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# A model for the emergence of cooperation, interdependence, and structure in evolving networks

^{*}Centre for Theoretical Studies and

^{¶}Physics Department, Indian Institute of Science, Bangalore 560 012, India;

^{†}Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501; and

^{‡}Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560 064, India

^{§}To whom reprint requests should be addressed. E-mail: ude.efatnas@niaj, ni.tenre.csii.stc@niaj, or ni.tenre.csii.scisyhp@peednas.

## Abstract

Evolution produces complex and structured networks of interacting components in chemical, biological, and social systems. We describe a simple mathematical model for the evolution of an idealized chemical system to study how a network of cooperative molecular species arises and evolves to become more complex and structured. The network is modeled by a directed weighted graph whose positive and negative links represent “catalytic” and “inhibitory” interactions among the molecular species, and which evolves as the least populated species (typically those that go extinct) are replaced by new ones. A small autocatalytic set, appearing by chance, provides the seed for the spontaneous growth of connectivity and cooperation in the graph. A highly structured chemical organization arises inevitably as the autocatalytic set enlarges and percolates through the network in a short analytically determined timescale. This self organization does not require the presence of self-replicating species. The network also exhibits catastrophes over long timescales triggered by the chance elimination of “keystone” species, followed by recoveries.

Structured networks of interacting components are a hallmark of several complex systems, for example, the chemical network of molecular species in cells (1), the web of interdependent biological species in ecosystems (2, 3), and social and economic networks of interacting agents in societies (4–7). The structure of these networks is a product of evolution, shaped partly by the environment and physical constraints and partly by the population (or other) dynamics in the system. For example, imagine a pond on the prebiotic earth containing a set of interacting molecular species with some concentrations. The interactions among the species in the pond affect how the populations evolve with time. If a population goes to zero, or if new molecular species enter the pond from the environment (through storms, floods, or tides), the effective chemical network existing in the pond changes. We discuss a mathematical model that attempts to incorporate this interplay between a network, populations, and the environment in a simple and idealized fashion. The model [including an earlier version (8, 9)] was inspired by the ideas and results in refs. 10–18. Related but different models are studied in refs. 19–21.

## The Model

The system consists of *s* species labeled by the
index *i* = 1,2,… ,*s.* The
network of interactions between species is specified by the
*s* × *s* real matrix
*C* {*c*_{ij}}.
The network can be visualized as a directed graph whose nodes represent
the species. A nonzero *c*_{ij} is
represented by a directed weighted link from node *j* to
node *i*. If *c*_{ij} > 0,
then the corresponding link is a cooperative link: species
*j* catalyzes the production of species *i*.
If *c*_{ij} < 0, it is a destructive link:
the presence of *j* causes a depletion of *i*
(22).

### Population Dynamics.

The model contains another dynamical variable **x**
(*x*_{1},….
*x _{s}*), where

*x*stands for the relative population of the

_{i}*i*th species (0 ≤

*x*

_{i}≤ 1, ∑

_{i=1}

^{s}

*x*

_{i}= 1)). The time evolution of

**x**depends on the interaction coefficients

*C*, as is usual in population models. The specific evolution rule we consider is

where

This is a particularly simple idealization of catalyzed chemical
reaction dynamics in a well stirred reactor (representing, say, a
prebiotic pond). It is motivated from the following considerations: If
species *j* catalyzes the ligation of reactants *A*
and *B* to form the species *i, *, then the rate of growth of the
population *y _{i}* of species

*i*in a well stirred reactor will be given by

*(1 + ν*

_{i}= k*y*

_{j})

*n*

_{A}

*n*

_{B}−

*y*

_{i}, where

*n*,

_{A}*n*are reactant concentrations,

_{B}*k*is the rate constant for the spontaneous reaction, ν is the catalytic efficiency, and represents a common death rate or dilution flux in the reactor (23). Assuming the catalyzed reaction is much faster than the spontaneous reaction, and the concentrations of the reactants are large and fixed, the rate equation becomes

_{i}=

*cy*

_{j}−

*y*

_{i}, where

*c*is a constant. If species

*i*has multiple catalysts, we get

_{i}= ∑

_{j}

^{s}

*c*

_{ij}

*y*

_{j}−

*y*

_{i}. The first of Eqs. 1 follows from this on using the definition

*x*

_{i}=

*y*

_{i}/∑

_{j=1}

^{s}

*y*

_{j}. Note that the second (quadratic) term in

*f*follows automatically from the

_{i}_{i}equation and the nonlinear relationship between

*x*and

_{i}*y*. Physically, it is needed to preserve the normalization of the

_{i}*x*under time evolution. When negative links are permitted, the second of Eqs. 1 is needed in general to prevent relative populations from going negative. (With negative links, a more realistic chemical interpretation would be obtained if

_{i}_{i}were proportional to

*x*, but for simplicity we retain the form of Eq. 1 in this paper.) Eq. 1 may be viewed as defining an artificial chemistry in the spirit of refs. 13–17.

_{i}### Graph Dynamics.

The dynamics of* C* in turn depends on **x**, as
follows: Start with a random graph of *s* nodes:
*c _{ij}* is nonzero with probability

*p*and zero with probability 1 −

*p*. If nonzero,

*c*is chosen randomly in the interval [−1, 1] for

_{ij}*i*≠

*j*and [−1, 0] for

*i*=

*j*. Thus a link between two distinct species, if it exists, is just as likely to be cooperative as destructive, and a link from a species to itself can only be inhibitive, i.e., autocatalytic or self-replicating individual species are not allowed. The variable

**x**is initialized by choosing each

*x*randomly between 0 and 1 and then rescaling all

_{i}*x*uniformly such that ∑

_{i}_{i=1}

^{s}

*x*

_{i}= 1. The evolution of the network proceeds in three steps:

(*i*) Keeping the network fixed, the populations are evolved
according to Eq. 1 for a time* T*, which is large
enough for **x** to get reasonably close to its attractor. We
denote *X _{i}*

*x*(

_{i}*T*).

(*ii*) The set of nodes *i* with the least value of
*X _{i}* is determined. We call this the set of
“least fit” nodes, identifying the relative population of a
species in the attractor (or, more specifically, at

*T*) with its “fitness” in the environment defined by the graph. One of the least fit nodes is chosen randomly (say

*i*

_{0}) and removed from the system along with all its links, leaving a graph of

*s*− 1 species.

(*iii*) A new node is added to the graph so that it again has
*s* nodes. The links of the added node
(*c*_{ii0} and
, for
*i* = 1, … , *s*) are assigned randomly
according to the same rule as for the nodes in the initial graph. The
new species is given a small relative population
=
*x*_{0}, and the other populations are
rescaled to keep ∑_{i=1}^{s} *x*_{i} = 1.
This process, from step *i* onwards, is iterated many times.

The rules for the evolution of the network *C* are intended to
capture two key features of natural evolution, namely selection and
novelty. The species that has the least population in the attractor
configuration is the one most likely to be eliminated in a large
fluctuation in a possible hostile environment. Often, the least value
of *X _{i}* is zero. Thus the model implements
selection by eliminating from the network a species that has become
extinct or has the least chance of survival (18). [Relaxing in various
ways the assumption (18) that only the least populated species is
removed does not change the qualitative picture presented here; details
on the robustness of the model to various deformations will be
presented elsewhere.] Novelty is introduced in the network in the form
of a new species. This species has on average the same connectivity as
the initial set of species, but its actual connections with the
existing set are drawn randomly. E.g., if a storm brings into a
prebiotic pond a new molecular species from the environment, the new
species might be statistically similar to the one being eliminated, but
its actual catalytic and inhibitory interactions with the surviving
species can be quite different. Another common feature of natural
evolution is that populations typically evolve on a fast timescale
compared with the network. This is captured in the model by having the

*x*relax to their attractor before the network is updated. The idealization of a fixed total number of species

_{i}*s*is one that we hope to relax in future work.

The model described above differs from the one studied in refs. 8 and 9
in that it allows negative links and varying link strengths, and that
the population dynamics, given by Eq. 1 is no longer linear.
The earlier model had only fixed point attractors; here limit cycles
are also observed. Because* C* now has negative entries, the
formalism of nonnegative matrices no longer applies.

## Results

### Emergence of Cooperation and Interdependence.

Fig. Fig.11 shows a sample run. The same
qualitative behavior is seen in each of several hundred runs performed
for *p* values ranging from 0.00002 to 0.01 and for
*s* = 100, 150, 200. That the ratio of number of
cooperative to destructive links at first remains constant at unity
(statistically) and then increases by more than an order of magnitude
is evidence of the emergence of cooperation. Fig. Fig.11 also shows how a
measure of the mutual interdependence of the species changes with time.
This measure, “interdependency,” denoted
, is defined as
(1/s)∑_{i=1}^{s} *d*_{i}, where
*d _{i}* is the “dependency” of the

*i*th node.

*d*

_{i}∑

_{kj}|

*c*

_{kj}|

*h*

_{k}

^{i}, where

*h*

_{k}

^{i}is 1 if there exists a directed path from

*k*to

*i*and 0 otherwise.

*d*is the sum of (the absolute value of) the strengths of all links that eventually feed into

_{i}*i*along some directed path.

*d*describes not just the character of the “neighborhood” of the

_{i}*i*th species but also the long-range connections that affect its dynamics. The increase in by an order of magnitude is a quantitative measure of the increase of interdependence of species in the network. The increase in the total density of links (

*l*

_{+}+

*l*

_{−})/

*s*is another aspect of the increase of complexity of the system. Note that in the model selection rewards only “performance” as measured in terms of relative population; the rules do not select for higher cooperativity

*per se*. Because a new species is equally likely to have positive or negative links with other species, the introduction of novelty is also not biased in favor of cooperativity. That this behavior is not a consequence of any intrinsic bias in the model that favors the increase of cooperation and interdependence is evidenced by the flat initial region of all of the curves.

### Autocatalytic Sets.

The explanation for the above behavior lies in the formation and growth
of certain structures, autocatalytic sets (ACSs), in the graph. An ACS
is defined as a set of nodes such that each node has at least one
incoming positive link from a node in the set. Thus an ACS has the
property of catalytic closure, i.e., it contains a catalyst for each of
its members (24–26). The simplest example of an ACS is a cycle of
positive links. Every ACS is not such a cycle but it can be shown that
an ACS must contain a cycle of positive links (9). In Fig. Fig.1,1, there is
no ACS in the graph until *n* = 1,903. A small ACS (which
happens to be a cycle of positive links between two nodes) appears at
*n* *n*_{1} = 1,904,
exactly where the behavior of the *s*_{1}
curve changes. As time proceeds, this ACS becomes more complex and
enlarges until at *n*
*n*_{2} = 3,643, the entire graph becomes
an ACS. *l*_{+} and
exhibit an increase and
*l*_{−} a decrease as the ACS comes to
occupy a significant part of the graph. After the ACS first appears (at
*n* = *n*_{1}), the set of
populated nodes in the attractor configuration
(*s*_{1} in number) is always an ACS
(except for certain catastrophic events to be discussed later), which
we call the “dominant ACS.” The spontaneous appearance of a small
ACS at some *n* = *n*_{1},
its persistence (except for catastrophes), and its growth until it
spans the graph at *n* =
*n*_{2}, are seen in each of the several
hundred runs mentioned earlier. The growth of the ACS across the graph
between *n*_{1} and
*n*_{2} occurs exponentially (with
stochastic fluctuations),

This expression (derived below) agrees with simulations as shown
in Fig. Fig.2.2. The average timescale
τ_{a}*
**n*_{1} for the first appearance
of the ACS is given, for sufficiently small *p*, by
τ_{a} ≈
4/(*p*^{2}*s*) (=1,600 for
*p* = 0.005 and *s* = 100). This follows
from the fact that the probability that a graph not containing an ACS
will acquire a 2 cycle of positive links at the next update is
*p*^{2}*s*/4, with larger cycles
being much less likely to appear when *ps* 1.

_{g}on

*p*. Each data point shows the average of τ

_{g}over 5 different runs with

*s*= 100 and the given

*p*value. The error bars correspond to one standard deviation. The best fit line has slope −1.02 ±

**...**

Up to *n* = *n*_{1}, the
graph has no ACS. It has chains and trees of positive and negative
links and possibly loops containing negative links. These latter
structures are not robust. For example, consider a chain of two
positive links 1→2→3. Because catalytic links are pointing to node
3, it will do well populationally compared with nodes 1 and 2. However,
because 1 has no incoming catalytic links, its relative population will
decline to zero under Eq. 1, and it can be picked for
replacement in the next graph update. This can disrupt the chain and
hence erode the “well being” of node 3 until eventually, after
some graph updates, the latter can also join the ranks of the least
fit. Species 3 gets eliminated eventually because it does not feed back
into and “protect” species 1 and 2, on whom its “well
being” depends. In a graph without an ACS, no structure is protected
from disruption. Because every node is liable to be replaced sooner or
later, the graph remains as random as the initial graph (we have
checked that the probability distribution of the number of incoming and
outgoing links at a node remains the appropriate binomial for* n* < *n*_{1}). This explains why
*s*_{1},
*l*_{±}, and
hover around their initial values.

The picture changes the moment a small ACS appears in the graph. The
key point is that by virtue of catalytic closure, members of the ACS do
well *collectively* in the population dynamics governed by Eq.
1. An ACS is a collective self replicator and beats chains,
trees, and other non-ACS structures in the population game, reducing
their* X _{i}* to zero when it appears. Thus,
because graph update proceeds by replacing one of the nodes with

*X*= 0 (if present) with a new one, such a replacement being outside the dominant ACS can cause no damage to the links that constitute the ACS. That is why the ACS structure, once it appears, is much more robust than the non-ACS structures discussed earlier. If the new node happens to get an incoming positive link from the dominant ACS, it becomes part of it. Thus the dominant ACS tends to expand in the graph as new nodes get attached to it (8, 9, 15), and

_{i}*s*

_{1}increases. In Δ

*n*graph updates, the average increase in

*s*

_{1}, which is the number of added nodes that will get a positive link from one of the

*s*

_{1}nodes of the dominant ACS, is Δ

*s*

_{1}≈ (

*p*/2)

*s*

_{1}Δ

*n*, for small

*p*. This proves Eq. 2. (Note that the exponential growth described by Eq. 2 is not to be confused with the exponential growth of populations

*y*of species that are part of the ACS. Eq. 2 reflects the growth of the ACS across the graph or the increase in the number of species that constitute the ACS.)

_{i}Because the dominant ACS grows by adding positive links from the existing dominant ACS, the number of positive links increases as the ACS grows. On the other hand, nodes receiving negative links usually end up being least fit, hence negative links get removed when these nodes are eliminated. Which novelty is captured thus depends on the existing “context”; the network evolves by preferentially capturing links and nodes that “latch on” cooperatively to the existing ACS and by disregarding those that do not. The “context” itself arises when the ACS structure first appears; this event transforms the nature of network evolution from random to “purposeful” (in this case directed toward increasing cooperation). Before the ACS appears, nothing interesting happens even though selection is operative (the least populated species are being eliminated). It is only after the ACS topological structure appears that selection for cooperation and complexity begins. Initially the ACS is small, and its impact on links and interdependency is not visible. As it grows and comes to occupy a significant part of the graph, the latter quantities depart significantly from their initial random graph values.

### Inevitability of Autocatalytic Sets.

Note that the appearance of an ACS, although a chance event, is
inevitable. For *sp* 1, the probability that a graph not
containing a 2 cycle will acquire one at the next time step is
*p*^{2}*s*/4
*q*. Because the probability of occurrence of 3 cycles, etc.,
is much smaller, the probability distribution of arrival times
*n*_{1} is approximated by
*P*(*n*_{1}) = ,
whose mean τ_{a} is 1/*q*.
Because this probability declines exponentially after a timescale
1/*q*, the appearance of an ACS is inevitable, even for
arbitrarily small (but finite) *p*.

Occasionally in a graph update, *s*_{1} can
decrease for various reasons. If the new node forms an ACS of its own
with nodes outside the dominant ACS, and the new ACS has a higher
population growth rate (as determined by Eq. 1) than the old
ACS, it drives the species of the latter to extinction and becomes the
new dominant ACS. Alternatively, the new node could be a
“destructive parasite:” it receives one or more positive links
from and gives one or more negative links to the dominant ACS. Then
part or whole of the ACS may join the set of least-fit nodes.
Structures that diminish the size of the dominant ACS or destroy it
appear rarely. For example, in Fig. Fig.1,1, destructive parasites appeared 6
times at* n* = 3,388, 3,478, 3,576, 3,579, 3,592, and
3,613. In each case, *s*_{1} decreased by
1.

### Emergence of Structure.

At *n* = *n*_{2}, the whole
graph becomes an ACS; the entire system can collectively self replicate
despite the explicit absence of individual self replicators. Such a
fully autocatalytic set is a very nonrandom structure. Consider a graph
of *s* nodes and let the probability of a positive link
existing between any pair of nodes be *p**. Such a graph has
on average *m** = *p**(*s* − 1)
incoming or outgoing positive links per node. For the entire graph to
be an ACS, each node must have at least one incoming positive link,
i.e., each row of the matrix *C* must contain at least one
positive element. Hence the probability, *P*, for the entire
random graph to be an ACS is

*P* =probability that every row has at
least one positive entry =[probability that a row has at least
one positive entry]^{s} =[1 − (probability that every
entry of a row is ≤ 0)]^{s} =[1 − (1 −
*p**)^{s−}^{1}]^{s} =[1 − (1 −
*m**/(*s* − 1))^{s−}^{1}]^{s}.

For large *s* and *m** ~
*O*(1),

where α is positive and *O*(1). At *n* =
*n*_{2}, we find in all our runs that
*l*_{+}(*n*_{2})
*l** is greater than *s* but of order
*s*, i.e., *m** ≈ *O*(1). Thus dynamical
evolution in the model via the ACS mechanism converts a random
organization into a highly structured one that is exponentially
unlikely to appear by chance. In the displayed run at *n*
= *n*_{2}, the graph had 117 positive
links. The probability that a random graph with *s* = 100
nodes and *m** = 1.17 would be an ACS is given by Eq.
3 to be
≈10^{−}^{16}.

Such a structure would take an exponentially long time to arise
by pure chance. The reason it arises inevitably in a short timescale in
the present model is the following: a *small* ACS can appear
by chance quite readily and, once appeared, it grows exponentially fast
across the graph by the mechanism outlined earlier. The dynamical
appearance of such a structure may be regarded as the emergence of
“organizational order.” The appearance of “exponentially
unlikely” structures in the prebiotic context has been a puzzle.
That in the present model such structures inevitably form in a short
time may be relevant for the origin of life problem.

### The Self-Organization Timescale in a Prebiotic Scenario.

We now speculate on a possible application to prebiotic chemical
evolution. Imagine the molecular species to be small peptide chains
with weak catalytic activity in a prebiotic pond alluded to earlier.
The pond periodically receives an influx of new molecular species being
randomly generated elsewhere in the environment through tides, storms,
or floods. Between these influxes of novelty, the pond behaves as a
well stirred reactor where the populations of existent molecular
species evolve according to (a realistic version of) Eq. 1
and reach their attractor configuration. Under the assumption that the
present model captures what happens in such a pond, the growth
timescale (Eq. 2) for a highly structured almost fully
autocatalytic chemical organization in the pond is
τ_{g} = 2/*p* in units of the graph
update time step. In this scenario, the latter time unit corresponds to
the periodicity of the influx of new molecular species, hence it ranges
from 1 day (for tides) to 1 year (for floods). Further, in the present
model *p*/2 is the probability that a random small peptide
will catalyze the production of another (26), and this has been
estimated in ref. 12 as being in the range
10^{−}^{5}–10^{−}^{10}.
With *p*/2 ≈
10^{−}^{8}, for example, the
timescale for a highly structured chemical organization to grow in the
pond would be estimated to be of the order of
10^{6}–10^{8} years. It is
believed that life originated on Earth in a few hundred million years
after the oceans condensed. These considerations suggest that it might
be worthwhile to empirically pin down the “catalytic probability”
*p* (introduced in ref. 26) for peptides, catalytic RNA,
lipids, etc., on the one hand, and explore chemically more realistic
models on the other.

### Catastrophes and Recoveries in Network Dynamics.

After *n* = *n*_{2}, the
character of the network evolution changes again. For the first time,
the least-fit node will be one of the ACS members. Most of the time
elimination of the node does not affect the ACS significantly, and
*s*_{1} fluctuates between *s* and
*s* − 1. Sometimes the least-fit node could be a
“keystone” species, which plays an important organizational role
in the network despite its low population. When such a node is
eliminated, many other nodes can get disconnected from the ACS,
resulting in large dips in *s*_{1} and
and subsequently large fluctuations in
*l*_{+} and
*l*_{−}. These large “extinction
events” can be seen in Fig. Fig.3.
3.
Occasionally, the ACS can even be destroyed completely. The system
recovers on the timescale τ_{g} after large
extinctions if the ACS is not completely destroyed; if it is, and the
next few updates obliterate the memory of previous structures in the
graph, then again a time on average τ_{a} elapses
before an ACS arises, and the self-organization process begins anew. It
may be of interest (especially in ecology, economics, and finance) that
network dynamics based on a fitness selection and the
“incremental” introduction of novelty, as discussed here, can by
itself cause catastrophic events without the presence of large external
perturbations.

## Discussion

We have described an evolutionary model in which the dynamics of species' populations (fast variables) and the graph of interactions among them (slow variables) are mutually coupled. The network dynamics displays self organization seeded by the chance but inevitable appearance of a small cooperative structure, namely an ACS. In a dynamics that penalizes species for low population performance, the collective cooperativity of the ACS members makes the set relatively robust against disruption. New species that happen to latch on cooperatively to this structure preferentially survive, further enlarging the ACS in the process. Eventually the graph acquires a highly nonrandom structure. We have discussed the time evolution of quantitative measures of cooperation, interdependence, and structure of the network, which capture various aspects of the complexity of the system.

It is noteworthy that collectively replicating ACSs arise even though individual species are not self replicating. Thus the present mechanism is different from the hypercycle (27), where a template is needed to produce copies of existing species. Unlike the hypercycle, the ACS is not disrupted by parasites and short circuits and grows in complexity, as evidenced in all our runs. It can be disrupted, however, when it loses a “keystone” species.

It is also worth mentioning one departure from ref. 12, in that we find
that a fully autocatalytic system (or percolating ACS) is not needed
*a priori* for self organization. In the present model, a
small ACS, once formed, typically expands (see also ref. 15) and
eventually percolates the whole network dynamically. This dynamical
process might be relevant for economic takeoff and technological growth
in societies.

## Acknowledgments

We thank J. D. Farmer, W. C. Saslaw, and anonymous referees for helpful comments on the manuscript. S.J. acknowledges the Associateship of the Abdus Salam International Centre for Theoretical Physics.

## Abbreviation

- ACS
- autocatalytic set

## Footnotes

Article published online before print: *Proc. Natl. Acad. Sci. USA*,
10.1073/pnas.021545098.

Article and publication date are at www.pnas.org/cgi/doi/10.1073/pnas.021545098

## References

**National Academy of Sciences**

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