Search

The garbage disposal game involves a finite set of individuals, each of whom updates their garbage by either receiving from or dumping onto others. We examine the case where only social neighbors, whose garbage levels differ by a given threshold, can offload an equal proportion of their garbage onto others. Remarkably, in the absence of this threshold, the garbage amounts of all individuals converge to the initial average on any connected social graph that is not a star., Comment: 1 figure

The \((n,k)\) game models a group of \(n\) individuals with binary opinions, say 1 and 0, where a decision is made if at least \(k\) individuals hold opinion 1. This paper explores the dynamics of the game with heterogeneous agents under both synchronous and asynchronous settings. We consider various agent types, including consentors, who always hold opinion 1, rejectors, who consistently hold opinion 0, random followers, who imitate one of their social neighbors at random, and majority followers, who adopt the majority opinion among their social neighbors. We investigate the likelihood of a decision being made in finite time. In circumstances where a decision cannot almost surely be made in finite time, we derive a nontrivial bound to offer insight into the probability of a decision being made in finite time., Comment: 6 pages, 1 figure

In the Hegselmann-Krause model, an agent updates its opinion by averaging with others whose opinions differ by at most a given confidence threshold. With agents' initial opinions uniformly distributed on the unit interval, we provide a straightforward proof that establishes the critical value is up to one-half. This implies that the probability of consensus approaches one as the number of agents tends to infinity for confidence thresholds larger than or equal to one-half., Comment: 4 pages

The mixed Hegselmann-Krause (HK) model covers the synchronous Hegselmann-Krause model, the asynchronous Hegselmann-Krause model and the Deffuant model. Previous study~\cite{mHK, mHK2} deals with the mixed HK model on finite graphs. In the study, we discuss the mixed HK model on infinite graphs which also covers the HK model and the Deffuant model on infinite graphs. We investigate conditions under which asymptotic stability holds or any two vertices in the same component approaches each other after some finite time., Comment: 7 pages

A money transfer involves a buyer and a seller. A buyer buys goods or services from a seller. The money the buyer decreases is the same as that the seller increases. At each time step, a pair of socially connected agents are selected and transact in agreed money. We evolve the Deffuant model to a money exchange system and study circumstances under which asymptotic stability holds, or equal wealth can be achieved., Comment: 5 pages

The original Leader-Follower (LF) model partitions all agents whose opinion is a number in $[-1,1]$ to a follower group, a leader group with a positive target opinion in $[0,1]$ and a leader group with a negative target opinion in $[-1,0]$. A leader group agent has a constant degree to its target and mixes it with the average opinion of its group neighbors at each update. A follower has a constant degree to the average opinion of the opinion neighbors of each leader group and mixes it with the average opinion of its group neighbors at each update. In this paper, we consider a variant of the LF model, namely the mixed model, in which the degrees can vary over time, the opinions can be high dimensional, and the number of leader groups can be more than two. We investigate circumstances under which all agents achieve a consensus. In particular, a few leaders can dominate the whole population., Comment: 5 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2109.10212

The voter model consists of a set of agents whose opinion is a binary variable. At each time step, an agent along with a social neighbor is selected and the agent imitates the social neighbor at the next time step. In this paper, we study a variant of the voter model: an imitation model based on the majority. The agent imitates the selected social neighbor if and only if the number of neighbors following the opinion of the selected social neighbor is more than that following the agent. We study circumstances under which a consensus can not be achieved and the probability of consensus on a finite connected social graph., Comment: 4 pages

The Bonabeau model is a competing model where agents fight to maintain or change their positions. Originally studied on a finite lattice, in this model, one agent is randomly selected to move to a neighboring site chosen at random. If the neighboring site is vacant, the agent moves there. However, if the site is occupied, a fight ensues. If the agent wins, they switch places with the other agent; otherwise, they remain in their original position. We investigate the Bonabeau model on fully occupied site graphs and derive a critical bound for the stability of the egalitarian state applicable to all fully occupied connected site graphs. Furthermore, we develop a competing model where all fights end in finite time on all site graphs., Comment: 6 pages, 1 figure

In a money exchange process involving a seller and a buyer, we develop a straightforward model encompassing conservative, non-conservative, and systems with or without debt. Our model integrates the Fermi function to capture the behavior of buyers and sellers. Under certain circumstances, we identify an equation that marks the phase transition between a stable equal wealth state across all connected social graphs and an unstable equal wealth state in some connected social graph., Comment: 5 pages

The mixed Hegselmann-Krause (HK) model consists of a finite number of agents characterized by their opinion, a vector in $\mathbf{R^d}$. For the deterministic case, each agent updates its opinion by the rule: decide its degree of stubbornness and mix its opinion with the average opinion of its neighbors, the agents whose opinion differs by at most some confidence threshold from its opinion at each time step. The mixed model is studied deterministically in \cite{mHK}. In this paper, we study it nondeterministically and involve a social relationship among the agents which can vary over time. We investigate circumstances under which asymptotic stability holds. Furthermore, we indicate the mixed model covers not only the HK model but also the Deffuant model., Comment: 7 pages

The original Deffuant model consists of a finite number of agents whose opinion is a number in $[0,1]$. Two socially connected agents are uniformly randomly selected at each time step and approach each other at a rate $\mu\in [0,1/2]$ if and only if their opinions differ by at most some confidence threshold $\epsilon>0$. In this paper, we consider a variant of the Deffuant model, namely the mixed model, where the convergence parameter $\mu$ and the social relationship can vary over time. We investigate circumstances under which asymptotic stability holds or a consensus can be achieved. Also, we derive a nontrivial lower bound for the probability of consensus which is independent of the number of agents., Comment: 6 pages

This paper is concerned with the probability of consensus in a multivariate, spatially explicit version of the Hegselmann-Krause model for the dynamics of opinions. Individuals are located on the vertices of a finite connected graph representing a social network, and are characterized by their opinion, with the set of opinions $\Delta$ being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold $\tau$, two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold $\tau$. Each vertex $x$ updates its opinion at rate the number of its compatible neighbors on the social network, which results in the opinion at $x$ to be replaced by a convex combination of the opinion at $x$ and the nearby opinions: $\alpha$ times the opinion at $x$ plus $(1 - \alpha)$ times the average opinion of its compatible neighbors. The main objective is to derive a lower bound for the probability of consensus when the opinions are initially independent and identically distributed with values in the opinion set $\Delta$., Comment: 13 pages, 1 figure

The original Leader-Follower model, proposed in \cite{zhao2018understanding}, categorizes agents with opinions in $[-1,1]$ into a follower group, a leader group with a positive target opinion in $[0,1]$, and a leader group with a negative target opinion in $[-1,0]$. Leaders maintain a constant attraction to their target, blending it with the average opinion of their group neighbors at each update. Followers, on the other hand, have a constant attraction to the average opinion of their leader group's opinion neighbors, also integrating it with their group neighbors' average opinion. This model was numerically studied in \cite{zhao2018understanding}. This paper extends the Leader-Follower model to include a social relationship, variable degrees over time, high-dimensional opinions, and a flexible number of leader groups. We theoretically investigate conditions for asymptotic stability or consensus, particularly in scenarios where a few leaders can dominate the entire population., Comment: 13 pages

The original Hegselmann-Krause (HK) model is composed of a finite number of agents characterized by their opinion, a number in $[0,1]$. An agent updates its opinion via taking the average opinion of its neighbors whose opinion differs by at most $\epsilon$ for $\epsilon>0$ a confidence bound. An agent is absolutely stubborn if it does not change its opinion while update, and absolutely open-minded if its update is the average opinion of its neighbors. There are two types of HK models--the synchronous HK model and the asynchronous HK model. The paper is about a variant of the HK dynamics, called the mixed model, where each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors at all times. The mixed model reduces to the synchronous HK model if all agents are absolutely open-minded all the time, and the asynchronous HK model if only one uniformly randomly selected agent is absolutely open-minded and the others are absolutely stubborn at all times. In \cite{mhk}, we discuss the mixed model deterministically. Point out some properties of the synchronous HK model, such as finite-time convergence, do not hold for the mixed model. In this topic, we study the mixed model nondeterministically. List some properties of the asynchronous model which do not hold for the mixed model. Then, study circumstances under which the asymptotic stability holds., Comment: 7 pages

The original Hegselmann-Krause (HK) model comprises a set of $n$ agents characterized by their opinion, a number in $[0,1]$. Agent $i$ updates its opinion $x_i$ via taking the average opinion of its neighbors whose opinion differs by at most $\epsilon$ from $x_i$. In the article, the opinion space is extended to $\mathbf{R^d}.$ The main result is to derive bounds for the probability of consensus. In general, we have a positive lower bound for the probability of consensus and demonstrate a lower bound for the probability of consensus on a unit cube. In particular for one dimensional case, we derive an upper bound and a better lower bound for the probability of consensus and demonstrate them on a unit interval., Comment: 16 pages, 9 figures

The original Hegselmann-Krause (HK) model consists of a set of~$n$ agents that are characterized by their opinion, a number in~$[0, 1]$. Each agent, say agent~$i$, updates its opinion~$x_i$ by taking the average opinion of all its neighbors, the agents whose opinion differs from~$x_i$ by at most~$\epsilon$. There are two types of~HK models: the synchronous~HK model and the asynchronous~HK model. For the synchronous model, all the agents update their opinion simultaneously at each time step, whereas for the asynchronous~HK model, only one agent chosen uniformly at random updates its opinion at each time step. This paper is concerned with a variant of the~HK opinion dynamics, called the mixed~HK model, where each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors at each update. The degree of the stubbornness of agents can be different and/or vary over time. An agent is not stubborn or absolutely open-minded if its new opinion at each update is the average opinion of its neighbors, and absolutely stubborn if its opinion does not change at the time of the update. The particular case where, at each time step, all the agents are absolutely open-minded is the synchronous~HK model. In contrast, the asynchronous model corresponds to the particular case where, at each time step, all the agents are absolutely stubborn except for one agent chosen uniformly at random who is absolutely open-minded. We first show that some of the common properties of the synchronous~HK model, such as finite-time convergence, do not hold for the mixed model. We then investigate conditions under which the asymptotic stability holds, or a consensus can be achieved for the mixed model., Comment: 14 pages, 0 figure

The Deffuant model is a spatial stochastic model for the dynamics of opinions in which individuals are located on a connected graph representing a social network and characterized by a number in the unit interval representing their opinion. The system evolves according to the following averaging procedure: pairs of neighbors interact independently at rate one if and only if the distance between their opinions does not exceed a certain confidence threshold, with each interaction resulting in the neighbors' opinions getting closer to each other. All the mathematical results collected so far about this model assume that the individuals are located on the integers. In contrast, we study the more realistic case where the social network can be any finite connected graph. In addition, we extend the opinion space to any bounded convex subset of a normed vector space where the norm is used to measure the level of disagreement or distance between the opinions. Our main result gives a lower bound for the probability of consensus. Interestingly, our proof leads to a universal lower bound that depends on the confidence threshold, the opinion space~(convex subset and norm) and the initial distribution, but not on the size or the topology of the social network., Comment: 12 pages, 1 figure