A homoclinic route to asymptotic full cooperation in adaptive networks and its failure

Zschaler, Gerd; Traulsen, Arne; Groß, Thilo

doi:10.1088/1367-2630/12/9/093015

Cited by 41 publications

(49 citation statements)

References 42 publications

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“…The ODE system (1)- (3) can now be closed using a pair-approximation [28][29][30], where triplets and quadruplets are given by…”

Section: Analytical Descriptionmentioning

confidence: 99%

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Adaptive-network models of swarm dynamics

Huepe¹,

Zschaler

et al. 2011

New J. Phys.

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We propose a simple adaptive-network model describing recent swarming experiments. Exploiting an analogy with human decision making, we capture the dynamics of the model by a low-dimensional system of equations permitting analytical investigation. We find that the model reproduces several characteristic features of swarms, including spontaneous symmetry breaking, noise-and density-driven order-disorder transitions that can be of first or second order, and intermittency. Reproducing these experimental observations using a nonspatial model suggests that spatial geometry may have a lesser impact on collective motion than previously thought.

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“…The ODE system (1)- (3) can now be closed using a pair-approximation [28][29][30], where triplets and quadruplets are given by…”

Section: Analytical Descriptionmentioning

confidence: 99%

“…The factor κ = ( k 2 − k )/ k 2 relates the second and first moments of the degree distribution. Because our network dynamics will yield an unknown, randomly evolving topology, we use a random graph approximation, setting κ = 1 as in [18,21,[28][29][30].…”

Section: Analytical Descriptionmentioning

confidence: 99%

Adaptive-network models of swarm dynamics

Huepe¹,

Zschaler

et al. 2011

New J. Phys.

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“…The bifurcation can play the role of an epidemic threshold in a small parameter space, where it occurs close to a Hopf bifurcation, such that very large simulations are needed to see the limit cycle. A homoclinic bifurcation was also found to enable a transition to full cooperation in a game theoretical model, but required global information transfer between agents 18 .…”

Section: Introductionmentioning

confidence: 95%

Network inoculation: Heteroclinics and phase transitions in an epidemic model

Yang

Rogers

Groß

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In epidemiological modelling, dynamics on networks, and in particular adaptive and heterogeneous networks have recently received much interest. Here we present a detailed analysis of a previously proposed model that combines heterogeneity in the individuals with adaptive rewiring of the network structure in response to a disease. We show that in this model qualitative changes in the dynamics occur in two phase transitions. In a macroscopic description one of these corresponds to a local bifurcation whereas the other one corresponds to a non-local heteroclinic bifurcation. This model thus provides a rare example of a system where a phase transition is caused by a non-local bifurcation, while both micro-and macro-level dynamics are accessible to mathematical analysis. The bifurcation points mark the onset of a behaviour that we call network inoculation. In the respective parameter region exposure of the system to a pathogen will lead to an outbreak that collapses, but leaves the network in a configuration where the disease cannot reinvade, despite every agent returning to the susceptible class. We argue that this behaviour and the associated phase transitions can be expected to occur in a wide class of models of sufficient complexity.

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“…The crucial point in this procedure is to choose an appropriate motif set and to check the validity of the 18 closure. The motif set and the approximations discussed here have been used in many applications, including adaptive networks [72,[81][82][83][84][85]. However, there are situations where this kind of approximation fails, because relevant heterogeneities and correlations are neglected.…”

Section: Reduction Of the State Spacementioning

confidence: 99%

Emergence and persistence of diversity in complex networks

Böhme

2013

Eur. Phys. J. Spec. Top.

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Complex networks are employed as a mathematical description of complex systems in many different fields, ranging from biology to sociology, economy and ecology. Dynamical processes in these systems often display phase transitions, where the dynamics of the system changes qualitatively. In combination with these phase transitions certain components of the system might irretrievably go extinct. In this case, we talk about absorbing transitions. Developing mathematical tools, which allow for an analysis and prediction of the observed phase transitions is crucial for the investigation of complex networks.In this thesis, we investigate absorbing transitions in dynamical networks, where a certain amount of diversity is lost. In some real-world examples, e.g. in the evolution of human societies or of ecological systems, it is desirable to maintain a high degree of diversity, whereas in others, e.g. in epidemic spreading, the diversity of diseases is worthwhile to confine. An understanding of the underlying mechanisms for emergence and persistence of diversity in complex systems is therefore essential. Within the scope of two different network models, we develop an analytical approach, which can be used to estimate the prerequisites for diversity.In the first part, we study a model for opinion formation in human societies. In this model, regimes of low diversity and regimes of high diversity are separated by a fragmentation transition, where the network breaks into disconnected components, corresponding to different opinions. We propose an approach for the estimation of the fragmentation point. The approach is based on a linear stability analysis of the fragmented state close to the phase transition and yields much more accurate results compared to conventional methods.In the second part, we study a model for the formation of complex food webs. We calculate and analyze coexistence conditions for several types of species in ecological communities. To this aim, we employ an approach which involves an iterative stability analysis of the equilibrium with respect to the arrival of a new species. The proposed formalism allows for a direct calculation of coexistence ranges and thus facilitates a systematic analysis of persistence conditions for food webs.In summary, we present a general mathematical framework for the calculation of absorbing phase transitions in complex networks, which is based on concepts from percolation theory. While the specific implementation of the formalism differs from model to model, the basic principle remains applicable to a wide range of different models.

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A homoclinic route to asymptotic full cooperation in adaptive networks and its failure

Cited by 41 publications

References 42 publications

Adaptive-network models of swarm dynamics

Adaptive-network models of swarm dynamics

Network inoculation: Heteroclinics and phase transitions in an epidemic model

Emergence and persistence of diversity in complex networks

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