Complex Adaptive Systems —Resilience, Robustness, and Evolvability: Papers from the AAAI Fall Symposium (FS-10-03)
Weaving the Social Fabric: The Past, Present and Future of Optimization
Problem Solving with Culture Algorithms
Xiangdong CHE
Computer Science Department, Wayne State University,
Detroit, MI 48202 , U.S.A
Mostafa ALI
Department of Computer Information Systems, Jordan University of Science & Technology,
Irbid, Jordan 22110
Robert G. REYNOLDS
Computer Science Department, Wayne State University,
Detroit, MI 48202, U.S.A
ABSTRACT
that describes how knowledge is exchanged between the first
two components. The population space can support any
population-based computational model, such as Genetic
Algorithms, Evolutionary Programming, etc. The basic
framework is shown in Figure 1.
In this paper we investigate the performance of Cultural
Algorithms over the complete range of system
complexities, from fixed to chaotic. In order to apply the
Cultural Algorithm over all complexity classes we
generalize on its co-evolutionary nature to keep the
variation in the population across all complexities. Based
on previous cultural algorithm approaches, we were to
extend the existing models to produce a more general one
that could be applied across all complexity classes. We
produced a new version of the Cultural Algorithms
Toolkit, CAT 2.0, which supported a variety of coevolutionary features at both the Knowledge and
Population levels. We then applied the system to the
solution of a 150 randomly generated problems that
ranged from simple to chaotic complexity classes. As a
result we were able to produce the following conclusions:
No homogeneous Social Fabric tested was dominant over
all categories of complexity. As the complexity of
problems increased, so did the complexity of the Social
Fabric that was need to deal with it efficiently. In other
words, there was experimental evidence that social
structure can be related to the frequency and complexity
type of the problems that presented to a cultural system.
Figure 1 Cultural Algorithm Framework
Keywords: Cultural Algorithm, Optimization, Social
Fabric, Co-evolution
A Cultural Algorithm is a dual inheritance system that
characterizes evolution in human culture at both the macroevolutionary level, which takes place within the Belief Space,
and at the micro-evolutionary level, which occurs in the
population space. Knowledge produced in the population space
at the micro-evolutionary level is selectively accepted or passed
to the Belief Space and used to adjust the knowledge structures
there. This knowledge can then be used to influence the changes
made by the population in the next generation. The basic
pseudo code of Cultural Algorithm is shown in Figure 2 where
Pt represents the Population component at time t, and Bt for the
Belief Space at time t.
The algorithm begins by initializing both the Population and
the Belief Space. Then it enters the evolution loop until the
termination condition is satisfied. At the end of each generation,
individuals in the Population Space are first evaluated with a
performance function, obj(). An acceptance function, accept(),
is then used to determine which individuals will be allowed to
update the Belief Space. Experiences of those chosen elite
1. INTRODUCTION
If we are in a system, it is hard for us to see or understand how
the system itself works. Since we all live within a culture, how
can we see the culture from an unbiased perspective? It would
be worthwhile and interesting to look at the culture from an
external point of view. Cultural Algorithms (Reynolds 1986;
Reynolds 1994) was developed by Reynolds to do just that assess a cultural system in terms of its performance relative to
problems presented to it by its physical and social environment.
Cultural Algorithms can provide a flexible framework in
which to study the emergence of organizational complexity in a
multi-agent system (MAS). The Cultural Algorithm (CA) is a
class of computational models derived from observing the
cultural evolution process in nature. CA has three major
components: a population space, a belief space, and a protocol
44
Begin
t = 0;
initialize Pt
initialize Bt
repeat
evaluate Pt {obj()}
update(Bt, accept(Pt))
generate(Pt, influence(Bt))
t = t + 1;
select Pt from Pt- 1
until (termination condition achieved)
End
Figure 2 Basic Pseudo-code for Cultural Algorithm
individuals are then added to the contents of the Belief Space
via function update(). Next, knowledge from the Belief Space is
allowed to influence the selection of individuals for the next
generation of the population through the influence() function.
The two feedback paths of information, one through the
accept() and influence() functions, and the other through
individual experience and the obj() function create a system of
dual inheritance of both the population and the belief spaces.
The Cultural Algorithm repeats this process for each generation
until the pre-specified termination condition is met. In this
way, the population component and the Belief Space interact
with, and support each other, in a manner analogous to the
evolution of human culture (Barkow et al. 1992; Johnson and
Earle 2000; Richerson and Boyd 2005).
Previously Peng (Peng 2005) found that the similarities in
social structures that emerge in similar cultures are produced as
a result of the integration of different knowledge sources in the
solving process. Peng proposed a biologically motivated
approach to integrating the application of these knowledge
sources based upon the Marginal Value Theorem (Charnov
1976) to drive the problem solving process and showed that
certain social structures emerged from the Cultural System as a
result of this approach. Peng implemented her system with
JAVA and used MATLAB as a tool for visualizing the results
of the experiments and graphing issues. In her system, each
individual agent in the population was associated with a single
knowledge source that influenced it and there was no exchange
of information between agents in the population(Peng and
Reynolds 2004).
Next, Ali(Ali 2008) expanded the ability of a knowledge
source to influence a population through the notion of a social
fabric. He embedded the Cultural Algorithms framework within
the Recursive Porous Agent Simulation Tool (Repast) (North et
al. 2005). He produced a toolkit that called Cultural Algorithms
Simulation Toolkit (CAT) (Reynolds and Ali, 2008). The social
fabric represents the extent to which the influence of a
knowledge source can spread through a population. The
interconnections between agents in the population were viewed
metaphorically as a social fabric, created by the interactions
between agents. Knowledge sources in the Belief Space select
individual from the population to influence individual agents in
the population as illustrated in Figure 3.
Figure 3 Embedded Social Fabric component in CAT
Each knowledge source in the Belief Space is color-coded and
its color is applied to the individual agent in a network that it is
to influence at that time step. The network shown in the
diagram is a homogeneous square network. Each of those
agents has a particular position on the landscape. Ali used three
homogeneous networks in his version: lbest (degree of two for
each agent), square (degree of 4 for each agent), and gbest
(degree of n-1 nodes for each agent). Lbest and gbest are show
in Figure 4.
In order to demonstrate the power of the social fabric to
solve optimization problems, Ali regenerated the graph
randomly each time for the configuration being tested. In other
words, he wanted to see whether just the presence of a network
would improve the optimization process - even if the network is
dynamically regenerated in a random fashion each time. Ali
was able to show that just having a social fabric to distribute
influence in the population was sufficient to improve
performance in the Cultural Algorithm over a variety of
different engineering design problems.
In this paper we extend the social fabric approach by
allowing the networks to have a memory. That is, the network
organization was established at the beginning of the run and the
connections between individual agents remained constant over
the course of the problem solving process even though the
agents were able to move to different positions on the
landscape. In addition, we add a variety of other configurations
in order to assess the relationship between graph degree and
problem complexity. This will allow us to test variations on the
following general hypothesis: Does the organization of the
social structure for a culture reflect the nature and number of
problems presented to it by its environment?
For example, it has been suggested by researchers such as
Barabási (Albert and Barabasi 2002) that the prevalence of
small world networks in biological systems may reflect an
evolutionary advantage of such an architecture. One possibility
is that small world networks are more robust to perturbations
than other network architectures. If this were the case, it would
provide an advantage to biological systems that are subject to
damage by mutation or viral infection.
45
generates a dynamic cones-world in two steps: first specify a
baseline static landscape of the desired morphological
complexity, and then add the desired dynamics. The base
landscape is given by:
n
f (< x 1,x 2 ,...,x n >) = max (H j R j *
j=1,k
(a) lbest topology.
( x C
i
j, i )
2
)
i=1
where:
k : the number of cones,
n : the dimensionality,
Hj: height of cone j,
Rj : slope of cone j, and
Cj,i: coordinate of cone j in dimension i.
The values for each cone (Hj, Rj, and Cj,i) are randomly assigned
based on the following user specified ranges:
Hj (Hbase, Hbase + Hrange)
Rj (Rbase, Rbase + Rrange)
Cj,i (-1, 1)
(b) gbest topology.
Figure 4 Topologies used in the Social Fabric
model(Reynolds and Ali 2008)
2. COMPLEX SYSTEMS AND SOCIAL FABRIC
A complex system usually has the following characteristics
(Holland 1992):
-Relationships in complex system are non-linear, i.e., effects are
not directly proportional to causes.
-Complex systems contain feedback loops.
-Complex systems are open, i.e. usually far from equilibriums,
but may form pattern stability.
-Complex systems have memory, i.e. history matters.
-Complex system may produce emergent phenomena.
Reynolds (Reynolds, Whallon et al. 2006) defines a complex
system as a system made up of an organized group of
heterogeneous independent agents who interact with their
environment and each other, and adapt based on feedback from
the environment. The separate behaviors of the individuals
aggregated together can cause higher-level behaviors to emerge
from the group as a whole that work to solve problems facing
the group at the group level. In this section we briefly introduce
the complex systems environment that we will be using to test
our hypotheses regarding the basic weaving of the social fabric.
Each of these independently specified cones are “blended”
together using the max function, i.e., if two cones overlap, the
height at a point is the height of the cone with the largest value
at that point. We use 2-dimensional landscapes throughout the
remainder of this paper for simplicity and for visualization
purposes, although the patterns and rules apply to more than
two-dimensional scenarios. An example landscape with k = 15,
Hbase = 1, Hrange = 9, Rbase = 8, and Rrange = 12 is given in
Figure 5 below.
2.1 The Cone’s World Generator
The Cones World that will be used in this paper was originally
developed by De Jong and his students (Morrison et al. 1999) in
order to examine the ability of evolutionary problem solving
approaches to solve randomly generated problems of arbitrary
complexity. This was proposed as an alternative to the
traditional approach of comparing algorithms on a small set of
benchmark problems. It was felt that by focusing on a small set
of problems, investigators might attempt to sacrifice algorithm
generality in order to produce good results on the specific
problems.
Peng (Peng and Reynolds 2004; Reynolds, Peng and Che,
2005) coined the term Cones World when she employed it to
test various Cultural Algorithm configurations. When Ali(Ali
2008) extended Peng’s Cultural Algorithm framework in his
CAT system, he employed the Cones World as one of the
problems available to system users along with the traditional
benchmark problems.
The Cones World Generator produces a problem landscape,
in which a field of resource cones of different heights and
different slopes that are randomly scattered across a multi-dimensional landscape. Basically, the Cones-World algorithm
Figure 5 An Example Landscape In two-dimensional space
x (-1.0, 1.0), y (-1.0, 1.0) with n = 50, H(1, 10), and R
(8, 20)
The second step is to specify the dynamics. For each cone j,
its every single parameter (every dimension Cj,i, height Hj, and,
and slope Rj) can be changed individually and independently. In
order to control the complexity of a landscape, we use the
logistics function given as:
Yi = A * Yi-1 * (1 - Yi-1)
where A is a constant and Yi is the value at iteration i.
A bifurcation map of this function is provided in Figure 6.
This figure shows the values of Y that can be generated by each
iteration of the logistic function given values of A between 1.0
and 4.0. The particular value of A specifies whether the
46
Periodic – For problems of this nature the cells need to switch
from one set of rules to another depending on the number of
bifurcations.
Chaotic – Problems for which the number of bifurcations is so
large the system is inherently chaotic. Thus, there are no
specific rule sets that apply.
Langton suggested that there was no sharp demarcation
between Periodic and Chaotic. He called this transition “the
edge of chaos”. In our situation this suggests that a fixed
communication or topology will be "more" effective for
problems of low entropy, A near 1 in our model than an A value
near 4. As A value increases the entropy increases and the
network needs to integrate the information acquired by its
agents. When A increases to more than 3, the system becomes
periodic. This will mean that the network will need to handle
two different types of changes. We can view this as an
environment that changes from one state to another in a
seasonal fashion. As we approach 4.0 the system becomes
increasingly chaotic, containing so many transitions that it is
hard to sufficiently gather information about any of them.
In order to test how these problem categories are reflected
by changes in the corresponding social organization of the
cultural systems needed to solve them, we will generate an
example set of landscapes from each of the three basic classes –
fixed, periodic and chaotic. The corresponding A values will be
1.01, 3.35, and 3.99 respectively. Our task will be to discern
how changes in problem complexity affect the structure of the
topology needed to solve them.
movement will be in small same-sized steps, large same-sized
steps, steps of few different sizes, or chaotically changing step
sizes.
We are particularly interested in characteristic points in
terms of the complexity in the cones world as shown in Figure
6. Here we pick A = 1.01, 3.35 and 3.99 for our test
environment complexity as marked with vertical mark line in
green. From left to right we have A = 1.01, 3.35, 3.99
corresponding to one step change, two steps change and totally
chaotic step size change. Each of these represents one of the
computational classes proposed by Langton as discussed below.
This cone landscape generator function can be specified for
any number of dimensions. Each time the generator is called, it
produces a randomly generated real-valued surface in which
random values for each cone are assigned based on userspecified ranges. By applying the logistic function to the
parameter of the cone’s generator, we are able to control the
complexity of the generated landscape by changing the A value
of the logistics function. Therefore, given that we can generate
problem landscapes at different levels of complexity we can
study the relationship between the social network and different
categories of computational problems. This enables us to
evaluate our model in a more flexible and systematic way. It
also is a reasonable facsimile of how resources are spread out
within natural environments: from information theory point of
view, the problem environment carrying certain complexity of
information could be represented by entropy; the more complex
that an environment is, the higher its entropy is.
2.2 Neighborhood Topology:
The concept is illustrated schematically in Figure 7 with 5
different networks given as color-coded vertical lines, one for
each of the five knowledge sources. Individuals are given as
horizontal lines with a node representing a possible
participation in each network. The node is blank if the
individual does not participate in that particular network. It is
darkened in with the networks color if it participates sometimes,
and darkened and circled if it is a frequent participant. The
individuals are ordered from highest participation to lowest
participation of the five networks. The topologies of the
networks are not shown here, just the extent to which the
networks are woven together by the participation of the
individuals.
Figure 6 Logistic Function with characteristic A values
Langton (Langton 1992) suggests that as the A value, which
relates to entropy of the system, increases, the system becomes
more unpredictable. He demonstrated this in terms of rules
needed by a Cellular Automata to solve problems at different
points along the curve in Figure 6. In particular he stated that
the amount of mutual information that cells in the space needed
to know about their neighbors increased as the entropy or
amount of information in the landscape increased. Based upon
that he established several basic computational classes as
follows:
Fixed - For problems of low entropy a fixed set of rules can be
given to each cell in order to allow them to exchange the
information needed to solve the problem. In our case this is
equivalent to having a fixed topology over which information is
exchanged, around 1.
Figure 7 Social Fabric
47
Notice that some groups, such as the “red” group, are
characterized by a small but active set of participants, while the
“turquoise” group is one where everyone participates
somewhat. For those individuals who participate in more than
one group, activities in one group can constrain activities in
another. Thus, a knowledge source can influence an individual
and its influence can be spread to the individuals’ neighbors
with modification.
The networks that comprise the social fabric can emanate
from either the Belief Space or the population space or both. In
terms of the population, the network can reflect a kinship
network or an economic network for example. In terms of the
Belief Space, the network can be the Internet, or a local area
network, or some other network directly accessible to the
knowledge sources. Previously, Ali employed 3 different
neighborhood topologies: lbest, square and global. Here, we
added more topologies in order to investigate in more detail
how topology will impact the optimization performance for
different landscapes.
We deployed a series of typical
topologies including lbest, square, hexagon, Octagon,
Hexadecagon (in this paper, it is simplified as “16-gon” in the
tables), and global. By introducing more network topologies,
we now have a more complete spectrum of topology and more
reasonable neighborhood selection.
Equation 1
Reynolds (Reynolds 1978) defined the following vector
voting model: “In the vector voting model each individual can
propose a direction based on the region he has explored. Each
vector is treated as a unit vector and then the vectoral sum is
taken. The resultant vector is then categorized according to the
quadrant in which it lies.” Vector-voting mode specifies a
linear threshold directional function as shown in Equation 1:
2.3 Agent Decision Making
Figure 8 The Vector Voting Paradigm
Each individual in the network now has several choices as to
which knowledge source will influence it. In Peng’s original
model individuals were independent, so that the individual was
influenced directly by the knowledge source selected. When Ali
added the Social Fabric topology each individual received a
direct influence plus that of its immediate neighbors. There
needed to a mechanism by which one of the Knowledge
Sources was selected from this set of alternatives. He employed
an un-weighted majority win scenario. The KS with the most
votes became the KS that was used by an individual. In case of
a tie then the direct influence for the individual agent was used
as the default.
In this paper we modify the decision-making approach to
allow for an incentive based scheme. Each of the votes received
by an individual is associated with a weight. The selected
Knowledge Source is the one with the most total weight. This
approach was selected because it supported the co-evolutionary
focus here in that a less frequently used knowledge source
might produce a new result that exceeds the Knowledge Source
that is used most frequently. The modification is based upon a
fundamental voting technique used in the earliest Cultural
Algorithms (Reynolds 1979) and again reflecting the predatorprey approach to co-evolution used here. The approach is called
vector voting and it is discussed below.
One of the goals of our research is to assess the impact of
Culture on decision making behavior in a complex social
system such as Hunter-Gatherer system(Goodhall 2002). A
Hunter-Gatherer society using vector voting model is viewed as
the context in which early cultural system arise. Specifically we
are concerned with the question of whether the emergence of
human culture provided humans with an adaptive advantage
over their non-human primate counterparts in terms of huntergathering capabilities. Reynolds’ vector voting model serves as
our theoretical bases in a series of research work on HunterGatherer/Predator-Prey simulation.
Each x is a local directional function mapping Z into
QUAD U(0,0). And QUAD maps the region R into quadrants.
In other words, each individual in the vector-voting model can
vote on a direction based on the information he has collected
from the region he has explored. Each vector is treated as a unit
vector and then the weighted vector sum is taken. The weight
of an individual is reflecting their status. The resultant vector is
then the direction of the group takes. This vector-voting model
is illustrated in Figure 8. Individuals in a band can have various
preferences regarding which direction to go in the next step
based on their individual experiences. Those experiences are
associated with different icons on the landscape. The resultant
direction of the group will be a weighted sum of all the
individuals’ desired directions.
The vector-voting model has been used to simulate a group
of individuals making a group decision concerning their
direction to travel, assuming the individuals can communicate
their desired direction and their level of determination for that
direction without knowing each other’s underlining rationale.
This model has been proven to be able to choose the direction
which will yield the most resources for the group under certain
assumptions including: static environment without interference
from other groups; sufficient individuals to provide complete
coverage over the whole area; no error in information
perception; and all individuals behave rationally.
2.4 Incentive based Majority Win
Here we introduce a new decision making schema called
Incentive based Majority Win. It is based on a Vector Voting
model employed in the earliest version of Cultural Algorithms
as discussed above. When each individual calls the influence
function, the influence function will have a “direct” knowledge
source designated to this individual by spinning the knowledge
48
wheel. At the same time, this individual can also receive
information from its neighbors regarding controlling knowledge
sources, as shown in Figure 9.
In this figure, we have individual A0 whose direct
controlling KS is S which stands for Situational KS. A0 has 8
neighbors: they are A1 through A8, each of them has a
controlling KS.
Here T stands for topographical KS, D stands for Domain
KS, N stands for Normative KS. H stands for History KS. In
population space, the previous CAT system used majority win
based decision making in order to decide which Knowledge
Source to select from the current social fabric. Figure 10 shows
the majority win process.
Figure 10 Majority Win in Belief Space through Social
Network.
Reynolds introduced vector voting (Reynolds 1986) as a
decision-making model for hunter-gatherer society simulation.
To introduce more innovation and maintain genetic diversity at
the population level we employed the weighted approach here.
We use the current average fitness of each Knowledge Sources
as weight of each Knowledge Sources count, and then do
majority win based on the weighted count as shown in Figure
11.
Figure 9 Knowledge Source Interactions at the Population
Level
At each time step, every individual is influenced by one of
the knowledge sources. In this version, Knowledge Sources do
not know anything about the network and the selected
individuals’ position in it and vice versa. The process is a
double blind. The individual then transmits the name of the
influencing Knowledge Source to its neighbors through as
many hops as specified. Next, each node counts up the number
of Knowledge source bids that it collects. It will have the direct
influence from the Knowledge Source that selected it, plus the
IDs of the Knowledge Sources transmitted to it by its neighbors.
The Knowledge Source that has the most votes is the winner
and will direct the individual for that time step. In case of a tie,
there are several tie breaking rules implemented in CAT.
In Figure 10, Individual A0 has following count of votes:
•
3 neighbors (including itself) votes for S;
•
2 vote for D;
•
1 votes for T;
•
1 votes for N;
•
1 votes for H.
So according to Majority Win Schema, S wins the vote.
Figure 11 Weighted Majority Win in Belief Space through
Social Network
As you can see after the count adjustment based on the
weight, shown beside the arrow, Domain knowledge becomes
the winner. This modified majority-win could be thought as a
vector voting variation; each knowledge source is a vector and
wants to decide where the individual needs to go. The average
fitness of the current generation is the key to win in this bidding
game. If a less used knowledge source suddenly finds a good
solution its average can rise quickly, and therefore the vector
voting approach will tend to magnify its influence in the
network.
3. SUMMARY RESULTS AND ANALYSIS
3.1 Introduction
In this section we compare the several topologies for the Social
Fabric in terms of their performance within each of the three
complexity classes. We demonstrate that there is indeed a
relationship between the complexity of the problem and the
Social Fabric used by the Cultural System. We start by
49
than the others. The mean number of generations needed to find
a solution increases markedly for all of the topologies with a
smaller in-degree here. Since a square topology was sufficient
for one generator, it makes sense that for two generators each
might use four links which gives us 8. The mean time to a
solution for the configurations with in-degree greater than 8
also increase. The motivation for four relates to the four
cardinal directions on the landscape over which cone slope can
be computed.
Table 2 Performance comparison of the topology A =3.35
50 224
168.7 38
11
339
137
72.6
square
50 213
152.5 42
9
358
158
93.3
Hexagon 50 229
169.5 40
40
491
161
112.3
Octagon 50 243
175.7 37
7
426
152
99.1
16-gon
50 219
160.9 41
44
477
158
100.9
global
50 238
186.4 35
10
361
126
83.0
Std. Deviation
Mean
Generation_Used
Found Maximum
Generation_Used
Minimum
Generation_Used
#of
Runs w/ Solution
Found
Std. Deviation
Found
Mean
Generation_Used
Runs w/ Solution
Found Maximum
Generation_Used
Minimum
Generation_Used
#of
Runs w/ Solution
Found
Std. Deviation
#ofTotalRuns
Overall Mean
Generation_Used
Topology
lBest
Std. Deviation
Overall Mean
Generation_Used
While all of the topologies tested were homogeneous they were
reasonably robust across the different landscapes. Table 1
though 3 give the statistical comparison of the topologies over
the three complexity classes respectively. One thing to note is
that even the fixed category was composed of complex
landscapes produced by the random combination of 500 cones.
In only one instance did a topology fail to get 50% or more of
the problems solved. In the best case the topology was
associated with an 84% solution ratio.
Table 1 Performance comparison of topology for A =1.01
#ofTotalRuns
3.2 Overall Performance Comparison
Topology
comparing their overall problem solving performances, and
then examine the differences in terms of how the knowledge
sources make use of the Social Fabric in the problem solving
process.
The experimental framework that we use is as follows. Each
Complexity (1.01, 3,35, 3,99) has five randomly generated
example Landscape (#1,#2,#3,#4,,#5) , We have 6 Topologies,
Lbest, Square, Hexagon, Octagon, Hexadecagon, and global.
For each landscape/topology combination we did 10 runs on
each of the 5 landscapes. So each complexity has a total of 50
runs for each topology. The maximum number of generations
for each run is 500.
lBest
50 337
159.8 29
69
448
219
102.3
square
50 308
171.8 33
60
496
209
125.1
Hexagon 50 286
176.3 33
33
431
177
105.3
Octagon 50 243
168.7 39
22
489
170
110.4
16-gon
50 287
181.0 32
51
459
167
103.7
global
50 371
170.4 20
62
448
178
97.4
For the chaotic landscape class all of the topologies find
almost the same number of solutions. The statistical results are
given in Table 3. Their number of solutions generated for each
varies only between 33 and 35. However, the average number
of generations used for all 50 runs by all of the topologies has
increased from the periodic class. The clear winning topology
here, as might be expected, is the global one. While it achieves
about the same success rate as the others, the average time to
solution and the standard deviation is less than the others.
In summary, as the complexity goes up, so the amount of
time needed to solve the associated optimization problems for
the topologies in general. Also, the connectivity or in-degree of
the best topology increases from 4, to 8, and finally to a global
connection as we move from fixed to periodic and finally
chaotic.
Table 3 Performance comparison of the topology A =3.99
Overall Mean
Generation_Used
Std. Deviation
Std. Deviation
Mean
Generation_Used
Found Maximum
Generation_Used
Minimum
Generation_Used
#of
Runs w/ Solution
Found
#ofTotalRuns
50
Topology
For the fixed category of problems, Table 1, the square
topology solved 84% of the problems and used the fewest mean
number of generations overall relative to the 50 runs. On the
other hand, a global topology had the fewest % of solved
problems, and greatest number of generations used on average
for any landscape. Yet the global network used fewer
generations in producing a solved problem and the lowest
standard deviation for them. That means when it solved a
problem it not only solved a problem faster, but more
predictably given the reduced variation in solution times. This
suggests that perhaps a non-homogeneous network might be
useful in solving some problems efficiently, a network that
supports both local and global connections such as a small
world network.
For the periodic case, given in Table 2, where the generator
function bifurcates, we notice that the success rate for all
topologies except the octagon drop. This reflects the higher
entropy associated with the addition of a second generator.
Here, the octagon topology finds 6 more solutions out of 50
lBest
50
301
159.0 35
83
487 216
107.9
square
50
302
160.8 35
54
464 217
112.6
Hexagon 50
300
163.7 33
34
481 197
95.1
Octagon 50
296
155.5 35
44
420 209
93.9
16-gon
50
291
172.0 34
10
441 193
113.5
global
50
289
172.4 33
38
400 181
99.5
Morrison, R., De Jong, K., Inc, G., & Vienna, V. (1999). A test
problem generator for non-stationary environments.
4. CONCLUSIONS AND FUTURE WORK
North, M. J., Howe, T. R., Collier, N. T., & Vos, J. R. (2005,
October 13–15). The Repast Simphony Development
Environment. Paper presented at the Proceedings of the Agent
2005 Conference on Generative Social Processes, Models, and
Mechanisms, Argonne National Laboratory and The University
of Chicago.
In this paper we investigated the performance of Cultural
Algorithms over the complete range of system complexities,
from fixed to chaotic. In order to apply the Cultural Algorithm
over all complexity classes it was necessary that we generalize
on its co-evolutionary nature in order to keep the variation in
the population across all complexities. In order to do this we
surveyed the history of Cultural Algorithms development. As a
result we produced a new version of the Cultural Algorithms
Toolkit, CAT 2.0, which supported a variety of co-evolutionary
features at both the Knowledge and Population levels. We then
applied the system to the solution of a 150 randomly generated
problems. As a result we were able to produce the following
conclusions.
Firstly, no homogeneous Social Fabric tested was dominant
over all categories of complexity. Secondly, as the complexity
of problems increased so did the complexity of the Social
Fabric that was needed to deal with it efficiently. So a fabric
that was good for fixed problems would be less adequate for
periodic problems, and chaotic ones. Thirdly, a cultural system
can, in fact, solve problems over all categories of complexity.
But the most efficient topology varies from one category to the
next. Therefore, a single system would probably need to support
both local connections and more global ones to solve problems
over all classes. As the problems facing an organization
increases, so must the computational power of the organization.
In future work we will investigate the synthesis of
heterogeneous network that can be applied to a broad class of
problem successfully.
Peng, B. (2005). Knowledge Swarms in Cultural Algorithms for
Dynamic Environment. University Microfilms International, P.
O. Box 1764, Ann Arbor, MI, 48106, USA.
Peng, B., & Reynolds, R. (2004). Cultural algorithms:
knowledge learning in dynamic environments.
Reynolds, R. (1979). An adaptive computer model for the
evolution of plant collecting and early agriculture in the eastern
valley of Oaxaca. Guila Naquitz: Archaic foraging and early
agriculture in Oaxaca, Mexico, 439–500.
Reynolds, R., & Ali, M. (2008). Computing with the social
fabric: The evolution of social intelligence within a cultural
framework. IEEE Computational Intelligence Magazine, 3(1),
18-30.
Reynolds, R. G. (1978). On Modeling the Evolution of HunterGatherer decision-Making Systems. Geographical Analysis,
10(1), 31-46.
Reynolds, R. G. (1986). An Adaptive Computer Model for the
Evolution of Plant Collecting and Early Agriculture in the
Eastern Valley of Oaxaca. Unpublished Ph.D. dissertation,
University of Michigan, Ann Arbor, MI.
5. REFERENCES
Reynolds, R. G. (1994). An introduction to Cultural Algorithms.
Proceedings of the Third Annual Conference on Evolutionary
Programming, 131-139.
Albert, R., & Barabasi, A. (2002). Statistical mechanics of
complex networks. Reviews of modern physics, 74(1), 47-97.
Reynolds, R. G., Peng, B., & Che, X. (2005). Knowledge
Swarms: Generating Emergent Social Structure in Dynamic
Environments. Proceedings of the Agent 2005 Conference on
Generative Social Processes, Models, and Mechanisms.
Ali, M. (2008). Using cultural algorithms to solve optimization
problems with a social fabric approach. University Microfilms
International, P. O. Box 1764, Ann Arbor, MI, 48106, USA.
Barkow, J., Cosmides, L., & Tooby, J. (1992). The adapted
mind: Evolutionary psychology and the generation of culture:
Oxford University Press, USA.
Reynolds, R. G., Whallon, R., Ali, M. Z., & Zadegan, B. M.
(2006, July 16-21). Agent-Based Modeling of Early Cultural
Evolution. Paper presented at the 2006 IEEE Congress on
Evolutionary Computation, Vancouver, BC, Canada.
Charnov, E. (1976). Optimal foraging, the marginal value
theorem. Theoretical population biology, 9(2), 129.
Richerson, P., & Boyd, R. (2005). Not by genes alone: How
culture transformed human evolution: University of Chicago
Press.
Goodhall, S. J. (2002). Multi-agent simulation of huntergatherer behavior. Unpublished MS thesis, Wayne State
University, Detroit, MI.
Holland, J. (1992). Adaptation in natural and artificial systems:
MIT Press Cambridge, MA, USA.
Johnson, A., & Earle, T. (2000). The evolution of human
societies: from foraging group to agrarian state: Stanford Univ
Pr.
Kennedy, J., & Eberhart, R. (1995). Particle Swarm
Optimization. Paper presented at the Proceedings of the 1995
IEEE International Conference on Neural Networks, Perth,
Australia.
Langton, C. (1992). Life at the edge of chaos. Artificial life II,
10, 41–91.
51