4 An Introduction to Emergence Dynamics in Complex Systems
143
4.2.3 Networks: Topology and Dynamics
The blossom of network science is absolutely a great milestone in the exploration
of complexity. Early studies of complex networks started from the graph theory in
mathematics. One can infer from the early work of Euler on the Konisburg bridge
problem. Graph theory proposed a number of useful concepts and laws in analyzing
the sets composed of vertices and edges. Although the studies of networks can be
traced back to the development of graph theory in mathematics, the scope of today’s
network science is an interdisciplinary field covering extensive subjects from physics,
chemistry, biology, economy, and even social science. The topics of network science
thus are explosive with the development of science and technology, the contributions
from various subjects vividly enrich the network science.
The important mission of network science is to give a common understanding
of various complex systems from the viewpoint of topology. Behind this mission
the important issue relates to the reduction of complexity based on network theory
and dynamics. Nowadays we can easily get a knowledge and big data via various
measure techniques from complex systems. How to gain useful information from
these data is essentially a process of reduction, i.e. to get the truth by getting rid of
redundant information. To perform such an effective reduction of a complex system
from its microdynamics, the very important starting point is to identify an appropriate
microscopic description. Two ingredients are indispensable when one studies the
microdynamics, i.e. unit dynamics and the coupling patterns of units in a system.
Now let me discuss these two points.
Different systems are composed of units with different properties. This is the
fundamental viewpoint of reductionism. For example, a drop of water is in fact
composed of ~10
23 H 2 O molecules, the human brain is composed of ~10
11 neurons,
and a heart tissue is composed of ~10
10 cardiac cells. Apparently, these units work
with different mechanisms and are described by distinct dynamics. It has been a
central topic in exploring the mechanism of these units. For example, a single neuron
or a cardiac tissue works in an accumulating-firing manner and can be modeled by a
mimic nonlinear electronic circuit. This dynamical feature is described by a number
of excitable models, e.g. the Hodgkin-Huxley model, the Fitzhugh-Nagumo model,
and so on.
The second ingredient is the modelling of interactions among units. At the particle
level, the forces among quarks, elementary particles, atoms, molecules are quite
different. It is an important task for physicists to explore these interactions. The interactions out of physics are also system dependent. For example, the relations of individuals in a society are complicated, depending strongly on the type of information
that is exchanged between two individuals.
It becomes astonishing when there are too many individuals in a system, when the
forms of unit dynamics and coupling functions sometimes matter, while in many cases
they are not essential. A typical situation happens when the coupling topology is more
important than the specific form of coupling function. The behaviors of emergence
form when the macroscopic behaviors appear for a complex system while they do not
143
4.2.3 Networks: Topology and Dynamics
The blossom of network science is absolutely a great milestone in the exploration
of complexity. Early studies of complex networks started from the graph theory in
mathematics. One can infer from the early work of Euler on the Konisburg bridge
problem. Graph theory proposed a number of useful concepts and laws in analyzing
the sets composed of vertices and edges. Although the studies of networks can be
traced back to the development of graph theory in mathematics, the scope of today’s
network science is an interdisciplinary field covering extensive subjects from physics,
chemistry, biology, economy, and even social science. The topics of network science
thus are explosive with the development of science and technology, the contributions
from various subjects vividly enrich the network science.
The important mission of network science is to give a common understanding
of various complex systems from the viewpoint of topology. Behind this mission
the important issue relates to the reduction of complexity based on network theory
and dynamics. Nowadays we can easily get a knowledge and big data via various
measure techniques from complex systems. How to gain useful information from
these data is essentially a process of reduction, i.e. to get the truth by getting rid of
redundant information. To perform such an effective reduction of a complex system
from its microdynamics, the very important starting point is to identify an appropriate
microscopic description. Two ingredients are indispensable when one studies the
microdynamics, i.e. unit dynamics and the coupling patterns of units in a system.
Now let me discuss these two points.
Different systems are composed of units with different properties. This is the
fundamental viewpoint of reductionism. For example, a drop of water is in fact
composed of ~10
23 H 2 O molecules, the human brain is composed of ~10
11 neurons,
and a heart tissue is composed of ~10
10 cardiac cells. Apparently, these units work
with different mechanisms and are described by distinct dynamics. It has been a
central topic in exploring the mechanism of these units. For example, a single neuron
or a cardiac tissue works in an accumulating-firing manner and can be modeled by a
mimic nonlinear electronic circuit. This dynamical feature is described by a number
of excitable models, e.g. the Hodgkin-Huxley model, the Fitzhugh-Nagumo model,
and so on.
The second ingredient is the modelling of interactions among units. At the particle
level, the forces among quarks, elementary particles, atoms, molecules are quite
different. It is an important task for physicists to explore these interactions. The interactions out of physics are also system dependent. For example, the relations of individuals in a society are complicated, depending strongly on the type of information
that is exchanged between two individuals.
It becomes astonishing when there are too many individuals in a system, when the
forms of unit dynamics and coupling functions sometimes matter, while in many cases
they are not essential. A typical situation happens when the coupling topology is more
important than the specific form of coupling function. The behaviors of emergence
form when the macroscopic behaviors appear for a complex system while they do not