Local Certification of Majority Dynamics

In majority voting dynamics, a group of $n$ agents in a social network are asked for their preferred candidate in a future election between two possible choices. At each time step, a new poll is taken, and each agent adjusts their vote according to the majority opinion of their network neighbors. After $T$ time steps, the candidate with the majority of votes is the leading contender in the election. In general, it is very hard to predict who will be the leading candidate after a large number of time-steps. We study, from the perspective of local certification, the problem of predicting the leading candidate after a certain number of time-steps, which we call Election-Prediction. We show that in graphs with sub-exponential growth Election-Prediction admits a proof labeling scheme of size $\mathcal{O}(\log n)$. We also find non-trivial upper bounds for graphs with a bounded degree, in which the size of the certificates are sub-linear in $n$. Furthermore, we explore lower bounds for the unrestricted case, showing that locally checkable proofs for Election-Prediction on arbitrary $n$-node graphs have certificates on $\Omega(n)$ bits. Finally, we show that our upper bounds are tight even for graphs of constant growth

Busca evolutiva por redes booleanas na tarefa de classificação de densidade

Redes Booleanas são compostas por nós que representam variáveis binárias computadas em função dos valores representados por nós adjacentes. Esta computação local leva a comportamentos globais, como a convergência para um estado fixo da rede. Tal comportamento é utilizado na tarefa de classificação de densidade, onde procura-se a convergência dos valores de todos os nós para um ponto fixo que reflete o estado predominante presente na configuração inicial da rede, ou seja, um objetivo global restrito a ações de caráter local. Neste trabalho são efetuadas buscas evolutivas de modo a encontrar regras e topologias de redes Booleanas com boa performance na classificação de densidade. Consideram-se exclusivamente vizinhanças irregulares e bidirecionais para todos os nós, representando inicialmente a função Booleana da rede através da regra da maioria da vizinhança. Primeiramente, efetuam-se buscas evolutivas por topologias de redes guiadas pela métrica ω, esta referente à classificação de redes de mundo pequeno, e em seguida, efetuam-se buscas evolutivas no espaço de possíveis funções Booleanas utilizando as topologias de redes encontradas anteriormente.Boolean networks consist of nodes that represent binary variables, which are computed as a function of the values represented by their adjacent nodes. This local processing entails global behaviors, such as the convergence to _xed points, a behavior found in the context of the density classi_cation problem, where the aim is the network's convergence to a fixed point of the prevailing node value in the initial global configuration of the network; in other words, a global decision is targeted, but according to a constrained, non-global action. In this work, we rely on evolutionary searches in order to _nd rules and network topologies with good performance in the task. All nodes' neighborhoods are assumed to be de_ned by non-regular and bidirectional links, and the Boolean function of the network initialized by the local majority rule. Firstly, is carried out a search in the space of network topologies, guided by the ω metric, related to the "small-worldness" of the networks, and then, in the space of Boolean functions, but constraining the network topologies to the best family identified in the previous experiment..Coordenação de Aperfeiçoamento de Pessoal de Nível Superio

On the complexity of two-dimensional signed majority cellular automata

International audienceWe study the complexity of signed majority cellular automata on the planar grid. We show that, depending on their symmetry and uniformity, they can simulate different types of logical circuitry under different modes. We use this to establish new bounds on their overall complexity, concretely: the uniform asymmetric and the non-uniform symmetric rules are Turing universal and have a P-complete prediction problem; the non-uniform asymmetric rule is in-trinsically universal; no symmetric rule can be intrinsically universal. We also show that the uniform asymmetric rules exhibit cycles of super-polynomial length, whereas symmetric ones are known to have bounded cycle length