of real-world ES reveals possible improvements in their design leading to lower susceptibility. Additionally, our simulation framework allows computation of popular indexes, as the Gallagher index or the effective number of parties, in different scenarios. Our work provides a new tool for dealing with modern threats to democracy that could destabilize voting processes. Furthermore, our results add an important argument in a long-standing discussion on evaluation of ES