4,040 research outputs found
From continuous to discontinuous transitions in social diffusion
Models of social diffusion reflect processes of how new products, ideas or
behaviors are adopted in a population. These models typically lead to a
continuous or a discontinuous phase transition of the number of adopters as a
function of a control parameter. We explore a simple model of social adoption
where the agents can be in two states, either adopters or non-adopters, and can
switch between these two states interacting with other agents through a
network. The probability of an agent to switch from non-adopter to adopter
depends on the number of adopters in her network neighborhood, the adoption
threshold and the adoption coefficient , two parameters defining a Hill
function. In contrast, the transition from adopter to non-adopter is
spontaneous at a certain rate . In a mean-field approach, we derive the
governing ordinary differential equations and show that the nature of the
transition between the global non-adoption and global adoption regimes depends
mostly on the balance between the probability to adopt with one and two
adopters. The transition changes from continuous, via a transcritical
bifurcation, to discontinuous, via a combination of a saddle-node and a
transcritical bifurcation, through a supercritical pitchfork bifurcation. We
characterize the full parameter space. Finally, we compare our analytical
results with Montecarlo simulations on annealed and quenched degree regular
networks, showing a better agreement for the annealed case. Our results show
how a simple model is able to capture two seemingly very different types of
transitions, i.e., continuous and discontinuous and thus unifies underlying
dynamics for different systems. Furthermore the form of the adoption
probability used here is based on empirical measurements.Comment: 7 pages, 3 figure
Network Inoculation: Heteroclinics and phase transitions in an epidemic model
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.Comment: 26 pages, 11 figure
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