A survey on the fusion process in opinion dynamics

In opinion dynamics with continuous opinion, bounded confidence is a critical parameter. Agents can interact with each other only when the opinion difference between them is less than the bounded confidence. Larger bounded confidence always leads to fewer opinion clusters. Stubbornness characterizing the insistence of an agent on her own opinion is thought to only affect the transition time. In this work, a modified Hegselmann–Krause model with heterogeneous population is investigated, where agents in different/same subpopulation have different/same bounded confidence and stubbornness. We find that, due to the interaction among subpopulations, increasing the stubbornness of agents in the subpopulation with the largest bounded confidence favors fewer opinion clusters and the expansion of the largest cluster. We also find that increasing the bounded confidence of a subpopulation leads to fewer clusters and a larger largest cluster provided that all the others have large bounded confidence. While one subpopulation is with a small bounded confidence, there exist an optimal bounded confidence of another subpopulation for the smallest number of opinion clusters and that for the largest size of the largest cluster.

Opinion clusters in a modified Hegselmann-Krause model with heterogeneous bounded confidences and stubbornness

Homophily between agents and structural balance in connected triads of agents are complementary mechanisms thought to shape social groups leading to, for instance, consensus or polarization. To capture both processes in a unified manner, we propose a model of pair and triadic interactions. We consider $N$ fully connected agents, where each agent has $G$ underlying attributes, and the similarity between agents in attribute space (i.e., homophily) is used to determine the link weight between them. For structural balance we use a triad-updating rule where only one attribute of one agent is changed intentionally in each update, but this also leads to accidental changes in link weights and even link polarities. The link weight dynamics in the limit of large $G$ is described by a Fokker-Planck equation from which the conditions for a phase transition to a fully balanced state with all links positive can be obtained. This ``paradise state'' of global cooperation is, however, difficult to achieve requiring $GgO({N}^{2})$ and $pg0.5$, where the parameter $p$ captures a willingness for consensus. Allowing edge weights to be a consequence of attributes naturally captures homophily and reveals that many real-world social systems would have a subcritical number of attributes necessary to achieve structural balance.

Homophily Based on Few Attributes Can Impede Structural Balance.

Proof of work and proof of stake (PoS) are commonly used in the current permissionless blockchains. These consensus protocols can be abstracted into a random process of selecting a node for accounting in a blockchain ledger. However, they are generally faced with resource consumption and vulnerability issues. We present proof of credit (PoC), a fair blockchain protocol based on the PoC blockchain protocol. It is a special PoS protocol where the credit is a special kind of stake quantifying whether the node’s activity is beneficial to the system. Any nodes cannot change their credits arbitrarily. We demonstrate that our PoC protocol satisfies the security properties, including common prefix, chain quality, and chain growth, under the assumption that the total credit the honest held is majority. In addition, we propose a self-audit mechanism and a hybrid incentive mechanism to enhance the security and stability. Finally, we explain the method by which the PoC protocol resists the double-spending attacks and the selfish mining attacks.

A Fair Blockchain Based on Proof of Credit

Predicting and utilizing the evolution trend of hot topics is critical for contingency management and decision-making purposes of government bodies and enterprises. This paper proposes a model named online opinion dynamics (OODs) where any node in a social network has its unique confidence threshold and influence radius. The nodes in the OOD are mainly affected by their neighbors and are also randomly influenced by unfamiliar nodes. In the traditional opinion model, however, each node is affected by all other nodes, including its friends. Furthermore, many traditional opinion evolution approaches are reviewed to see if all nodes (participants) can eventually reach a consensus. On the contrary, OOD is more focused on such details as concluding the overall trend of events and evaluating the support level of each participant through numerical simulation. Experiments show that OOD is superior to the improvement of the original Hegselmann–Krause (HK) model, HK-13 and HK-17, with respect to qualitative predictions of the evolution trend of an event. The quantitative predictions of the HK model cannot be used to make decisions, whereas the results of the OOD model are proved to be acceptable.

Predicting the Evolution of Hot Topics: A Solution Based on the Online Opinion Dynamics Model in Social Network

In this paper, we propose a new model for binary opinion dynamics in a (fully connected) structurally balanced network. In a structurally balanced network, agents are classified into two clusters and two agents in the same cluster (resp. different clusters) are connected with a positive (resp. negative) edge. Initially, every agent is assigned with one of the two opinions randomly. In every time slot, three agents are randomly selected to have their opinions updated. If the three agents belong to the same cluster, the majority rule (MR) is used to update their opinions. On the other hand, if the three agents belong to two different clusters, with probability $p$ , a consensus is reached by the MR, and with probability $1-p$ , a polarization (in line with the signs of the three edges) is reached. The probability $p$ , called the rationality probability, plays a significant role for measuring how rational the agents in a network behave when they encounter different opinions. By applying a fluid limit theorem for jump Markov processes, we derive a system of differential equations for the density functions of opinions for large networks. We show that the equilibrium points corresponding to consensus and polarization are the only stable equilibrium points. All other equilibrium points are all unstable. As such, as time goes on, the network eventually reaches a consensus or a polarization, depending on the rationality probability and the initial state of the network.

Ying Liu

Papers

Consensus and Polarization of Binary Opinions in Structurally Balanced Networks