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# A natural class of robust networks

^{*}The James Franck Institute and

^{‡}Institute for Biophysical Dynamics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637

^{†}To whom correspondence should be addressed. E-mail: ude.ogacihcu.lortnoc@onimixam.

## Abstract

As biological studies shift from molecular description to system analysis we need to identify the design principles of large intracellular networks. In particular, without knowing the molecular details, we want to determine how cells reliably perform essential intracellular tasks. Recent analyses of signaling pathways and regulatory transcription networks have revealed a common network architecture, termed scale-free topology. Although the structural properties of such networks have been thoroughly studied, their dynamical properties remain largely unexplored. We present a prototype for the study of dynamical systems to predict the functional robustness of intracellular networks against variations of their internal parameters. We demonstrate that the dynamical robustness of these complex networks is a direct consequence of their scale-free topology. By contrast, networks with homogeneous random topologies require fine-tuning of their internal parameters to sustain stable dynamical activity. Considering the ubiquity of scale-free networks in nature, we hypothesize that this topology is not only the result of aggregation processes such as preferential attachment; it may also be the result of evolutionary selective processes.

Complex protein and genetic networks are large systems of interacting molecular elements that control the propagation and regulation of various biological signals (1–3). They constitute essential classes of biological computations reflecting vital cellular processes such as the regulation of the cell cycle and gene expression. Because some intracellular processes are crucial for the survival of the cell they need to be achieved with reliability. For example, the variation of the concentration of network components in a metabolic network may affect numerous processes. The intricate architecture of such networks raises the question of the stability of their functioning. How can such complex dynamical systems achieve important cellular tasks and remain stable against the variations of their internal parameters?

Three decades ago, Savageau (4, 5) hypothesized that robustness was an essential property of some genetic networks whose functioning would be preserved even if some of their components were produced in various quantities. More recently, several compelling theoretical (6, 7) and experimental (8, 9) studies demonstrated that key processes of specific intracellular networks exhibited a robust behavior to variations of biochemical parameters. Robustness has emerged as a fundamental concept for the characterization of the dynamical stability of biological systems.

Are there universal design principles that would determine whether a given class of networks achieves important intracellular processing with reliability (10)? It would then be possible to predict dynamical properties underlying cell functions without a full knowledge of the molecular details. In particular, it would be important to characterize dynamical robustness as the ability for the network to perform a sequence of biological tasks in the presence of perturbations (3). Driven by such considerations, we analyzed when dynamical robustness may be a direct consequence of the network's architecture.

As a test-bed for the study of the dynamical properties of complex
biological networks we chose to model molecular activities by a simple
two-state model closely related to the one proposed by Kauffman
(11,
12). This model was originally
created for the study of the dynamics of genetic networks, but was later
applied to evolution and social models and became a prototype for the study of
dynamical systems. In this model, the network's architecture follows a random
topology in which every element interacts, on average, with *K* other
elements. Each network element has two functional states, active and inactive.
The state of a given element is determined by its interactions with other
elements of the network (see *Appendix*). A generic biochemical
parameter for the whole system is the probability ρ that an arbitrary
element of the network becomes active after interacting with other elements
(13). This probability can be
inferred in principle from the reaction rates and the relative concentration
of each element. The nature of the dynamical process performed by a given
network is determined by its topological parameter *K* and its
biochemical parameter ρ. Under the simple assumption of a random network
topology, a very rich and unexpected dynamical behavior of the network was
found (11,
12,
14). During the time evolution
of the system, the network elements pass through different states until they
reach a cyclic behavior. Different cycles are possible. Each cycle represents
a variety of intracellular tasks. Two regimes of network activity exist:
chaotic and robust. In the chaotic regime a perturbation in the state of a
single element can make the system jump from one cycle to another. In the
robust regime all such perturbations die out over time. The transition from
one regime to the other is controlled by the parameters *K* and ρ
and is determined by the equation

Fig. 1*a* defines the
two regimes of network activity with drastically different dynamical
properties. In the chaotic regime, which entirely dominates the parameter
space (*K*-ρ), small variations of the initial concentration of
only one component would ruin the network function. Conversely, the robust
regime would give rise to a reliable dynamics for which the network activity
would be insensitive to perturbations in the initial state of the network
elements. With a random network topology this regime is attained only for a
narrow range of the parameters *K* and ρ.

*a*) The network dynamics exhibit both chaotic and robust behaviors, depending on the value of the parameters

*K*and ρ. The parameter space

*K*-ρ is then divided into two distinct

**...**

Unfortunately, this model is inadequate to account for the functioning of
biological networks that have heterogeneous architecture. Such topological
heterogeneity is present in the human genome. For example, the expression of
the β-globin gene is controlled by >20 regulatory proteins. The
fibroblast or the platelet-derived growth factor activation induces the
activation of >60 other genes
(15). Also, the zinc-finger
protein Sp1 controls the expression of >300 genes
(16). Under an assumption of a
random homogeneous architecture with a mean connectivity *K* = 20,
robust behavior would be virtually attained only for ≈5% of the possible
values of ρ. Fig.
1*b* shows the fraction of the interval (0,1) for which the
parameter ρ gives rise to networks with a robust behavior. It is apparent
that for a random topology only the fine-tuning of the parameters *K*
and ρ in a narrow interval would allow networks to operate with a robust
behavior. An alternative approach is to substitute the random network topology
for a more realistic one
(17).

In recent years, the analyses of the topology of large intracellular networks have revealed a common architecture (18–20). In these natural networks not all elements are equivalent. A small, but significant, fraction of the elements are highly connected, whereas the majority of the elements are poorly connected. This architecture is called scale-free topology and was found to be ubiquitous in networks as diverse as social (21), ecological (22), and genetic, protein networks (18, 23), and the World Wide Web (24, 25). Although the effect of deleterious perturbations on the topology of these networks has been thoroughly studied (23, 26), their dynamical properties need to be explored.

In view of the ubiquity of scale-free networks in nature, we chose to
implement the scale-free topology into the two-state model. This topology is
defined such that the probability *P*(*k*) that an arbitrary
element of the networks interacts with exactly *k* other elements
follows the power law *P*(*k*) =
[*Z*(γ)*k*^{γ}]^{–}^{1}.
The parameter γ is called the scale-free exponent, and
*Z*(γ) is the normalization factor. When γ increases both
the fraction of highly connected elements and the mean connectivity of the
network decrease. In the limit γ → ∞ the network becomes a
homogeneous random network with *K* = 1.

Our approach aims to characterize how the network architecture shapes its
dynamical properties and how one particular topology favors a class of
networks with robust dynamical behavior
(10). Although the average
connectivity *K* is a relevant parameter to characterize the
homogeneous topology of random networks, it becomes irrelevant to describe the
highly heterogeneous scale-free topology. Thus, the only relevant parameter
that characterizes the architecture of scale-free networks is the scale-free
exponent γ.

For networks with scale-free topology, the transition from chaotic to
robust dynamics is determined by the equation (see *Appendix*)

The values of γ and ρ that satisfy this transcendental equation are
plotted on Fig. 2*a*.
The results of our model reported in the diagram reveal a large class of
robust networks. This class is defined for networks with topological
parameters γ > 2. These networks can operate with a robust dynamical
behavior for which the mean biochemical parameter ρ spans over a large
range of values. In particular, networks with a parameter γ > 2.5
would display a robust dynamics for any value of ρ. The transition from
the chaotic to the robust regimes occurs for values of γ in the interval
[2,2.5] and different values of ρ. Fig.
2*a* reveals another class of networks with a topological
parameter γ < 2 that exhibit chaotic behavior. This dynamical
behavior is characterized by an extreme sensitivity to small variations of the
initial states of the network elements. Our model shows that the emergence of
a large class of robust networks is a direct consequence of the network
architecture.

*a*) The mean connectivity

*K*is not a relevant parameter to characterize the network topology for a scale-free network. The dynamics is then characterized in terms of the parameters

**...**

It is interesting to note that experimental values of γ, extracted
from real intracellular networks, range in the interval [2, 2.5]. To
illustrate this general remark, we indiscriminately report on a histogram 46
published values of scale-free exponents
(Fig. 2*c*). These
values were obtained not only from real biological systems but also from many
other scale-free networks (20,
23,
25). Remarkably, the majority
of these exponents fall in the predicted interval where the transition from
chaotic to robust behavior occurs.

This observation leads us to explore the dynamical response of the network
to perturbations in the three different regimes: chaotic, critical, and
robust. In a scale-free network not all elements are equivalent. We do not
expect that perturbing highly connected elements have the same effect on the
network dynamics than perturbing poorly connected elements. To test this idea
we analyzed numerically how a perturbation produced on only one element
affects the dynamics of the network. We computed the overlap (see
*Appendix*) between two trajectories. One trajectory is computed with
one element σ_{i} with *k*_{i}
connections being perturbed at every time step of the temporal evolution of
the network. The second trajectory is computed from the same initial condition
but with no perturbation. Such overlap is a measure of the dynamical stability
of the system under sustained perturbations.
Fig. 3 shows that the stability
of the network decreases with the connectivity *k*_{i}
of the perturbed element σ_{i}. Remarkably, we found
that networks operating in the robust regime are sensitive to perturbations
applied to the highly connected elements. In contrast, perturbations produced
on arbitrary elements, which in general are poorly connected, will have little
effect on the network functioning. The heterogeneity of scale-free networks
operating in the robust regime allows different dynamical responses to
perturbations depending on the particular element being perturbed. This
behavior cannot be observed in homogeneous random networks in which all of the
elements are equivalent.

_{i}with

**...**

For a homogeneous network topology, stable behavior was achieved only for
very restricted values of the parameters ρ and *K*. Conversely, the
scale-free topology does not require a fine-tuning of the system parameters to
attain robustness. This topology allows the existence of robust dynamics in
heterogeneous networks. Furthermore, the dynamics of scale-free networks can
change if the highly connected elements are perturbed. Thus, scale-free
networks operating in the robust regime exhibit both, robustness and
sensitivity to perturbations. The above results suggest that the dynamical
robustness of a wide variety of biological processes may be a direct
consequence of the network architecture. The ubiquity of scale-free networks
in nature with a topological parameter 2 < γ < 3 could be the
result of evolutionary processes
(27). Such a large class of
networks, whose key dynamical processes are robust, may have an essential
evolutionary advantage in possessing a larger parameter space to adapt to new
environmental conditions (27,
28). In our opinion, it would
be surprising if nature did not exploit this convenient dynamical property of
the scale-free topology.

This work describes the dynamics of networks where complex molecular activities are idealized with a two-state model (29). There exists a fair number of intracellular regulatory systems with components obeying the rules of a two-state model. For example, in signal transduction networks, the signal is processed through phosphorylation cascades where signaling molecules can be active or inactive upon phosphorylation. It is also common to describe regulatory transcription networks in this simple language: A transcription factor regulates positively or negatively the transcription of a given gene. Moreover, available data on the activity of molecular components from large intracellular networks are produced by high-throughput experiments such as two-hybrid assays and C-DNA microarrays. Because of technical constraints, the readout of these experiments is fashioned accordingly to a two-state format that facilitates the applicability of our approach.

The identification of a relationship between topology and dynamical properties of large biological networks primarily motivated our work. But it seems appropriate to think that in view of the significance of scale-free networks in other fields such as sociology and economics our results could be of some interest beyond the scope of biology.

## Acknowledgments

We thank Leo P. Kadanoff, Leo Silbert, Tao Pan, Tamara Griggs, and Calin Guet for useful comments. This work was supported by the Materials Research Science and Engineering Center program of the National Science Foundation under Award 9808595 and National Science Foundation Grant DMR 0094569. M.A. also acknowledges the Santa Fe Institute of Complex Systems for partial support through the David and Lucile Packard Foundation Program in the Study of Robustness.

## Appendix

A Boolean network is represented by a set of *N* elements,
{σ_{1}, σ_{2},..., σ_{N}}.
Each element has two possible states: active (1) or inactive (0). The value of
each σ_{i} is controlled by
*k*_{i} other elements of the network (see
Fig. 4), where
*k*_{i} is a random variable chosen from a probability
distribution *P*(*k*). This probability distribution determines
the topology of the network. We define *K* as the mean value of
*P*(*k*). Let {σ_{i}_{1},
σ_{i}_{2},...,
σ_{i}_{ki}} be the set of controlling
elements of σ_{i}. We then assign to each
σ_{i} a Boolean function
*f*_{i}(σ_{i}_{1},
σ_{i}_{2},...,
σ_{i}_{ki}). For each configuration of
the controlling elements,
*f*_{i}(σ_{i}_{1},
σ_{i}_{2},...,
σ_{i}_{ki}) = 1 with probability ρ
and *f*_{i}(σ_{i}_{1},
σ_{i}_{2},...,
σ_{i}_{ki}) = 0 with probability 1
– ρ. Once the controlling elements and the Boolean functions have
been assigned to every element in the network, the dynamics of the system are
given by

We will denote as Σ(*t*) the configuration of the entire system
at time *t*: Σ(*t*) = {σ_{1}(*t*),
σ_{2}(*t*),...,
σ_{N}(*t*)}.

The dynamical robustness of the network is analyzed by considering the
trajectories Σ(0) → Σ(1) → · · ·
→ Σ(*t*) and
→
→ · ·
· → produced by
two slightly different initial configurations, Σ(0) and
, respectively. These
trajectories can always be different under the temporal evolution of the
system, or they can eventually converge to the same trajectory. An important
quantity that reveals the robustness of the dynamics against perturbations in
the initial configuration is the overlap between Σ(*t*) =
{σ_{1}(*t*), σ_{2}(*t*),...,
σ_{N}(*t*)} and
,
defined as

where · represents the average over all possible initial
configurations and network realizations. If
lim_{t}_{→}_{∞}
*x*(*t*) = 1, all perturbations in a given initial configuration
die out over time (robust behavior). In contrast, if
lim_{t}_{→}_{∞}
*x*(*t*) ≠ 1 even small perturbations in the initial
configuration propagate across the entire system and never disappear (chaotic
behavior).

It has been shown in ref.
17 that for a large system
(*N* → ∞), the temporal evolution of the overlap is given by

In the limit *t* → ∞ the above equation becomes a
fixed-point equation for the stationary value of the overlap *x* =
lim_{t}_{→}_{∞}*x*(*t*).
It follows from Eq. **A.2** that if 2ρ(1 – ρ)*K* ≤
1, the only stable fixed point is *x* = 1 [*K* is mean value of
*P*(*k*)]. In this case the system is in the robust regime. On
the other hand, if 2ρ(1 – ρ)*K* > 1 there is a stable
fixed point *x* ≠ 1 and the dynamics of the network are chaotic. The
critical value *K*_{c} of the mean connectivity at
which the transition from robust to chaotic dynamics occurs is given as a
function of ρ by the equation

The values of *K*_{c} and ρ that satisfy this
equation are plotted on Fig. 1.
It is interesting to note that the transition from robust to chaotic dynamics
is governed by *K*, the mean value of *P*(*k*). However,
this parameter is meaningful only when the distribution *P*(*k*)
has a well-defined variance. In this case the fluctuations in the number of
connections of the individual elements around the mean connectivity *K*
are bounded. This is actually the case when *P*(*k*) is a
Poissonian or a Gaussian distribution. For these distributions the mean
connectivity *K* is the relevant parameter that characterizes the
network topology. Originally Kauffman
(11,
12) proposed this model under
the assumption that *K* is a relevant parameter. However, the variance
of the scale-free distribution *P*(*k*) =
[*Z*(γ)*k*^{γ}]^{–}^{1}
is infinite for γ ≤ 3. This means that the actual number of
connections of the individual elements can vary from 1 up to *N*.
Consequently, the mean value of the connectivity is no longer a meaningful
parameter to characterize the network topology. For a highly heterogeneous
network with scale-free topology, the only relevant parameter that
characterizes the network topology is the scale-free exponent γ itself.
For these kinds of networks it is better to represent the mean connectivity
*K* as a function of the scale-free exponent γ, which gives
*K* = *Z*(γ – 1)/*Z*(γ). In the above
expression
is the Riemann Zeta function. With this representation, the transition from
robust to chaotic behavior is then given by

The values of γ and ρ for which this transcendental equation is satisfied are plotted on Fig. 2.

The network that we have considered is a directed graph. The value of each
element σ_{i} is controlled by a set of
*k*_{i} elements. But σ_{i} can
in turn control the values of a number of other elements. Therefore, each
element has a set of input connections and a set of output connections. The
number of inputs and outputs of each element are not necessarily distributed
with the same probability distribution *P*(*k*). However, the
total number of inputs in the network equals the total number of outputs, so
that on average, every element has the same number of inputs than outputs.
Because the phase transition is governed only by the first moment of
*P*(*k*), it follows that Eq. **A.4** is valid if either the
distribution of input connections or that of the output connections (or both)
is scale free.

## Notes

This paper was submitted directly (Track II) to the PNAS office.

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**National Academy of Sciences**

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- A natural class of robust networksA natural class of robust networksProceedings of the National Academy of Sciences of the United States of America. Jul 22, 2003; 100(15)8710PMC

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