4 An Introduction to Emergence Dynamics in Complex Systems
155
In this section we study the occurrence of sustained oscillation depending on the
linking probability on excitable ER random networks, and find that the minimum
Winfree loop (MWL) is the intrinsic mechanism in determining the emergence of
collective oscillations. Furthermore, the emergence of sustained oscillation is optimized at an optimal connection probability (OCP), and the OCP is found to form a
one-to-one relationship with the MWL length. This relation is well understood that
the connection probability interval and the OCP for supporting the oscillations in
random networks are exposed to be determined by the MWL. These three important
quantities can be approximately predicted by the network structure analysis, and
have been verified in numerical simulations [70].
One adopts the Bär-Eiswirth model (1) on ER networks with N nodes. Each pair of
nodes are connected with a given probability P, and the total number of connections
is PN (N − 1)/2. By manipulating P, one can produce a number of random networks
with different detailed topologies for a given P.
We introduce the oscillation proportion
P os = N os /N ALL
(4.38)
as the order parameter to quantitatively investigate the influence of system parameters
on self-sustained oscillations in random networks, where N ALL is the total number
of tests starting from random initial conditions for each set of parameters, and N os is
the number of self-sustained oscillations counted in N ALL dynamical processes.
In Figs. 4.6a–d, the dependence of the oscillation proportion P os on the connection
probability P for different parameters a, b, ε and D on ER random networks with
N = 100 nodes are presented. It is shown from all these curves that the system can
exhibit self-sustained oscillation in a certain regime of the connection probability,
and no oscillations are presented at very small or very large P. Moreover, an OCP
P = P OCP for supporting self-sustained oscillations can be expected on ER random
networks. The number of self-sustained oscillations increases as the parameter a is
increased (see Fig. 4.6a), while P os decreases as b is increased as shown in Fig. 4.6b.
Moreover, the OCP for supporting self-sustained oscillations is independent of the
parameters a and b. Figure 4.6c reveals the dependence of P os on the relaxation
parameter ε. It is shown from Fig. 4.6c that as ε is increased, P os decreases remarkably.
Increasing the coupling strength D is shown to enhance the sustained oscillation (see
Fig. 4.6d).
The non-trivial dependences of collective oscillations on various parameters such
as the connection probability P are very interesting. As discussed above, the excitable
wave propagating along an excitable loop can form a 1D Winfree loop, which serves
as the oscillation source and maintain self-sustained oscillation in excitable complex
networks. Figure 4.7a presents the dependence of the sustained-oscillation period T
of the 1D Winfree loop on the loop length, where a shorter/longer period is expected
for a shorter/longer loop length. However, due to the existence of the refractory
period of excitable dynamics, a too short 1D Winfree loop cannot support sustained
oscillations, implying a minimum Winfree loop (MWL) length L min for a given set
155
In this section we study the occurrence of sustained oscillation depending on the
linking probability on excitable ER random networks, and find that the minimum
Winfree loop (MWL) is the intrinsic mechanism in determining the emergence of
collective oscillations. Furthermore, the emergence of sustained oscillation is optimized at an optimal connection probability (OCP), and the OCP is found to form a
one-to-one relationship with the MWL length. This relation is well understood that
the connection probability interval and the OCP for supporting the oscillations in
random networks are exposed to be determined by the MWL. These three important
quantities can be approximately predicted by the network structure analysis, and
have been verified in numerical simulations [70].
One adopts the Bär-Eiswirth model (1) on ER networks with N nodes. Each pair of
nodes are connected with a given probability P, and the total number of connections
is PN (N − 1)/2. By manipulating P, one can produce a number of random networks
with different detailed topologies for a given P.
We introduce the oscillation proportion
P os = N os /N ALL
(4.38)
as the order parameter to quantitatively investigate the influence of system parameters
on self-sustained oscillations in random networks, where N ALL is the total number
of tests starting from random initial conditions for each set of parameters, and N os is
the number of self-sustained oscillations counted in N ALL dynamical processes.
In Figs. 4.6a–d, the dependence of the oscillation proportion P os on the connection
probability P for different parameters a, b, ε and D on ER random networks with
N = 100 nodes are presented. It is shown from all these curves that the system can
exhibit self-sustained oscillation in a certain regime of the connection probability,
and no oscillations are presented at very small or very large P. Moreover, an OCP
P = P OCP for supporting self-sustained oscillations can be expected on ER random
networks. The number of self-sustained oscillations increases as the parameter a is
increased (see Fig. 4.6a), while P os decreases as b is increased as shown in Fig. 4.6b.
Moreover, the OCP for supporting self-sustained oscillations is independent of the
parameters a and b. Figure 4.6c reveals the dependence of P os on the relaxation
parameter ε. It is shown from Fig. 4.6c that as ε is increased, P os decreases remarkably.
Increasing the coupling strength D is shown to enhance the sustained oscillation (see
Fig. 4.6d).
The non-trivial dependences of collective oscillations on various parameters such
as the connection probability P are very interesting. As discussed above, the excitable
wave propagating along an excitable loop can form a 1D Winfree loop, which serves
as the oscillation source and maintain self-sustained oscillation in excitable complex
networks. Figure 4.7a presents the dependence of the sustained-oscillation period T
of the 1D Winfree loop on the loop length, where a shorter/longer period is expected
for a shorter/longer loop length. However, due to the existence of the refractory
period of excitable dynamics, a too short 1D Winfree loop cannot support sustained
oscillations, implying a minimum Winfree loop (MWL) length L min for a given set