Problems of consensus in multi-agent systems are often viewed as a series of independent, simultaneous local decisions made between a limited set of options, all aimed at reaching a global agreement. Key challenges in these protocols include estimating the likelihood of various outcomes and finding bounds for how long it may take to achieve consensus, if it occurs at all. To date, little attention has been given to the case where some agents have no initial opinion. In this paper, we introduce a variant of the consensus problem which includes what we call `agnostic' nodes and frame it as a combination of two known and well-studied processes: voter model and rumour spreading. We show (1) a martingale that describes the probability of consensus for a given colour, (2) bounds on the number of steps for the process to end using results from rumour spreading and voter models, (3) closed formulas for the probability of consensus in a few special cases, and (4) that the computational complexity of estimating the probability with a Markov chain Monte Carlo process is $O(n^2 \log n)$ for general graphs and $O(n\log n)$ for Erd\H{o}s-R\'enyi graphs, which makes it an efficient method for estimating probabilities of consensus. Furthermore, we present experimental results suggesting that the number of runs needed for a given standard error decreases when the number of nodes increases.

Consider an undirected graph G, representing a social network, where each node is blue or red, corresponding to positive or negative opinion on a topic. In the voter model, in discrete time rounds, each node picks a neighbour uniformly at random and adopts its colour. Despite its significant popularity, this model does not capture some fundamental real-world characteristics such as the difference in the strengths of individuals connections, individuals with neutral opinion on a topic, and individuals who are reluctant to update their opinion. To address these issues, we introduce and study a generalisation of the voter model. Motivating by campaigning strategies, we study the problem of selecting a set of seeds blue nodes to maximise the expected number of blue nodes after some rounds. We prove that the problem is NP- hard and provide a polynomial time approximation algorithm with the best possible approximation guarantee. Our experiments on real-world and synthetic graph data demonstrate that the proposed algorithm outperforms other algorithms. We also investigate the convergence properties of the model. We prove that the process could take an exponential number of rounds to converge. However, if we limit ourselves to strongly connected graphs, the convergence time is polynomial and the period (the number of states in convergence) divides the length of all cycles in the graph.

We establish PAC learnability of influence functions for three common influence models, namely, the Linear Threshold (LT), Independent Cascade (IC) and Voter models, and present concrete sample complexity results in each case. Our results for the LT model are based on interesting connections with neural networks; those for the IC model are based an interpretation of the influence function as an expectation over random draw of a subgraph and use covering number arguments; and those for the Voter model are based on a reduction to linear regression. We show these results for the case in which the cascades are only partially observed and we do not see the time steps in which a node has been influenced. We also provide efficient polynomial time learning algorithms for a setting with full observation, i.e.

Previous explanations for the persistence of polarization of opinions have typically included modelling assumptions that predispose the possibility of polarization (e.g.\ repulsive interactions). An exception is recent research showing that polarization is stable when agents form their opinions using reinforcement learning. We show that the polarization observed in this model is not stable, but exhibits consensus asymptotically with probability one. By constructing a link between the reinforcement learning model and the voter model, we argue that the observed polarization is metastable. Finally, we show that a slight modification in the learning process of the agents changes the model from being non-ergodic to being ergodic. Our results show that reinforcement learning may be a powerful method for modelling polarization in opinion dynamics, but that the tools appropriate for analysing such models crucially depend on the properties of the resulting systems. Properties which are determined by the details of the learning process.

Collective decision-making enables multi-robot systems to act autonomously in real-world environments. Existing collective decision-making mechanisms suffer from the so-called speed versus accuracy trade-off or rely on high complexity, e.g., by including global communication. Recent work has shown that more efficient collective decision-making mechanisms based on artificial neural networks can be generated using methods from evolutionary computation. A major drawback of these decision-making neural networks is their limited interpretability. Analyzing evolved decision-making mechanisms can help us improve the efficiency of hand-coded decision-making mechanisms while maintaining a higher interpretability. In this paper, we analyze evolved collective decision-making mechanisms in detail and hand-code two new decision-making mechanisms based on the insights gained. In benchmark experiments, we show that the newly implemented collective decision-making mechanisms are more efficient than the state-of-the-art collective decision-making mechanisms voter model and majority rule.

We show PAC learnability of influence functions for three common influence models, namely, the Linear Threshold (LT), Independent Cascade (IC) and Voter models, and present concrete sample complexity results in each case. Our results for the LT model are based on interesting connections with neural networks; those for the IC model are based an interpretation of the influence function as an expectation over random draw of a subgraph and use covering number arguments; and those for the Voter model are based on a reduction to linear regression. We show these results for the case in which the cascades are only partially observed and we do not see the time steps in which a node has been influenced. We also provide efficient polynomial time learning algorithms for a setting with full observation, i.e.

Autonomous robot swarms must be able to make fast and accurate collective decisions, but speed and accuracy are known to be conflicting goals. While collective decision-making is widely studied in swarm robotics research, only few works on using methods of evolutionary computation to generate collective decision-making mechanisms exist. These works use task-specific fitness functions rewarding the accomplishment of the respective collective decision-making task. But task-independent rewards, such as for prediction error minimization, may promote the emergence of diverse and innovative solutions. We evolve collective decision-making mechanisms using a task-specific fitness function rewarding correct robot opinions, a task-independent reward for prediction accuracy, and a hybrid fitness function combining the two previous. In our simulations, we use the collective perception scenario, that is, robots must collectively determine which of two environmental features is more frequent. We show that evolution successfully optimizes fitness in all three scenarios, but that only the task-specific fitness function and the hybrid fitness function lead to the emergence of collective decision-making behaviors. In benchmark experiments, we show the competitiveness of the evolved decision-making mechanisms to the voter model and the majority rule and analyze the scalability of the decision-making mechanisms with problem difficulty.

Strongly opinionated minorities can have a dramatic impact on the opinion dynamics of a large population. Two factions of inflexible minorities, polarised into two competing opinions, could lead the entire population to persistent indecision. Equivalently, populations can remain undecided when individuals sporadically change their opinion based on individual information rather than social information. Our analysis compares the cross-inhibition model with the voter model for decisions between equally good alternatives, and with the weighted voter model for decisions among alternatives characterised by different qualities. Here we show that cross-inhibition, differently from the other two models, is a simple mechanism, ubiquitous in collective biological systems, that allows the population to reach a stable majority for one alternative even in the presence of asocial behaviour. The results predicted by the mean-field models are confirmed by experiments with swarms of 100 locally interacting robots. This work suggests an answer to the longstanding question of why inhibitory signals are widespread in natural systems of collective decision making, and, at the same time, it proposes an efficient mechanism for designing resilient swarms of minimalistic robots.

Opinion diffusion is a crucial phenomenon in social networks, often underlying the way in which a collective of agents develops a consensus on relevant decisions. The voter model is a well-known theoretical model to study opinion spreading in social networks and structured populations. Its simplest version assumes that an updating agent will adopt the opinion of a neighboring agent chosen at random. The model allows us to study, for example, the probability that a certain opinion will fixate into a consensus opinion, as well as the expected time it takes for a consensus opinion to emerge. Standard voter models are oblivious to the opinions held by the agents involved in the opinion adoption process. We propose and study a context-dependent opinion spreading process on an arbitrary social graph, in which the probability that an agent abandons opinion $a$ in favor of opinion $b$ depends on both $a$ and $b$. We discuss the relations of the model with existing voter models and then derive theoretical results for both the fixation probability and the expected consensus time for two opinions, for both the synchronous and the asynchronous update models.

We investigate opinion dynamics in a fully-connected system, consisting of $n$ identical and anonymous agents, where one of the opinions (which is called correct) represents a piece of information to disseminate. In more detail, one source agent initially holds the correct opinion and remains with this opinion throughout the execution. The goal for non-source agents is to quickly agree on this correct opinion, and do that robustly, i.e., from any initial configuration. The system evolves in rounds. In each round, one agent chosen uniformly at random is activated: unless it is the source, the agent pulls the opinions of $\ell$ random agents and then updates its opinion according to some rule. We consider a restricted setting, in which agents have no memory and they only revise their opinions on the basis of those of the agents they currently sample. As restricted as it is, this setting encompasses very popular opinion dynamics, such as the voter model and best-of-$k$ majority rules. Qualitatively speaking, we show that lack of memory prevents efficient convergence. Specifically, we prove that no dynamics can achieve correct convergence in an expected number of steps that is sub-quadratic in $n$, even under a strong version of the model in which activated agents have complete access to the current configuration of the entire system, i.e., the case $\ell=n$. Conversely, we prove that the simple voter model (in which $\ell=1$) correctly solves the problem, while almost matching the aforementioned lower bound. These results suggest that, in contrast to symmetric consensus problems (that do not involve a notion of correct opinion), fast convergence on the correct opinion using stochastic opinion dynamics may indeed require the use of memory. This insight may reflect on natural information dissemination processes that rely on a few knowledgeable individuals.