=Paper=
{{Paper
|id=Vol-1885/210
|storemode=property
|title=On the Optimal Setting
of Spreading Activation Parameters to Calculate Recommendations on the
Knowledge Graph
|pdfUrl=https://ceur-ws.org/Vol-1885/210.pdf
|volume=Vol-1885
|authors=László Grad-Gyenge
|dblpUrl=https://dblp.org/rec/conf/itat/Grad-Gyenge17
}}
==On the Optimal Setting
of Spreading Activation Parameters to Calculate Recommendations on the
Knowledge Graph==
https://ceur-ws.org/Vol-1885/210.pdf
J. Hlaváčová (Ed.): ITAT 2017 Proceedings, pp. 210–217
CEUR Workshop Proceedings Vol. 1885, ISSN 1613-0073, c 2017 L. Grad-Gyenge
On the Optimal Setting of Spreading Activation Parameters to Calculate
Recommendations on the Knowledge Graph
László Grad-Gyenge
Eötvös Loránd University, Budapest, Hungary
laszlo.grad-gyenge@inf.elte.hu,
WWW home page: http://laszlo.grad-gyenge.com
Abstract: This paper presents the analysis on the optimal Although the optimal parameter settings of the method
settings of the spreading parameters of the spreading acti- should have been further investigated. In order to fill this
vation technique. The method is applied on the knowledge gap, this paper presents the results of our numerical exper-
graph, an information representation technique that com- iments conducted to identify the optimal values for the re-
bines collaborative and content-based information. The lax parameters of the technique. Our results show that the
evaluation of the recommendation technique is based on method delivers a stable performance regarding the dif-
recommendation lists. The involved measures are preci- ferent parameter settings. The quality of the recommen-
sion, recall and normalized discounted cumulative gain. dations calculated by the method become low if the relax
The numerical experiments are conducted on the Movie- parameters are set to extreme values.
Lens 1M dataset. The evaluation results show that spread- The rest of the paper is organized as follows. Section 2
ing activation delivers a stable performance regarding to discusses the work related to graph based information rep-
the evaluation measures over different parameter settings. resentation and the application of spreading activation as a
The quality of the recommendations degrade in the case, recommendation technique. Section 3 presents the repre-
the method parameters are set to extreme values. sentation technique and the dataset the method is evaluated
on. Section 4 describes the recommendation technique.
Section 5 discusses the evaluation technique and presents
1 Introduction
the evaluation results. Section 6 concludes the paper and
As discussed in detail by Grad-Gyenge et al. [1], in con- gives insight into our plans for the future.
trast to the paradigm of user preference, the paradigm of
relatedness provides a more general and effective approach
to the design and development of the recommendation al- 2 Related Work
gorithms. The essence of the paradigm is to focus on the
relations between the entities of the recommendation sce- Next to matrix factorization methods [3][4][5], the graph
nario instead of involving models emphasizing the user based information representation technique has proven
item interactions. The primary outcome of the applica- its effectiveness. Several research projects utilize the
tion of the paradigm is the potential to involve transitivity general information representation capability of heteroge-
into the recommendation techniques. Transitivity is one neous graphs.
possibility to define the relation between the users and the As the state of the art results of the research on recom-
recommended items in a more general way. mender systems illustrate, the application of graph based
The application of spreading activation in the field of information representation techniques gaining attention
recommender systems is an illustrative example of the uti- nowadays. Graphs are powerful tools of knowledge repre-
lization of the paradigm. The method is applied on the sentation. An example of the involvement of a hybridiza-
knowledge graph [2], which is a general information rep- tion technique at the information representation level, Lee
resentation technique. The entities of the recommendation at al. [6] introcude a heterogeneous graph based technique
scenario and also the relatations between them are gen- to combine collaborative and content based information.
eralized. The users, items and their attribute values are Further investigating the topic, Lee et al. [7] analyse the
representated with nodes. User preferences on items and correlations of the entities found in the graph. As the work
content based information are represented with edges. The presented by Tiroshi et al. [8] illustrate, the graph based
transitive relation in this case means the possibilty to in- representation is straightforward technique in the case of
volve paths of heterogeneous types between two nodes. modeling social relations.
The advantage of the paradigm is that it does not restrict In addition to ontologies, the involvement of social net-
the type of the individual edges on the path between two works into the recommendation process is an intensively
nodes, thus the relations between the entities can be treated researched field. Typically, there are two classes of the
as generalized and transitive. approaches modeling social networks, the asymmetric and
The past research on the evaluation of spreading acti- the symmetric case. The asymmetric case can be described
vation to generate recommendations showed its potential. as the follow and the trust relationships. The symmetric
On the Optimal Setting of Spreading Activation Parameters to Calculate Recommendations on the Knowledge Graph 211
case can be described as the friendship relationship. Influ- 3 Dataset
encing results conducted on trust networks are published
by Guha et al. [9], Massa et al. [10], Ziegler et al. [11], 3.1 Representation Technique
and Jøsang et al. [12]. Important works calculating recom-
mendations with the help of the social network are pub- To represent the data, a graph based technique described
lished by Guy et al. [13], Konstas et al. [14], and He et by Grad-Gyenge et al [2] is involved. The advantage of
al. [15]. The layered graph technique is a typical repre- this method is its ability to represent various information
sentation method found in the early works in this field. sources by alloying both collaborative and content based
Kazienko et al. [16] also derive recommendations with the information. The essence of the method is to represent
help of this technique. Cantador et al. [17] present a clus- the information in a labeled, heterogeneous graph. Each
tering technique based method on the layered graph based user and each item is represented with a dedicated node.
representation. The explicit or implicit interaction between the users and
the items are modeled with relations annotated by the spe-
cific type of the interaction. In addition, the content-based
information, namely the user and the item attributes are
An important feature of the knowledge graph is its ca- represented with the so-called attribute value nodes. In
pability to represent heterogeneous information. Proba- this case, a dedicated node is created for each attribute
bly, this is the reason why the technique is intensively re- value. The node representing the user or item is bound to
searched as the more the involved information sources are, the specific attribute value node indicating then the partic-
the less the cold start cases occur. Hu et al. [18] gener- ular attribute value. This representation technique leads to
ate leads with a label propagation technique. Catherine a knowledge graph containing heterogeneous information
et al. [19] introduce a probabilistic logic method to cal- at the informaiton representation level.
culate recommendations with the help of the knowledge The information is represented in a labeled, undirected,
graph. Yu et al. [20] investigate the observed and the po- weighted graph. The definition of the graph is extended
tential paths of the knowledge graph. To have a numerical with labels, thus types can be assigned to the nodes and
measure, they involve the PathSim measure for path com- edges of the graph. Although parallel edges are allowed,
parison. Kouki et al. [21] describe a hybridization tech- in practical applications it is recommended to add only one
nique with the help of a probabilistic framework. Burke et edge per type between two specific nodes in order to re-
al. [22] present a recommendation technique based on the duce the computational complexity. Equation 1 presents
k-Nearest Neighbours method applied to various matrices the definition of the knowledge graph.
as the user-tag matrix, user-resource matrix, resource-tag
matrix and the resource-user matrix.
K = (Tn , Te , N, E,tn ,te , re ), (1)
where Tn stands for the node types, Te stands for the edge
types. N stands for the nodes of the graph. E ⊆ {{u, v}|u ∈
The spreading activation technique was originally ap-
N ∧ v ∈ N ∧ u 6= v} stands for the set of edges, thus the
plied to ontologies and is recently involved in different
graph is undirected. The function tn ⊂ N × Tn specifies the
domains to derive recommendations. The primary goal
type of the nodes. The function te ⊂ E × Te specifies the
of Hussein et al. is to close the gap between context-
type of the edges. The function re ⊂ Eu × R specific the
awareness and self-adaptation [23]. To perform this task,
rating value of the appropriate edges. The function re does
SPREADR, a spreading activation based recommendation
not assign rating to all edges of the graph, thus re is partial.
method is defined. Gao et al. [24] propose an ontology
based approach to model both user interests and items in
the same knowledge base. Jiang et al. [25] define a user 3.2 MovieLens
model with the help of an ontology and calculate rec-
ommendations with the help of the spreading activation. To analyse the performance of the various method config-
Blanco-Fernandez et al. [26] argue that spreading activa- urations, the MovieLens 1M dataset is involved [30]. The
tion is to be involved to avoid overspecialisation. They advantage of the MovieLens 1M dataset over the other
present a semantic model of the preferences of the users published MovieLens dataset is that in addition to user
and apply spreading activation to proceed content based preferences on items it also contains user and item at-
reasoning. Codina et al. [27] define an item score based tributes. This allows us to utilize both the collaborative
on the weighted average of concepts related to each other and content-based information to come to a lower cold
in their model. In their work, they estimate user ratings start rate than the pure collaborative filtering technique
with a semantic recommendation technique treating it a has.
reasoning method. Troussov et al. [28] investigate decay To give an insight into the information representation
configurations over a tag aware representation. Alvarez et technique, Figure 1 presents a detailed part of the knowl-
al. [29] introduce ONTOSPREAD in the field of medical edge graph in the case of the MovieLens 1M dataset. To
systems. illustrate the different information source types, the nodes
212 L. Grad-Gyenge
technique. For each attribute value, a node of the appro-
priate type is created and bound to the item with a corre-
sponding edge. The node types introduced in this expri-
ment are as follows.
• Type YearOfRelease annotates the years of release.
• Type Genre annotates the genres. In the case of the
MovieLens 1M dataset, multiple genres can be as-
signed to a movie. It means the a node representing
an item can be connected to multiple nodes represent-
ing a genre.
Figure 1: A detailed view of the MovieLens dataset repre- The item attribute values are bound to the items with
sented in the knowledge graph. edges of the corresponding type. The edge types defined
are ItemGenre and ItemYearOfRelease, respectively.
The advantage of the MovieLens 1M dataset is that
and the relations are coloured to show the type of informa- it contains both collaborative and content-based informa-
tion they represent. tion. The content-based information is described above.
Colour light blue annotates the items to recommend, The collaborative information in this case is basically
i.e., Jaws, Forrest Gump and Chasing Amy. Colour the 1 000 209 known user rating events contained in the
light brown annotates the movie genres, i.e., Romance and dataset. Each rating event consists of a user, an item, a
Drama. Colour orange annotates the release year of a preference value and a time-stamp. The preference values
movie, i.e., 1997. Colour lilac annotates the users, i.e., are in the [1, 5] interval. In the current experiment, a lin-
Person 1, Person 2 and Person 3. Colour grey an- ear normalization is conducted on this value to the [0.2, 1]
notates the gender of a person, i.e., Male. Colour blue interval.
annotates the occupation of a person, i.e., Scientist. In order to represent the known rating in the dataset, the
Colour green annotates the ZIP region of a person, i.e., edge type ItemRating is introduced. Adding a known
ZIP region 2. Colour red annotates the age category, rating to the knowledge graph means the creation of the
i.e., 25-34. edge of the appropriate type (ItemRating) between the
In order to represent the users of the recommendation particular user and item. The concrete value of the rating
screnario, a node with the appropriate type (User) is cre- is assigned to the particular edge by the re partial function.
ated in the knowledge graph. The user attribute values are Table 1 presents the amount of information contained
represented with the attribute node technique. For each at- in the knowledge graph in two subtables. Table 1a and
tribute value, a node of the appropriate type is created and Table 1b contain the amount of nodes and edges per type,
bound to the user with a corresponding edge. The node respectively. The knowledge graph contains 10 062 nodes
types introduced in this expriment are as follows. in total. Not counting the edges of type ItemRating, the
knowledge graph contains 34 451 edges in total.
• Type Gender annotates the genders of the users.
• Type AgeCategory annotates the age categories. 4 Recommendation Technique
• Type Occupation annotates the occupations.
According to Grad-Gyenge et al. [1], the application of
• Type ZipCodeRegion annotates the ZIP code re- spreading activation on the knowledge graph described in
gions. The ZIP code region is derived from the ZIP Section 3.1 to estimate user preferences on items leads to
code by using only its first digit representing the U.S. high quality recommendations. As described in several
region. works in the past [29, 28, 24, 26, 25, 23, 27], spreading
activation is an iterative technique to calculate a similarity
The user attribute values are bound to the users between a source node and the other nodes of a particular
with edges of the corresponding type. The edge graph. As already discussed by Grad-Gyenge et. al [1],
types defined are PersonGender, PersonAgeCategory, the advantage of the method is that the similarity value
PersonOccupation and PersonZipCodeRegion, re- incorporates both the distance between two nodes and the
spectively. parallel paths in between.
Analogously to the user attribute values, the items are As already mentioned, spreading activation is an itera-
represented with a dedicated node of the appropriate type tive method. In this experiment, similarly to Grad-Gyenge
(Item) is created in the knowledge graph. The item at- et al. [1], the iteration is conducted until a pre-specified
tribute values are also represented with the attribute node iteration step (c) limit is reached. The method assigns
On the Optimal Setting of Spreading Activation Parameters to Calculate Recommendations on the Knowledge Graph 213
Table 1: Count of node and edge types in the MovieLens 5 Evaluation
dataset.
5.1 Evaluation Technique
(a) Count of node types.
The methods are evaluated with an iterative process as also
Node type Count
described by Grad-Gyenge et al. [2]. Before starting the
Person 6 040
AgeCategory 7
iteration, all the user preference information is removed
Gender 2 from the knowledge graph, thus it contains no edges rep-
Occupation 21 resenting ratings. In the initial step, the knowledge graph
ZipCodeRegion 10 is filled with content-based information, meaning that it
Item 3 883 contains the nodes representing the users, items and their
Genre 18 attribute values. The relations between users and user at-
YearOfRelease 81 tribute values, items and item attributes values are also
present in the knowledge base.
(b) Count of edge types. Having the knowledge base initialized, the evaluation
Edge type Count process is basically an iteration over the known rating val-
PersonAgeCategory 6 040 ues. This iteration is conducted in an ascending order by
PersonGender 6 040 the timestamp of the known rating. It also means that the
PersonOccupation 6 040 knowledge graph is filled with collaborative information
PersonZipCodeRegion 6 040 during the evaluation process. Each iteration step consist
ItemGenre 6 408 of the following operations.
ItemYearOfRelease 3 883
ItemRating 1 000 209 1. Generate recommendations with the evaluated
method.
2. Record the evaluation measures.
activation values to the nodes of the graph as defined in
3. Add the known rating to the knowledge graph.
Equation 2.
The advantage of this evaluation method is that in the
a(i) ⊂ N × R, (2) beginning the methods are analysed in a cold start environ-
ment. As the knowledge based is being filled with collab-
where a denotes the activation function.
orative information, the analsys tends to be conducted in
In the initial step of the iteration, the activation of the
an information dense environment. According to our past
source node is set to 1 (a(0) (ns ) = 1). The source node (ns )
experiments [1, 2], in order to represent the problem, the
represents the user the recommendations are generated for.
experiments are to be conducted on the first 10 000 known
The activation of all the other nodes are set to 0 (a(0) (n) =
ratings. This is the amount of information, the perfor-
0, n ∈ N, n 6= ns ).
mance indicators of the methods typically stabilize, thus
In each iteration step, the activation of the nodes is
the methods can be interpreted.
maintained. A part of the activation of each node is prop-
The analysis of the spreading parameters is basically a
agated to its neighbour nodes and a part of the activation
greedy search method. All the combinations of the rs and
of each node is being kept at the node. The distribution of
the ra parameters are evaluated in the interval [0, 1]. The
the propagated activation values is conducted based on the
fidelity of the analysis is 0.1. The experiment covers all
weight of the edges. In our case the graph is not weighted,
possible combinations of the parameter values and does
in other words, the edges are weighted basically equally.
not restrict to the rs + ra = 1 cases.
In order to control the propagation process, two param-
To evaluate the methods, the precision, recall and the
eters are introduced. The parameter spreading relax (rs )
nDCG measure is recorded at each evaluation step for a
controls the amount of the distributed activation. The pa-
concrete method configuration. The recorded measure val-
rameter activation relax (ra ) controls the amount of acti-
ues are then averaged and are presented. The measures are
vation kept at each node. The function to maintain the
calculated on the first 10 items of the list of recommended
activation of the nodes is defined in Equation (3).
items. The relevance of an item is defined as follows. If
a(i) (m) the known rating value of a specific item is 4 or 5 then the
a(i+1) (n) = ra a(i) (n) + rs ∑ , (3) item is treated relevant. In other words, relevance is de-
m∈Mn |Mn |
fined as the threshold 0.8 on the normalized rating value.
where n ∈ N, i ≥ 0. Mn stands for the neighbour nodes of
n. Mn = {m|{m, n} ∈ E}. 5.2 Evaluation Results
Having the iteration step count (c) reached, the prefer-
ence value of each node regarding the source node is de- Figure 2 contains the results of the evaulation in three sub-
fined as the value of the activation function. figures. Subfigure 2a, 2b and 2c presents the precision,
214 L. Grad-Gyenge
recall and nDCG of the methods, respectively. On each be generated as due to the setting of the parameter the ac-
figure, the horizontal axis represents the setting of the ac- tivation can not spread from the source node to neighbour-
tivation relax (ra ) parameter, the vertical axis represents ing nodes, thus the evaluation measures of these cases are
the setting of the spreading relax (rs ) parameter. The color 0. This presentation allows us to investigate the evaluation
of each cell represents the average value of the respective measures more refined.
evaluation measure as also indicated in the legend of the The figures show that the low values of ra lead to a de-
figure. crease in the recommendation quality. Especially if the
setting of rs is high. Taking a closer look, it can be read
Mean of Precision from the figures that the difference in the quality occurs
from the second or the third digit depending on the con-
1.00
crete evaluation measure. It means that although the qual-
ity of the recommendations may decrease in some cases
Spreading Relax
0.75 but the method shows a quite stable performance on this
dataset. The reason behind this stabily can be found in its
0.650
0.50
0.625
0.600
network science background. We assume that the fact that
0.575 the method operates on the network is more important than
its actual configuration.
0.25
Table 2 presents the result of the evaluation in numbers
in its subtables. Subtable 2a, 2b and 2c presents the preci-
0.00 0.25 0.50 0.75 1.00
sion, recall and nDCG values respectively. The columns of
Activation Relax the table represent different activation relax (ra ) parameter
(a) Precision settings. The rows of the table represent different spread-
ing relax (rs ) parameter settings. The value of each table
Mean of Recall cell contain the actual value of the appropriate evaluation
1.00 measure.
The results give a deeper insight into the performance
of the methods than the figure based presentation. The ta-
Spreading Relax
0.75
ble shows that setting the rs to zero leads to unproducible
0.072
0.068 recommendation lists. Regarding precision and recall, the
0.50 0.064
0.060 highest recommendation quality can be achieved by set-
ting rs to 0.1 and setting ra to higher than 0.5. It means that
0.25 in the case of precision and recall, low spreading and high
activation values lead to higher performance. Looking at
nDCG, the configurations leading to the highets quality for
0.00 0.25 0.50
Activation Relax
0.75 1.00 (rs , ra ) are (0.4, 0.3) and (0.5, 0.4). In this case a less ex-
treme and a more common setting of the parameters can
(b) Recall be adequate.
Mean of nDCG At the macro level, the numerical presentation of the
1.00 values shows a consistent view to the figure based repre-
sentation. The results show that the extreme setting of the
method configuration parameters (rs =0) leads to a signifi-
Spreading Relax
0.75
cant decrease in the recommendation quality. In addition,
0.90
setting ra to a low value while setting rs to a high value is
0.50 0.88 not recommended.
0.86
To summarize the results, in the case there is no re-
0.25
search capacity available to analyise the optimal setting
of the spreading parameters, a good practice can be to set
these parameters to a moderate value, e.g., to the mean of
0.00 0.25 0.50 0.75 1.00 the value interval.
Activation Relax
(c) nDGC
6 Conclusion
Figure 2. Evaluation measures at different spreading parameter settings.
The paper discusses the analysis of the configuration of the
In order to easier interpret the results visually, the fig- spreading activation method. The technique is applied on
ures do not present the evaluation measures resulted in the the knowledge graph to generate recommendations. Pa-
rs = 0 case. This is the case when no recommendation can rameters spreading and activation relax are investigated
On the Optimal Setting of Spreading Activation Parameters to Calculate Recommendations on the Knowledge Graph 215
Table 2. The evaluation measures of the method configuration over various spreading parameter settings.
(a) Precision.
S/A 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.0 0.5592 0.5922 0.6326 0.6495 0.6568 0.6616 0.6655 0.6669 0.6697 0.6716 0.6718
0.9 0.5592 0.5981 0.6370 0.6530 0.6594 0.6642 0.6664 0.6693 0.6715 0.6718 0.6719
0.8 0.5592 0.6042 0.6426 0.6558 0.6616 0.6661 0.6685 0.6714 0.6718 0.6719 0.6723
0.7 0.5592 0.6117 0.6476 0.6585 0.6647 0.6673 0.6710 0.6718 0.6720 0.6723 0.6726
0.6 0.5592 0.6210 0.6530 0.6616 0.6664 0.6706 0.6718 0.6721 0.6725 0.6726 0.6728
0.5 0.5592 0.6326 0.6568 0.6655 0.6697 0.6718 0.6722 0.6726 0.6727 0.6729 0.6732
0.4 0.5592 0.6426 0.6616 0.6685 0.6718 0.6723 0.6726 0.6729 0.6732 0.6734 0.6734
0.3 0.5592 0.6530 0.6664 0.6718 0.6725 0.6728 0.6732 0.6733 0.6733 0.6734 0.6735
0.2 0.5592 0.6616 0.6718 0.6726 0.6732 0.6734 0.6734 0.6735 0.6735 0.6736 0.6736
0.1 0.5592 0.6718 0.6732 0.6734 0.6735 0.6736 0.6737 0.6737 0.6737 0.6737 0.6737
0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
(b) Recall.
S/A 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.0 0.0574 0.0632 0.0693 0.0712 0.0719 0.0723 0.0726 0.0727 0.0729 0.0730 0.0730
0.9 0.0574 0.0641 0.0698 0.0716 0.0721 0.0725 0.0727 0.0728 0.0730 0.0730 0.0730
0.8 0.0574 0.0650 0.0705 0.0718 0.0723 0.0726 0.0728 0.0730 0.0730 0.0730 0.0730
0.7 0.0574 0.0663 0.0710 0.0721 0.0725 0.0727 0.0729 0.0730 0.0730 0.0731 0.0731
0.6 0.0574 0.0678 0.0716 0.0723 0.0727 0.0729 0.0730 0.0730 0.0731 0.0731 0.0731
0.5 0.0574 0.0693 0.0719 0.0726 0.0729 0.0730 0.0730 0.0731 0.0731 0.0731 0.0731
0.4 0.0574 0.0705 0.0723 0.0728 0.0730 0.0730 0.0731 0.0731 0.0731 0.0731 0.0731
0.3 0.0574 0.0716 0.0727 0.0730 0.0731 0.0731 0.0731 0.0731 0.0731 0.0731 0.0731
0.2 0.0574 0.0723 0.0730 0.0731 0.0731 0.0731 0.0731 0.0731 0.0731 0.0731 0.0731
0.1 0.0574 0.0730 0.0731 0.0731 0.0731 0.0731 0.0732 0.0732 0.0732 0.0732 0.0732
0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
(c) nDCG.
S/A 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.0 0.8508 0.8694 0.8897 0.9008 0.9075 0.9124 0.9145 0.9166 0.9168 0.9166 0.9165
0.9 0.8508 0.8717 0.8927 0.9040 0.9099 0.9134 0.9162 0.9169 0.9166 0.9165 0.9165
0.8 0.8508 0.8743 0.8960 0.9065 0.9124 0.9154 0.9168 0.9167 0.9165 0.9165 0.9163
0.7 0.8508 0.8785 0.9003 0.9087 0.9140 0.9165 0.9167 0.9165 0.9164 0.9163 0.9163
0.6 0.8508 0.8843 0.9040 0.9124 0.9162 0.9167 0.9165 0.9164 0.9163 0.9163 0.9162
0.5 0.8508 0.8897 0.9075 0.9145 0.9168 0.9165 0.9164 0.9163 0.9162 0.9162 0.9161
0.4 0.8508 0.8960 0.9124 0.9168 0.9165 0.9163 0.9163 0.9162 0.9161 0.9161 0.9161
0.3 0.8508 0.9040 0.9162 0.9165 0.9163 0.9162 0.9161 0.9160 0.9161 0.9161 0.9160
0.2 0.8508 0.9124 0.9165 0.9163 0.9161 0.9161 0.9161 0.9160 0.9160 0.9160 0.9160
0.1 0.8508 0.9165 0.9161 0.9161 0.9160 0.9160 0.9160 0.9159 0.9159 0.9159 0.9159
0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
216 L. Grad-Gyenge
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