150
Z. Zheng
three nodes whose dynamical time series labeled by green/dashed, blue/dotted and
red/dash-dotted curves, respectively. The green/dashed curve exhibits a lagged oscillation to the reference curve, therefore one calls it the phase-lagged oscillation. The
blue/dotted and red/dash-dotted curves provide the drivings for exciting the reference
node, so these curves are identified as PAD. The blue/dotted one presents the earliest
oscillation and makes the most significant contribution, thus it is the DPAD.
Figure 4.4b gives an illustration of a DPAD structure consisting of one loop and the
nodes outside the loop radiated from the loop. For excitable node dynamics, as shown
below, the red nodes form a unidirectional loop that acts as the oscillatory source,
and yellow nodes beyond the loop form paths for the propagation of oscillations.
The DPAD structure reveals the dynamical relationship between different nodes.
Based on this functional structure, we can identify the loops as the oscillation source,
and illustrate the wave propagation along various branches. All the above ideas
are generally applicable to diverse fields for self-sustained oscillations of complex
networks consisting of individual non-oscillatory nodes.
4.3.3 The Functional-Weight (FW) Approach
The above DPAD approach provides a way in analyzing the phase relations embedded
in dynamical data to unveil information on unit connections. In some cases, it is
possible to get the detailed dynamics of coupled systems. We start from the following
dynamical system composed of N units labeled as x = (x 1 , x 2 , . . . , x N ):
dx i
dt
= λ i x i + f i (x),
(4.30)
where x i denotes the state variable of the i-th node in a network (for simplicity
one adopts the one-dimensional dynamics on the nodes). We separate the linear
component from the nonlinear coupling function f i (x) at the right-hand side of (4.30).
It is our motivation here to explore the topological mechanism of collective oscillation
in networks of coupled non-oscillatory units. Therefore the linear coefficients λ i are
negative, and we set λ i = −1 without losing generality.
The functional weight (FW) approach comes from a simple but solidly standing
idea: as all the nodes in the networks cannot oscillate individually, the oscillation of
any node i (the target node) is due to the interactions from its neighbors represented
by f i (x) in Eq. (4.30). However, all the inputs from the neighbors to the target node
are mixed together in f i (x) by nonlinear functions. To measure the importance of
different neighbors to the oscillation of the target node, contributions from all these
neighbors should be separated. The cross differential force between the i-th and the
j-th nodes can be measured by.
Z. Zheng
three nodes whose dynamical time series labeled by green/dashed, blue/dotted and
red/dash-dotted curves, respectively. The green/dashed curve exhibits a lagged oscillation to the reference curve, therefore one calls it the phase-lagged oscillation. The
blue/dotted and red/dash-dotted curves provide the drivings for exciting the reference
node, so these curves are identified as PAD. The blue/dotted one presents the earliest
oscillation and makes the most significant contribution, thus it is the DPAD.
Figure 4.4b gives an illustration of a DPAD structure consisting of one loop and the
nodes outside the loop radiated from the loop. For excitable node dynamics, as shown
below, the red nodes form a unidirectional loop that acts as the oscillatory source,
and yellow nodes beyond the loop form paths for the propagation of oscillations.
The DPAD structure reveals the dynamical relationship between different nodes.
Based on this functional structure, we can identify the loops as the oscillation source,
and illustrate the wave propagation along various branches. All the above ideas
are generally applicable to diverse fields for self-sustained oscillations of complex
networks consisting of individual non-oscillatory nodes.
4.3.3 The Functional-Weight (FW) Approach
The above DPAD approach provides a way in analyzing the phase relations embedded
in dynamical data to unveil information on unit connections. In some cases, it is
possible to get the detailed dynamics of coupled systems. We start from the following
dynamical system composed of N units labeled as x = (x 1 , x 2 , . . . , x N ):
dx i
dt
= λ i x i + f i (x),
(4.30)
where x i denotes the state variable of the i-th node in a network (for simplicity
one adopts the one-dimensional dynamics on the nodes). We separate the linear
component from the nonlinear coupling function f i (x) at the right-hand side of (4.30).
It is our motivation here to explore the topological mechanism of collective oscillation
in networks of coupled non-oscillatory units. Therefore the linear coefficients λ i are
negative, and we set λ i = −1 without losing generality.
The functional weight (FW) approach comes from a simple but solidly standing
idea: as all the nodes in the networks cannot oscillate individually, the oscillation of
any node i (the target node) is due to the interactions from its neighbors represented
by f i (x) in Eq. (4.30). However, all the inputs from the neighbors to the target node
are mixed together in f i (x) by nonlinear functions. To measure the importance of
different neighbors to the oscillation of the target node, contributions from all these
neighbors should be separated. The cross differential force between the i-th and the
j-th nodes can be measured by.