4 An Introduction to Emergence Dynamics in Complex Systems
159
and p i is the expression level of gene i,0 < p i < 1. The adjacency matrices α, β
determine the network structure, which are defined in such a way that α ij = 1 if gene j
activates gene i, β ij = 1 if gene j inhibits gene i and α ij ·β ij = 0 for no dual-regulation
of gene i by gene j. act i (rep i ) represents the sum of active (repressive) transcriptional
factors to node i. The regulated expression of genes is represented by Hill functions
with cooperative exponent h and the activation coefficient K, characteristic for many
real genetic systems [66].
4.3.5.2 Skeletons and Cores
Based on the known oscillation data of GRNs, we can explicitly compute all the
functional weights w ij between any two genes based on Eqs. (4.33)–(4.35) and draw
the FW maps for the oscillatory networks. Figure 4.9a shows a simple example
of 6-node GRN with the dynamics given by Eq. (4.39), and computer simulations
indicate that this GRN possesses an oscillatory attractor for a set of parameters
given in Ref. [66]. From the oscillation data, we can draw the FW map in Fig. 4.9b.
An interesting as well as valuable feature of Fig. 4.9b is that the w ij distribution is
strongly heterogeneous, i.e. some weights w ij ≈ 1 indicate a significant control while
many others w ij ≈ 0 represent weak functional links. The heterogeneity allows us to
explore the self-organized functional structures supporting oscillatory dynamics by
the strongly weighted links. In Fig. 4.9c, we remove all the less important links with
w ij ≤w th , where the threshold value w th = 0.30. By removing any interaction and
node in the network, we always mean to delete their oscillatory parts while keep the
average influence of the deleted parts as control parameters in Eq. (4.34). The reduced
subnetwork shown in Fig. 4.9c retaining only strongly weighted interactions is called
the skeleton of the oscillatory GRN. Furthermore, we can reduce the skeleton network
by removing the nodes without output one by one, and finally obtain an irreducible
subnetwork where each node has both input and output which is defined as the core
of the oscillation.
For the network Fig. 4.8b the core is given in Fig. 4.8d. In Fig. 4.8e, f, we plot
the oscillation orbits of the original GRN and those of the core and skeleton in 2D
(p 1 , p 2 ) and (p 4 , p 6 ) phase planes, respectively. It is found that the dynamical orbits
of the original network given in Fig. 4.8a can be well reproduced by the reduced
structures in Fig. 4.8c, d, conforming that the skeleton and core defined by the highly
weighted links can dominate the essential dynamics of the original GRN. Considering
Fig. 4.8d–e, c–f it is clear that the core Fig. 4.8d serves as the oscillation source of
the GRN while the skeleton Fig. 4.8c plays the role of main signal propagation paths
from the core throughout the network.
The analysis presented in Fig. 4.8 can be well applied to more complicated GRNs
with larger numbers of nodes and links to fulfill the reduction of network complexity
for understanding and controlling network dynamics. Numerical works for a large
network with many nodes and links indicates that the comparisons of the dynamics
of the core and skeleton with those of the original network are in good agreement.
159
and p i is the expression level of gene i,0 < p i < 1. The adjacency matrices α, β
determine the network structure, which are defined in such a way that α ij = 1 if gene j
activates gene i, β ij = 1 if gene j inhibits gene i and α ij ·β ij = 0 for no dual-regulation
of gene i by gene j. act i (rep i ) represents the sum of active (repressive) transcriptional
factors to node i. The regulated expression of genes is represented by Hill functions
with cooperative exponent h and the activation coefficient K, characteristic for many
real genetic systems [66].
4.3.5.2 Skeletons and Cores
Based on the known oscillation data of GRNs, we can explicitly compute all the
functional weights w ij between any two genes based on Eqs. (4.33)–(4.35) and draw
the FW maps for the oscillatory networks. Figure 4.9a shows a simple example
of 6-node GRN with the dynamics given by Eq. (4.39), and computer simulations
indicate that this GRN possesses an oscillatory attractor for a set of parameters
given in Ref. [66]. From the oscillation data, we can draw the FW map in Fig. 4.9b.
An interesting as well as valuable feature of Fig. 4.9b is that the w ij distribution is
strongly heterogeneous, i.e. some weights w ij ≈ 1 indicate a significant control while
many others w ij ≈ 0 represent weak functional links. The heterogeneity allows us to
explore the self-organized functional structures supporting oscillatory dynamics by
the strongly weighted links. In Fig. 4.9c, we remove all the less important links with
w ij ≤w th , where the threshold value w th = 0.30. By removing any interaction and
node in the network, we always mean to delete their oscillatory parts while keep the
average influence of the deleted parts as control parameters in Eq. (4.34). The reduced
subnetwork shown in Fig. 4.9c retaining only strongly weighted interactions is called
the skeleton of the oscillatory GRN. Furthermore, we can reduce the skeleton network
by removing the nodes without output one by one, and finally obtain an irreducible
subnetwork where each node has both input and output which is defined as the core
of the oscillation.
For the network Fig. 4.8b the core is given in Fig. 4.8d. In Fig. 4.8e, f, we plot
the oscillation orbits of the original GRN and those of the core and skeleton in 2D
(p 1 , p 2 ) and (p 4 , p 6 ) phase planes, respectively. It is found that the dynamical orbits
of the original network given in Fig. 4.8a can be well reproduced by the reduced
structures in Fig. 4.8c, d, conforming that the skeleton and core defined by the highly
weighted links can dominate the essential dynamics of the original GRN. Considering
Fig. 4.8d–e, c–f it is clear that the core Fig. 4.8d serves as the oscillation source of
the GRN while the skeleton Fig. 4.8c plays the role of main signal propagation paths
from the core throughout the network.
The analysis presented in Fig. 4.8 can be well applied to more complicated GRNs
with larger numbers of nodes and links to fulfill the reduction of network complexity
for understanding and controlling network dynamics. Numerical works for a large
network with many nodes and links indicates that the comparisons of the dynamics
of the core and skeleton with those of the original network are in good agreement.