On Bifurcations in Nonlinear Consensus Networks

Srivastava, Vaibhav; Moehlis, Jeff; Bullo, Francesco

doi:10.1007/s00332-011-9103-4

Cited by 28 publications

(18 citation statements)

References 23 publications

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“…where k ij is an arbitrary function from R to R. This form implies that the interaction function is, for each edge, an odd function of (x j − x i ), which is a popular choice in the study of non-linear consensus [26]. Within the language of non-linear consensus, this model belongs to the family of relative non-linear flow.…”

Section: A Symmetries and Quasi-linearitymentioning

confidence: 99%

Multibody interactions and nonlinear consensus dynamics on networked systems

2020

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Multi-body interactions can reveal higher-order dynamical effects that are not captured by traditional two-body network models. In this work, we derive and analyse models for consensus dynamics on hypergraphs, where nodes interact in groups rather than in pairs. Our work reveals that multi-body dynamical effects that go beyond rescaled pairwise interactions can only appear if the interaction function is non-linear, regardless of the underlying multi-body structure. As a practical application, we introduce a specific non-linear function to model three-body consensus, which incorporates reinforcing group effects such as peer pressure. Unlike consensus processes on networks, we find that the resulting dynamics can cause shifts away from the average system state. The nature of these shifts depends on a complex interplay between the distribution of the initial states, the underlying structure and the form of the interaction function. By considering modular hypergraphs, we discover state-dependent, asymmetric dynamics between polarised clusters where multi-body interactions make one cluster dominate the other.

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Section: A Symmetries and Quasi-linearitymentioning

confidence: 99%

Multibody interactions and nonlinear consensus dynamics on networked systems

2020

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“…However, the decision to accept the overall information received, y j (t) = i y j (t; i), is based on the collective behaviour of the whole neighbourhood. This step is implemented by applying a non-linear transformation to y j (t), in order to capture how the accumulated information received from all the neighbours transforms into a change for the state of v j , which is reminiscent of the mechanisms of the LT model, but also of non-linear models for opinion dynamics [39]. Specifically, we assume that there is a lower bound b t in the model, corresponding to the critical mass to trigger the diffusion, s.t.…”

Section: Model Descriptionmentioning

confidence: 99%

Unifying information propagation models on networks and influence maximisation

Tian,

Lambiotte

2021

Preprint

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Information diffusion in social networks is a central theme in computational social sciences, with important theoretical and practical implications, such as the influence maximisation problem for viral marketing. Two widely adopted diffusion models are the independent cascade model where nodes adopt their behaviour from each neighbour independently, and the linear threshold model where collective effort from the whole neighbourhood is needed to influence a node. However, both models suffer from certain drawbacks, including a binary state space, where nodes are either active or not, and the absence of feedback, as nodes can not be influenced after having been activated. To address these issues, we consider a model with continuous variables that has the additional advantage of unifying the two classic models, as the extended independent cascade model and the extended linear threshold model are recovered by setting appropriate parameters. For the associated influence maximisation problem, the objective function is no longer submodular, a feature that most approximation algorithms are based on but is arguably strict in practice. Hence, we develop a framework, where we formulate the influence maximisation problem as a mixed integer nonlinear programming and adopt derivative-free methods. Furthermore, we propose a customised direct search method specifically for the proposed diffusion model, with local convergence. We also show that the problem can be exactly solved in the case of linear dynamics by selecting nodes according to their Katz centrality. We demonstrate the rich behaviour of the newly proposed diffusion model and the close-to-optimal performance of the customised direct search numerically on both synthetic and real networks.

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“…Our statement of bifurcation phenomenon can also be generalized to a setting with heterogeneous agents. For more discussions on bifurcation analysis in multi-agent systems, see [33], [34].…”

Section: Theoremmentioning

confidence: 99%

Opinion Behavior Analysis in Social Networks Under the Influence of Coopetitive Media

Xue

Hirche

Cao

2020

IEEE Trans. Netw. Sci. Eng.

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Both interpersonal communication and media contact are important information sources and play a significant role in shaping public opinions of large populations. In this article, we investigate how the opinion-forming process evolves over social networks under the media influence. In addition to being affected by the opinions of their connected peers, the media cooperate and/or compete mutually with each other. Networks with mixed cooperative and competitive interactions are said to be coopetitive. In this endeavor, a novel mathematical model of opinion dynamics is introduced, which captures the information diffusion process under consideration, makes use of the community-based network structure, and takes into account personalized biases among individuals in social networks. By employing port-Hamiltonian system theory to analyze the modeled opinion dynamics, we predict how public opinions evolve in the long run through social entities and find applications in political strategy science. A key technical observation is that as a result of the port-Hamiltonian formulation, the mathematical passivity property of individuals' self-dynamics facilitates the convergence analysis of opinion evolution. We explain how to steer public opinions towards consensus, polarity, or neutrality, and investigate how an autocratic media coalition might emerge regardless of public views. We also assess the role of interpersonal communication and media exposure, which in itself is an essential topic in mathematical sociology.

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On Bifurcations in Nonlinear Consensus Networks

Cited by 28 publications

References 23 publications

Multibody interactions and nonlinear consensus dynamics on networked systems

Multibody interactions and nonlinear consensus dynamics on networked systems

Unifying information propagation models on networks and influence maximisation

Opinion Behavior Analysis in Social Networks Under the Influence of Coopetitive Media

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