Complex Adaptive Systems —Resilience, Robustness, and Evolvability: Papers from the AAAI Fall Symposium (FS-10-03)
Emergence of Self-Sustaining Activation
in Dynamically Growing Networks
Joshua D. Ebner1 and Mirsad Hadzikadic2
1
2
University of North Carolina, Charlotte, NC
College of Computing and Informatics, University of North Carolina, Charlotte, NC
jdebner@uncc.edu
mirsad@uncc.edu
The Model
Abstract1
We constructed a spreading activation model in
NetLogo consisting of nodes and links based on an existing
NetLogo network model (Stonedahl and Wilensky 2008).
The model has two components, nodes and links. Nodes
are connected to each other by links creating a network.
Each node exists in one of three states: on, off, or
resetting. Nodes that are off are considered inactive, and
susceptible to activation via neighboring active nodes. If a
node becomes activated at time t, it remains active for one
time step, and will activate all inactive neighboring nodes
(nodes to which it is linked) at time t+1. This activation of
neighboring nodes at subsequent time steps causes
activation to cascade across the network.
After becoming active a node will enter a refractory
period during which it is "resetting" and is neither
considered active or inactive (i.e. it can neither spread
activation nor become re-activated during the refractory
period).
Here we present a network model in which self-sustaining
recurrent activation emerges from simple cascades of
activation. It is demonstrated that the ability to support such
self-sustaining activation in our model depends on network
connectivity as well as the ability to grow new links over
time. Additionally, we explore how the probability of
emergence of self-sustaining activity can be modulated by
changing various network parameters and suggest potential
applications of our findings.
Introduction
The last decade has seen an increase in research devoted
to networks, yielding insights into phenomena in physics,
biology, social science, and complex systems more
generally. Previous work in this area has explored network
connectivity and structure (Watts and Strogatz 1998;
Barabási and Albert, 1999), as well as cascades of
activation across a network (Watts 2002).
In this paper, we will explore the relationship between
network connectivity, random growth, and cascading
activation. Three primary results are identified: 1) selfsustaining activation can emerge from simple cascades of
activation when networks are allowed to generate new
links over time; 2) a relationship exists between the
connectivity of a network (specifically, the average degree
of the nodes, z) and the ability to support such selfsustaining activity; and, 3) both the probability of selfsustaining behavior as well as the critical z value can be
manipulated by altering various network parameters related
to node activation and link growth.
Parameters and Model Initialization
The model can be initialized with different values for the
following parameters:
- Node count, n: The initial number of nodes. New nodes
are not created after initialization.
- Refractory period, The duration of the reset time nodes
experience after activation.
- Link growth switch: Boolean value indicating whether
links are allowed to grow or not. In model initializations
with link growth switch = off, no new links are generated.
- New link rate, : Number of time steps between growth
phases, during which new links may be formed if link
growth switch is set to on.
1
Copyright © 2010, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.
52
- New link radius, : The radius in which a node may look
for potential new nodes to link to, if link growth switch is
set to on.
- Percent new link, : The percent of total nodes n allowed
to create new links during growth phases, if link growth
switch is set to on.
nodes with refractory time , network growth is necessary
but not sufficient for supporting self-sustaining behavior.
Upon initialization, a single node is randomly activated,
and as the model runs activation is allowed to spread to
neighbor nodes on subsequent time steps. If on a
subsequent time step all nodes become inactive, another
node is randomly selected and activation is allowed to
spread.
Given some parameters, the model exhibits selfsustaining behavior; that is, the model never reaches a state
in which all nodes become inactive, and instead the model
enters something akin to a stable attractor.
Table 2: Comparison of self-sustaining behavior with link growth
allowed and disallowed.
Link Growth Switch
ON
OFF
To further examine the conditions under which selfsustaining behavior emerges, we conducted a series of
other experiments in which the model was initialized with
zero links and link growth was allowed.
In each
experiment and all parameters were held constant except
for one. Varying one parameter at a time in this way
helped clarify how each parameter in question effects
model behavior.
Simulation Results
Experiment Two
To explore the nature of this self-sustaining behavior,
four simulation experiments were performed. First, a
parameter sweep was executed to explore the wide range
of model behavior and to seek basic conditions under
which self-sustaining behavior emerges. Experiments two
through four examined the emergence of self-sustaining
behavior in networks with link growth switch set to on and
allowing one parameter to change while keeping others
constant.
In experiment two, all parameters were held constant
(=12, =10, =4) except for node count. Node count n
was varied with the following values: n {10, 30, 50}.
Note that these values are much lower than values used in
later experiments, and were selected to emphasize how
model behavior varies with node count.
For each set of parameters the model was executed 200
times. On each execution, the model was allowed to run
until the ratio of links (l) to nodes (n) were such that the
average node degree of the network (z=2l/n) met a
maximum value, zmax, at which time the model execution
ceased and it was recorded whether or not the model run
exhibited self-sustaining behavior. The parameter z was
selected both because the average degree of a network is a
common metric used to describe undirected networks
Experiment One
In simulation experiment one, the model was run in a
"parameter sweep" wherein the model was initialized and
allowed to run (for 2000 time steps) 30 times for every
combination of input parameters (table 1).
Parameter
n
link growth switch
Range
30 – 300
10 – 90
3–7
3 – 12
20
[ON, OFF]
% self-sustaining
71.9%
0.0%
Increment
30
20
1
3
n/a
n/a
Table 1: List of values used in parameter sweep for experiment 1.
Critically, no self-sustaining behavior was detected in
trials without link growth. However, in trials where link
growth was set to true, self-sustaining behavior was
detected in 71.9% of trials (table 2).
It must be pointed out that while self-sustaining behavior
was detected in a high proportion of trials when link
growth switch was set to on, growth of new links was not
found to be sufficient for the emergence of self-sustaining
activation. Indeed, even with link growth allowed, some
parameter sets exhibited self-sustaining activation on 0%
of trials. Thus, for this cascading activation model using
Figure 1: Graph of percent of trials becoming self-sustaining
as a function of zmax for n=10 (+), n=30 (x) and n=9
(diamond).
53
was varied with the following values: {3, 6, 9}.
For each set of parameters the model was executed 200
times. On each execution, the model was allowed to run
until the average degree of the network z grew to a
maximum allowed value, zmax, at which time the model
execution ceased and it was recorded whether or not the
model run exhibited self-sustaining behavior.
Like experiment one, the model exhibited an increase in
probability of self-sustaining behavior at critical values of
z. Unlike experiment one, variation of did slightly alter
the value of zcrit at which self-sustaining behavior began to
emerge. For =3, the value zcrit at which self-sustaining
behavior became probable was roughly 2.8. zcrit increased
to roughly 3.2 for both =6 and =9, but curiously the
overall curve for =6 is shifted farther rightward than the
curve for =9. It seems that increases in do not create
linear changes in the critical value of z at which the model
becomes self-sustaining.
Experiment four continued the examination of effects of
single parameter shifts by varying .
(Barabási and Oltvai 2004) and also in order to parallel
previous network theory work; specifically z serves as a
proxy for the ratio of links to nodes discussed elsewhere
(Kauffman 1993).
The model exhibited an increase in probability of selfsustaining behavior when the model was allowed to grow
to a critical value of z, zcrit (for this parameter set, zcrit 2).
While n did not effect the critical value of zcrit at which
self-sustaining behavior began to emerge, n did effect the
maximum percent of trials that exhibited self-sustaining
behavior (Fig. 1).
Figure 1 displays two important findings of the model.
First, for given set of input parameters , , and , a certain
number of nodes are needed in order to support selfsustaining behavior. Further, because nodes exist in a
world with dimensions x and y, this suggests that a certain
node density is needed to support self-sustaining behavior.
If the node count n is too low (e.g. n=10 in Figure 1), selfsustain behavior rarely occurs.
Second, Figure 1 shows that given appropriate input
parameters – and if links are allowed to grow over time – a
critical value of z must be reached for self-sustaining
behavior to become probable. Said differently, given
appropriate input parameters, dynamically growing
network models undergo a phase transition at critical
values of z, causing the model to change from cascading
behavior to recurrent, self-sustaining behavior.
Experiment three explores this relationship between
connectivity z, parameter values, and self-sustaining
activation by examining the effect of varying new link
radius, .
Experiment Four
In experiment four, all parameters were held constant
(n=200, =10, =12) except for refractory period, . was
allowed vary with the following values: {3, 6, 9}. For
each set of parameters the model was executed 200 times.
On each execution, the model was allowed to run until the
average node degree met a maximum value, at which time
the model execution ceased and it was recorded whether or
not the model run exhibited self-sustaining behavior.
Experiment Three
In experiment three, all parameters were held constant
(n=200, =10, =4) except for new link radius, . Note
that in contrast to experiment one, experiment two used a
higher value for node count, n, in order to emphasize
changes in model behavior resulting from changes in .
Figure 3: Graph of percent of trials becoming self-sustaining as
a function of zmax for =3 (square), =6 (triangle) and =9
(circle).
Alteration of showed a clear effect on the percent of
trials that become self-sustaining (Figure 3), with increases
in refractory period leading to decreases in the
percentage of trials that became self-sustaining. This
manipulation of refractory time suggests that for selfsustaining behavior to occur, node refractory times must be
Figure 2: Graph of percent of trials becoming self-sustaining as
a function of zmax for =3 (square), =6 (triangle) and =9
(circle).
54
sufficiently low. Yet, for values of refractory time that are
too low the model either never exhibits self-sustaining
behavior or always does (for values of = 1, 2
respectively). Collectively, these findings suggest that
moderate values of are necessary for non-trivial selfsustaining behavior.
Future Work
Additional work remains to be completed. Open
questions include 1) how model behavior changes if
different "types" of nodes are allowed (such that type 1 can
only link to type 2, type 2 to type 3, type 3 to type 1, etc);
2) how model behavior changes if more than one node is
randomly activated at initialization; and 3) how nonuniform values of might effect the emergence of selfsustaining phenomena.
In addition to these empirical studies, we aim to ground
this research more firmly in existing network theory.
There are indications that this phase transition to recurrent
activation
may
be
describable
mathematically.
Specifically, Erdös and Rényi described a phase transition
in the size of the largest component as links are added to a
random graph (Erdös and Rényi 1959). That we also find a
phase transition with the addition of new links suggests a
relationship to Erdös and Rényi's random graph work.
Thus, grounding the phase transition described in this
manuscript requires two developments: i) examining if this
phase transition occurs on the special case of a random
network (i.e. the case where new links are not constrained
by ), and ii) examining if the probabilistic/combinatorial
methods of Erdös and Rényi can be applied to this
particular phase transition on a random graph, as has been
done in other network science research (Watts 2002).
Work has been begun on (i) and evidence shows that
indeed, this phenomena does occur on networks based on
random graphs.
Finally, while we have offered suggestions for how this
abstract network model could be applied as a model of real
world networks, more work is required for this application
to be complete. Certainly, we need to clarify how our
model parameters map to real world networks. We also
need to clarify how events like "link growth" in our model
might map to events in real networks. Given the
complexity of real world networks, executing this mapping
will require consultation with subject matter experts in
each of the areas we have outlined above. That said,
opportunities are being explored to implement our model
as a model of the phenomena above and validate it against
existing data in those areas.
Potential Applications
While the model presented here is an abstract model of
networks generally, it could be implemented in the future
as a model of real world networks. Thus, the results
offered here may provide insight into the relationship
between connectivity, node behavior, and self-sustaining
activity in real-world phenomena. Three potential areas in
which the model could be applied are the brain, innovation,
and the origins of life.
Recurrent Activation Models of Short-Term
Memory and Goal Maintenance in the Pre-frontal
Cortex
Psychologists
and
neuroscientists
exploring
computational models of short-term memory (STM) and
goal maintenance have explored ways in which networks
could support such phenomena in the form of recurrent
networks (Hopfield 1982; Zipser 1991). In such neural
network models, nodes are networked together in loops
that allow for "recurrent" activation; activation that occurs
in loops that self-sustain.
While these models have explored how recurrent or selfsustaining activation could occur in networks, they have
not explored how such recurrent activity could emerge;
that is, recurrence was built into these models in the form
of directed links. The model under discussion here could
provide some insight into how self-sustaining activity
could emerge in neural systems without external influence.
Emergence of Innovation Among Networks of
Agents
Treating nodes as individuals and links as means of
communicating between individuals, this model could
elucidate the conditions under which innovation occurs.
Using cascading networks to explore the subject of
innovation has been performed in the past (Watts 2002),
and the present model may clarify how sustained
innovative activity can emerge out of systems that may
have otherwise only exhibited cascades (e.g. fads, etc) that
simply die out.
References
Barabási, A.-L., and Albert, R. 1999. Emergence of
Scaling in Random Networks. Science 286: 509-512.
Origins of Life
Due to the way in which self-sustaining network
activation emerges out cascading activation and random
wiring of nodes, it seems plausible that this model may
have applications to the origin of life similar to those
proposed in the area of autocatalytic sets and random
boolean networks (Kauffman 1993).
Barabási, A.-L., and Oltvai, Z. N. 2004. Network Biology:
Understanding the Cell's Functional Organization. Nature
Reviews Genetics 5: 101-113.
55
Erdös, P., and Rényi, A. 1959. On Random Graphs.
Publicationes Mathematicae, 6: 290-297.
Hopfield, J. J. 1982. Neural Networks and Physical
Systems with Emergent Collective Computational
Abilities. Proceedings of the National Academy of Sciences
of the USA 79: 2554-58.
Kauffman, S. A. 1993. The Origins of Order. New York,
NY: Oxford University Press.
Stonedahl, F., and Wilensky, U. 2008. NetLogo Virus on a
Network model.
http://ccl.northwestern.edu/netlogo/models/VirusonaNetwo
rk. Center for Connected Learning and Computer-Based
Modeling, Northwestern University, Evanston, IL.
Watts, D. J. 2002. A Simple Model of Global Cascades on
Random Networks. Proceedings of the National Academy
of Sciences of the USA 99(9): 5766-5771.
Watts, D. J., and Strogatz, S. H. 1998. Collective
Dynamics of 'Small-world' Networks. Nature 393 (6684):
409–10.
Wilensky, U. 1999. NetLogo.
http://ccl.northwestern.edu/netlogo/. Center for Connected
Learning and Computer-Based Modeling, Northwestern
University, Evanston, IL.
Zipser, D. 1991. Recurrent Network Model of the Neural
Mechanism of Short-term Active Memory.
Neural
Computation 3: 179-193.
56