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EVOLUTIONARY NEURAL GAS (ENG):  A MODEL OF SELF ORGANIZING 
NETWORK FROM INPUT CATEGORIZATION. 
Ignazio Licata (a) ↑ , Luigi Lella (b) 
(a) Ixtucyber for Complex Systems, Marsala, TP and Institute for Scientific Methodology, 
Palermo, Italy; 
(b) A.R.C.H.I. - Advanced Research Center for Health Informatics, Ancona, Italy 
ABSTRACT  
Despite their claimed biological plausibility, most self organizing networks have strict 
topological constraints and consequently they cannot take into account a wide range of external 
stimuli. Furthermore their evolution is conditioned by deterministic laws which often are not 
correlated with the structural parameters and the global status of the network, as it should happen in 
a real biological system. In nature the environmental inputs are noise affected and “fuzzy”. Which 
thing sets the problem to investigate the possibility of emergent behaviour in a not strictly 
constrained net and subjected to different inputs. 
It is here presented a new model of Evolutionary Neural Gas (ENG) with any topological 
constraints, trained by probabilistic laws depending on the local distortion errors and the network 
dimension. The network is considered as a population of nodes that coexist in an ecosystem sharing 
local and global resources. 
Those particular features allow the network to quickly adapt to the environment, according to its 
dimensions. The ENG model analysis shows that the net evolves as a scale-free graph, and justifies 
in a deeply physical sense- the term “gas” here used. 
Key-words: Self-Organizing Networks; Neural Gas; Scale-Free Graph; Information in Network 
Functional Specialization. 
1. INTRODUCTION 
Self organizing networks are systems widely used in categorization tasks. A network can be 
seen as a set A={c1, c2,… ,cn} of units with associated reference vectors wc∈R
n where Rn is the 
same space where inputs are defined. Each unit (or node) can establish connections with the other 
ones, the units belonging to the same clusters are subjected to similar modification affecting their 
reference vectors.  
Self organizing networks can automatically adapt to input distributions without supervision by 
means of training algorithms that are simple sequences of deterministic rules. Competitive hebbian 
learning and neural gas are the most important strategies used for their training. 
Neural gas algorithm (Martinetz T.M. and Schulten K.J., 1991) sorts the network units according to 
the distance of their reference vector to each input. Then the reference vectors are adapted so that 
the ones related to the first nodes in the rank order are moved more close than the others to the 
considered input. 
Competitive hebbian learning (Martinetz and Schulten, 1991; Martinetz, 1993) consists in 
augmenting the weight of the link connecting the two units whose reference vectors are closest to 
the considered input (the two most activated units). 
Both strategies are examples of deterministic rules. As we know there are other rules that constrain 
the topology of the network which has a fixed dimensionality. That’s the case of Self Organizing 
Maps (Kohonen, 1982) and Growing Cell Structures (Fritzke, 1994). 
                                                
↑ Corresponding author: Ignazio.licata@ejtp.info  
In other cases the network structures haven’t topological constraints, they take a well ordered 
distribution by exactly adapting to the manifold inputs. For example TRN (Martinetz and Schulten, 
1994) and GNG are networks whose final structure is similar to a Delaunay Triangulation 
(Delaunay, 1934).We have tried to define a new self organizing network that is trained by 
probabilistic rules avoiding any topological constraints.  
According to Jefferson (1995) life and evolution are structured at least into four fundamental 
levels:  molecular, cellular, organism and population. We propose a population level based on 
evolutionary algorithm where the network is seen as a population of units whose interactions are 
conditioned by the availability of resources in their ecosystem. The evolution of the population is 
driven by a selective process that favours the fittest units. This approach has a biological 
plausibility. As stated by recent theories (Edelman, 1987) human brain evolution is subjected to 
similar selective pressures. 
Obviously we are not interested in recreating the same structure as the human brain. Our work 
aims at finding innovative and effective solutions to the categorization problem adopting natural 
system strategies. So our system falls within the Artificial Life field (Langton, 1989). 
Our model is a complex system that shows emergent features. In particular its structure evolves as a 
scale free graph. In the training phase there arise clusters of units with a limited number of nodes 
that establish a great number of links with the others.  
Scale free graphs are a particular structure that is really common in natural systems. Human 
knowledge, for instance, seems to be structured as a scale free graph (Steyvers, Tenenbaum 2001). 
If we represent words and concepts as nodes, we’ll find that some of these are more connected than 
the others. 
Scale free graphs have three main features.The small world structure. It means there is a 
relatively short path between any couple of nodes (Watts, Strogatz, 1998).The inherent tendency to 
cluster that is quantified by a coefficient introduced by Watts and Strogatz. Given a node i of ki 
degree i.e. having ki edges which connect it to ki other nodes, if those make a cluster, they can 
establish ki(ki-1)/2 edges at best. The ratio between the actual number of edges and the maximum 
number gives the clustering coefficient of node i. The clustering coefficient of the whole network is 
the average of all the individual clustering coefficients.  
Scale free graphs are also characterized by a particular degree distribution that has a power-law 
tail P(k)~k n− . That’s why such networks are called “scale free” (Albert, Barabasi, 2000). 
The three previous features are quantified by three parameters: the average path length between 
any couple of nodes, the clustering coefficient and the exponent of the power law tail. We’ll show 
that the values of these parameters in our model seem to confirm its scale free nature. 
2. AN OUTLINE ON SELF-ORGANIZATION AND EVOLUTIONARY SYSTEMS 
Natural selection mechanism has been successfully used for a lot of industrial applications 
spanning from projecting to real-time control and neural networks training. 
It was in the 60s that Genetic Algorithms based on the Evolution Theory’s three main 
mechanisms - reproduction, mutation and fitness – were first used in dealing with optimization 
problems. Although the solution is reached by a population of individuals, systems based on this 
approach are not considered self organizing because their dynamics depend on the external 
constraint of the fitness function. 
In the 80s a new approach to the study of living systems which mixed together self organization 
and evolutionary systems came out (Rocha, 1997). Its success was due to the studies on the way 
how biological systems work (metabolism, adaptability, autonomy, self repairing, growth, evolution 
etc.). The hybrid systems make us possible to get a better simulation both of the evolutionary 
optimization processes and the internal structure modification to reach a greater biological 
plausibility in the fitness 
Neuroevolutionary systems are an example of this approach. In classic neuroevolutionary models 
the network parameters are genetically set, whereas the connection weights are modified according 
to a training strategy. This solution follows the classic vision of cerebral development where genes 
control the formation of synaptic connections while their reinforcement depends on neural activity. 
More recent neuroevolutionary systems are characterized by different forms of self organizing 
processes which are cooperative coevolution (Paredis, 1995; Smith, Forrest and Perelson, 1993) and 
synaptic Darwinism (Edelman, 1987). 
Cooperative co evolutionary systems offer a promising alternative to classic evolutionary 
algorithms when we face complex dynamical problems. The main difference with respect to classic 
EA is the fact that each individual represents only a partial solution of the problem. Complete 
solutions are obtained by grouping several individuals. The goal of each individual is to optimize 
only a part of the solution, cooperating with other individuals that optimize other parts of the 
solution. It is so avoided the premature convergence towards a single group of individuals. An 
example of such approach is given by the Symbiotic Adaptive Neuroevolution System (Moriarty 
and Miikkulainen, 1998) that operates on populations of neural networks. 
While in most neuroevolutionary systems each individual represents a complete neural network, 
in SANE each individual represents a hidden unit of a two-layered network. Units are continuously 
combined and the resulting networks are evaluated on the basis of the performances shown in a 
given task. The global effect is equal to schemas promoting in standard EAs. In fact during the 
evolution of the population the neural schemas having the highest fitness values are favoured and 
the possible mutations in the copies of these schemas don’t affect the other copies in the population. 
Other recent strategies focus on the evolution of connection schemas in the network. In the human 
brain the number of synapses established by a single neuron is always much lower than the overall 
number of neurons. That gives the network a sparsely connected aspect. In the last years several 
models have been proposed to emulate the mechanism involved in the selection of links without 
referring to the physical and chemical properties of neurons. 
The Chialvo and Bak model (Chialvo and Bak, 1999) is based on two simple and biological 
inspired principles. First, the neural activity is kept low selecting the activated units by a winner 
takes all strategy. Second, the external environment gives a negative feedback that inhibits active 
synapses if the network behaviour is not satisfying. With these simple rules the model operates in a 
highly adaptive state and in critical conditions (extreme dynamics). The fundamental difference of 
this strategy based on the synaptic inhibition with respect to the classic one based on synaptic 
reinforcement is that the reinforcement-based learning is a continuative process by definition, while 
the inhibition-based learning stops when the training goal is achieved. The synaptic inhibition is 
also biologically plausible. According to Young (Young, 1964; Young, 1966) learning is the result 
of the elimination of synaptic connections (closing of unneeded channels). Dawkins (Dawkins R., 
1971) stressed that pattern learning is achieved by synaptic inhibition. As stated by the neural 
groups’ selection theory developed by Edelman (Edelman, 1978; Edelman, 1987), brain 
development is characterized by generating a structural and dynamical variability within and 
between populations of neurons, by the interaction of the neural circuit with the environment and by 
the differential attenuation or amplification of synaptic connections. Research in neurobiology 
seems to confirm the validity of the negative feedback model and the fact that neural development 
follows the process of Darwinian evolution. 
The Chialvo and Bak model is a simple two-layered network. After the training each input pattern 
is associated with a single output unit leading to the formation of an associative map. When an input 
pattern is presented the most activated input unit i is selected. Then the neuron j from the hidden 
layer that establishes the most robust connection with i is selected. Finally the output neuron k that 
is the most strongly connected with j is selected. If k is not the desired output the two links 
connecting i with j and j with k are inhibited by a coefficient d that is the only parameter of the 
model. The iterative application of these rules leads to a rapid convergence towards any input-
output mapping. This selective process followed by an inhibitory one is the essence of the natural 
selection in the evolutionary context. The fittest individual is selected on the basis of a strategy that 
doesn’t reward the best but punishes the worst. That’s the reason why this model has been 
considered a particular kind of synaptic Darwinism. 
Our neuroevolutionary model is also based on a selection strategy. The structural information of 
our network is not codified by genes. We directly consider the entire network as a population of 
nodes that can establish connections, generate other units or die. The probability of these events 
depends on the presence of local and global resources. If there are few resources the population 
falls, if there is a lot of resources the population grows. Like in the Chialvo and Bak model we don’t 
select the fittest nodes reinforcing their links, but we simply remove the worst nodes when the 
ecosystem resources are low. This generates a selective process that indirectly rewards the units 
which can better model the input patterns. Our evolutionary strategy can be seen as a selective 
retention process (Heylighen, 1992) that removes those units which cannot reach a stable state, 
remaining associated with several input patterns. Even if the stability of a unit is quantified by the 
minimum distortion error related to it, this information mustn’t be considered to be environmental 
information. The minimum distortion error simply quantifies the difficulty encountered by the unit 
during the modelling of input patterns. 
3. THE EVOLUTIONARY ALGORITHM 
Research has confirmed (Roughgarden, 1979; Song and Yu, 1988) that in natural environments 
the population size along with competition and reproduction rates continuously changes according 
to some natural resources and the available space in the ecosystem. 
These mechanisms have been reproduced in some evolutionary algorithms, for example to optimize 
the evolution of a population of chromosomes in a genetic algorithm (Annunziato and Pizzuti, 
2000). We have tried to use a similar strategy for the evolution of a population of units in a self 
organizing network without using the string representation of genetic programming. 
In our model each node is defined by a vector of neighbouring units connected to it, a reference 
vector and a variable D that is the smallest distance between its reference vector and the closest 
modelled input. The value of this variable quantifies the debility degree of the unit.  The lower is D 
the higher are the chances for the unit to survive. At each presentation of the training input set, D is 
set to the maximum value. After the presentation of a given input x, if the reference vector w of the 
unit is modified, the resulting distance between the two vectors ||x-w|| is calculated. If this value is 
lower than D it becomes its new value. 
The  training algorithm here used can be subdivided in three phases: 
1) Winners are selected. For each input the unit having the closest reference vector is selected. 
2) The reference vectors of the winners and their neighbours are updated according to the 
following formula : 
(3.1) ( ) ( ) ( )( )1w t w t x w tα+ = + − . 
So the reference vectors w of the selected units are moved towards the relative inputs x of a certain 
fraction of the distances that separate them. For winners this fraction is two or three orders of 
magnitude higher than the one used for their neighbours. So winners have the reference vectors 
moving more quickly towards the inputs. 
3) The population of units evolves producing descendants, establishing new connections and 
eliminating the less performing units. All these events can occur with a well defined probability that 
depends on the availability of resources. 
These rules are iterated until a given goal is achieved. For example the minimization of the 
expected quantization error that is the mean of the distances between the winners and the K inputs 
they model: 
(3.2) 
D x wK
= −�  
If this value falls below a certain threshold Dmin, training is stopped. 
The first two phases can be considered a kind of winner takes all strategy, where only the most 
activated units are selected and enabled to modify their reference vectors. The third phase is the 
evolutionary phase (fig. 3.1). Each unit i, i=[1…N(t)] where N(t) is the actual population size can 
meet the closest winner j with probability Pm: 
Fig. 3.1 – The evolutionary phase of the algorithm. 
If meeting occurs, the two units establish a link and they can interact by reproducing with 
probability Pr. In this case two new units are created. One is closer to the first parent, the other to 
the second parent: 
(3.3) 
If reproduction doesn’t take place due to the lack of resources the weaker unit of the population, i.e. 
the one with the highest debility degree, is removed. 
If unit i doesn’t meet any winner it can interact with the closest node k with probability Pr 
establishing a connection and producing a new unit whose reference vector is set between the 
parents reference vectors: 
(3.4)  1 22 2
p pw ww
When we fix a maximum population size, the ratio between the actual size and the threshold 
N(t)/Nmax can be seen as a global resource of the ecosystem affecting the probabilities of the events. 
For example if the population size is low the reproduction rate should be high. So we can 
reasonably choose Pr = 1-N(t)/Nmax.  If the population size is high, the chance for the units to meet 
each other will be higher, so we can set Pm = N(t)/Nmax. 
We can also consider a local resource that is the ratio between the threshold Dmin and the 
debility degree Di of the unit i. Each unit i should meet a winner with a probability 
Pm=(N(t)/Nmax)(1-Dmin/Di) and Pr = 1 – Pm. In this way winners are not encouraged to migrate to 
other groups of nodes and weaker units don’t participate in reproduction activities. 
We can estimate the population grow rate in the following way: 
(3.5) 
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )
2 2 2
2min min
1 2 1 1 1
2 1 1 2 1 ( )
2 1 1 1 2 1 1 ( )
m r m r m m d
N t N t P P N t P P N t P N t P P N t
N t P N t
N t X t X t X t first model
N t D D
N t X t X t X t second model
M D D
+ = + − − + − − − =
= − =
= − � + = −� �
� � � �� � � �= − − � + = − −� � � �� � � �� �� �� � � �� �� �
where X(t) is the normalized size N(t)/Nmax. This is the quadratic-logistic map of Annunziato and 
Pizzuti(Annunziato and Pizzuti, 2000): 
(3.6) ( ) ( ) ( )( )21 1X t aX t X t+ = −  
They proved that by varying the parameter different chaotic regimes arise. For a<1.7 the behaviour 
is not chaotic, for 1.7<a<2.1 we have chaotic regimes with simple attractors localized in a fixed part 
of the plane of the phases. Theoretically for the first model we expect to obtain a chaotic regime 
that is described by a simple attractor. In the second model the factor (1 – Dmin/D) might reduce the 
influence of the negative feedback in the final part of network training. 
It is possible to demonstrate that during the evolution the population size converges to N(t) = 
0.72 Nmax. In this phase the probability that a unit establishes n connections with the other ones for 
the first model, considering only clusters of n units, is given by: 
(3.7) ( )
max0.72 1
max max
1max max
0.72 0.72
n i nN n
P n n
� � � �
= − =� � � �
� � � �
It has to be pointed out we have subtracted the probability that such n links developed within a 
cluster of more than a n unit. 
The coefficients α  and β  of the power law are considered constant at the end of the training. To 
compute their values, we can take into consideration the cases n=1 and n=0.72N Nmax-1 which 
correspond to the minimum and maximum number of connection at the end of the training. 
(3.8) ( )
max0.72 2
1 0.72 0.72 1
P βα α
= − = =�  
( ) ( )
0.72 2
0.72 1 1
max max
0.72 1 0.72 0.72 0.72 0.72 1
P N N
− = = − −� �
max max
0.72 2
0.72 1 0.72 1
0.72 0.72
� �� =
The distribution tail of the degrees tends to stretch when the maximum size of the population 
increases, it means that in wider networks there are more hubs with a higher degree. 
For the second model we can consider that at the end of the training (1-Dmin/D) ∼ε   
So the probability that a unit establishes n links becomes: 
(3.9) 
max0.72 1
max max
1max max
0.72 0.72
n i nN n
P n n
βε ε α
� � � �
= − =� � � �
� � � �
max max
0.72 2
0.72 1 0.72 1
0.72 0.72
� �� =
and the considerations made for the first model can be therefore extended to the second model. 
4. TRAINING THE NET: SIMULATIONS  
We have compared the performances of our networks with those of a Growing Neural Gas in 
categorizing bidimensional inputs.  
GNG is a self organizing network which thanks to both the competitive hebbian learning 
strategy and the neural gas algorithm can categorize inputs without altering their exact 
dimensionality.For the GNG, the parameters of the model are �����α  = 0.5, β  = 0.0005 and at each λ  = 
300 steps a new unit is inserted. The maximum age of the links is set to 88.  
For the two different ENG models, the parameters are α  = 0.05, β  = 0.0006 and the maximum 
size is set to Nmax = 120. 
As stopping criterion for both the algorithms we have chosen the minimization of the expected 
quantization error that is the average distance between the winners and the corresponding inputs. 
We have considered two different input domains. In the first case inputs are localized within four 
square regions, in the second one inputs are uniformly distributed in a ring region. 
As shown in fig.4.1 after the training, GNG reference vectors are all positioned in the input 
domain. In the Evolutionary Self Organizing Networks (fig.4.2a and fig.4.2b) some units fall 
outside the input domain, but in this way the network remains fully connected. The nodes’ 
distribution statistical analysis confirms what appears to be intuitively patent: the emerging network 
structure is a typical scale-free one, i.e. a structure where few hubs manage the links. 
Fig. 4.1 – Growing Neural Gas simulations. 
Fig. 4.2 a – Evolutionary Self Organizing Network simulations (first model). 
Fig. 4.2 b – Evolutionary Self Organizing Network simulations (second model). 
We trained 30 networks of each type obtaining the average degree distributions reported in fig.4.3-
4.5. In tab. 4.1 – 4.2 are reported the average values of the structural parameters of the two 
networks. 
y = 37,074x
-2,0371
1 2 3 4 5
degree
y = 68,961x
-2,9864
1 2 3 4 5 6
degree
Fig. 4.3 – Average degree distribution in GNG (two different input domains) 
y = 23,75x-1,149
1 2 3 4 5 6 7 8 9
degree
y = 22,069x
-1,1143
1 2 3 4 5 6 7 8
degree
Fig. 4.4 – Average degree distribution in ENG (first model, two different input domains) 
y = 24,415x-1,1411
1 2 3 4 5 6 7 8 9
degree
y = 21,98x
-1,1408
1 2 3 4 5 6 7 8
degree
Fig. 4.5 – Average degree distribution in ENG (second model, two different input domains) 
While GNG have a high value for the average path length and a low clustering coefficient, ENG 
have a short average path length and a high clustering coefficient which along with the power law 
tail of the degree distribution confirm its scale free graph features. 
 Average 
path 
length 
Clustering 
coefficient 
Power 
exponent 
GNG -  0.49 2.04 
ESON (1st) 3.82 0.64 1.15 
ESON(2nd) 3.92 0.63 1.14 
 Tab. 4.1 – Comparison of structural parameters (average values, first input domain) 
 Average 
path 
length 
Clustering 
coefficient 
Power 
exponent 
GNG 6.4 0.42 2.98 
ESON (1st) 3.61 0.58 1.11 
ESON(2nd) 3.67 0.59 1.14 
Tab. 4.2 – Comparison of structural parameters (average values, second input domain) 
Fig. 4.6 – 4.7 shows the population dynamics of the two ENG models. 
The structure shared by the two different ENG models is due to the fact that the winner units tend to 
establish the greatest number of connections. These are the favoured units with which each node try 
to establish a connection. If the probability depends also on the local distortion error as it happens in 
the second model, we obtain a final structure that is more similar to the GNG, which is to say more 
similar to a gas. In point of fact, the conditions to create a new link become more restrictive, 
reducing the interaction among each cluster and the whole network. The structure of connections 
seems to extend more uniformly in the regions where inputs are present as it can be seen in picture 
4.2b (more evident in the circular distribution). 
Picture 4.7 shows the dynamics of the populations in the two different models of ENG. In the first 
model the population size seems to converge to the final value of 0.72Nmax, confirming the 
experimental results of Annunziato and Pizzuti. As it can be noticed in fig. 4.6, since the d value 
gradually diminishes during the training, the influence of the factor (1-Dmin/d) grows reducing the 
effects of the negative feedback which characterizes the quadratic logistic map. This justifies the 
sudden growth of the population at the end of the training in the second model. 
1 3 5 7 9 1
epochs
1 2 3 4 5 6 7 8 9 10 11
epochs
Fig. 4.6 – Network size evolution of the two ENG models (first input domains). 
At the end of training new units connect with the winner units which have a lower d, while the 
subgroups of units become more isolated. Considering the function (X(t),X(t+1)) the attractor 
becomes more marked in the second model. This means that the system tends to converge more 
toward a precise final state with a lower interaction among the groups of units. 
Fig. 4.7 – population dynamics (X(t),X(t+1)) of the two Esonet models (first input domain). 
5. THE ROLE OF INFORMATION IN FUNCTIONAL SPECIALIZATION AND 
INTEGRATION 
We can classify a system as complex when it is made up of different parts heterogeneously 
interacting. In addition, its behaviour and its structure have to be neither completely casual (as it 
happens in a gas) nor too regular (as it happens in a crystal). In Nature we generally observe the co-
existence of functionally highly specialized integrated areas. 
That’s what happens in the brain, where different areas and groups of neurons interact to give 
rise to an integrated and unitary cognitive scenario (G. M. Edelman, G. Tononi, 2000). 
Edelman has introduced the integration, reciprocal information and complexity concepts in 
order to mathematically define the functional organization of the cerebral structures. 
Within a complex system, a subset of elements can be defined an integrated process if – on a 
given temporal scale – the elements interact more strongly with each other than with the system. In 
a neural net or in a self-organizing one it means that the units of an integrated group will tend to 
simultaneously activate themselves. 
When the units in a subset are independent, the system’s entropy reaches its maximum value 
which is the sum of the entropies of the single elements (local entropies). On the contrary, when any 
kind of interaction occurs, the global entropy decreases so becoming lower than the sum of the local 
entropies. The integration measure is, therefore, a natural indicator of the system informational 
“capacity”. 
So the integration of a subset of network units can be calculated by deducting the sum of the 
entropies of each single component ( )ix  from the entropy of the system considered as a whole. If 
each unit can only take two states (activated/not-activated), the amount of the possible activation 
patterns of a subset with N units is N2 . So the system maximum entropy is: 
(5.1) ( ) ( )max 2 2 2
1 2 1
log log 1 log 2
i i NN
i i i
H X H x p N
� � � �= = = = =� �� �
� �� �
� �  
and the integration will be: 
(5.2) ( ) ( ) ( )
I X H x H X
= −�  
for the self-organized net here considered, the integration of a sub-group of units takes the 
following expression: 
(5.3) ( )
I X N P
� �� �
= − � �� �+� � � �
where Pi is the probability for a node to establish i connections. The overall number of the 
system’ states is equal to the total number of possible groups of i+1 units. Groups of units having 
the same dimension (groups of i+1 units) give the same contribution to the entropy of the system. 
If we choose the WTA strategy as activation modality, for each presented input only a single 
unit (the winner) and the 1<i<N-1 i units will activate themselves. All the other ones remain not-
activated. 
The probability for a node to create connections is ruled by the power law βα −= kPi , with α and 
β depending on 1) the network dimension, 2) the local distortion errors (for the second model) and 
3) the particular evolution of the network structure, i.e. the dynamic behaviour of ( )tα  and ( )tβ . 
So the integration of the two self-organizing network here presented is: 
(5.4) ( ) ( )
I X N i
� � � �= − � �� �+ � �� �
The integration can be seen as a measure of the statistic dependency within a subset of units. 
The stronger their interactions are, the higher their integration. 
In order to measure the statistic dependency between a subset and the whole system, Edelman 
introduced the concept of mutual information. Given an n subset made up of k elements ( )knX  and 
its complement in the system ( )knXX − , the mutual information is: 
(5.5) ( ) ( ) ( ) ( );k k k kn n n nIR X X X H X H X X H X− = + − −  
The mutual information is essential to evaluate the differentiation degree of a system, i.e. it is a 
significant index of the system’ “resolution” degree, calculated on the subdividable and distinct 
states.  
In order to measure the information of an integrated activation pattern, we calculate how the 
states of a given subset can differentiate them from the whole system ones. Which thing, following 
Edelman, is equivalent to considering the whole system as the observer of itself. In fact, if entropy 
measures the variability of a system according to an external observer evaluation, the mutual 
information measures the system variability according to an observer ideally placed within the 
system itself. 
The overall measure of the differentiation degree of a complex system is given by the mutual 
information average between each subset and the whole system: 
(5.6) ( ) ( )
C X IR X X X
= −�  
Edelman defined such measure as complexity and its value is high if each subset can averagely take 
many different states which are statistically depending on the whole system’s ones, so it shows how 
the system is differentiated. High complexity values correspond to an optimal synthesis of 
functional specialization and functional integration. Systems whose elements are not integrated 
(such as a gas) or not specialized ( such as an homogeneous crystal) have a minimum complexity. 
In the evolutionary neural gas case, the WTA strategy limits the integration among the 
activation patterns. So the mutual information between any activation pattern and the other possible 
patterns is equal to zero. It justifies the use of the term “gas”, since the patterns behave like isles of 
information weakly interacting each other. 
If there were selected more winner units for the same input signal in the early training phase, we 
could get a given system status characterized by i + 1 activated units not only by the activation of 
just a single winner and its related i units, but also by the activation of more winners. therefore  we 
should also take into consideration all the possible subgroups with j+1 elements. 
The mutual information formula between a subgroup with k activated units and the system is 
given by: 
(5.7) 
( ) ( ) ( )
( ) ( )
k ii jk
n ii ji j
H X i j
= = − −
� �+	 
� � � � � �= +� �� � � � +� �+ + � �� � � �
 � +� �� �+� �� �
( ) ( ) ( )
( ) ( )
k ii jk
n ii ji j
H X X i j
= = − −
� �+	 
� � � � � �− = + +� �� � � � +� �+ + � �� � � �
 � +� �� �+� �� �
( )( ) ( )
( )( ) ( )
1 1 log
kk jj
−−− −
−−−= −
� �	 
� � � � � �+ − − + +� �� �� � � � � �+ � �� � � �� �
 � − +� �� �+� �� �
( ) ( ) ( )
N ii j j
N i i
i j j
i j k
β β βα α α
− − −
+ +� �� � � � � �� �+ + + − ⋅� �� � � � � �� �+ +� � � � � �� �� �
� �  
( ) ( ) ( )
ii j j
i j j
β β βα α α− − −
+ +� �� �� � � �
+ + −� �� �� � � �+� � � �� �� �
( ) ( ) ( )
( ) ( )
N ii j
ii ji j
H X i j
= = − −
� �+	 
� � � � � �= +� �� � � � +� �+ + � �� � � �
 � +� �� �+� �� �
To provide the system with a greater level of complexity, in order to favouring the integration 
among the network unit subgroups, it is, therefore, necessary adopting a strategy different from the 
WTA in the early training phases so as to select more winner units. 
6) CONCLUSIONS AND FUTURE WORKS 
The here presented self-organizing network can be considered as an example of autopoietic 
system which evolves by means of a closed network of interactions and based upon the production 
of components (the categorization units). In the course of the reproductive dynamics, those ones 
produce other components, also belonging to the system (i.e. other categorization units) which 
maintain the system identity over time with respect to the experimental task.  
In particular, it has to be noticed that they are not just the environmental information to lead the 
evolution of the network of connections, but rather the network internal status, which is 
individuated globally by the size that the population has reached and locally by the values of the 
parameters of the units. The latter show the difficulties that the units encounter in modelling the 
presented input, such difficulty is directly proportional to the amount of variations their reference 
vectors are subjected to. 
Learning and the capability to model the system external inputs, therefore, emerges more by 
means of the population internal dynamics than by means of a learning algorithm. 
The appearing of a scale-free structure emerging from the choice of the population dynamics is 
peculiarly significant for the model’s biological plausibility. Which thing describes a quite phase-
transition-like status where cluster “float” as informational “isles” in a “gaseous” configuration. It is 
worthy noticing that the WTA strategy and the environmental noise (probabilistic laws) suffice to 
create a kind of basic informational skeleton around which more interconnected functional 
structures can then aggregate. In the nervous system, it plausibly happens according to an 
essentially genetic design. Such kind of neural dynamics guarantees flexibility and redundancy to 
the informational nuclei which are ready to synchronize and connect through signals. Actually, 
what we tried here to describe is a proto-neural scenario with low integration of clusters which are 
specialized in easy categorization tasks. 
Developing the ENG model requires to investigate different synchronization scenarios among 
clusters and their ensuing functional integration to execute more complex tasks. In particular, it is 
necessary to modify the evolutive dynamics so as to mane the connections among units active. In 
this way, it should be possible to create a dynamic neural topology susceptible of hierarchical 
organization. 
Everything seems to confirm not only the deep reasons for the scale-free structures recurring in 
nature (Z. Toroczkai, K. E. Bassler, 2004), but also the fundamental lesson associating complexity 
with a thin border zone between integration and differentiation among the functional modules of a   
system. 
Acknowledgements: The authors thank Eliano Pessa and Graziano Terenzi for their precious 
suggestions. 
REFERENCES 
Albert R. and Barabasi A.(2000) Topology of evolving networks: Local events and universality. 
Physical Review Letters vol.85, p.5234. 
Annunziato M. and Pizzuti S.(2000), Adaptive Parametrization of Evolutionary Algorithms Driven 
by Reproduction and Competition, in Proceedings of ESIT 2000, Aachen, Germany. 
Chialvo D.R. and Bak P. (1999) Learning from mistakes, in  Neuroscience Vol.90, No.4, pp.1137-
1148. 
Dawkins R. (1971), Selective neurone death as a possible memory mechanism, in Nature n.229, 
pp.118-119. 
Delaunay B. (1934), Bullettin of the Academy of Sciences USSR, vol.7, pp. 793-800. 
Edelman G.M. (1978), Group selection and phasic reentrant signaling: a theory of higher brain 
function, in The Mindful Brain (eds Edelman G.M. and Mountcastle V.), pp. 51-100, MIT, 
Cambridge. 
Edelman G.M. (1987), Neural Darwinism: The Theory of Neuronal Group Selection, Basic Books, 
New York. 
Edelman G.M., Tononi G. (2000), Un universo di coscienza, Biblioteca Einaudi, Torino. 
Fritzke B. (1994), Growing Cell Structures. A Self-Organizing Network for Unsupervised and 
Supervised Learning. in Neural Networks, 7(9), pp. 1441-1460. 
Heylighen F. (1992), Principles of Systems and Cybernetics: an evolutionary perspective, in: 
Cybernetics and Systems ’92, R. Trappl (ed.), World Science, Singapore, pp. 3-10. 
Langton C.G. (1989), Artificial Life: The Proceedings of an Interdisciplinary Workshop on the 
Synthesis and Simulation of Living Systems,  Addison-Wesley. 
Jefferson D. and Taylor C. (1995), Artificial Life as a Tool for Biological Inquiry, in Artificial Life: 
an Overview, edited by C.G. Langton, MIT press, pp.1-10. 
Kohonen T. (1982), Self-Organized Formation of Topologically Correct Feature Maps, in 
Biological Cybernetics, n.43, pp.59-69. 
Martinetz T.M. (1993), Competitive Hebbian Learning Rule Forms Perfectly Topology Preserving 
Maps, in ICANN’93, International Conference on Artificial Neural Networks, Springer, pp. 427-
434. Amsterdam.  
Martinetz T.M. and K.J. Schulten, (1991), A Neural Gas Network Learns Topologies, In Artificial 
Neural Networks, T.Kohonen, K. Makisara, O. Simula, and J. Kangas, eds, , pp. 397-402. North-
Holland, Amsterdam. 
Martinetz T.M. and Schulten K.J. (1994), Topology Representing Networks, in Neural Networks, 
7(3), pp. 507-522. 
Moriarty D.E. and R.Miikkulainen (1998), Forming Neural Networks Through Efficient and 
Adaptive Coevolution, in Evolutionary Computation, 5(4), pp.  373-399. 
Paredis J. (1995), Coevolutionary Computation, in Artificial Life, 2, pp.355-375. 
Rocha L.M. (1997) Evolutionary Systems and Artificial Life, Lecture Notes. Los Alamos, NM 
87545 
Roughgarden J.,(1979), Theory of Population Genetics and Evolutionary Ecology, Prentice-Hall. 
Smith R.E., Forrest S., Perelson A.S. (1993), Searching for Diverse Cooperative Populations with 
Genetic Algorithms, in  Evolutionary Computation, 1(2), 127-149. 
Song J. and Yu J. (1988), Population System Control, Springer-Verlag. 
Steyvers M. and Tenenbaum J., 2001. The Large-Scale structure of Semantic Networks. Working 
draft submitted to Cognitive Science. 
Toroczkai,Z. and Bassler,K.E. (2004), Jamming is Limited in Scale-Free Systems, in Nature, 428 , 
p.716 
Watts D.J., Strogatz S.H. , 1998. Collective dynamics of ‘small-world’ networks. Nature, vol. 393, 
pp. 440-442. 
Young J.Z. (1964), A Model of the Brain, Clarendon, Oxford. 
Young J.Z. , (1966), The Memory System of the Brain, University of California Press, Berkeley.

  Despite their claimed biological plausibility, most self organizing networks
have strict topological constraints and consequently they cannot take into
account a wide range of external stimuli. Furthermore their evolution is
conditioned by deterministic laws which often are not correlated with the
structural parameters and the global status of the network, as it should happen
in a real biological system. In nature the environmental inputs are noise
affected and fuzzy. Which thing sets the problem to investigate the possibility
of emergent behaviour in a not strictly constrained net and subjected to
different inputs. It is here presented a new model of Evolutionary Neural Gas
(ENG) with any topological constraints, trained by probabilistic laws depending
on the local distortion errors and the network dimension. The network is
considered as a population of nodes that coexist in an ecosystem sharing local
and global resources. Those particular features allow the network to quickly
adapt to the environment, according to its dimensions. The ENG model analysis
shows that the net evolves as a scale-free graph, and justifies in a deeply
physical sense- the term gas here used.

Microsoft Word - ENG-EJTP.doc
EVOLUTIONARY NEURAL GAS (ENG):  A MODEL OF SELF ORGANIZING 
NETWORK FROM INPUT CATEGORIZATION. 
Ignazio Licata (a) ↑ , Luigi Lella (b) 
(a) Ixtucyber for Complex Systems, Marsala, TP and Institute for Scientific Methodology, 
Palermo, Italy; 
(b) A.R.C.H.I. - Advanced Research Center for Health Informatics, Ancona, Italy 
ABSTRACT  
Despite their claimed biological plausibility, most self organizing networks have strict 
topological constraints and consequently they cannot take into account a wide range of external 
stimuli. Furthermore their evolution is conditioned by deterministic laws which often are not 
correlated with the structural parameters and the global status of the network, as it should happen in 
a real biological system. In nature the environmental inputs are noise affected and “fuzzy”. Which 
thing sets the problem to investigate the possibility of emergent behaviour in a not strictly 
constrained net and subjected to different inputs. 
It is here presented a new model of Evolutionary Neural Gas (ENG) with any topological 
constraints, trained by probabilistic laws depending on the local distortion errors and the network 
dimension. The network is considered as a population of nodes that coexist in an ecosystem sharing 
local and global resources. 
Those particular features allow the network to quickly adapt to the environment, according to its 
dimensions. The ENG model analysis shows that the net evolves as a scale-free graph, and justifies 
in a deeply physical sense- the term “gas” here used. 
Key-words: Self-Organizing Networks; Neural Gas; Scale-Free Graph; Information in Network 
Functional Specialization. 
1. INTRODUCTION 
Self organizing networks are systems widely used in categorization tasks. A network can be 
seen as a set A={c1, c2,… ,cn} of units with associated reference vectors wc∈R
n where Rn is the 
same space where inputs are defined. Each unit (or node) can establish connections with the other 
ones, the units belonging to the same clusters are subjected to similar modification affecting their 
reference vectors.  
Self organizing networks can automatically adapt to input distributions without supervision by 
means of training algorithms that are simple sequences of deterministic rules. Competitive hebbian 
learning and neural gas are the most important strategies used for their training. 
Neural gas algorithm (Martinetz T.M. and Schulten K.J., 1991) sorts the network units according to 
the distance of their reference vector to each input. Then the reference vectors are adapted so that 
the ones related to the first nodes in the rank order are moved more close than the others to the 
considered input. 
Competitive hebbian learning (Martinetz and Schulten, 1991; Martinetz, 1993) consists in 
augmenting the weight of the link connecting the two units whose reference vectors are closest to 
the considered input (the two most activated units). 
Both strategies are examples of deterministic rules. As we know there are other rules that constrain 
the topology of the network which has a fixed dimensionality. That’s the case of Self Organizing 
Maps (Kohonen, 1982) and Growing Cell Structures (Fritzke, 1994). 
                                                
↑ Corresponding author: Ignazio.licata@ejtp.info  
In other cases the network structures haven’t topological constraints, they take a well ordered 
distribution by exactly adapting to the manifold inputs. For example TRN (Martinetz and Schulten, 
1994) and GNG are networks whose final structure is similar to a Delaunay Triangulation 
(Delaunay, 1934).We have tried to define a new self organizing network that is trained by 
probabilistic rules avoiding any topological constraints.  
According to Jefferson (1995) life and evolution are structured at least into four fundamental 
levels:  molecular, cellular, organism and population. We propose a population level based on 
evolutionary algorithm where the network is seen as a population of units whose interactions are 
conditioned by the availability of resources in their ecosystem. The evolution of the population is 
driven by a selective process that favours the fittest units. This approach has a biological 
plausibility. As stated by recent theories (Edelman, 1987) human brain evolution is subjected to 
similar selective pressures. 
Obviously we are not interested in recreating the same structure as the human brain. Our work 
aims at finding innovative and effective solutions to the categorization problem adopting natural 
system strategies. So our system falls within the Artificial Life field (Langton, 1989). 
Our model is a complex system that shows emergent features. In particular its structure evolves as a 
scale free graph. In the training phase there arise clusters of units with a limited number of nodes 
that establish a great number of links with the others.  
Scale free graphs are a particular structure that is really common in natural systems. Human 
knowledge, for instance, seems to be structured as a scale free graph (Steyvers, Tenenbaum 2001). 
If we represent words and concepts as nodes, we’ll find that some of these are more connected than 
the others. 
Scale free graphs have three main features.The small world structure. It means there is a 
relatively short path between any couple of nodes (Watts, Strogatz, 1998).The inherent tendency to 
cluster that is quantified by a coefficient introduced by Watts and Strogatz. Given a node i of ki 
degree i.e. having ki edges which connect it to ki other nodes, if those make a cluster, they can 
establish ki(ki-1)/2 edges at best. The ratio between the actual number of edges and the maximum 
number gives the clustering coefficient of node i. The clustering coefficient of the whole network is 
the average of all the individual clustering coefficients.  
Scale free graphs are also characterized by a particular degree distribution that has a power-law 
tail P(k)~k n− . That’s why such networks are called “scale free” (Albert, Barabasi, 2000). 
The three previous features are quantified by three parameters: the average path length between 
any couple of nodes, the clustering coefficient and the exponent of the power law tail. We’ll show 
that the values of these parameters in our model seem to confirm its scale free nature. 
2. AN OUTLINE ON SELF-ORGANIZATION AND EVOLUTIONARY SYSTEMS 
Natural selection mechanism has been successfully used for a lot of industrial applications 
spanning from projecting to real-time control and neural networks training. 
It was in the 60s that Genetic Algorithms based on the Evolution Theory’s three main 
mechanisms - reproduction, mutation and fitness – were first used in dealing with optimization 
problems. Although the solution is reached by a population of individuals, systems based on this 
approach are not considered self organizing because their dynamics depend on the external 
constraint of the fitness function. 
In the 80s a new approach to the study of living systems which mixed together self organization 
and evolutionary systems came out (Rocha, 1997). Its success was due to the studies on the way 
how biological systems work (metabolism, adaptability, autonomy, self repairing, growth, evolution 
etc.). The hybrid systems make us possible to get a better simulation both of the evolutionary 
optimization processes and the internal structure modification to reach a greater biological 
plausibility in the fitness 
Neuroevolutionary systems are an example of this approach. In classic neuroevolutionary models 
the network parameters are genetically set, whereas the connection weights are modified according 
to a training strategy. This solution follows the classic vision of cerebral development where genes 
control the formation of synaptic connections while their reinforcement depends on neural activity. 
More recent neuroevolutionary systems are characterized by different forms of self organizing 
processes which are cooperative coevolution (Paredis, 1995; Smith, Forrest and Perelson, 1993) and 
synaptic Darwinism (Edelman, 1987). 
Cooperative co evolutionary systems offer a promising alternative to classic evolutionary 
algorithms when we face complex dynamical problems. The main difference with respect to classic 
EA is the fact that each individual represents only a partial solution of the problem. Complete 
solutions are obtained by grouping several individuals. The goal of each individual is to optimize 
only a part of the solution, cooperating with other individuals that optimize other parts of the 
solution. It is so avoided the premature convergence towards a single group of individuals. An 
example of such approach is given by the Symbiotic Adaptive Neuroevolution System (Moriarty 
and Miikkulainen, 1998) that operates on populations of neural networks. 
While in most neuroevolutionary systems each individual represents a complete neural network, 
in SANE each individual represents a hidden unit of a two-layered network. Units are continuously 
combined and the resulting networks are evaluated on the basis of the performances shown in a 
given task. The global effect is equal to schemas promoting in standard EAs. In fact during the 
evolution of the population the neural schemas having the highest fitness values are favoured and 
the possible mutations in the copies of these schemas don’t affect the other copies in the population. 
Other recent strategies focus on the evolution of connection schemas in the network. In the human 
brain the number of synapses established by a single neuron is always much lower than the overall 
number of neurons. That gives the network a sparsely connected aspect. In the last years several 
models have been proposed to emulate the mechanism involved in the selection of links without 
referring to the physical and chemical properties of neurons. 
The Chialvo and Bak model (Chialvo and Bak, 1999) is based on two simple and biological 
inspired principles. First, the neural activity is kept low selecting the activated units by a winner 
takes all strategy. Second, the external environment gives a negative feedback that inhibits active 
synapses if the network behaviour is not satisfying. With these simple rules the model operates in a 
highly adaptive state and in critical conditions (extreme dynamics). The fundamental difference of 
this strategy based on the synaptic inhibition with respect to the classic one based on synaptic 
reinforcement is that the reinforcement-based learning is a continuative process by definition, while 
the inhibition-based learning stops when the training goal is achieved. The synaptic inhibition is 
also biologically plausible. According to Young (Young, 1964; Young, 1966) learning is the result 
of the elimination of synaptic connections (closing of unneeded channels). Dawkins (Dawkins R., 
1971) stressed that pattern learning is achieved by synaptic inhibition. As stated by the neural 
groups’ selection theory developed by Edelman (Edelman, 1978; Edelman, 1987), brain 
development is characterized by generating a structural and dynamical variability within and 
between populations of neurons, by the interaction of the neural circuit with the environment and by 
the differential attenuation or amplification of synaptic connections. Research in neurobiology 
seems to confirm the validity of the negative feedback model and the fact that neural development 
follows the process of Darwinian evolution. 
The Chialvo and Bak model is a simple two-layered network. After the training each input pattern 
is associated with a single output unit leading to the formation of an associative map. When an input 
pattern is presented the most activated input unit i is selected. Then the neuron j from the hidden 
layer that establishes the most robust connection with i is selected. Finally the output neuron k that 
is the most strongly connected with j is selected. If k is not the desired output the two links 
connecting i with j and j with k are inhibited by a coefficient d that is the only parameter of the 
model. The iterative application of these rules leads to a rapid convergence towards any input-
output mapping. This selective process followed by an inhibitory one is the essence of the natural 
selection in the evolutionary context. The fittest individual is selected on the basis of a strategy that 
doesn’t reward the best but punishes the worst. That’s the reason why this model has been 
considered a particular kind of synaptic Darwinism. 
Our neuroevolutionary model is also based on a selection strategy. The structural information of 
our network is not codified by genes. We directly consider the entire network as a population of 
nodes that can establish connections, generate other units or die. The probability of these events 
depends on the presence of local and global resources. If there are few resources the population 
falls, if there is a lot of resources the population grows. Like in the Chialvo and Bak model we don’t 
select the fittest nodes reinforcing their links, but we simply remove the worst nodes when the 
ecosystem resources are low. This generates a selective process that indirectly rewards the units 
which can better model the input patterns. Our evolutionary strategy can be seen as a selective 
retention process (Heylighen, 1992) that removes those units which cannot reach a stable state, 
remaining associated with several input patterns. Even if the stability of a unit is quantified by the 
minimum distortion error related to it, this information mustn’t be considered to be environmental 
information. The minimum distortion error simply quantifies the difficulty encountered by the unit 
during the modelling of input patterns. 
3. THE EVOLUTIONARY ALGORITHM 
Research has confirmed (Roughgarden, 1979; Song and Yu, 1988) that in natural environments 
the population size along with competition and reproduction rates continuously changes according 
to some natural resources and the available space in the ecosystem. 
These mechanisms have been reproduced in some evolutionary algorithms, for example to optimize 
the evolution of a population of chromosomes in a genetic algorithm (Annunziato and Pizzuti, 
2000). We have tried to use a similar strategy for the evolution of a population of units in a self 
organizing network without using the string representation of genetic programming. 
In our model each node is defined by a vector of neighbouring units connected to it, a reference 
vector and a variable D that is the smallest distance between its reference vector and the closest 
modelled input. The value of this variable quantifies the debility degree of the unit.  The lower is D 
the higher are the chances for the unit to survive. At each presentation of the training input set, D is 
set to the maximum value. After the presentation of a given input x, if the reference vector w of the 
unit is modified, the resulting distance between the two vectors ||x-w|| is calculated. If this value is 
lower than D it becomes its new value. 
The  training algorithm here used can be subdivided in three phases: 
1) Winners are selected. For each input the unit having the closest reference vector is selected. 
2) The reference vectors of the winners and their neighbours are updated according to the 
following formula : 
(3.1) ( ) ( ) ( )( )1w t w t x w tα+ = + − . 
So the reference vectors w of the selected units are moved towards the relative inputs x of a certain 
fraction of the distances that separate them. For winners this fraction is two or three orders of 
magnitude higher than the one used for their neighbours. So winners have the reference vectors 
moving more quickly towards the inputs. 
3) The population of units evolves producing descendants, establishing new connections and 
eliminating the less performing units. All these events can occur with a well defined probability that 
depends on the availability of resources. 
These rules are iterated until a given goal is achieved. For example the minimization of the 
expected quantization error that is the mean of the distances between the winners and the K inputs 
they model: 
(3.2) 
D x wK
= −�  
If this value falls below a certain threshold Dmin, training is stopped. 
The first two phases can be considered a kind of winner takes all strategy, where only the most 
activated units are selected and enabled to modify their reference vectors. The third phase is the 
evolutionary phase (fig. 3.1). Each unit i, i=[1…N(t)] where N(t) is the actual population size can 
meet the closest winner j with probability Pm: 
Fig. 3.1 – The evolutionary phase of the algorithm. 
If meeting occurs, the two units establish a link and they can interact by reproducing with 
probability Pr. In this case two new units are created. One is closer to the first parent, the other to 
the second parent: 
(3.3) 
If reproduction doesn’t take place due to the lack of resources the weaker unit of the population, i.e. 
the one with the highest debility degree, is removed. 
If unit i doesn’t meet any winner it can interact with the closest node k with probability Pr 
establishing a connection and producing a new unit whose reference vector is set between the 
parents reference vectors: 
(3.4)  1 22 2
p pw ww
When we fix a maximum population size, the ratio between the actual size and the threshold 
N(t)/Nmax can be seen as a global resource of the ecosystem affecting the probabilities of the events. 
For example if the population size is low the reproduction rate should be high. So we can 
reasonably choose Pr = 1-N(t)/Nmax.  If the population size is high, the chance for the units to meet 
each other will be higher, so we can set Pm = N(t)/Nmax. 
We can also consider a local resource that is the ratio between the threshold Dmin and the 
debility degree Di of the unit i. Each unit i should meet a winner with a probability 
Pm=(N(t)/Nmax)(1-Dmin/Di) and Pr = 1 – Pm. In this way winners are not encouraged to migrate to 
other groups of nodes and weaker units don’t participate in reproduction activities. 
We can estimate the population grow rate in the following way: 
(3.5) 
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )
2 2 2
2min min
1 2 1 1 1
2 1 1 2 1 ( )
2 1 1 1 2 1 1 ( )
m r m r m m d
N t N t P P N t P P N t P N t P P N t
N t P N t
N t X t X t X t first model
N t D D
N t X t X t X t second model
M D D
+ = + − − + − − − =
= − =
= − � + = −� �
� � � �� � � �= − − � + = − −� � � �� � � �� �� �� � � �� �� �
where X(t) is the normalized size N(t)/Nmax. This is the quadratic-logistic map of Annunziato and 
Pizzuti(Annunziato and Pizzuti, 2000): 
(3.6) ( ) ( ) ( )( )21 1X t aX t X t+ = −  
They proved that by varying the parameter different chaotic regimes arise. For a<1.7 the behaviour 
is not chaotic, for 1.7<a<2.1 we have chaotic regimes with simple attractors localized in a fixed part 
of the plane of the phases. Theoretically for the first model we expect to obtain a chaotic regime 
that is described by a simple attractor. In the second model the factor (1 – Dmin/D) might reduce the 
influence of the negative feedback in the final part of network training. 
It is possible to demonstrate that during the evolution the population size converges to N(t) = 
0.72 Nmax. In this phase the probability that a unit establishes n connections with the other ones for 
the first model, considering only clusters of n units, is given by: 
(3.7) ( )
max0.72 1
max max
1max max
0.72 0.72
n i nN n
P n n
� � � �
= − =� � � �
� � � �
It has to be pointed out we have subtracted the probability that such n links developed within a 
cluster of more than a n unit. 
The coefficients α  and β  of the power law are considered constant at the end of the training. To 
compute their values, we can take into consideration the cases n=1 and n=0.72N Nmax-1 which 
correspond to the minimum and maximum number of connection at the end of the training. 
(3.8) ( )
max0.72 2
1 0.72 0.72 1
P βα α
= − = =�  
( ) ( )
0.72 2
0.72 1 1
max max
0.72 1 0.72 0.72 0.72 0.72 1
P N N
− = = − −� �
max max
0.72 2
0.72 1 0.72 1
0.72 0.72
� �� =
The distribution tail of the degrees tends to stretch when the maximum size of the population 
increases, it means that in wider networks there are more hubs with a higher degree. 
For the second model we can consider that at the end of the training (1-Dmin/D) ∼ε   
So the probability that a unit establishes n links becomes: 
(3.9) 
max0.72 1
max max
1max max
0.72 0.72
n i nN n
P n n
βε ε α
� � � �
= − =� � � �
� � � �
max max
0.72 2
0.72 1 0.72 1
0.72 0.72
� �� =
and the considerations made for the first model can be therefore extended to the second model. 
4. TRAINING THE NET: SIMULATIONS  
We have compared the performances of our networks with those of a Growing Neural Gas in 
categorizing bidimensional inputs.  
GNG is a self organizing network which thanks to both the competitive hebbian learning 
strategy and the neural gas algorithm can categorize inputs without altering their exact 
dimensionality.For the GNG, the parameters of the model are �����α  = 0.5, β  = 0.0005 and at each λ  = 
300 steps a new unit is inserted. The maximum age of the links is set to 88.  
For the two different ENG models, the parameters are α  = 0.05, β  = 0.0006 and the maximum 
size is set to Nmax = 120. 
As stopping criterion for both the algorithms we have chosen the minimization of the expected 
quantization error that is the average distance between the winners and the corresponding inputs. 
We have considered two different input domains. In the first case inputs are localized within four 
square regions, in the second one inputs are uniformly distributed in a ring region. 
As shown in fig.4.1 after the training, GNG reference vectors are all positioned in the input 
domain. In the Evolutionary Self Organizing Networks (fig.4.2a and fig.4.2b) some units fall 
outside the input domain, but in this way the network remains fully connected. The nodes’ 
distribution statistical analysis confirms what appears to be intuitively patent: the emerging network 
structure is a typical scale-free one, i.e. a structure where few hubs manage the links. 
Fig. 4.1 – Growing Neural Gas simulations. 
Fig. 4.2 a – Evolutionary Self Organizing Network simulations (first model). 
Fig. 4.2 b – Evolutionary Self Organizing Network simulations (second model). 
We trained 30 networks of each type obtaining the average degree distributions reported in fig.4.3-
4.5. In tab. 4.1 – 4.2 are reported the average values of the structural parameters of the two 
networks. 
y = 37,074x
-2,0371
1 2 3 4 5
degree
y = 68,961x
-2,9864
1 2 3 4 5 6
degree
Fig. 4.3 – Average degree distribution in GNG (two different input domains) 
y = 23,75x-1,149
1 2 3 4 5 6 7 8 9
degree
y = 22,069x
-1,1143
1 2 3 4 5 6 7 8
degree
Fig. 4.4 – Average degree distribution in ENG (first model, two different input domains) 
y = 24,415x-1,1411
1 2 3 4 5 6 7 8 9
degree
y = 21,98x
-1,1408
1 2 3 4 5 6 7 8
degree
Fig. 4.5 – Average degree distribution in ENG (second model, two different input domains) 
While GNG have a high value for the average path length and a low clustering coefficient, ENG 
have a short average path length and a high clustering coefficient which along with the power law 
tail of the degree distribution confirm its scale free graph features. 
 Average 
path 
length 
Clustering 
coefficient 
Power 
exponent 
GNG -  0.49 2.04 
ESON (1st) 3.82 0.64 1.15 
ESON(2nd) 3.92 0.63 1.14 
 Tab. 4.1 – Comparison of structural parameters (average values, first input domain) 
 Average 
path 
length 
Clustering 
coefficient 
Power 
exponent 
GNG 6.4 0.42 2.98 
ESON (1st) 3.61 0.58 1.11 
ESON(2nd) 3.67 0.59 1.14 
Tab. 4.2 – Comparison of structural parameters (average values, second input domain) 
Fig. 4.6 – 4.7 shows the population dynamics of the two ENG models. 
The structure shared by the two different ENG models is due to the fact that the winner units tend to 
establish the greatest number of connections. These are the favoured units with which each node try 
to establish a connection. If the probability depends also on the local distortion error as it happens in 
the second model, we obtain a final structure that is more similar to the GNG, which is to say more 
similar to a gas. In point of fact, the conditions to create a new link become more restrictive, 
reducing the interaction among each cluster and the whole network. The structure of connections 
seems to extend more uniformly in the regions where inputs are present as it can be seen in picture 
4.2b (more evident in the circular distribution). 
Picture 4.7 shows the dynamics of the populations in the two different models of ENG. In the first 
model the population size seems to converge to the final value of 0.72Nmax, confirming the 
experimental results of Annunziato and Pizzuti. As it can be noticed in fig. 4.6, since the d value 
gradually diminishes during the training, the influence of the factor (1-Dmin/d) grows reducing the 
effects of the negative feedback which characterizes the quadratic logistic map. This justifies the 
sudden growth of the population at the end of the training in the second model. 
1 3 5 7 9 1
epochs
1 2 3 4 5 6 7 8 9 10 11
epochs
Fig. 4.6 – Network size evolution of the two ENG models (first input domains). 
At the end of training new units connect with the winner units which have a lower d, while the 
subgroups of units become more isolated. Considering the function (X(t),X(t+1)) the attractor 
becomes more marked in the second model. This means that the system tends to converge more 
toward a precise final state with a lower interaction among the groups of units. 
Fig. 4.7 – population dynamics (X(t),X(t+1)) of the two Esonet models (first input domain). 
5. THE ROLE OF INFORMATION IN FUNCTIONAL SPECIALIZATION AND 
INTEGRATION 
We can classify a system as complex when it is made up of different parts heterogeneously 
interacting. In addition, its behaviour and its structure have to be neither completely casual (as it 
happens in a gas) nor too regular (as it happens in a crystal). In Nature we generally observe the co-
existence of functionally highly specialized integrated areas. 
That’s what happens in the brain, where different areas and groups of neurons interact to give 
rise to an integrated and unitary cognitive scenario (G. M. Edelman, G. Tononi, 2000). 
Edelman has introduced the integration, reciprocal information and complexity concepts in 
order to mathematically define the functional organization of the cerebral structures. 
Within a complex system, a subset of elements can be defined an integrated process if – on a 
given temporal scale – the elements interact more strongly with each other than with the system. In 
a neural net or in a self-organizing one it means that the units of an integrated group will tend to 
simultaneously activate themselves. 
When the units in a subset are independent, the system’s entropy reaches its maximum value 
which is the sum of the entropies of the single elements (local entropies). On the contrary, when any 
kind of interaction occurs, the global entropy decreases so becoming lower than the sum of the local 
entropies. The integration measure is, therefore, a natural indicator of the system informational 
“capacity”. 
So the integration of a subset of network units can be calculated by deducting the sum of the 
entropies of each single component ( )ix  from the entropy of the system considered as a whole. If 
each unit can only take two states (activated/not-activated), the amount of the possible activation 
patterns of a subset with N units is N2 . So the system maximum entropy is: 
(5.1) ( ) ( )max 2 2 2
1 2 1
log log 1 log 2
i i NN
i i i
H X H x p N
� � � �= = = = =� �� �
� �� �
� �  
and the integration will be: 
(5.2) ( ) ( ) ( )
I X H x H X
= −�  
for the self-organized net here considered, the integration of a sub-group of units takes the 
following expression: 
(5.3) ( )
I X N P
� �� �
= − � �� �+� � � �
where Pi is the probability for a node to establish i connections. The overall number of the 
system’ states is equal to the total number of possible groups of i+1 units. Groups of units having 
the same dimension (groups of i+1 units) give the same contribution to the entropy of the system. 
If we choose the WTA strategy as activation modality, for each presented input only a single 
unit (the winner) and the 1<i<N-1 i units will activate themselves. All the other ones remain not-
activated. 
The probability for a node to create connections is ruled by the power law βα −= kPi , with α and 
β depending on 1) the network dimension, 2) the local distortion errors (for the second model) and 
3) the particular evolution of the network structure, i.e. the dynamic behaviour of ( )tα  and ( )tβ . 
So the integration of the two self-organizing network here presented is: 
(5.4) ( ) ( )
I X N i
� � � �= − � �� �+ � �� �
The integration can be seen as a measure of the statistic dependency within a subset of units. 
The stronger their interactions are, the higher their integration. 
In order to measure the statistic dependency between a subset and the whole system, Edelman 
introduced the concept of mutual information. Given an n subset made up of k elements ( )knX  and 
its complement in the system ( )knXX − , the mutual information is: 
(5.5) ( ) ( ) ( ) ( );k k k kn n n nIR X X X H X H X X H X− = + − −  
The mutual information is essential to evaluate the differentiation degree of a system, i.e. it is a 
significant index of the system’ “resolution” degree, calculated on the subdividable and distinct 
states.  
In order to measure the information of an integrated activation pattern, we calculate how the 
states of a given subset can differentiate them from the whole system ones. Which thing, following 
Edelman, is equivalent to considering the whole system as the observer of itself. In fact, if entropy 
measures the variability of a system according to an external observer evaluation, the mutual 
information measures the system variability according to an observer ideally placed within the 
system itself. 
The overall measure of the differentiation degree of a complex system is given by the mutual 
information average between each subset and the whole system: 
(5.6) ( ) ( )
C X IR X X X
= −�  
Edelman defined such measure as complexity and its value is high if each subset can averagely take 
many different states which are statistically depending on the whole system’s ones, so it shows how 
the system is differentiated. High complexity values correspond to an optimal synthesis of 
functional specialization and functional integration. Systems whose elements are not integrated 
(such as a gas) or not specialized ( such as an homogeneous crystal) have a minimum complexity. 
In the evolutionary neural gas case, the WTA strategy limits the integration among the 
activation patterns. So the mutual information between any activation pattern and the other possible 
patterns is equal to zero. It justifies the use of the term “gas”, since the patterns behave like isles of 
information weakly interacting each other. 
If there were selected more winner units for the same input signal in the early training phase, we 
could get a given system status characterized by i + 1 activated units not only by the activation of 
just a single winner and its related i units, but also by the activation of more winners. therefore  we 
should also take into consideration all the possible subgroups with j+1 elements. 
The mutual information formula between a subgroup with k activated units and the system is 
given by: 
(5.7) 
( ) ( ) ( )
( ) ( )
k ii jk
n ii ji j
H X i j
= = − −
� �+	 
� � � � � �= +� �� � � � +� �+ + � �� � � �
 � +� �� �+� �� �
( ) ( ) ( )
( ) ( )
k ii jk
n ii ji j
H X X i j
= = − −
� �+	 
� � � � � �− = + +� �� � � � +� �+ + � �� � � �
 � +� �� �+� �� �
( )( ) ( )
( )( ) ( )
1 1 log
kk jj
−−− −
−−−= −
� �	 
� � � � � �+ − − + +� �� �� � � � � �+ � �� � � �� �
 � − +� �� �+� �� �
( ) ( ) ( )
N ii j j
N i i
i j j
i j k
β β βα α α
− − −
+ +� �� � � � � �� �+ + + − ⋅� �� � � � � �� �+ +� � � � � �� �� �
� �  
( ) ( ) ( )
ii j j
i j j
β β βα α α− − −
+ +� �� �� � � �
+ + −� �� �� � � �+� � � �� �� �
( ) ( ) ( )
( ) ( )
N ii j
ii ji j
H X i j
= = − −
� �+	 
� � � � � �= +� �� � � � +� �+ + � �� � � �
 � +� �� �+� �� �
To provide the system with a greater level of complexity, in order to favouring the integration 
among the network unit subgroups, it is, therefore, necessary adopting a strategy different from the 
WTA in the early training phases so as to select more winner units. 
6) CONCLUSIONS AND FUTURE WORKS 
The here presented self-organizing network can be considered as an example of autopoietic 
system which evolves by means of a closed network of interactions and based upon the production 
of components (the categorization units). In the course of the reproductive dynamics, those ones 
produce other components, also belonging to the system (i.e. other categorization units) which 
maintain the system identity over time with respect to the experimental task.  
In particular, it has to be noticed that they are not just the environmental information to lead the 
evolution of the network of connections, but rather the network internal status, which is 
individuated globally by the size that the population has reached and locally by the values of the 
parameters of the units. The latter show the difficulties that the units encounter in modelling the 
presented input, such difficulty is directly proportional to the amount of variations their reference 
vectors are subjected to. 
Learning and the capability to model the system external inputs, therefore, emerges more by 
means of the population internal dynamics than by means of a learning algorithm. 
The appearing of a scale-free structure emerging from the choice of the population dynamics is 
peculiarly significant for the model’s biological plausibility. Which thing describes a quite phase-
transition-like status where cluster “float” as informational “isles” in a “gaseous” configuration. It is 
worthy noticing that the WTA strategy and the environmental noise (probabilistic laws) suffice to 
create a kind of basic informational skeleton around which more interconnected functional 
structures can then aggregate. In the nervous system, it plausibly happens according to an 
essentially genetic design. Such kind of neural dynamics guarantees flexibility and redundancy to 
the informational nuclei which are ready to synchronize and connect through signals. Actually, 
what we tried here to describe is a proto-neural scenario with low integration of clusters which are 
specialized in easy categorization tasks. 
Developing the ENG model requires to investigate different synchronization scenarios among 
clusters and their ensuing functional integration to execute more complex tasks. In particular, it is 
necessary to modify the evolutive dynamics so as to mane the connections among units active. In 
this way, it should be possible to create a dynamic neural topology susceptible of hierarchical 
organization. 
Everything seems to confirm not only the deep reasons for the scale-free structures recurring in 
nature (Z. Toroczkai, K. E. Bassler, 2004), but also the fundamental lesson associating complexity 
with a thin border zone between integration and differentiation among the functional modules of a   
system. 
Acknowledgements: The authors thank Eliano Pessa and Graziano Terenzi for their precious 
suggestions. 
REFERENCES 
Albert R. and Barabasi A.(2000) Topology of evolving networks: Local events and universality. 
Physical Review Letters vol.85, p.5234. 
Annunziato M. and Pizzuti S.(2000), Adaptive Parametrization of Evolutionary Algorithms Driven 
by Reproduction and Competition, in Proceedings of ESIT 2000, Aachen, Germany. 
Chialvo D.R. and Bak P. (1999) Learning from mistakes, in  Neuroscience Vol.90, No.4, pp.1137-
1148. 
Dawkins R. (1971), Selective neurone death as a possible memory mechanism, in Nature n.229, 
pp.118-119. 
Delaunay B. (1934), Bullettin of the Academy of Sciences USSR, vol.7, pp. 793-800. 
Edelman G.M. (1978), Group selection and phasic reentrant signaling: a theory of higher brain 
function, in The Mindful Brain (eds Edelman G.M. and Mountcastle V.), pp. 51-100, MIT, 
Cambridge. 
Edelman G.M. (1987), Neural Darwinism: The Theory of Neuronal Group Selection, Basic Books, 
New York. 
Edelman G.M., Tononi G. (2000), Un universo di coscienza, Biblioteca Einaudi, Torino. 
Fritzke B. (1994), Growing Cell Structures. A Self-Organizing Network for Unsupervised and 
Supervised Learning. in Neural Networks, 7(9), pp. 1441-1460. 
Heylighen F. (1992), Principles of Systems and Cybernetics: an evolutionary perspective, in: 
Cybernetics and Systems ’92, R. Trappl (ed.), World Science, Singapore, pp. 3-10. 
Langton C.G. (1989), Artificial Life: The Proceedings of an Interdisciplinary Workshop on the 
Synthesis and Simulation of Living Systems,  Addison-Wesley. 
Jefferson D. and Taylor C. (1995), Artificial Life as a Tool for Biological Inquiry, in Artificial Life: 
an Overview, edited by C.G. Langton, MIT press, pp.1-10. 
Kohonen T. (1982), Self-Organized Formation of Topologically Correct Feature Maps, in 
Biological Cybernetics, n.43, pp.59-69. 
Martinetz T.M. (1993), Competitive Hebbian Learning Rule Forms Perfectly Topology Preserving 
Maps, in ICANN’93, International Conference on Artificial Neural Networks, Springer, pp. 427-
434. Amsterdam.  
Martinetz T.M. and K.J. Schulten, (1991), A Neural Gas Network Learns Topologies, In Artificial 
Neural Networks, T.Kohonen, K. Makisara, O. Simula, and J. Kangas, eds, , pp. 397-402. North-
Holland, Amsterdam. 
Martinetz T.M. and Schulten K.J. (1994), Topology Representing Networks, in Neural Networks, 
7(3), pp. 507-522. 
Moriarty D.E. and R.Miikkulainen (1998), Forming Neural Networks Through Efficient and 
Adaptive Coevolution, in Evolutionary Computation, 5(4), pp.  373-399. 
Paredis J. (1995), Coevolutionary Computation, in Artificial Life, 2, pp.355-375. 
Rocha L.M. (1997) Evolutionary Systems and Artificial Life, Lecture Notes. Los Alamos, NM 
87545 
Roughgarden J.,(1979), Theory of Population Genetics and Evolutionary Ecology, Prentice-Hall. 
Smith R.E., Forrest S., Perelson A.S. (1993), Searching for Diverse Cooperative Populations with 
Genetic Algorithms, in  Evolutionary Computation, 1(2), 127-149. 
Song J. and Yu J. (1988), Population System Control, Springer-Verlag. 
Steyvers M. and Tenenbaum J., 2001. The Large-Scale structure of Semantic Networks. Working 
draft submitted to Cognitive Science. 
Toroczkai,Z. and Bassler,K.E. (2004), Jamming is Limited in Scale-Free Systems, in Nature, 428 , 
p.716 
Watts D.J., Strogatz S.H. , 1998. Collective dynamics of ‘small-world’ networks. Nature, vol. 393, 
pp. 440-442. 
Young J.Z. (1964), A Model of the Brain, Clarendon, Oxford. 
Young J.Z. , (1966), The Memory System of the Brain, University of California Press, Berkeley.
