effects of adaptive social networks on the robustness of evolutionary algorithms

8/6/2019 EFFECTS OF ADAPTIVE SOCIAL NETWORKS ON THE ROBUSTNESS OF EVOLUTIONARY ALGORITHMS

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1

JAMES M. WHITACRE

Birmingham University, School of Computer Science

Edgbaston, Birmingham, B15 2TT, UK

[email protected]

RUHUL A. SARKER

University of New South Wales at the Australian Defence Force Academy

School of Information Technology and Electrical Engineering,

Canberra 2600, Australia

[email protected]

Q. TUAN PHAM

University of New South Wales, School of Chemical Sciences and Engineering,

Sydney, 2052 Australia

[email protected]

AbstractBiological networks are structurally adaptive and take on non-random topological properties

that influence system robustness. Studies are only beginning to reveal how these structural features

emerge, however the influence of component fitness and community cohesion (modularity) have attracted

interest from the scientific community. In this study, we apply these concepts to an evolutionary

algorithm and allow its population to self-organize using information that the population receives as it

moves over a fitness landscape. More precisely, we employ fitness and clustering based topological

operators for guiding network structural dynamics, which in turn are guided by population changes

taking place over evolutionary time. To investigate the effect on evolution, experiments are conducted on

six engineering design problems and six artificial test functions and compared against cellular genetic

algorithms and panmictic evolutionary algorithm designs. Our results suggest that a self-organizing

topology evolutionary algorithm can exhibit robust search behavior with strong performance observed

over short and long time scales. More generally, the coevolution between a population and its topology

may constitute a promising new paradigm for designing adaptive search heuristics.

Keywords: Evolutionary Algorithms; Network Evolution; Optimization; Adaptive Population

Topology; Self-Organization

1. INTRODUCTION

Local interaction constraints have a strong influence on the global dynamics of complex

systems. Restricting interactions in population-based evolutionary simulations has been foundto promote robustness against parasitic invasion

1,2, enhance speciation rates

3, sustain

population diversity in rugged fitness landscapes4, facilitate the emergence of cooperative

EFFECTS OF ADAPTIVE SOCIAL NETWORKS ON THE ROBUSTNESS OF EVOLUTIONARY

ALGORITHMS


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2 James Whitacre, Ruhul Sarker, and Tuan Pham

behavior5, enhance robustness towards local failures

6, and may influence system evolvability,

i.e. a systems propensity to adapt7.Parallel developments have taken place in population based search heuristics such as

evolutionary algorithms, where restricting interactions in the competition and mating of

individuals in a population has been found to influence many facets of algorithm behavior.

This has been reported in several seemingly disparate studies involving age restrictions in

genetic algorithms8, genealogical and phenotypic restrictions through Deterministic Crowding

(DC)9, limited interactions between heterogeneous subpopulations

10, and explicit static

topologies for constraining interactions in cellular genetic algorithms (cGA)11-16

.

1.1. Population Networks for Evolutionary Algorithms

Defining an EA population on a network modifies an EA by localizing its genetic operators,

i.e. restricting mating and selection to occur only among individuals directly connected or

near each other within the network. Three types of population structures commonly studied in

EAs are shown on the top row ofFig. 1. The fully connected graph in Fig. 1a represents the

canonical EA design, which we refer to as the panmictic EA (PEA). In PEA, each individual

(represented by a node in the graph) can interact with every other and no definition of locality

is possible. The network in Fig. 1b represents an island model where individuals reside in

panmictic subgroups or islands. In Fig. 1b, the large arrows represent migrations between

islands that occur every few generations. As a consequence of this topology, locality is

specified on a scale that can be considerably larger than the individual. The final EA

structure shown in Fig. 1c is referred to as a cellular Genetic Algorithm (cGA). In cGA, the

network of interactions takes on a lattice structure with interactions constrained by the

dimensionality of the lattice space. The ring topology in Fig. 1c is an example of a one

dimensional lattice with periodic boundary conditions. With the cGA, each individual has aunique environment defined by its own set of links, i.e. a neighborhood.


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Adaptive Networks and Robustness in Evolutionary Algorithms 3

Fig. 1: Examples of networks. The networks on the top row represent common EA population structures and are

known as (from left to right) panmictic, island model, and cellular population structures. Networks on the bottom

row have been developed with one or more characteristics of biological networks and are classified as (from left to

right) Self-Organizing Networks (presented here), Hierarchical Networks17, and Small World Networks18. Fig. 1e is

reprinted with permission from AAAS.

The ratio of neighborhood size (i.e. number of connections per node) to system size (i.e. total

number of nodes) provides one measurement of locality that decreases in the networks from

left to right on the top row of Fig. 1. However, these networks also share important

similarities. Within each network (Fig. 1a-c), nodes have the same number of interactions and

the same types of interactions, i.e. regular graphs, and each of the networks are static and

predefined. These properties are notably distinct from those of biological networks. As seen

in metabolic pathways, cell signaling, protein-protein interactions, and gene regulation, most

biological networks have evolved several similar topological characteristics19

and some of

these have been found to support robustness towards certain types of perturbations1,2,6

.

While the structure of biological networks has developed slowly over evolutionary time, at

shorter timescales it also supports robust autonomous responses to internal and environmental

perturbations, e.g. through the dynamic formation of modular units. Such evolutionarily-

constrained structural plasticity is observed at every scale in biology including protein

interactions (e.g. molecular assemblies), cellular functions (e.g. lymphocyte avidity and

formation of the immunological synapse), neural rewiring in the brain, morphological

plasticity of multi-cellular organisms, and food web rewiring within ecosystems (e.g. adaptive

foraging). Structural adaptation in these networks changes how information is processed from

the environment and subsequently alters the system-wide traits that emerge from the

integrated actions of their constituent elements. In this study, we investigate whether

mimicking the structural plasticity of biological systems can influence the performance

characteristics of an evolutionary search process.

1.2. Robustness in Biology and Search Heuristics

In systems biology, robustness typically refers to the capacity to maintain system integrity,

functionality, or phenotypic traits in the face of component changes or failures or in the face

of changes in the external environment. This does not imply that the biological system is

static or in equilibrium but only that the measured system property that is robust has displayed

little sensitivity to encountered perturbations20

. Thus we can speak of the robustness of

development in multi-cellular organisms (i.e. developmental canalization21

) or the robustness

of an organisms fitness post-development that is achieved through adaptive phenotypic

plasticity22

. In either case, aspects of a systems morphological structure are driven to new

configurations, which are partly guided by feedback from the external environment and act to

confer stability within higher level traits.Within the context of a stochastic search process such as an evolutionary algorithm, our

interest is in robustness to what one might call intermittent imperfections in search bias.

These intermittent imperfections arise at several distinct scales within optimization. For


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instance, robustness measures are sometimes used to quantify the sensitivity of a solution

towards noise or errors in fitness evaluations, the sensitivity of a search process towards localattractors within a fitness landscape, sensitivity towards initial conditions of the population, or

more generally, the sensitivity of algorithms performance over multiple runs, i.e.

performance reliability. Finally, a robust algorithm framework might also be described as one

that is reliable across problems with somewhat unique fitness landscape properties. Proxies

for many of these types of robustness are evaluated in this study.

1.3. SOTEA

In this paper, we investigate evolutionary algorithms with a population topology that changes

in response to interactions between the population and fitness landscape; what we have

referred to previously as Self-Organizing Topology Evolutionary Algorithms (SOTEA)4.

Although some studies have investigated the search characteristics of EAs with non-regular

population topologies14-16

, few have investigated the behavior of EAs that evolve on an

adaptive network. One exception is seen in23

where the grid shape of a cellular GA adapts in

response to performance data using a predefined adaptive strategy. In that system, structural

changes are globally controlled using statistics on system behavior and topological changes

do not deviate from a lattice structure. In contrast, the algorithms in this study adapt to

(topologically) local conditions through a coevolution of network states and network

structure.

Previous SOTEA research: In previous research4, we developed a SOTEA model using

simple rules that allowed a populations structure to coevolve with EA population dynamics.

Structural modifications were driven by a contextual definition of fitness ranking and the

topological changes were designed to loosely mimick the process of gene duplication and

divergence in genetic evolution. This resulted in population topologies exhibiting somecharacteristics that were similar to biological networks and more importantly, a capacity to

sustain genetic diversity within rugged fitness landscapes. An example of a network which

evolved using this algorithm is shown in Fig. 1d.

This earlier SOTEA algorithm was developed to explore theoretical topics related to

evolution on rugged fitness landscapes and was not easily modified for practical optimization

purposes. For instance, the genetic diversity observed in the first SOTEA did not persist in

correlated fitness landscapes (a prominent feature in optimization problems)4

and the

algorithm did not appear to be easily amenable to sexual reproduction. In contrast, the

present study focuses on improving the optimization search characteristics of evolutionary

algorithms with an adaptable population topology. What we report in this paper is the

development of an evolutionary algorithm framework that achieves robust performance

characteristics through the creation and exploitation of structural properties.In the next section, we briefly review common topological properties of complex networks as

well as network models that can recreate some of these properties in silico. Section 3.

presents the SOTEA adaptive network and Section 4. describes our experiments including


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pseudocode and a summary of the SOTEA algorithm. Results are provided in Sections 5.

and 6. with discussion and conclusions in Sections 7. and 8. .

2. Structural Characteristics of Complex Networks

2.1. Properties of real networks

Many natural and manmade systems consist of large networks of interacting components as

seen in biology (e.g. gene regulatory networks, food webs, neural networks), social systems

(e.g. co-authorship, personal relationships, organizations) and manmade systems (e.g. internet,

power grids). Despite the considerable simplifications needed to create network

representations of these systems and despite their inherent differences in scale, environmental

context and functionality, many real networks have surprising similarities in their topological

properties. These similarities include small characteristic path lengths, high clusteringcoefficients, fat-tailed degree distributions (e.g. power law), degree correlations, and low

average connectivity. Each of these features are notably distinct from random graphs and

regular lattices. Below we describe and formally define these topological properties, and in

Section 6. we use these properties to characterize the networks evolved in this study.

Comprehensive descriptions of these properties can also be found in19,24,25

.

2.2. Topological Metrics

Networks are represented by an adjacency matrixJof sizeN, such that individual nodes i and

j are connected (not connected) when Jij=1 (Jij=0). All networks discussed in this study are

unweighted and undirected (symmetric).

Characteristic Path Length: The path length is the shortest distance between two nodes in a

network. The characteristic path length L is the average path length over all node pair

combinations in a network. Generally,L grows very slowly with increasing system size (e.g.

population size)Nin complex systems. For instance, networks exhibiting the Small World

property, such as the network in Fig. 1f, haveL proportional to logN26

.

Degree Average: The degree ki is the number of connections that node i has with other

nodes in the network. The degree average kave is simply k averaged over all nodes in the

network. The degree average is expected to remain small, even for large networks24

.

=

=

N

j

jii Jk1

, (1)

Degree Distribution: The degree distribution has been found to closely approximate a

power law for many biological systems with power law and exponential distributions often

fitting abiotic complex systems25

. Networks which display a power law k distribution are

often referred to as scale free networks in reference to the scale invariance ofk.


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Clustering Coefficient: Many complex biological systems have high levels of modularity

which is typically indicated by the clustering coefficient. The clustering coefficient for a nodeci is a measure of how well the neighbors of a given node are locally interconnected. More

specifically, ci is defined as the ratio between the number of connections ei among the ki

neighbors of node i and the maximum possible number of connections between these

neighbors which is ki(ki-1)/2. The clustering coefficient for a networkc is simply the average

ci value.

( )1

2

=

ii

i

ikk

ec (2)

Although in practice, more efficient calculation methods are used, ei can be formally defined

using the adjacency matrixJas shown in eq. (3).

kjiJJJeN

j

N

k

jkikiji

=

= =

,1 1

(3)

Clustering-Degree Correlations: A common feature of biological and social systems is the

existence of a hierarchical architecture. Such an architecture is believed to require that

sparsely connected nodes form tight modular units or clusters and communication paths

between these modular units are maintained via the presence of a few highly connected

hubs26. Fig. 1e shows a network with these hallmark signs of modularity and hierarchy which

was grown using the deterministic models presented in17

.

The existence of hierarchy in a network is typically measured by evaluating the correlation

between the clustering coefficient and the node degree. Based on the description given above,

a hierarchical network is expected to exhibit higher connectivity for nodes with low clustering(i.e. hubs) and vice versa. Furthermore, for the feature of hierarchy to be a scale invariant

property of the system, c should have a power law dependence on k.

Degree-Degree Correlations: For many complex networks, there exist degree correlations

such that the probability that a node of degree k is connected to another node of degree k`

depends on k. This correlation is typically measured by first calculating the average nearest

neighbors degree kNN,i.

=

=

N

j

jji

i

iNN kJk

k1

,,

1(4)

Networks are classified as assortative ifkNNincreases with kor disassortative ifkNNdecreases

with k. Degree correlations are often reported as the value of the slope for kNN as a linearfunction ofk.

Random Networks: Thus far, only qualitative statements have been given regarding the

topological properties of complex networks. In many cases, when topological properties are


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mentioned as being large or small (as has been mentioned above), the statements are referring

to property values in relation to those values observed in random graphs and particularly themodels developed by Erds and Rnyi

27,28

. As reviewed in19

, random graphs have i) a

characteristic path length LRand similar to that observed in complex networks and

approximated by eq. (5), ii) a Poisson degree distribution (as opposed to the fat tailed degree

distribution in complex networks), and iii) a clustering coefficient cRandgiven by eq. (6) which

is orders of magnitude smaller than what is typically seen in complex networks18

. Random

graphs also do not exhibit any degree correlations or correlations between the degree and the

clustering coefficient.

( )( )Ave

Randk

NL

ln

ln (5)

N

kc Ave

Rand =

(6)

2.3. Network Growth Models

In many man-made and biological systems, it is generally understood that network

development occurs through a process of constrained growth and coevolution with the

environment. Over the last decade, progress has been made in the design of network growth

models which evolve to display characteristics found in real-world complex systems.

Exemplars of this success can be seen in the Barabasi-Albert (BA) Model

29

, the Duplicationand Divergence (DD) Model

30, the intrinsic fitness models in

31and the stochastic walk models

in32

. The emergence of important topological properties often occurs through the use of

simple, locally defined rules that constrain structural dynamics and are guided by state

properties of the nodes. In other words, the connections in the network change and nodes are

added or removed with a bias derived by property values that are assigned or calculated for

each node. Properties that have been used in models include the degree of a node k29

,

measures of node modularity33

, as well as measures of node fitness31

.

3. SOTEA Model Description

Our model aims to adapt an EA population topology in a manner that is inspired by biological

phenomena but also that is relevant in an optimization context. Like the models mentioned in

the previous section, the topological changes in SOTEA are driven by properties that can be

calculated locally within the network. One property that we focus on is modularity; a


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structural feature that often contributes to the robustness of natural systems. Importantly, the

dynamic construction of modularity can alter the behavior of constituent elements to be basedlargely on interactions with other members. This not only encourages specialization and

efficiency, it also can protect other parts of a system, e.g. from error propagation. In a search

process, dynamically constructed modularity may help to focus individuals on promising

regions of a solution space while reducing sensitivity to local attractors at the population

level. In other words, dynamically constructed modularity may help to facilitate both

exploitation and exploration within a distributed search process. To encourage modularity, we

use a combination of fitness measures and measures of network clustering (described in

Topological Driving Forces). The network dynamics are implemented by rewiring local

regions of the network (described in Topological Operators).

3.1. Topological Driving Forces

The SOTEA network is represented by an adjacency matrix Jsuch that individuals i andj are

connected (not connected) whenJij=1 (Jij=0). This study only deals with undirected networks

such that Jij = Jji. The terms individual and node are used interchangeably to refer to

members of the EA population situated within a network. Also, the terms links and

connections are used interchangeably to refer to directly connected nodes, i.e. individuals that

are neighbors in the population.

Topological driving forces encourage the emergence of (partially) isolated

clusters that are integrated with the broader population through high fitness

hubs. This is done by: 1) encouraging high fitness nodes to be highly

connected and 2) by encouraging clustering between solutions that are not of

high fitness.

3.1.1. fitness- degree correlations

High fitness nodes are driven to achieve higher connectivity kin the following manner. First,

an adaptive set point KSet establishes a nodes desired number of links as defined for node i in

eq. (7). The value for KSet is defined in eq. (8) as a quadratic function of fitness ranking with

a lower bound ofKMin = 3 and an upper bound KMax.

The equations (7) and (8) drive node connectivity ki to high values for nodes with

exceptionally good fitness. Enforcing a lower bound ofKMin = 3 ensures clustering is feasible

in the lowest fit nodes while the quadratic form of the set point KSet helps ensure that only the

most highly fit nodes are able to attain high connectivity, i.e. hub positions. We treat KMax as

a parameter of the model, which can be used to control the extent that high fitness nodes are

able to influence network dynamics. As seen in the experimental results, the optimal settingof this parameter changes depending on the problem.

iSeti KkMin , (7)


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( )

+=

2

,N

RankNKKKK iMinMaxMiniSet (8)

3.1.2. weighted clustering coefficient

To encourage high modularity amongst lower fitness nodes, network rewiring is driven to

maximize a weighted version of the clustering coefficient c as defined for node i in eq. (9).

The more common definition of c was provided in Section 2. . In eq. (9), a connections

contribution to c is weighted to give less importance to connections involving nodes of higher

fitness. This weighting factor W is defined in (11) and alters the ei term of the clustering

coefficient in (10). This weighting factor is identical to the intrinsic fitness measure used in

the network growth models in31.

( )1

2 **

=

ii

ii

kk

ecMax (9)

kjiWJJJeN

k

jkjkik

N

j

iji = ==

,11

*(10)

2N

RankRank

W

kj

jk

= (11)

3.2. Topological Operators

Section A described the driving forces for network structural dynamics. To respond to these

forces, changes to the network take place involving the addition, removal, and transfer of

links. Below we define the rules (topological operators) for executing these structural

changes and also illustrate their implementation in Fig. 2. The add link rule and the remove

link rule are topological operators that allow the k value for each node to reach KSet. The

transfer link rule allows for the improvement of local clustering within the network, but only

if this does not impede upon the desired ksettings for each node.

Add Link Rule: Starting with a selected nodeN1, a two step random walk is taken, moving

from nodeN1 to nodeN2 to nodeN3. IfN1 wants to increase its number of links (kN1 < KSet)


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andN3 wants to increase its number of links (kN3 < KSet) then a link is added betweenN1 and

N3.Remove Link Rule: For a selected nodeN1 with kN1 > KSet, a two step random walk is taken,

moving fromN1 toN2 toN3. IfN3 is already connected toN1 (JN1,N3=1) and kN3 > KSet then

remove the link betweenN1 andN3. Notice the presence ofN2 withJN2,N1=JN2,N3= 1 ensures

that connections removed using this rule do not result in network fragmentation.

Transfer Link Rule: For a selected nodeN1 a two step random walk is taken, moving from

N1 toN2 toN3. IfkN3 < KSet, then the connection betweenN1 andN2 is transferred to now be

betweenN1 andN3 (i.e.JN1,N2= 1,JN1,N3= 0 changes toJN1,N2= 0,JN1,N3= 1). To determine if

the transfer will be kept, the local modularity is calculated using (9) for N1, N2 andN3 both

BEFORE and AFTER the connection transfer occurs. If ( )* 3*

2

*

1 NNNccc ++ increases after the

connection transfer then the transfer is kept, otherwise it is reversed. In this way connections

are only added which strengthen the weighted clustering metric and dont cause a net increase

in KSetviolations.

Fig. 2 Topological Operators: A selected nodeN1 will attempt to add, remove or transfer its connections based on

the satisfaction of constraints and the improvement of properties. Add Rule: The dotted line represents a feasible

new connection in the network assuming nodesN1 andN3 both would like to increase their number of connections.

Remove Rule: The gray dotted line represents a feasible connection to remove in the network assuming nodesN1

andN2 both have an excess of connections. Transfer Rule: The connection betweenN1 andN2 (gray dotted line) is

transferred to now connectN1 andN3 (black dotted line) if this action results in an overall improvement to local

clustering. There are several constraints that each rewiring rule must satisfy in order to be executed. Consequently,

in each instance of rule usage, we make up to ten attempts to satisfy the conditions for executing a rule, i.e. ten

stochastic walks starting from a nodeN1.

The topological operators determine how connections are added and removed in the network.

These operators were developed based on several considerations. First, unlike systems that

operate in a physical space, there are no a priori constraints on topological changes and it was

thus necessary to determine how stochastic interactions between nodes should take place.

When defining operators for modifying a network topology, we felt it was important to: 1)

maintain the notion of locality that is implied by the network (i.e. prohibit long-range


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interactions) 2) ensure that the network does not fragment into disconnected sub-networks and

3) keep the rules as simple as possible. These were the primary considerations that guided thedevelopment of these topological operators.

4. Experimental Setup

4.1. Algorithm Designs

4.1.1. SOTEA

A high level pseudocode for SOTEA is provided below. The algorithm starts by defining the

initial population P on a ring topology with each node connected to exactly two others (e.g.

Fig. 1c). For a given generation t, each node N1 is subjected to both topological and genetic

operators. Once the topological operators are executed (defined in Section 3.2. ), N1 is

selected as a parent and a second parent N2 is selected by conducting a two step stochastic

walk across the network. An offspring is created using these parents and a single search

operator that is selected at random from Tab. 1. The better fit between the offspring andN1 is

stored in a temporary list Temp(N1) while the topological and genetic operators are repeated

on the remaining nodes in the population. The population is then updated with the temporary

list to begin the next generation. This sequence of steps is repeated until a stopping criterion

is met. In all experiments, the stopping criterion is set as a maximum 150,000 objective

function evaluations.

The two-step stochastic walk mating scheme is used to maintain consistency with the

topological operators. This both simplifies our model and allows for a more intuitive

understanding of system dynamics. This mating scheme is expected to generate a weak

selection pressure in most EAs, however this is not necessarily the case for SOTEA. Because

high fitness nodes are driven towards increased connectivity, they are more likely to be

encountered in a stochastic walk across the network. Hence, the selection pressure becomes a

locally defined property that can be much stronger than stochastic walk mating would

otherwise create for panmictic or cellular EAs.

Pseudocode for SOTEA

t=0

Initialize P(t) (at random)

Initialize population topology (ring structure) [Fig. 1c]

Evaluate P(t)

Do

For each N1 in P(t)

Add Link Rule(N1) [Section III.B]Remove Link Rule(N1) [Section III.B]

Transfer Link Rule(N1) [Section III.B]

Select N1 as a first parent


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Select parent N2 by conducting a two step stochastic walk from N1

Select Search Operator (at random from Tab. 1)Create and Evaluate offspring

Temp(N1) = Best_of(offspring, N1)

Next N1

t++

P(t) = Temp()

Loop until stopping criteria

4.1.2. cellular GA

SOTEA is compared with cellular and panmictic EAs. The cellular GA used in these

experiments is identical to SOTEA except for two design changes (see pseudocode). First,

the cGA does not implement any topological operators and maintains a static ring topology.The second change is that during mating, the second parent N2 is selected among all

neighbors within a radiusR fromN1 using linear ranking selection. This additional departure

from SOTEA was made based on experimental evidence that it enhances the performance of

the cGA. In experiments where mating took place using random walks of length R (i.e. the

mating scheme in SOTEA), the cGA displayed exceptionally poorer performance across all

problems in this study. Moreover, in a thorough study on the performance of distributed and

non-distributed GA designs34

, the cGA we use (referred to in34

as ci) frequently exhibited

the best performance.

Pseudocode for cGA

t=0

Initialize P(t) (at random)

Initialize population topology (ring structure) [Fig. 1c]Evaluate P(t)

Do

For each N1 in P(t)

Select N1 as first parent

Select N2 from Neighborhood(N1,R)

Select Search Operator (at random from Tab. 1)

Create and evaluate offspring

Temp(N1) = Best_of(offspring, N1)

Next N1

t++

P(t) = Temp()



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4.1.3. Panmictic EAs

SOTEA is compared against 16 distinct Panmictic EA designs that in some cases vary

significantly from the SOTEA and CGA algorithms. The variety of PEA designs

corresponded with a variety of performance outcomes. In particular, the variance in

(performance-based) ranking of PEA designs was typically greater than in SOTEA or cGA

and the best performing PEA typically depended on the problem.

Two PEA design frameworks were considered: one where selection schemes are applied

during mating and one where selection schemes are applied through a culling process. These

are labeled as Evolution Strategies (ES) and Genetic Algorithm (GA) designs respectively.

The core of the ES style Panmictic EA is given by the pseudocode below. For this

pseudocode, the parent population of size at generation tis defined by P(t). For each new

generation, an offspring population P`(t) of size is created through variation operators and is

evaluated to determine fitness values for each offspring. As was also the case in cGA andSOTEA, offspring are created by selecting a single operator at random from Tab. 1. The

parent population for the next generation is then selected from P`(t) and Q, where Q is subset

ofP(t). Q is derived from P(t) by selecting those in the parent population with an age less

than .

Pseudocode for Panmictic EA (ES)

t=0

Initialize P(t)

Evaluate P(t)

Do

For i=1 to {

{p1, p2} = Select randomly from P;

c = Create an offspring from{p1, p2};Add c to P'(t);

Next i

P`(t) = Variation(P`(t))

Evaluate (P`(t))

P(t+1) = Select(P`(t) Q)

t=t+1


Eight ES designs are tested which vary by the use of Generational (with elitism) vs. Pseudo

Steady State population updating, the use of Binary Tournament Selection vs. Truncation

Selection, and by the number of search operators. Details are given below for each of the

design conditions.Population updating: The generational EA design (with elitism for retaining the best parent)

has the parameter settings N==2, =1 (= for best individual). The pseudo steady state

EA design has the parameter settingsN==, =.


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Selection: Selection occurs by either binary tournament selection (without replacement) or by

truncation selection.Search Operators: For each EA design, an offspring is created by using a single search

operator. Two designs were considered: i) a seven search operator design and ii) a two search

operator design. For the seven operator case, an offspring is created by an operator that is

selected at random from the list in Tab. 1. For the two operator case, uniform crossover is

used with probability = 0.95 and single point random mutation is used with probability = 0.05.

Tab. 1: The seven search operators used in the cellular GA, SOTEA, and selected Panmictic EA designs are listed

below. More information on each of the search operators can be found in35.

Search Operators

Wrights Heuristic Crossover

Simple Crossover

Extended Line CrossoverUniform Crossover

BLX-

Differential Evolution

Operator

Single Point Random Mutation

GA Designs: The previous algorithmic framework invokes selection after offspring are

generated and in this way is most similar to evolution strategies. To include experiments with

the more commonly used genetic algorithm, we use the pseudocode below. In this case, =,

=1 (Steady State) and selection from P occurs using either Linear Ranking (Lin) or Binary

Tournament Selection.

Pseudocode for Panmictic EA (GA)

Initialize P;Evaluate P;

Do{

P' ={};

For i=1 to {

{p1, p2} = Select individuals from P;

c = Create an offspring from{p1, p2};

Add c to P';

Next i

P = replacement (P`(t) U Q)

Evaluate P;


Constraint Handling: Each of the engineering design case studies involve nonlinearinequality constraints. Solution feasibility is addressed by defining fitness using the

stochastic ranking method presented in36

. Parameter settings for stochastic ranking were

taken from recommendations found in36

.


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5. Performance Results

The experimental results are evaluated using several metrics and statistical tests in order to

gain a clearer picture of the strengths and weaknesses of SOTEA. In concluding the section,

we summarize these results and relate them back to different concepts of algorithm

robustness. A summary of our methods for analyzing algorithm performance is given below

followed by a summary of results for each problem.

Performance profiles: Performance profiles comparing SOTEA and cGA are provided in

Figure 3. Each algorithm searches for up to a maximum 150,000 objective function

evaluations. Experiments with SOTEA test different settings ofKmax while the cellular GA

was run with different settings of neighborhood radius R. Performance for each EA is

reported as the median objective function value over 30 runs. The caption text in Figure 3

includes optimal (Fopt) or best known (Fbest) objective function values for each problem.

Statistical Tests: To compare performance between specific algorithm designs that aretuned for a particular problem, we take the best algorithm from each class and calculate the

confidence in algorithm performance superiority using a non-parametric statistical test (i.e.

the Mann-Whitney U-Test). To compare algorithm classes, U tests are conducted using all of

the performance results from each class. Tab. 3 provides p values for these tests with

confidence levels under 99% (p>0.01) listed as statistically insignificant.

5.1. Engineering Design Performance Results

This section presents algorithm performance results for six engineering design problems (for

formal problem definitions, see35

) that have proven difficult to solve to optimality and have

been used frequently in other studies. For several of these problems, the best known solution

has steadily improved over time, however the optimal solution remains unknown. We

compare the best solutions from previous studies with those obtained in this study. We should

point out however that conclusions derived from comparisons between different studies

should be made with caution. Because the results reported within each study utilize different

computational resources, it is not entirely appropriate to make direct comparisons. With this

caveat in mind, our intention is to compare the best final solutions from amongst the many

algorithms within each study in order to get a sense of the potential capabilities of these best

in study algorithms. Importantly, there were a number of engineering problems where

SOTEA obtained the best result ever reported for the problem and we felt that this was useful

information to provide as it demonstrates our algorithms potential utility. In the current

study, the maximum number of function evaluations was chosen in order to be similar to these

other studies yet also consistent across all experiments.


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Figure 3 Performance profiles for the pressure vessel (Fopt=5850.38), alkylation process (Fopt=1772.77), heat

exchanger network (Fopt=7049.25), gear train (Fbest=2.70E-12, reported in37

), tension compression spring(Fbest=0.01270, reported in

38), and welded beam (Fbest=1.7255, reported in39) design problems.


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Tab. 2: Performance results for six engineering design problems are shown for twelve Evolutionary Algorithms run

for 3000 generations with algorithm designs varying by the use of Generational (Gen) or Pseudo Steady State (SS)

population updating, the use of Binary Tournament Selection (Tour) or Truncation Selection (Trun), and the number

of search operators (Nops). Performance is presented as the single best objective function value found in 30 runs

FBestas well as the average objective function value over 30 runs FAve. All EAs listed below obtained a feasible

solution within 3000 generations. The single best fitness values found for each problem are in bold.

EA Gen Sel Nops Pressure Vessel Heat Exchanger Alkylation Process

FBest FAve FBest FAve FBest FAve

ES SS Tour 7 6059.70 6190.31 7053.47 7109.20 1771.35 1750.38

ES SS Trun 7 6059.73 6214.31 7056.09 7179.02 1760.77 1630.90

ES Gen Tour 7 5953.06 6123.22 7116.72 7213.38 1711.00 1667.34

ES Gen Trun 7 5964.23 6174.55 7186.97 7250.82 1641.47 1495.13

ES SS Tour 2 5867.87 6382.61 7070.57 7233.18 1756.00 1708.38

ES SS Trun 2 5857.39 6449.57 7093.12 7269.02 1748.95 1661.17

ES Gen Tour 2 6144.69 6340.23 7235.69 7412.11 1621.77 1510.93

ES Gen Trun 2 6188.86 6391.15 7184.51 7398.23 1501.24 1343.48

GA SS Tour 7 5903.55 6418.48 7092.00 7399.75 1767.22 1649.42

GA SS Lin 7 5853.21 6390.27 7050.31 7303.13 1759.20 1533.20

GA SS Tour 2 6091.55 6491.42 7063.97 7290.57 1764.93 1675.21

GA SS Lin 2 6074.73 6617.18 7094.76 7332.24 1751.35 1554.77

Gear Train Tension Compression Welded Beam


ES SS Tour 7 2.70E-12 2.62E-10 0.012665 0.012758 1.72485 1.74602

ES SS Trun 7 2.70E-12 7.70E-10 0.012665 0.012778 1.72494 1.80945

ES Gen Tour 7 2.70E-12 2.70E-12 0.012679 0.012710 1.75465 1.77920

ES Gen Trun 7 2.70E-12 1.09E-11 0.012687 0.012725 1.76485 1.79732

ES SS Tour 2 2.70E-12 1.12E-09 0.012701 0.013861 1.73570 1.96193

ES SS Trun 2 2.31E-11 1.81E-09 0.012804 0.015078 1.73060 2.06087

ES Gen Tour 2 2.70E-12 4.74E-12 0.012739 0.013035 1.83742 1.93124

ES Gen Trun 2 2.70E-12 2.70E-12 0.012694 0.012864 1.75302 1.88472

GA SS Tour 7 2.31E-11 1.12E-09 0.012665 0.012969 1.72599 1.96120

GA SS Lin 7 2.70E-12 6.39E-10 0.012665 0.012906 1.72673 1.89600

GA SS Tour 2 2.31E-11 2.98E-09 0.012879 0.015302 1.72830 2.06871GA SS Lin 2 2.70E-12 3.14E-09 0.013073 0.015830 1.82331 2.21587

Tab. 3 Mann-Whitney U tests comparing best algorithms from each design class (first entry) and comparing all data

from design classes (second entry). For best in class comparisons (first entry), the best algorithm from a design class

is determined based on median performance after 150,000 evaluations. Winner of test is indicated along with p

value. insig indicates p > 0.05.

Problem PEA vs. SOTEA cGA vs. SOTEA PEA vs. cGA

Pressure Vessel

SOTEA (p


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Alkylation Proc.

SOTEA (p


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reported in41. Results are also reported for39 however their solution violates integer constraints for the 3rd and 4th

parameters making their final solution infeasible. It should also be mentioned that equations for defining the

problem have errors in38 and41. The best solution found in these experiments was (F,x1, x2, x3, x4) = (5850.37,

38.8601, 221.365, 12, 6).

Reference Fitness Ranking

Sandgren, 1990 0 8129.80 11

Fu, 1991 2 8084.62 10

Kannan and Kramer, 1994 3 7198.04 9

Cao, 1997 4 7108.62 8

Deb, 1997 5 6410.38 7

Lin 199937 6370.70 6

Coello, 199938 6288.74 5

Zeng et al., 200239 5804.39 --

Li et al., 2002 1 5850.38 3

SOTEA (This Work) 5850.37 1

cGA (This Work) 5850.37 1

Panmictic EA (This Work) 5853.21 4

5.3. Alkylation Problem

The alkylation process design problem, originally defined in46

, has the goal of improving the

octane number of an olefin feed stream through a reaction involving isobutene and acid. The

reaction product stream is distilled with the lighter hydrocarbon fraction recycled back to the

reactor. The objective function considers maximizing alkylate production minus the material

(i.e. feed stream) and operating (i.e. recycle) costs. Design parameters all take on continuous

values and include the olefin feed rate x1 (barrels/day), acid addition rate x2 (thousands of

pounds/day), alkylate yieldx3 (barrels/day), acid strengthx4 (wt. %), motor octane numberx5,

external isobutene to olefin ratiox6, and the F-4 performance numberx7.

Fig. 5 Simplified diagram of an alkylation process (recreated from47)

Results: All but one of the SOTEA algorithms outperformed all cGA designs (Figure 3) andthe best tuned algorithm was a SOTEA design (Tab. 3). For this problem there was no clear

trend between performance and network connectivity. PEA algorithms performed relatively

poorly on this problem (Tab. 2). Comparisons to studies from previous authors (see Tab. 5)


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highlight the strong performance of the distributed EAs. Of the stochastic search methods

described in the five studies referenced in47 including their own differential evolutionalgorithms, none reached the fitness values obtained by the distributed EA designs employed

here. However, two BB (Branch and Bound non-linear programming) algorithms were cited

that did find the global optimum and did so more consistently than SOTEA or cGA.

Tab. 5 Comparison of results for the alkylation process design problem (maximization problem). Results from other

authors were reported in47. The best solution found in these experiments was (F,x1, x2, x3, x4, x5, x6, x7) = (1772.77,

1698.18, 53.66, 3031.3, 90.11, 95, 10.5, 153.53).


Bracken and McCormick, 1968 8 1769 6

Maranas and Floudas, 1997 9 1772.77 1

Adjiman et al., 199850 1772.77 1

Edgar and Himmelblau, 200151 1768.75 7

Babu and Angira, 2006 7 1766.36 8SOTEA (This Work) 1772.77 1

cGA (This Work) 1772.77 1


5.4. Heat Exchanger Network (HEN) Problem

The Heat Exchanger Network design problem, originally defined by52

, has the goal of

minimizing the total heat exchange surface area for a network consisting of one cold stream

and three hot streams. As shown in Fig. 6, there are eight design parameters consisting of the

heat exchanger areas (x1, x2, x3), intermediate cold stream temperatures (x4, x5) and hot stream

outlet temperatures (x6, x7, x8). The problem is presented below in a reformulated form taken

from53

where a variable reduction method has been used to eliminate equality constraints.

Fig. 6 Heat Exchanger Network Design involves 1 cold stream that exchanges heat with three hot streams.

Parameters to optimize include heat exchange areas (x1, x2, x3) and stream temperatures (x4, x5, x6, x7, x8).

Results: All of the SOTEA algorithms outperformed the cGA designs (Figure 3) and the best

tuned algorithm was a SOTEA design (Tab. 3). Performance tended to improve as network


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Cao and Wu, 1997

4

2.36 x10

-9

5Lin et al. 199937 2.70 x10-12 1

SOTEA (This Work) 2.70 x10-12 1

cGA (This Work) 2.70 x10-12 1

Panmictic EA (This Work) 2.70 x10-12 1

5.6. Tension Compression Spring Design Problem

The Tension Compression Spring problem, shown in Fig. 7, has the goal of minimizing the

weight of a tension/compression spring subject to constraints on minimum deflection, shear

stress, surge frequency, and dimensional constraints38

. There are three design parameters to

optimize consisting of the mean coil diameterD, the wire diameter dand the number of active

coilsN.

Fig. 7 Diagram of Tension Compression Spring. Parameters of the problem include the mean coil diameterD, the

wire diameter dand the number of active coilsNwhich is represented by the number of loops of wire in the

diagram. Forces acting on the spring are shown as P. This figure is taken out of38 and is reprinted with permission

from IEEE ( 1999 IEEE).

Results: All but one of the distributed EA designs converge to similar values (Figure 3).Comparing the results from previous studies, we find strong performance from both

distributed EAs. Of the three studies referenced in and including38

, no previous method has

been able to find the solutions reported in this study.

Tab. 8 Comparison of results for the tension compression spring problem (minimization problem). Results from

other authors were reported in38. The best solution found in these experiments was (F,x1, x2, x3) = (0.0126652303,

0.051689, 0.356732, 11.2881).


Belegundu,198254 0.0128334375 6

Arora, 198955 0.0127302737 5

Coello, 199938 0.0127047834 4


cGA (This Work) 0.0126652303 1Panmictic EA (This Work) 0.0126652593 3


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5.7. Welded Beam Design Problem

The Welded beam design problem has the goal of minimizing the cost of a weight bearing

beam subject to constraints on shear stress , bending stress , buckling load on the bar Pc,

and dimensional constraints38

. There are four design parameters to optimize consisting of the

dimensional variables h, l, t, and b shown in Fig. 8.

Fig. 8: Diagram of a welded beam. The beam load is defined as P with all other parameters shown in the diagram

defining dimensional measurements relevant to the problem. This figure is taken out of38 and is reprinted with

permission from IEEE ( 1999 IEEE).

Results: Each of the distributed EA designs converge to similar values (Figure 3) and both

strongly outperformed the PEA (Tab. 2). Comparisons to work from previous authors

highlight the strong performance of both of the distributed EAs. Of the three studies

referenced in and including39

, no previous method has been able to find the solutions reported

in this study.

Tab. 9 Comparison of results for the welded beam design problem (minimization problem). Results from other

authors were reported in39. The best solution found in these experiments was (F,x1, x2, x3, x4) = (1.72485,

0.205729, 3.47051, 9.03662, 0.2057296).


Deb, 1991 5 2.43311600 6

Coello, 199938 1.74830941 5

Zeng et al. 200239 1.72553637 4


cGA (This Work) 1.72485217 1


5.8. Artificial Test Function Results

This section presents results from experiments conducted on six artificial test functions. This

suite of problems was chosen in order to evaluate performance over a broad range of fitness


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landscapes. Information regarding the fitness landscape properties of these problems as well

as formal problem definitions can be found in35

.

Fig. 9 Performance for FM (Fopt=0), ECC (shifted from Fopt=0.067416 to Fopt=0), system of linear equations (Fopt=0),

Rastrigin (Fopt=0), Griewangk (Fopt=0), and Watsons (Fopt=0.01714) test functions.

Tab. 10: Performance results for all six artificial test problems are shown for twelve Evolutionary Algorithms run

for 3000 generations with algorithm designs varying by the use of Generational (Gen) or Pseudo Steady State (SS)


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population updating, the use of Binary Tournament Selection (Tour) or Truncation Selection (Trun), and the number

of search operators (Nops). Performance is presented as the single best objective function value found in 20 runs

FBestas well as the average objective function value over 20 runs FAve.

EA Gen Sel Nops Freq. Mod. Error Correcting Code Sys. of Lin. Eq.


ES SS Tour 7 0.00 15.36 3.53E-03 4.32E-03 8.53E-14 2.12E-05

ES SS Trun 7 6.69 18.28 3.68E-03 4.29E-03 3.16E-05 1.32

ES Gen Tour 7 23.07 26.95 2.47E-03 3.75E-03 10.90 14.58

ES Gen Trun 7 22.87 25.97 3.44E-03 4.13E-03 2.45 5.27

ES SS Tour 2 8.98 15.87 2.70E-07 3.84E-03 1.67 3.54

ES SS Trun 2 0.55 16.49 3.43E-03 3.96E-03 4.26 5.90

ES Gen Tour 2 23.35 26.33 4.18E-03 4.77E-03 50.21 74.11

ES Gen Trun 2 21.95 26.77 2.70E-07 3.17E-03 35.69 51.75

GA SS Tour 7 9.02 16.23 4.03E-03 4.47E-03 0.03 1.88

GA SS Lin 7 0.68 17.74 3.49E-03 4.30E-03 0.04 2.32

GA SS Tour 2 0.22 15.92 3.90E-03 4.55E-03 2.41 4.78

GA SS Lin 2 3.04 16.44 3.59E-03 4.43E-03 3.97 6.25Rastigrin Griewangk Watson


ES SS Tour 7 1.25E-10 1.65E-06 0.012 0.052 1.716E-02 2.025E-02

ES SS Trun 7 4.24E-02 1.26E-01 0.049 0.158 1.728E-02 2.922E-02

ES Gen Tour 7 6.33E-01 9.17E-01 0.615 0.751 1.778E-02 1.941E-02

ES Gen Trun 7 8.82E-02 1.96E-01 0.348 0.508 1.730E-02 1.828E-02

ES SS Tour 2 3.10E-02 6.92E-02 0.131 0.216 1.804E-02 4.887E-02

ES SS Trun 2 1.64E-01 2.83E-01 0.154 0.366 1.829E-02 4.369E-02

ES Gen Tour 2 7.82 10.51 1.476 2.729 2.444E-02 5.673E-02

ES Gen Trun 2 4.89 7.53 1.474 2.199 2.205E-02 4.111E-02

GA SS Tour 7 8.99E-02 2.79E-01 0.046 0.212 1.716E-02 4.406E-02

GA SS Lin 7 9.38E-03 1.52E-01 0.089 0.167 1.730E-02 2.957E-02

GA SS Tour 2 1.54E-01 2.93E-01 0.212 0.407 1.901E-02 6.413E-02

GA SS Lin 2 1.00E-01 1.99E-01 0.236 0.431 1.821E-02 5.189E-02

Tab. 11 Mann-Whitney statistical tests comparing best algorithms from each design class (first entry)

and comparing all data from design classes (second entry). For best in class comparisons (first entry),

the best algorithm from a design class is determined based on median performance after 150,000

evaluations. Winner of test is indicated along with p value. insig indicates p > 0.05.

Problem PEA vs. SOTEA cGA vs. SOTEA PEA vs. cGA

ECC PEA (p=0.002) , insig insig, SOTEA (p=0.01) PEA (p=0.0008) , insig

Freq. Mod. insig, SOTEA (p


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Frequency Modulation: SOTEA designs are found to be both the best and worst performers

(compared to the cGA) throughout the optimization runs (Fig. 9).ECC: Both SOTEA and the cGA designs are able to make steady progress toward the

optimal solution with little difference between the two designs (Fig. 9). One PEA was found to

be the best tuned algorithm as seen in Tab. 11 (this is the only artificial test function where a

PEA dominates).

System of Linear Equations: SOTEA designs strongly outperform the cGA (Fig. 9).

Comparison with results in Tab. 10 finds that both distributed EA designs were able to strongly

outperform the PEAs.

Rastrigin: SOTEA designs strongly outperform the cGA and the PEA. Although both

distributed EA designs have significantly better median performance than the PEA designs,

there is some indication that the PEA can occasionally find good solutions (Tab. 10).

Griewangk: SOTEA designs are very similar in performance to the cellular GA as seen in

Fig. 9 and Tab. 11. Both distributed EA designs perform better than the PEA designs (Tab.

11).

Watson: SOTEA designs strongly outperform the cGA (Fig. 9 and Tab. 11). Both distributed

EA designs perform better than the Panmictic EA designs (Tab. 11).

5.9. General Performance Statistics

Algorithm performance has thus far been evaluated through comparisons between

algorithms on individual optimization problems. Here we investigate whether more general

conclusions can be made about the EA design classes (PEA, cGA, SOTEA) using metrics

presented in Tab. 12. The first statistic (Tab. 12, column two) measures the proportion of runs

where an EA design class found the best known solution. This value is averaged over all test

problems and indicates the tendency of an algorithm class to converge to the optimal (or best

known) solution. The first statistic provides a measure of run consistency and can thus be seenas a proxy for an algorithms robustness to initial conditions. The second statistic (Tab. 12,

column three) measures the proportion of runs where an EA design class finds a solution that

ranks in the top 5% of all solutions found in these experiments. The purpose with this metric

is to relax the criteria from the previous statistic and get a broader sense of EA design class

performance. The third statistic (Tab. 12, column four) is a p value for the Mann-Whitney U-

test where the statistical hypothesis is that the given EA design class is superior to the other

two EA design classes.

Tab. 12 Overall performance statistics for the Panmictic EA, the cellular GA, and SOTEA. Statistics in columns 1-3

are an average value over all test problems.

EA Design % of runs where EA U-Test % of problems where EA

found best was top 5% p


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The last two statistics in Tab. 12 are confined to the best implementations of an EA design

class and thus indicate algorithm effectiveness after parameter tuning. For instance, the fourthstatistic (Tab. 12, column five) measures the proportion of problems where the algorithm

obtained the best median objective function value. This indicates the likelihood of preferring

a given algorithm when it can only be run a small number of times on a problem. The final

statistic (Tab. 12, column six) measures the proportion of problems where the algorithm was

able to find the best known solution at least one time. This indicates likely algorithm

preference when repeated optimization runs are possible. For each of the statistics, and in the

context of the selected test problems, SOTEA is found to be better than any of the other

algorithm design classes. Particularly noteworthy are the results in column five which

indicate that a tuned SOTEA design was the best EA design in about 80% of the problems

tested. Moreover, we have greater than 95% confidence that SOTEA is a superior search

method for the problems considered in this study.

5.10. Summary of Performance Results

The aim of this section was to evaluate several aspects of SOTEA performance robustness.

For a single run of a search algorithm, robustness can be related directly to the search process

and reflect the competitiveness of an algorithm over different timescales. In a rugged fitness

landscape for instance, this requires the capacity to both exploit new information and also

explore the fitness landscape. Robustness can also be defined at other resolutions as well.

Over multiple optimization runs, the robustness of a search process could refer to

performance consistency, which is reflected by a small variance in final solution quality. At

yet a higher resolution, robustness might refer to the ability of a search algorithm to achieve

the previously mentioned forms of robustness but in different fitness landscapes and with

minimal changes to algorithm design parameters. In this section we have found SOTEA to

be highly robust based on each of these definitions.

6. Topological Analysis

To understand the basis by which SOTEA establishes a robust search process requires a

deeper understanding of the spatio-temporal dynamics of SOTEA and how these are

influenced by fitness landscape properties. With this in mind, we conducted a genealogical

analysis using tools described in56

and a topological analysis reported here. The genealogical

analysis evaluated gene takeover dynamics across a population, however these tests did not

provide clear insights into SOTEA search behavior and the results are not presented. In this

section, we report the structural characteristics of SOTEA and compare this with the cellular

GA, Panmictic EA, and values observed in biological systems. Here we find that, unlike

standard EA population topologies, SOTEA obtains several topological characteristicsobserved in biological systems that are in some cases potentially useful to a search process.


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6.1. SOTEA Topological Analysis

Methods: In SOTEA, network structural changes are driven by node fitness, however

because node fitness is constantly evolving (due to population dynamics), the SOTEA

network never converges to a stable structure. In order to make general statements about

topological characteristics, measurements are therefore averages taken every 50 generations

for SOTEA run 10 times over 1000 generations. To consider the impact of system size,

topological properties for population sizes ofN= 50, 100 and 200 have been measured with

results shown in Fig. 10. Here it is seen that most properties show little dependency on the

population size except for L which is generally smaller for smaller systems. Fig. 10 also

indicates that the topological properties of SOTEA are sensitive to the setting ofKMax which is

the only parameter of the SOTEA design. The topological property values for SOTEA with

N=50 are reported in Tab. 13, which are taken as an average over all KMax settings considered

in this study (KMax = 3, 5, 7, 9).

0

10

20

30

40

0 5 10

Kmax

L

a)

-10

-8

-6

-4

-2

0

0 5 10

Kmax

c-k

b)

0

10

20

30

0 5 10

Kmax

v

c)

0.4

0.8

0 5 10

Kmax

c

(ave)

d)

2

3

4

5

0 5 10

Kmax

k

(ave)

e)

Fig. 10 Topological properties for SOTEA with different values ofKMax and population sizes ofN= 50 (), 100( ),

and 200( ). Characteristics include a) the characteristic path length (L), b) the correlation between c and k(c-k), c)

the slope of the degree correlation (), d) the average clustering coefficient cave and e) the degree average kave.

Tab. 13: Topological characteristics for the Panmictic EA, cGA, and SOTEA. The topological characteristics for

biological systems are taken from24 and references therein. In column five, refers to the exponent for k

distributions that fit a power law. Two values for are given for the metabolic network and refer to the in/out-

degree exponents (due to this being a directed network). Results for degree correlations are given as the slope of

kNNvs k. Nis the population size and R is a correlation coefficient for the stated proportionalities.


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System N L kave k dist. cave (crand) c-k k-kNN

Panmictic EA 50 L = 1 kave = N-1 k = N-1 1 (1_new) no nocellular GA 50 L ~ N kave = 2 k = 2 0 (0.04) no no

SOTEA 50 5.97 3.6 Poisson 0.687 (0.07) c = -4.75k = 11.8

Complex

Networks

Large L ~ log N kave 0

or < 0

Protein 2,115 2.12 6.80 Power Law, =

2.4

0.07 (0.003) Power Law < 0

Metabolic 778 7.40 3.2 Power Law, =

2.2/2.1

0.7 (0.004) Power Law < 0

6.1.1. Topological Properties of SOTEA

Here we comment on some of the topological properties of SOTEA and discuss potential

causes. Some topological properties such as the assortative character of the SOTEA networks( > 0) and the linear relation between c and k are not discussed as they are not easily

interpreted within the context of algorithm search behavior.

Characteristic Path Length L: The total distance genetic material must travel across the

network is always small as indicated by small L. Although this seems to imply that the

population will be more tightly coupled, we provide arguments below as to why this is not the

case.

Clustering Coefficient: The clustering coefficient is an order of magnitude larger than what

is observed in random networks which indicates that the SOTEA driving forces were

successful in achieving this topological property. This topological feature encourage random

walk interactions (e.g. for mating) to remain within clusters; a behavior that would be

straightforward to confirm using the network analysis methods described in57

. A related

consequence of this topological feature may be that it acts to slow down communicationbetween clusters, which could dampen the rate of genetic transfer that would otherwise be

observed in undirected random networks with similar values for L.

Degree Average: The low value for kave suggests the SOTEA network maintains a sparsely

connected architecture with high levels of locality similar to that of the cellular GA.

Degree distribution: kapproximates a Poisson distribution which is not similar to the fat

tailed distributions observed in complex systems or the distributions observed in the first

SOTEA algorithm developed in4. The distributions results suggest relatively little

heterogeneity in kis present such that the level of locality is fairly uniform within the system.

Previous studies, reviewed in19

, have found that placing upper bounds on kcan result in strong

deviations from a power law. This SOTEA model introduces tight constraints on the values

of k (e.g. upper and lower bounds, quadratic set point) so the k distribution results are not

unexpected.


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6.1.2. SOTEA Scaling

A visual analysis can often provide useful insights into network structure. Fig. 11 shows

SOTEA networks after 500 generations of evolution with varying population sizes (N=50,

100, 200) and with either KMax = 7 or KMax= 5. One noticeable consequence of the SOTEA

model is that many nodes are found in four neighborhood clusters and in particular, there

appears to be a kite motif present in the network. It is expected that this is in part due to the

degree lower bound of KMin = 3 in SOTEA. In the network visualizations, node size is

increased to reflect individuals with better fitness. Because fitness can change with each

generation, we see that nodes with high fitness are not always associated with hub positions.

More generally, the emergence of these properties is limited by the fact that population

members are always in flux and thus the driving forces for cluster and hub formation are also

subject to change over time. As population size increases, one can also notice residual ring-

like structures in the network, even after 500 generations. This indicates that initialtopological bias continues to impact the network over long periods of time for larger

populations. Further investigation is needed to determine how this historical structural bias

(in conjunction with initial genetic bias) can influence algorithm search behavior.

SOTEA (KMax= 7) SOTEA (KMax= 5)


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Fig. 11 SOTEA Network Visualizations with population sizesN= 50 (top),N= 100 (middle), andN= 200

(bottom).

7. Discussion

7.1. SOTEA Network Model

Network dynamics were modeled in SOTEA based on a few guiding principles. First, we felt

it was necessary to have topological changes guided by interactions with the fitness landscape,

with structural changes enacted on local regions of the network. This local restructuring not

only occurs for many real-world complex systems, it is also necessary for efficient parallel

implementation of the algorithm. This led to the use of network rewiring rules based on short

stochastic walks and node property values that are based on local information.

Second, we wanted to couple the structural dynamics of the network to the dynamics of theEA population in a way that could promote modularity and allow for high levels of

exploration and exploitation in different regions of the population. To be effective, such

modularity could not be imposed on the population but instead needed to emerge and adapt

based on information gathered during the search process. SOTEAs superior performance

across most problems provides evidence that coevolution between a population and its

topology is a readily exploitable feature of natural systems and can be effectively utilized in

nature-inspired population based search heuristics.

7.2. Distributed EA research

Considerable research efforts have been devoted to the study of distributed evolutionary

algorithms. These efforts include the study of fine-grained (e.g. cellular grids), coarse-

grained (e.g. island models), and hybrid structures (e.g. hierarchical). The highly modular

topology of the SOTEA model combined with short stochastic walk interactions within the

system are likely to create virtual islands in the system where interactions within a cluster are

much more frequent compared to interactions between clusters. Quantifying the prevalence of

such behavior is possible by calculating the characteristic residence time of random walks on

local regions of the network, e.g. using methods described in Section 2.3 in57

. Assuming that

clusters do become relatively isolated from other clusters, this would allow for a more nature-

inspired approach to the integration of fine-grain and coarse-grain structures within an EA

population (compared with explicitly defined hierarchical topologies). Of course, how these

clusters form using fitness information contained in the population will greatly influence

algorithm behavior. With the topological operators presented in this study, SOTEA appears

to generate a robust search process that evolves as an emergent property of the system

(through its interaction with the problem). This should be contrasted with surrogates of

robustness (e.g. diversity preservation, niching, crowding) that have been incorporated into


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EAs in previous studies based on apriori knowledge about a problems fitness landscape

properties.Some studies have suggested that a population that is spatially distributed over a static

topology can enhance some types of robustness in an EA (e.g. see34

). How this occurs has not

been entirely determined, however intuition suggests that a distributed population topology

influences population dynamics by creating a weaker coupling across the population. A

weaker coupling can attenuate fast systemic responses to local attractors and may allow for a

more diffuse and explorative search to take place. On the other hand, the use of a static

topology is itself a global and inflexible approach to achieving robustness to local attractors.

Moreover, it is expected to reduce the speed by which any information can be exploited since

it establishes a global predefined tradeoff between exploration and exploitation in the system.

Alternatively, a topology that adapts in response to local attractors has the potential to allow

for qualitative differences in search behavior for different segments of the population.

8. Conclusions

SOTEA Network Model: A Self-Organizing Topology Evolutionary Algorithm (SOTEA)

has been presented with a distributed population structure that coevolves with EA population

dynamics; the first known optimization algorithm with such a coevolving state-structure

relationship. Based on the results of this study as well as theoretical issues raised in the

introduction, we feel that the coevolution of states and structure provides a unique and

interesting extension to the design of search algorithms.

The general framework that allows for this coevolution to be implemented is straightforward.

With the population defined on a network, rules are used to modify the network topology

based on the current state of the population. In particular, structural changes are initiated by a

dynamic state value in each node, e.g. individual fitness. Node state dynamics are a simpleconsequence of the genetic operators implemented within the evolutionary search process.

The SOTEA model presented in this paper was designed to structurally adapt to the fitness

landscape based on local network information and local topological changes; features that

were motivated by both practical implementation concerns and theoretical motivations.

Network dynamics were driven by i) an adaptive connectivity where higher fitness individuals

were encouraged to obtain higher levels of connectivity and ii) an adaptive definition of

community that encourages high levels of clustering amongst nodes with low fitness.

Topological Analysis: Self-organization of the population network topology resulted in high

levels of clustering, small characteristic path length, and correlations between the clustering

coefficient and a nodes degree. Each of these characteristics are approximately similar to

what is observed in biological networks.

Performance: A number of engineering design problems and artificial test functions wereselected to evaluate the robustness of the new SOTEA algorithm compared with another

distributed design, the cellular GA. Results indicate the SOTEA algorithm often had better

performance and more consistent results compared with the cGA. Both of the distributed


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