1) The document describes an evolving network model and a modified version that incorporates preferential attachment. In the original model by Jain and Krishna, crashes often occurred due to core-shifts or complete crashes as the network evolved. 2) The modified model with preferential attachment leads to faster formation of the first autocatalytic set and transition to an organized phase. It also makes crashes extremely rare. 3) Networks in the organized phase of the preferential attachment model have denser cores with more fundamental loops, resulting in higher robustness against crashes compared to the original model.

1. Preferential attachment renders an
evolving network of populations robust
against crashes
Areejit Samal
Max Planck Institute for Mathematics in the Sciences
Inselstr. 22 04103 Leipzig Germany
Email: samal@mis.mpg.de

2. Outline
 Jain and Krishna (JK) evolving network model
 Interesting features of the model
 Structural changes leading to crashes in the model
 Modified model with preferential attachment scheme
 Results

3. ( Jain and Krishna, PRL 1998, PNAS 2001, PNAS 2002 )
100-500 million years
Jain and Krishna (JK) model is motivated by the
origin of life problem
The model tries to address some of the puzzles behind the emergence of prebiotic chemical
organization.

4. Variables of the model
1ijc =
1ijc =1
2
4
8
5
3
7
6
A graph of interacting molecular species
An arrow from node j to i implies that j is a catalyst for
the production of i, and then
The absence of an arrow from j to i implies that
0
The s x s matrix C = (cij ) is the adjacency matrix of the
graph
s is the number of molecular species
Each species i has a population yi or a relative population xi
The variables x and C characterize the chemical organization in the pond and
they change with time.

5. Dynamical rules
Initialization:
Cij = 1 with probability p,
= 0 with probability 1-p
p is the “catalytic
probability”.
xi are chosen randomly.
Relative population of new node is
set to x0, a small constant.
All other xi are perturbed randomly.
x → X: Attractor
C fixed
Population Dynamics
(step 1)
( Jain and Krishna, PRL 1998, PNAS 2001, PNAS 2002 )

6. Population Dynamics
,
( , )i i ij j i jk k
j j k
x f x c c x x c x
•
= = −∑ ∑
= population of species i
ix = relative population of species i =
iy
Ywhere i
i
Y y= ∑
0 1ix≤ ≤Therefore, and
1
1
s
i
i
x
=
=∑
1ijc = implies i j¬ ⇔ j is a catalyst for the production of i
.
Interpretation of the chemical rate equation
i j iy Ky yφ= −
.
how efficient j is in catalyzing
i, reactant concentrations etc.
dilution flux
(2)
Eq. (1) implies Eq. (2). Note that Eq. (2) does not contain φ.
iy
Y1
s
i ij j i
j
y c y yφ
=
= −∑
.
(1)

7. Auto Catalytic Set (ACS)
An ACS is a subgraph, each of whose nodes has at least one incoming link
from a node belonging to the same subgraph.
This definition of ACS was introduced in the context of a set of catalytically
interacting molecules where it was defined to be a set of molecular species
that contains within itself a catalyst for each of the member species.
Examples of graph structures that are ACS
Sandeep Krishna, PhD Thesis (2003)

8. Core and Periphery of an ACS
An ACS can be further subdivided into core and periphery.
The core is the irreducible subgraph of the ACS, i.e., every node in the core
has access to every other node in the directed subgraph.
The periphery consists of nodes which can be reached from the core nodes
while the core nodes cannot be reached from the periphery nodes in the
directed subgraph.
In the ACS shown below, nodes 1 and 2 form the core and node 3 forms the
periphery.
Sandeep Krishna, PhD Thesis (2003)

9. Three phases during network evolution
There are three phases observed during the graph evolution
process:
 Random phase
 Growth phase
 Organized phase

10. Random phase
n = 1 n = 78 n = 2853
The random phase is characterized by a graph where there is no ACS. The
onset of the growth phase is triggered by the appearance of the first ACS
in the graph.
Sandeep Krishna, PhD Thesis (2003)

11. Growth phase
n = 2854 n = 3022 n = 3386
The growth phase starts with the appearance of the first ACS in the graph.
The growth phase is characterized by a dramatic increase in the number of
links in the graph with the ACS accreting more and more nodes into it.
Sandeep Krishna, PhD Thesis (2003)

12. n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
NumberofLinks
0
20
40
60
80
100
120
140
160
n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Numberofpopulatednodes(s
1
)
0
20
40
60
80
100
120
n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
λ
1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
The creation of the first ACS is
marked by the increase in the
Perron-Frobenius eigenvalue to a
value larger than or equal to 1. In
the growth phase, the number of
links in the graph increases
dramatically.

13. Organized phase
n = 3880 n = 4448 n = 5041
The growth phase culminates with the ACS spanning the whole graph. This
marks the start of the organized phase where at least s-1 nodes are
populated.
The system remains in the organized phase until a crash occurs after
which the system ends in the growth or the random phase.
Sandeep Krishna, PhD Thesis (2003)

14. n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
NumberofLinks
0
20
40
60
80
100
120
140
160
n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Numberofpopulatednodes(s
1
)
0
20
40
60
80
100
120
n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
λ
1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
In the organized phase, the
number of populated nodes is
greater than equal to s-1.
There is always an ACS in the
graph in the growth and organized
phase with Perron-Frobenius
eigenvalue larger than or equal to
1.

15. Random phase
n = 8500 n = 10000
The system eventually returns to the random phase from the organized
phase.
Sandeep Krishna, PhD Thesis (2003)

16. n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
NumberofLinks
0
20
40
60
80
100
120
140
160
n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Numberofpopulatednodes(s
1
)
0
20
40
60
80
100
120
n
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
λ
1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
After a crash, the number of
populated nodes suddenly
decreases.

17. Crashes
In the context of the JK model, a crash has been
defined as a graph update event where a significant
fraction of the nodes (arbitrarily chosen as 50%)
become extinct.
It has been shown that the crashes usually occur as a
result of two different types of structural changes in the
graph:
1. Core-shifts
2. Complete crashes
Jain and Krishna, PRE (2002)

18. Core-shifts
n = 5041 n = 5042
Most crashes in the model are a result of core-shifts.
A core-shift is a graph update event which results in a graph at the present
time step with a core that has a zero overlap with the core of the graph at
the previous time step.
In this case, a new irreducible subgraph takes over as the core of the ACS
after the graph update event.

19. Complete crash
n = 8232 n = 8233
After a complete crash, there is no ACS in the graph.
Such crashes are rare in the model.

20. Modified model with preferential attachment scheme
We have modified the evolving network model of Jain and Krishna (JK) and studied
its features in detail. Our model differs from the JK model in the graph update
scheme.
In JK model, at each time step, one of the nodes with the least population is
eliminated along with its links and a new node is added with links assigned randomly
to existing nodes. The average in-degree and out-degree of the incoming node is
equal to m.
In our modified model, at each time step, one of the nodes with the least population is
eliminated along with its links and a new node is added with links assigned
preferentially to existing nodes with higher degree. The new node will have an
outgoing link and an incoming link to an existing node j in the graph with the same
probability
1
/
s
j j
j
k m k
=
∑
where kj is the total degree of node j.
The average in-degree and out-degree of the incoming node is again equal to m for
the modified model.
Note that the average in-degree and out-degree is an average over many graph
update events.
We choose the number of nodes (s) equal to 100 and m=0.25 for our simulations of
both models.

21. Dynamical rules of the modified model
Initialization:
Cij = 1 with probability p,
= 0 with probability 1-p
p = m/(s -1)
xi are chosen randomly.
x → X: Attractor
C fixed
Population Dynamics
(step 1)
1
/
s
j j
j
k m k
=
∑
The new node
attaches
preferentially to high
degree nodes with
probability:

22. Three phases are also observed in the modified model
In both the models, there are three phases observed:
initial random phase, growth phase and organized phase.
Former model Modified model

23. Preferential attachment accelerates the creation
of the first ACS and the transition from growth to
organized phase
In the former model:
– Average time for the creation of the first ACS = 1107 time steps
– Average time of the first growth phase = 1600 time steps
In the modified model:
– Average time for the creation of the first ACS = 113 time steps
– Average time of the first growth phase = 491 time steps
The numbers presented here are an average over 1000 different
runs with different seeds.

24. Crashes are extremely rare in the modified model
We compared the number of crashes in both models in a larger data set
compiled from 25 different runs of 105
times steps each.
The number of crashes in the former model was equal to 1160.
The number of crashes in the modified model was equal to 6.
Crashes are extremely rare in the modified model.
Also, in the runs of the modified model, we observed that after the creation
of the first ACS there was always an ACS in the graph.
Former model Modified model

25. A typical graph in the organized phase of the modified model has a
much larger Perron-Frobenius eigenvalue
The Perron-Frobenius eigenvalue is related to the density of links in the
core of the graph.
In the modified model, the value of λ1 is much larger than 1 in the organized
phase.
Former model Modified model

26. The core of the graph in the modified model has many more
fundamental loops
The number of nodes in the core of the graph or core size in the former
model can become as large as that in the modified model but the number of
fundamental loops in the core is much larger in the modified model
compared to the former model.
This explains the much larger Perron-Frobenius eigenvalue λ1 in the
modified model.
The number of fundamental loops is given by the first Betti number or
cyclomatic number.

27. Number of links in the graph
There are much more links in the graph of the modified model compared to
the former model with same parameter values.

28. Typical graph in the organized phase of the modified model
The graph shows the dense structure of the core of the graph in the organized
phase of the modified model. The multiplicity of paths within the core of the
graph is an indicator of stability of the system in the modified model.

29. Degree distribution and clustering coefficient of typical graphs in
the organized phase
Both the degree distributions are power-like but it is not possible to uniquely
read of the power from the data.
The average value of clustering coefficient of graphs in the former model is
given by 0.02 and that of graphs in the modified model is given by 0.79. This
is a result of the dense architecture of the core in the modified model.
In-degree distribution Out-degree distribution

30. Both diversity and preferential attachment can enhance network
robustness
In the context of the former model, it was shown recently that the number of
crashes decreases with the increase in the number of nodes in the graph or
diversity.
Reference: Mehrotra, Soni and Jain, J. R. Soc. Interface (2008)
We have shown here an alternate mechanism, i.e., preferential attachment
mechanism which can also render the system robust against crashes.
Reference: Samal and Meyer-Ortmanns, Physica A (2009)

31. Classification of graph update events into different
categories of innovation explains enhanced robustness of
the modified model

32. Acknowledgement
Collaboration:
Hildegard Meyer-Ortmanns, Jacobs University, Bermen, Germany
Discussions:
Sanjay Jain, University of Delhi
Sandeep Krishna,NBI Copenhagen
Reference:
Preferential attachment renders an evolving network of populations
robust against crashes,
Areejit Samal and Hildegard Meyer-Ortmanns,
Physica A (2009) (MPI-MIS Preprint 77/2008).