Self-Organizing Map with Weighted Connections
Avoiding False-Neighbor Effects
Haruna MATSUSHITA∗ and Yoshifumi NISHIO∗∗ Department of Electrical and Electronic Engineering, Tokushima University

2–1 Minami-Josanjima, Tokushima 770–8506, Japan
Telephone: +81–88–656–7470, Fax: +81–88–656–7471,
Email: {haruna, nishio}@ee.tokushima-u.ac.jp
Abstract— This study proposes the Self-Organizing Map with
Weighted Connections avoiding false-neighbor effects (WCSOM). We investigate the effectiveness of WC-SOM in comparison with the conventional SOM, Growing Grid and FN-SOM. We
confirm that WC-SOM enables the most flexible self-organization
among the four algorithms and can obtain the effective map
reflecting the distribution state of the input data using fewer
neurons.

I. I NTRODUCTION
The Self-Organizing Map (SOM) is an unsupervised neuralnetwork [1] and has attracted attention for its clustering properties [2]. In the learning algorithm of SOM, a winner, which isa neuron closest to the input data, and its neighboring neurons
are updated, regardless of the distance between the input data
and the neighboring neurons. For this reason, if we apply
SOM to clustering of the input data including some clusters
located at distant locations, there are some inactive neurons
between clusters. Because the inactive neurons are on a partwithout the input data, we are misled into thinking that there
are some input data between clusters. Furthermore, because
the simulation time depends on the number of neurons, it isimportant to utilize the used neurons effectively by reducing
the inactive neurons.
In the real world, it is not always true that neighboringhouses are physically adjacent or close to each other. In
addition, the relationship between neighborhoods is not fixed,
but keeps changing with time. It is important to change theneighborhood relationship flexibly according to the situation.
Meanwhile, the synaptic strength is not constant in the brain.
So far, the Growing Grid network was proposed in 1985 [3].
Growing Grid increases the neighborhood distance between
neurons by increasing the number of neurons. However, there
is not much research changing the synaptic strength itself.
In our past study, we proposed the SOM with FalseNeighbor degree between neurons (called FN-SOM) [5].
False-neighbor degrees (FNDs) are allocated between adjacentrows and adjacent columns of FN-SOM. FNDs act as a burdenof the distance between map nodes when the weight vectors ofneurons are updated. FN-SOM can greatly reduce the inactive
neurons, however, the algorithm has following problems. All
the FNDs between the neurons on same line are increased
simultaneously and forcibly. It often produces the increase of978-1-4244-5309-2/10/$26.00 ©2010 IEEE

FNDs between correct-neighboring neurons, namely, FALSE
false-neighbor. Then, it produces the twist of the map.
In this study, we propose a new SOM algorithm, SOM with
Weighted Connections avoiding false-neighbor effects (WCSOM). In WC-SOM, all the connections between adjacentneurons are weighted to avoid false-neighbor effects unlike
FN-SOM. This weights are called as false-neighbor weights(FNWs). In the algorithm of WC-SOM, we find winlessneurons and its “false neighbors”, and FNWs between theseneurons are increased. The initial values of all of FNWs are
set to zero, however, they are increased with learning. In thisway, WC-SOM changes the neighborhood relationship more
flexibly according to the situation and the shape of data.
We compare WC-SOM with the conventional SOM, Growing Grid and FN-SOM, and effectiveness of WC-SOM areinvestigated by applying to various input data. We confirm
that WC-SOM can obtain the effective map reflecting the
distribution state of the input data using fewer neurons.
II. SOM WITH W EIGHTED C ONNECTIONS AVOIDINGFALSE - NEIGHBOR EFFECTS (WC-SOM)
The previous method, FN-SOM, has false-neighbor degrees(FNDs) between neurons. However, all the FNDs between
neurons at same line on the map are forcibly the same value.
It often produces the twist of the map.
In this study, we propose SOM with Weighted Connections
avoiding false-neighbor effects (WC-SOM). We explain WCSOM in detail in this section. WC-SOM consists of n×m neurons located at 2-dimensional rectangular grid. Each neuron i
has a d-dimensional weight vector wi = (wi1 , wi2 , · · · , wid )(i = 1, 2, · · · , nm). The range of the elements of the input
data xj = (xj1 , xj2 , · · · , xjd ) (j = 1, 2, · · · , N ) are assumed
to be from 0 to 1.
In WC-SOM, a false-neighbor weight (FNW) denoted by
nf (i,k) is allocated between directly-connected neurons i andk, and we propose a new neighborhood distance considering
FNWs. The initial values of all FNWs are set to zero, andthe initial values of all the weight vectors are given over the
input space at random. Moreover, a winning frequency γi is
associated with each neuron i and is set to zero initially.
Learning Step
(Step1) Input an input vector xj to all the neurons.

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exist, namely S = ∅, return to (Step1) without considering thefalse-neighbor.
(Step8) Choose a false-neighbor fq of each neuron q in S
from the set of direct topological neighbors of q denoted as
Nq 1 . fq is a neuron whose weight vector is most distant fromq:(6)
fq = arg max{wi − wq }, q ∈ S, i ∈ Nq 1 .

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Fig. 1. The connection weight df (c,i) between the winner c(= 7) andthe neuron i. df (c,i) is the minimum of sum-of-FNWs on the shortest-pathfrom c to i. Values in circles mean respective neuron numbers. Values between
neurons mean FNWs nf between directly-connected neurons. (a) Let i = 14.
The possible shortest path routes from c to i are [7 8 9 14], [7 8 13 14]
and [7 12 13 14], and its sum of FNWs are 8, 6 and 5, respectively. Then,df (7,9) = 5 which is a minimum value. (b) Let i = 9. Then, df (7,9) = 7
because the possible shortest-path route is only [7 8 9]. Note that a route [712 13 14 9] is NOT adopted because it is not the shortest-path route although
the sum of FNWs is 6 which is smaller than 7.

(Step2) Calculate distances between xj and all the weightvectors. Find a winner, denoted by c, which is the neuron
with the weight vector closest to the input vector xj .
c = arg min{wi − xj },i(1)where  ·  is Euclidean distance measure.
(Step3) Increase the winning frequency of the winner c byγc new = γc old + 1.
(Step4) Calculate the neighboring distance dis(c, i) between
the winner c and each neuron i by considering FNWs nf as
the following measure ;dis(c, i) = r i − r c 2 + df (c,i) ,(2)

where r i −r c  is the Euclidean distance between map nodes cand i on the map grid. df (c,i) is the connection weight betweenc and i, and it is defined as the minimum of sum-of-FNWs
on the shortest-path from c to i (as shown in Fig. 1).
(Step5) Update the weight vectors of all the neurons:
wi (t + 1) = wi (t) + hF c,i (t)(xj − wi (t)),(3)

where hF c,i (t) is the neighborhood function of WC-SOM:dis(c, i)hF c,i (t) = α(t) exp −.
(4)
2σ 2 (t)where α(t) is the learning rate, and σ(t) corresponds tothe width of the neighborhood function. Both α(t) and σ(t)decrease with time, in this study, we use following equations;α(t) = α0 (1 − t/tmax ),σ(t) = σ0 (1 − t/tmax ),(5)

where α0 and σ0 are initial values of α and σ, respectively,
and tmax isthe maximum number of the learning.
nm(Step6) If i=1 γi ≥ λ is satisfied, find the false-neighborand increase FNWs nf , according to steps from (Step7) to(Step10). If not, perform step (Step11).
Consideration of False-Neighbor(Step7) Find a set of neurons S which have never become
the winner: S = {i | γi = 0}. If the winless neuron does noti(Step9) Increase FNW nf (q,fq ) between each q and its falseneighbor fq as
nf (q,fq ) = nf (q,fq ) + 1,

(7)

where nf (q,fq ) = nf (fq ,q) .
(Step10) Reset the winning frequencies of all the neurons tozero: γi = 0.
(Step11) Repeat the steps from (Step1) to (Step10) for all the
input data.
III. E XPERIMENTAL R ESULTS
We apply WC-SOM to various input data and compare itwith the conventional SOM, Growing Grid and FN-SOM. For
all experiments, SOM, FN-SOM and WC-SOM have nm =100 neurons (10×10). Growing Grid starts learning with 2×2
neurons, and the maximum number of neurons is less than 100.
The parameters of the learning for WC-SOM are chosen as
α0 = 0.3, σ0 = 3, λ = 3000, where we use the same α0 for
all the method and the same λ for FN-SOM and WC-SOM.
For SOM, Growing Grid and FN-SOM, we use σ0 = 4.
A. For Chainlink data set
We considered a 3-dimensional input data which is Chainlink data set [6] with clustering problems of the linearlyinseparable. The total number of the input data N is 1000,
and the input data has two clusters. We repeated the learning
20 times for all the input data, namely tmax = 20000.
Learning results of the conventional SOM, Growing Gridare shown in Figs. 2(a) and (b). We can see that there are a
lot of inactive neurons between clusters. The other side, in thelearned maps of FN-SOM and WC-SOM shown in Figs. 2(c)and (d), there are only a few inactive neurons between clusters.
Furthermore, in order to compare the learning performanceof WC-SOM with the other SOMs numerically, we use thefollowing well-used two measurements.
Quantization Error Qe measures the average distance between each input vector and its winner [1]. The small value
Qe is more desirable.
Neuron Utilization U measures the percentage of neurons thatare the winner of one or more input vectors in the map [4].
Thus, U nearer 1.0 is more desirable.
We carry out 30 simulations with different initial state of theweight vectors and different order of inputting. The averagesof the two measurements over 30 independent runs are listed
in Table I. The quantization error Qe of WC-SOM is the bestvalue among the four algorithms and has improved by 12.3%

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Learning results of four algorithms for Chainlink data. (a) Conventional SOM. (b) Growing Grid. (c) FN-SOM. (d) WC-SOM.

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U-Matrices of simulation results for Chainlink data. (a) Conventional SOM. (b) Growing Grid. (c) FN-SOM. (d) WC-SOM.

TABLE IT WO MEASUREMENTS FOR C HAINLINK DATA .
Qe
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Fig. 4. Gray scale display of false-neighbor degrees and false-neighborweights after learning for Chainlink data. (c) False-neighbor degrees of FNSOM. (d) False-neighbor weights of WC-SOM.

compared to that of SOM. This is because the result of WCSOM has few inactive neurons, therefore, more neurons canself-organize the input data. This is confirmed by the neuron
utilization U . U of WC-SOM is the best value among the fouralgorithms. It means that 87.6% of the neurons of WC-SOM
are the winner of one or more input data, in other words, there
are only 12.4% inactive neurons.
In order to evaluate how well SOM preserves the topologyof the data set, we calculate U-Matrix [6] which visualizes
the cluster structure of the map by showing distances between
neighboring neurons. Figure 3 shows U-Matrices of the fouralgorithms. We can see that the boundary lines of FN-SOM
and WC-SOM are clearer than other two algorithms and it

is easy to distinguish between light areas (cluster) and darkareas (no input data) because there are few inactive neurons
in the results of FN-SOM and WC-SOM. However, it should
be noted that it is clear that U-Matrix of FN-SOM does not
visualize the cluster structure correctly. It means that FN-SOM
can not preserve the topology of the data set by increasing its
FNDs forcibly.
This can be confirmed by differences between FND of
FN-SOM and FNW of WC-SOM after learning. Figure 4
shows FND and FNW between the neurons displayed bygray-scale. From FNDs of FN-SOM as Fig. 4(a), we canclearly see that even FNDs between the correct-neighboring
neurons were increased. This is because that all the FNDs
between the neurons on same line are increased simultaneously
and forcibly, in FN-SOM algorithm. On the other hand, inFig. 4(b), WC-SOM can adapt its FNWs flexibly to the shapeof the input data.
From these results measured in terms of the quantization
error, the number of inactive neurons and the topographic
error, we can say that WC-SOM enables more flexible selforganization than FN-SOM and other SOMs.
B. For Target data set
Next, we considered Target data set [6] which has a clustering problem of outliers. The input data is 2-dimension andhas six clusters including 4 outliers, and the total number ofthe input data N is 770. We repeated the learning 26 times
for all the input data, namely tmax = 20020.
From the learning results of the four algorithms shown in
Fig. 5, it is clear that FN-SOM and WC-SOM can greatly
reduce the number of inactive neurons in comparison withthe conventional SOM. It is conformed by the measurements

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U-Matrices of simulation results for Target data. (a) Conventional SOM. (b) Growing Grid. (c) FN-SOM. (d) WC-SOM.

TABLE IIafter learning shown in Fig. 7, FNWs of WC-SOM has beincreased flexibly according to the shape of the input data incomparison with FN-SOM.

T WO MEASUREMENTS FOR TARGET DATA .
SOM
0.01960.8113Growing Grid
0.02320.8317

FN-SOM
0.01900.9263

WC-SOM
0.01750.8917

1.8

IV. C ONCLUSIONS
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Fig. 7. Gray scale display of false-neighbor degrees and false-neighborweights after learning for Target data. (c) False-neighbor degrees of FN-SOM.
(d) False-neighbor weights of WC-SOM.

This study has proposed the Self-Organizing Map with
Weighted Connections (WC-SOM). WC-SOM has falseneighbor weights (FNWs) allocated between connections ofadjacent neurons to avoid false-neighbor effects. In the algorithm, the calculation method of the neighborhood distance has
been proposed in accordance with FNWs. We have compared
WC-SOM with the conventional SOM, Growing Grid and FNSOM and have confirmed that WC-SOM enabled the mostflexible self-organization by increasing FNWs with learning.
Furthermore, we have confirmed visually and numericallythat WC-SOM had few inactive neurons. These results meanthat WC-SOM can obtain the effective map reflecting the
distribution state of the input data using fewer neuron.
R EFERENCES

listed in Table II. It should be noted that the quantization
error Qe of WC-SOM is better than FN-SOM although theneuron utilization U of FN-SOM is the best value. Thismeans that there are few inactive neurons in the result of FNSOM, however, FN-SOM has not self-organized the statisticalfeatures of the input data correctly. In other words, WC-SOM
can obtain the clustering map reflecting the distribution stateof the input data more effectively than FN-SOM.
This is backed by the U-Matrices shown in Fig. 6. The
boundary lines of FN-SOM is clear, however, we can not saythat it visualizes the cluster structure correctly. Meanwhile, UMatrix of WC-SOM visualizes the cluster structure effectively,
and its boundary lines are very clear. From FNDs and FNWs

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