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1
We demonstrate that the adoption process of a
dynamical system gets delayed not only due to the
size of mean of the initial distribution of the timers but
also due to the heterogeneity of the initial distribution
of the timer. As an example, one can observe in Fig.
1 that the timer model reaches steady state earlier in
Fig. 1a, where the timers are initially homogeneously
distributed, than in Fig. 1b, where the timers are
initially heterogeneously distributed. However, heterogeneous timers can accelerate the adoption process
depending on the way timers are distributed. We seek
to understand the relationship between the delay of the
adoption process and the initial distribution of timers,
and we derive a pair approximation that incorporates
a timer by modifying a pair approximation of the
Watts threshold model presented in [1], and it exhibits
good agreement with the numerical results of the Watts
thresholds model with a timer on random graphs. We
also examine this model on real networks.
Fraction of active nodes ρ
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Fraction of active nodes ρ
A great deal of effort has gone into trying to model
spreading phenomena, such as the spread of behaviour,
norms, and ideas. Binary-state dynamical models use
two states (e.g., state 0 and 1), and each node in a network is in one of the two states [1]. This makes it possible to describe spreading phenomena by observing
the spread of adoption of states: One investigates how a
node in one state adopts the other state by the influence
of neighbours and how the node’s change of state
leads changes in other nodes’ states. Some existing
binary-state models, such as the Watts threshold model
[3], can exhibit cascades, in which local adoption
becomes widespread throughout a network [2]. Those
models tend to assume that individuals instantaneously
react to their neighbours’ changes without any time
delay – which is often not true in social contexts,
where different agents have different response times or
different degrees of laziness. In our work, we present a
timer model, which includes the following mechanism:
if a state-0 (inactive) node i has a timer τi – which
is a non-negative integer – and meets its adoption
condition at time T , then the node adopts state 1 at
T + 1 + τi and becomes an active node. The update
can either be synchronous or asynchronous depending
on context; we use synchronous updating in our study.
Formulating our model in this way enables us to apply
the timer model to any existing deterministic socialinfluence models where nodes change their states by
influence from the other nodes [2]. Here we illustrate
the incorporation of a timer using the Watts threshold
model.
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Fig. 1: Numerical results (red squares) and the analytical results (blue curve) of the Watts threshold
model with a timer on Erdős-Rényi random graphs.
The timers are initially distributed as a Gaussian with
mean µτ = 3 and standard deviation (a) στ = 0 (i.e.,
homogeneous timers) and (b) στ = 10 (i.e., heterogeneous timers). The number of nodes is N = 1000,
the mean degree is 5, the fraction of seed nodes is
ρ0 = 0.001, and the thresholds are distributed as a
Gaussian with mean µφ = 0.3 and standard deviation
σφ = 0.2. We average over 100 realizations of an
Erdős-Rényi graph.
[1] James P Gleeson. Binary-state dynamics on complex networks: pair approximation and beyond. Physical Review X,
3(2):021004, 2013.
[2] Mason A Porter and James P Gleeson. Dynamical systems on
networks: A tutorial. arXiv preprint arXiv:1403.7663, 2014.
[3] Duncan J Watts. A simple model of global cascades on random
networks. Proceedings of the National Academy of Sciences
of the USA, 99(9):5766–5771, 2002.