Untangling Complex Systems, page 83
Systems are scale-free networks, containing hubs. Why are scale-free networks so common? There
are two main reasons. First, most real networks grow continuously, and new nodes join the system
over extended periods of time. Second, nodes prefer to connect to nodes that already have many links,
according to a process known as the preferential attachment (Barabási and Albert 1999). A new node,
joining a network, has a higher probability to connect with a node having, already, many links.
This probability П is, in fact, given by the ratio:
d
Π =
i
[12.3]
∑ Ndj
j=1
where:
di and dj are the degrees of node i-th and j-th, respectively,
N is the total number of nodes.
The node that has the largest degree increases further its connectivity and becomes a hub. “The rich
get richer.” Scale-free networks have two key features: robustness to random failure and vulner-
ability to hubs’ attack (Albert et al. 2000). The robustness of the scale-free networks is rooted in
their inhomogeneous connectivity. In fact, the power-law distribution guarantees that the majority
of nodes have only a few links and that there are just a few hubs. The probability that random failure
involves nodes with low connectivity is much higher. Therefore, the failure of nodes scarcely con-
nected does not affect the overall topology and the average length of the shortest paths. However,
the removal of a few key hubs splinters the network into small isolated clusters of nodes.
Finally, it has been discovered that real networks that are scale-free and devoid of topological
constraints, such as limitations on the link length, have their clustering coefficients that are power-
law functions of the degree d: C( d ) d−
~
1 (Ravasz and Barabási 2003). This evidence means that
many Complex Systems can be modelled as hierarchical networks. The best example is a living
cell. The structure and functions of a living cell depend on the complex web of the interactions
between the cell’s numerous constituents. A cell grounds on a network of networks, including
metabolic, signaling, transcription-regulatory, and protein-protein interaction modules. None of
these modules is independent. A major challenge of contemporary biology is to describe quantita-
tively the network of networks that rules the behavior of a cell. A recent study reported the success-
ful construction of a whole-cell computational model for the bacterium Mycoplasma genitalium
(Karr et al. 2012) in which the function of all its 525 genes has been simulated. Marcus Covert
and his colleagues at Stanford University have broken the Mycoplasma cell’s overall functions
down into 28 separate functional modules describing different sub-processes, like DNA replica-
tion, RNA transcription, protein folding, metabolism. Then, they modeled each module separately
based on the information available in literature. They assumed that the modules are approxi-
mately independent on short timescales (less than 1 s). Simulations were carried out by running
through a loop in which the modules were run independently at each time step but depended on
the values of the variables (playing like connecting nodes) determined by the other modules at
the previous time step. The simulated bacterium had many expected properties, in its metabolic
fluxes, global distribution of protein and RNA, metabolite amounts, and gene function but also
provided insights into many previously unobserved cellular behaviors, including in vivo rates of
Complex Systems
425
protein-DNA association, and an inverse relationship between the duration of DNA initiation and
that of replication. This successful research suggests an approach to model more complex cells
and even multicellular systems, tissues, ecosystems, and economy. It could become a foundational
platform for interpreting the behaviors of all kinds of Complex Systems, including climate and
geology of our planet. In the latter cases, nodes and links would be molecules and intermolecular
forces, respectively. For instance, convection starts when strong macroscopic thermal gradients
transform a regular dynamic network, which is the liquid described at the molecular level,9 into a modular network; modules are micrometer-sized clusters of molecules that move either upwards or
downwards depending on their mutual densities.
BOX 12.1 THE IMMUNE SYSTEM
The immune system of a human adult contains approximately l trillion T-cells (the T indi-
cates that they are lymphocytes developed by the thymus) and l trillion B-cells (the B means
that they are lymphocytes developed by the bone marrow), located in the lymphoid organs
and the blood. Moreover, there are approximately 10 billion antigen-presenting cells (that
take up antigens and present them to T-cells) located in the lymphoid organs (Bianconi et al.
2013). This collection of cells works together in an efficient way without any central control.
For maximizing the chances of encountering antigens, lymphocytes continually circulate
between the blood and specific lymphoid tissues. A given lymphocyte spends an average of
30 minutes per day in the blood and recirculates about 50 times per day between the blood
and lymphoid tissues. If lymphocytes encounter an antigen trapped by the antigen-presenting
cells of the lymphoid organs, lymphocytes with receptors specific to that antigen stop their
migration and settle to mount an immune response locally. As these lymphocytes accumu-
late in the affected lymphoid tissue, the tissue often becomes enlarged. The immune system
network involves many spatial scales: the molecular scale, through the dynamic interactions
occurring within cells; the cellular scale, through crosstalk among the many distinct immune
cell types; the tissue scale and, finally, the organism scale. The system-level network analyses
of the immune system allow a more profound understanding. However, we are still far from
applying network-based approaches to the construction of comprehensive predictive models
of the immune response to diverse perturbations (Subramanian et al. 2015). One reason is the
adaptive power of the immune system. There is an astronomical number of possible antigens,
which is much bigger than the number of receptors in lymphocytes. When a new kind of anti-
gen enters our body, there will be at least one receptor that binds to the intruder, though very
weakly. The activated lymphocyte migrates to a lymph node, where it divides rapidly, produc-
ing many daughter cells with mutations that alter the receptor shapes. These new daughter
cells are tested against the new antigen. The cells that do not bind die after a short time.
Those that bind more strongly are unleashed into the bloodstream, where they encounter the
antigens and bind to them more tightly than did their mother lymphocyte. It is a process of
Darwinian Natural Selection. The outcome is an emerging arsenal of antibodies and killer
T-cells that evolves through mutation, selection, and replication of the fittest ones to attack the
new antigen. Further investigation and advances in mathematical and computational tools are
needed for a deep comprehension of the immune system.
9 A crystal is a neat example of a regular static network. On the other hand, a liquid can be modeled as a regular dynamic network because the single particles are much free to move and change their positions. However, the overall system may be still described as a regular network. A rarefied gas looks more like a random dynamic network rather than a regular network.
426
Untangling Complex Systems
12.4.2 ouT-of-eQuilibrium sysTems
Complex Systems are networks that are out-of-equilibrium. They can be either open or closed or
isolated (for instance, our Universe is postulated to be an isolated thermodynamic system). They
are maintained out-of-equilibrium by external and/or internal gradients of intensive variables. For
example, Earth is out-of-equilibrium due to three principal contributions. The first is the gravitational
field generated by the sun and the moon. The second is the thermal energy released by the processes
of nuclear fissions involving unstable radionuclides that are beneath the terrestrial crust. The last
contribution is the electromagnetic radiation and the wind of particles and gamma rays that come
from the sun. The sun is maintained out-of-equilibrium by the nuclear fusion reactions occurring in
its inner core. Through the so-called proton-proton chain reactions, hydrogen converts to helium, and
vast amounts of thermal energy are unleashed. 10 The thermal energy produced within the core of our star migrates by irradiation and convection towards the external surface, the so-called photosphere.
The photosphere of our sun has an average temperature of 5,777 K (NASA website) and emits ther-
mal radiation whose frequencies belong to the UV, visible and near-IR regions of the electromagnetic
spectrum (see Figure 12.7). The solar thermal radiation is the primary power source for our planet (Kleidon 2010). For this reason, it is essential to know its thermodynamic properties.
12.4.2.1 The Thermodynamics of Thermal Radiation
The electromagnetic, thermal radiation, also called heat radiation, is due to the thermal motion of
charged particles that are at the foundation of any material. All the bodies emit thermal radiation:
our sun, our planet, ourselves, this book, and so on. The spectrum of thermal radiation depends
only on the temperature. It was Max Planck (1914), at the beginning of the twentieth century, who
derived the expression for the energy density of thermal radiation, as a function of the temperature
( T) and frequency (ν), after introducing the revolutionary quantum hypothesis:
8 h 3
1
u ν
π ν
( , T) =
[12.4]
c 3
( ehν/ kBT − )1
2.0
−1 )
nm−2 1.5
m
(W
1.0
tral irradiance 0.5
ecSp
0.0
500
1000
1500
2000
2500
3000
3500
Wavelength (nm)
FIGURE 12.7 The spectral irradiance of the sun outside of the atmosphere, i.e., at Air Mass 0. (data extracted from the website http://www.pveducation.org/).
10 A nuclear reaction releases an amount of energy that is of the order of MeV. On the other hand, in a chemical reaction, only the electrons of the outer electronic spheres of atoms and molecules participate, and the amounts of energy that are involved are of the order of eV.
Complex Systems
427
dΩ
ϑ
dA
φ
FIGURE 12.8 Definition of the angles ϑ and φ used to obtain the general equation [12.7].
In equation [12.4], h =
× −
6 626 0 34
.
Js is Planck’s constant, c =
×
−
2 998 108
1
.
ms is the speed of light,
and kB =
× −
−
1 381 10 23
1
.
JK is Boltzmann’s constant.
Let us consider a cavity with walls at a uniform and constant temperature. The elementary par-
ticles, constituting the walls, move, being powered by the thermal energy. They generate electro-
magnetic waves that fill the cavity. Energy is transferred from the walls to the electromagnetic field.
On the other hand, the electromagnetic waves hit the walls and are absorbed by the particles. The
system reaches the equilibrium when the energy transferred from the walls to the electromagnetic
field is equal to that transferred from the field to the walls. To know the properties of thermal radia-
tion within the cavity, we make a tiny hole in a wall of the cavity. dA is its infinitesimal area, which
is so small that does not perturb the equilibrium between the cavity and the radiation. The thermal
radiation, which would hit the area dA, escapes from the cavity. The energy emitted per unit of time,
in the frequency range ν and ν + dν, within the solid angle dΩ, which defines an angle ϑ with the normal to the surface (see Figure 12.8), is
d
U ν
( , T )( dν ) = u ν(, T )( dν )( dAcosϑ) Ω
c
[12.5]
π
4
The intensity of radiation, in the frequency range ν and ν + dν, is
d
I ν
( , T )( dν ) = u ν(, T )( dν )( cosϑ) Ω
c
[12.6]
π
4
Since dΩ = sin(ϑ) dϑ dϕ, by integrating ϑ between 0 and π/2, and φ between 0 and 2 π, we obtain11
u ν
( , T)( dν ) c
I ν
( , T)( dν ) =
[12.7]
4
The emissivity of a body k, E (
)
k T,ν dν, is defined as the radiation intensity emitted by that
body that is at temperature T and in the frequency range ν and ν + dν. The absorptivity, ak ν
( , T ) dν,
is defined as the fraction of the incident intensity I ν
( ) dν that is absorbed by the same body at
temperature T and in the frequency range ν and ν + dν. By considering the principle of energy
conservation, Kirchhoff (1860) formulated the law that brings his name (Kirchhoff’s law):
E (
)
( ) ( )
k T,ν
= ak ν, T I ν [12.8]
The absorptivity is a dimensionless absorption coefficient, and it assumes values between 0 and 1:
0 ≤ a
( )
k (ν , T ) ≤ 1. When the body k is in equilibrium with its thermal radiation, ak ν , T is equal to 1, k
11 The integrals are ∫π /2
2
2
si ϑ
n c ϑ
os ϑ
d
cos ϑ
π
/
/
2π
ϕ
d
π
0
= (−
2)
= 1/2
0
and ∫ 0
= 2 .
428
Untangling Complex Systems
is called blackbody. For a blackbody, E (
)
k T,ν equals I( v). This is a general rule and does not depend on
the chemical composition of the body. It is worthwhile noticing that if a (
)
( )
k ν , T = 0, then Ek T,ν
= 0.
This result means that a body transparent to a frequency ν is not able to thermally emit that particular
frequency. If we transform the energy density of the thermal radiation (defined by the Planck’s formula
[12.4]) in intensity by using equation [12.7], and we integrate over the entire range of frequencies, we
obtain the total emissive power of a blackbody:
∞
2π h
3
ν dν
2 5
π k 4
E ( )
4
4
k T =
=
T = ξ T [12.9]
2
∫
c
ehν / kT −1
c
15 2 h 3
0
Equation [12.9] is known as the Stephan-Boltzmann law. The constant is ξ =
× −
5 67 10 8
.
W m2K4.
From the application of the Second Law of Thermodynamics, it is possible to infer some prop-
erties of the spectral composition of the thermal radiation. First, it must be isotropic and uniform,
independent of position, within the cavity. If this were not the case, it would be possible to transfer
energy from one place to another of the cavity. But the energy transfer is impossible because the
system is in thermal equilibrium. Second, it must be independent of the chemical composition of
the cavity walls and the shape of the cavity. If this were not the case, it would be possible to transfer
energy between two cavities having the same temperature, but differing just in the chemical com-
position and/or shape.
Thermal radiation can be conceived as a gas of photons. Is it an ideal or a real gas? What kind
of gas is it? Which are its properties? The thermal radiation, contained in a cavity, exerts a pressure
on the walls of the cavity. To define this pressure, let N(ν, T ) be the number of photons of frequency ν contained in a cubic cavity of volume V and at temperature T. The momentum of each photon is
hν
p =
[12.10]
c
When a photon hits a wall of the cavity, it transfers a momentum 2 p to the wall, every time it is
reflected.12 Since the photons move randomly, at every instant of time, 1/6 of the photons will be moving in the direction of a wall. The number of photons with frequency ν, which will collide a
wall, per second, will be N ν
( , T) c / V
6 . The total momentum transferred to the wall per unit of time
and per unit of area corresponds to the pressure exerted by the photons of frequency ν:
