On Sympathy and Symphony: Network-Oriented
Modeling of the Adaptive Dynamics of Sympathy States
Ilze A. Auzina, Suzanne Bardelmeijer, Jan Treur (https://orcid.org/0000-0003-2466-9158)
Behavioural Informatics Group, Vrije Universiteit Amsterdam, the Netherlands
ilze.amanda.auzina@gmail.com
suzanne-bardelmeijer@live.nl j.treur@vu.nl
Abstract Social network analysis commonly focuses on the relationships be-
tween two actors that could represent either individuals or populations. The pre-
sent paper not only introduces a new concept of sympathy states to represent a
sympathy between two actors but also models how different sympathy states af-
fect each other in an adaptive manner taking into account who expresses the sym-
pathy and who receives it. The designed network model was designed with the
Eurovision Song Contest in mind and takes into account external political events
that affect the scores in this contest over the years. The properties of the model
were analyzed using social network analysis. The model represents a first attempt
in modeling sympathy states and their adaptive dynamics modulated by external
events by Network-Oriented Modeling based on adaptive temporal-causal net-
works.
Keywords: sympathy states, social network, European countries, Hebbian
learning, adaptive network
1 Introduction
Social Network Modeling or Analysis is used to model relationships between a set of
social actors. In a social network, the edges represent the connections, while the nodes
are the social actors [1]. Even though Social Network Analysis (SNA) often focuses on
the relationship between people, it is also used to capture connections between groups,
organizations or nations [1]. Particularly, the development of worldwide social plat-
forms, such as Facebook, has allowed modeling of social networks on a population
basis between countries [2]. Nonetheless, the influence of one relationship of two social
actors on another relationship of two actors has not been studied yet neither on an indi-
vidual nor on a population level. Therefore, the aim of the present study is to fill in this
research gap by using a Network-Oriented Modeling approach applying them to this
domain.
The newly designed model is based upon temporal-causal model principles. A cen-
tral role was assigned to nodes called sympathy states that represent how one actor feels
connected to another actor. In the current application of the network model, the actors
2
are based on European populations, where the sympathy states represent a sympathy
from the inhabitants of one European country to those of another country, as often ob-
served for the Eurovision Song Contest. For the scope of this project, the sympathy
states were created only for countries who participated in the final of Eurovision song
contest of the year 2013. This selection was based upon the fact that Eurovision song
contest voting system provides empirical data of sympathy between countries as a
score, and that the initial sympathy state selection is based on transnational Facebook
friendship data set of the year 2012 [4]. The model covers a time period of four years,
from 2013 to 2016, to make the results more robust. Moreover, external political events
are included in the model for each year that is modeled. These events affect the sympa-
thy states, thus creating an adaptive dynamic model that relates to the real-life situation.
In this paper, in Section 2 the network model is introduced. Section 3 explains the
principles used to determine the connection weights. In Section 4 the model is illus-
trated by example simulations. Analysis of the network model based on Social Network
Analysis is discussed in Section 5. Section 6 discusses the possibility of tuning the pa-
rameters to empirical data.
2 The Designed Temporal Causal Network Model
A dynamic modeling approach was used that enables to design complex high-level con-
ceptual representations of adaptive dynamic models in the form of temporal–causal
networks [8, 9, 16]. This approach can be considered as generic and is suitable to de-
scribe complex networks ranging from mental networks to social networks [16]. The
approach can be considered as a branch in the causal modeling area which has a long
tradition in AI; e.g., see [13, 14, 15]. It distinguishes itself by a dynamic perspective on
causal relations, according to which causal relations exert causal effects over time, and
these causal relations themselves can also change over time. The models, which can be
represented conceptually or numerically by a set of parameters, are declarative and
therefore not dependent on specific computational methods for simulation or analysis
[8, 9, 16]. The connections represent the causal impacts according to the chosen domain
which causes the state values to vary over time. When a state is affected by more than
one causal relation a specific combination function is used. For example, a logistic sum
is utilized to aggregate multiple impacts, where the threshold and steepness are used as
parameters to define the curvature. Other parameters take the form of connection
weights and speed factors. The connection weights show differences in the causal con-
nection strengths, whereas a state’s speed factor indicates the time necessary for the
state to change [8]. Together the connections weights, speed factors, and combination
functions define the structure of a temporal-causal network model. In the upper part of
Table 1 these concepts, their notation, and explanation are shown. The corresponding
numerical representation is obtained from the conceptual representation as shown in
the lower part of Table 1. The following difference and differential equations are ob-
tained:
Y(t+t) = Y(t) + Y [cY(X1,YX1(t), …, Xk,YXk(t)) - Y(t)] t (1)
3
dY(t)/dt = Y [cY(X1,YX1(t), …, Xk,YXk(t)) - Y(t)]
A variety of standard combination functions are available that can be used to deal
with multiple impacts on a state. This study uses the alogistic sum combination function
in which the parameters steepness σ and threshold τ can be adjusted. For example, the
model will show more abrupt behavior when high steepness values are applied. To in-
dicate the dependence of τ and σ these are used as subscripts:
(1+e-στ) (2)
In principle parameters such as connection weights may have specific constant val-
ues. However, in adaptive cases, these parameters may change over time as well [8].
Table 1: Conceptual and numerical representation of a temporal-causal network model
Therefore, in the adaptive network model introduced here the connection weights
ωX,Y(t) can be modeled as states with their own combination function. This study will
use an adaptive model based on Hebbian learning. Hebbian learning was originally in-
vented by Hebb [7] for the assumption that ‘neurons that fire together, wire together’
[7] [6]. In other words, when both states are often active simultaneously the connection
between these states becomes stronger, which is a useful effect in the present domain
of the model: if both countries exhibit high sympathy towards each other then the con-
nection between these countries should be strengthened. Therefore, in the present
model Hebbian learning is used for the connections between reciprocal sympathy
states; see Fig. 1. The numerical representation used for Hebbian learning is based on
the combination function
Concept
Representation
Explanation
States and connections
X, Y, X → Y
Represents the structure of a network via nodes and links
Connection weight
ωX,Y
A connection weight ωX,Y [-1, 1] denotes the strength of
the causal impact of state X on state Y
Aggregating multiple
impacts on a state
cY(..)
For each state Y a combination function cY(..) is chosen to
aggregate the causal impacts on state Y
Timing of causal effect
ηY
For each state Y a speed factor ηY ≥ 0 is used to describe
the speed of change of a state
Concept
Representation
Explanation
State values over time t
Y(t)
At each time point t each state Y in the model
has a real number value in [0, 1]
Single causal impact
impactX,Y(t) = X,Y X(t)
At t state X with a connection to state Y has an
impact on Y, using connection weight X,Y
Aggregating multiple
impacts
aggimpactY(t)
= cY(impactX1,Y(t),…, impactXk,Y(t))
= cY(X1,YX1(t), …, Xk,YXk(t))
The aggregated causal impact of
multiple states Xi on Y at t, is determined
using combination function cY(..)
Timing of the causal
effect
Y(t+t) = Y(t) +
Y [aggimpactY(t) - Y(t)] t
= Y(t) +
Y [cY(X1,YX1(t), …, Xk,YXk(t)) - Y(t)] t
The causal impact on Y is exerted over
time gradually, using speed factor Y;
here the Xi are all states with connec-
tions to state Y
4
c(X1, X2, W) = X1X2(1 − W) + W (3)
where X1 and X2 indicate the activation level of the two connected states, W the con-
nection weight, and the persistence factor; this entails the following difference and
differential equation for the connection weight:
ωX1,X2(t+t) = ωX1,X2(t) + ηωX1,X2 [X1(t)X2(t)(1 − ωX1,X2(t)) + ωX1,X2(t) - ωX1,X2(t)] t
dωX1,X2(t)/dt = ηωX1,X2 [X1(t)X2(t)(1 − ωX1,X2(t)) + ωX1,X2(t) - ωX1,X2(t)] (4)
Fig 1: The Hebbian learning principle for the connections between reciprocal sympathy states
Initially, twenty-six countries were selected to model sympathy states. This selection
was based on the countries who participated in the Eurovision song contest 2013 final.
Subsequently, a subset was selected from the transnational Facebook friendship data
set based on the twenty-six countries chosen. The Facebook transnational data set con-
tains information on the five countries to which people in the selected country are most
connected to in terms of border-crossing Facebook Friendships [4]. Accordingly, sym-
pathy states were created between ’sender’ and ’receiver’ countries where the obtained
score in the Facebook matrix was at least 1. This indicates that the receiver country is
the country with which sender country has the fifth-highest number of Facebook friend-
ships [4]. This resulted in 76 sympathy states, which contained the in- formation of the
sender and the receiver country. For a part of such a network model, see Fig. 2.
However, once the model was extended over multiple years, 2013, 2014, 2015 and
2016, respectively, three sympathy states had to be removed from the model due to lack
of empirical data over the years. Thus, the final model contained 73 sympathy states
(see Appendix A). The initial state values of these sympathy states were also based on
the Facebook Friendship data set. The data set contained cell values that range from 0
to 5, where 0 indicates that country is not mentioned, and 5 that the receiver country is
the country with which sender country has the highest number of Facebook friendships
[4]. As mentioned earlier, only connections that had a value from 1 to 5 were included
in the model. Consequently, the initial values were also based on this scoring system
but converted to a range within 0 to 1. Thus, the resulting initial values ranged from 0.1
to 0.5.
5
Fig. 2: Some sympathy states and their causal relations
3 Principles Determining the Nonadaptive Connection Weights
The values of the nonadaptive connection weights were based upon four basic elements:
country name, sender or receiver country, geographical location, and size. The first rule
stands for that country X’s sympathy to country Y would affect any other sympathy
state where also country X’ and/or Y is a participant. This rule determines all the con-
nections that should be established. The second rule is more elaborate as it takes into
account which country is the sender and which the receiver (see Fig. 3 column 1). Fur-
thermore, this rule was refined by implementing the location and size rule upon it (see
Fig. 3 column 2-5). Fig. 3 shows that the relationship of the connection is defined first.
In other words, every connection is classified based on the information of why this
connection is created. In total there are four categories: (1) receiving countries are the
same, (2) sending countries are the same, (3) receiving country in the sender node and
the sending country in the receiver node are the same, and (4) sending country in the
sender node and the receiving country in the receiver node are the same. This creates a
principle that each connection consist of three countries as one of the countries always
is repeating, thus creating the connection. An exception for this rule is when the con-
nection is reciprocal, meaning that both sympathy states consists of the same countries
but in a reversed way. Then a connection weight of 0.8 was assigned and the Hebbian
learning principle was implemented as discussed before.
For the remaining connections that consist of three countries it is checked whether
all three of these countries are bordering; for an overview, see Table 2. If this is true,
then a connection weight of 0.8 is assigned, based on the assumption that the sympathy
state of bordering countries could have a strong effect on the third bordering country’s
sympathy as well. If the countries are not bordering, it is checked if the three countries
are in the same neighborhood. If that is the case, then a connection strength value of
0.6 is assigned, meaning that the sympathy state still exhibits an influence but weaker
than before. Lastly, if none of the previous constraints holds true, a value of 0.2 is as-
signed.
6
Table 2: Connection Weight Principles Used
If the 3 countries are
bordering
If the 3 countries are in the
same neighbourhood
If from small to big country
(based on European population)
Else
Receiving countries are the
same
SR* to SR*
Georgia/Germany to
Finland/Germany
SR* to SR*
Hungary/Germany to
Romania/Germany
SR* to SR*
Hungary/Germany to
Greece/Germany
0.8
0.6
0.2
Sending countries are the
same
S*R to S*R
Finland/Norway to
Finland/Germany
S*R to S*R
Norway/Iceland to
Norway/Denmark
0.5
0.6
Receiving country and
sending country in new
node are the same
SR* to S*R
Estonia/Russia to
Russia/Azerbaijan
SR* to S*R
Georgia/Russia to
Russia/Georgia
SR* to S*R
Germany/Netherlands to
Netherlands/Belgium
SR* to S*R
Hungary/Romenia to
Romenia/Germany
SR* to S*R
Russia/Belarus to
Belarus/Italy
SR* to S*R
Netherlands/Belgium to
Belgium/Spain
0.8
0.8
0.6
0.2
0.2
0.2
Sending country and re-
ceiving country in new
node are the same
S*R to SR*
Azerbaijan/Russia to
Georgia/Azerbaijan
S*R to SR*
Italy/Spain to
Malta/Italy
S*R to SR*
Italy/Spain to
Belarus/Italy
0.8
0.6
0.2
An exception of these assignment principles is possible when the connection is
based on the combination of the size rule and the sender or receiver country rule
7
category three. If the sender country is relatively small based on its population as com-
pared to the receiver country in the sender sympathy node, for example, Estonia to
Russia, then this sympathy state would have a small connection weight value to the
sender node, 0.2 respectively. This is based upon the assumption that the proportion of
friendships of a small population towards a large population is relatively irrelevant for
the larger population’s friendships to other countries. The size ranking was based upon
the population of each country, where the larger countries were considered to be Russia,
Germany, France, United Kingdom, and Italy (population above 60 million) [10].
The model was built upon the assumption that the results of the Eurovision song
contest are influenced by political events that have happened in the preceding year, as
Eurovision happens in the first half of the year. Therefore, political events in the years
2012, 2013, 2014, and 2015 that had an influence on the foreign affairs between coun-
tries were identified. Consequently, this search resulted in 12 events across the time
span of 2012 to late 2015 that were thought to have an effect on the sympathy state
value (see Table 3). Information about these events was obtained from news articles
and European Foreign Policy Scorecards [5]. The events are independent of each other
and are not affected by the sympathy states.
Table 3: Political events included in the model
Event
Year
Event
Year
Common economic space
2010
Belgium royal family visits France
2014
Russia military support
2011
Crimean crisis
2015
Eastern Partnership
2012
Poster conflict
2015
Vladimir Putin re-elected
2012
Malta security operation
2015
DCFTA
2013
Paris attacks
2015
Restored border between Russia and Belarus
2014
Railway collaboration
2015
Implementing the events allows adjusting a sympathy state value at a certain year
by increasing it, positive effect, or decreasing it, negative effect. Consequently, the final
model consists in total of 85 states, where 73 of these states represent sympathy states
and the other 12 states represent events. A small example illustrating the final model
can be found in Fig. 3.
Fig 3: Example of sympathy states in relation to relevant events
8
4 Example Model Simulation
As mentioned before, the main components of the model are the sympathy states that
interact with each other, while the number of the events can be varied depending on the
number of years included as well as whether meaningful events can be identified in the
context of the sympathy states. In Fig. 4 a base scenario of the model is represented.
Specifically, one event is modeled (indicated by the arrow), which has both positive,
strengthening, and negative, weakening effects on specific sympathy states. In the pre-
sent scenario, the speed factors for all sympathy states are set to 0.3, while the speed
factor for the event is set to 0.06 to achieve that its effects only occur at a later time
point. In total 60 time points were simulated, that represent 60 months or approximately
5 years. As can be seen in Fig. 4 the model does not reach an equilibrium within the
considered time interval, which is a desirable effect as it is assumed that the interactions
between the sympathy states change over time. This example considers only one event;
by incorporating multiple events over time a realistic situation can be achieved.
Fig 4: Example simulation results
5 Social Network Analysis
To analyze the network the Social Network Analysis tool Gephi version 0.9.2. was used
[3]. The social network consists of 85 nodes and 797 edges, where the nodes represent
the sympathy states and events, and the edges represent the relationships between them.
It can be observed in Fig. 5 that sympathy nodes with high betweenness centrality are
located relatively more in the center of the network, meaning that these nodes are often
involved between other actors in the network. Table 3 shows the top five sympathy
nodes in regards to betweenness centrality, suggesting that these sympathy nodes have
a high influence over the other sympathy states. Furthermore, the number of relations
(edges) was analyzed. In particular, the average degree is 9.376, with a range from 1 to
9
36 degrees. Table 4 indicates the sympathy nodes with the highest degree value, mean-
ing that these nodes have the highest connections with other nodes. Lastly, the clusters
were analyzed in the network. Based on modularity analysis the number of communi-
ties in this network is 5 with a modularity of 0.55.
Sympathy state
Degree
Sympathy state
Betweenness centrality
Finland/Russia
36
Finland/Germany
630.13
Russia/Ukraine
36
Lithuania/Russia
493.36
Lithuania/Russia
35
Finland/Russia
480.67
Ukraine/Russia
35
Finland/Norway
362.22
Georgia/Russia
33
Romania/Germany
350.19
Table 4: Top 5 sympathy states with highest degree values and top 5 sympathy states with
highest betweenness centrality
Fig. 5: Social Network Analysis graph with labels
6 Best Fit to the Empirical Eurovision Voting Data
Matlab v2017a was used to simulate the numerical representation of the model. An
attempt has been made to exploit available empirical data based on voting in order to
get a good fit of the model parameters. It was not easy to use such data in a solid way.
The empirical data was obtained from Eurovision voting results from the years 2013,
10
2014, 2015 and 2016 [11]. The scoring system of Eurovision is based on a scale from
0 to 12, however, the model uses state values in the range between 0 and 1, thus the
empirical data was converted to match this range. Initially only final voting results were
taken into account, however, if the country did not participate in any of the finals after
the year 2013 then a score from a semi-final was used. In a situation where empirical
data could not be obtained in a certain year, because the sender and the receiver country
were in a different semi-final, the approximate value was estimated using logarithmic
trend-line equation using the other scoring values as reference points. Consequently, a
complete empirical data set was created for four time points. Subsequently, by interpo-
lation this data set was extended across all months of every year using Matlab, resulting
in a data set of 60 time points for all 85 states. The empirical data was compared with
the model data with a step size of ∆t = 1, representing a one-month interval, and the
time interval from a starting point at t = 17 and end point at t = 53. Such a selection was
chosen because the generated empirical data outside these boundaries was either greater
than 1 or smaller than 0, which is by default outside of the range of the model. To
improve the model’s fit to the empirical data sympathy state speed factors were tuned
using the simulated annealing algorithm in Matlab. Consequently, 73 parameters were
optimized and the implications can be seen in Fig. 6. The optimization was based on
decreasing the root mean square error (RMSE) between the model and the empirical
data. After optimization, the obtained RMSE was still 0.56.
Fig 6: Simulation results after parameter tuning
7 Discussion
The current model could be viewed as an expansion of the social influence theories. All
the previous social influence theories have discussed how one person influences the
release of the same behavior in others either via contagion, conformity, social facilita-
tion or other types of social influence [12]. This could be viewed in analogy with the
sympathy states of the present model: a certain sympathy from one country to another
could be viewed as an expressed social behavior on a population level. Moreover, the
vicinity of the actors that often plays a role in social influence theories is also taken into
account in the present model: if the participants of the sympathy states are closely lo-
cated then the influence of one sympathy state to another is stronger. However, where
11
the present model differs from the current theories is that rather than modeling whether
the initial behavior is replicated by the other sympathy state, the present model attempts
to show that, firstly, any behavior could have an influence on another behavior as long
as at least one of the participants is part of the new behavior, and, secondly, that these
behaviors can be modulated by external factors that are independent of the behaviors
themselves. Nonetheless, the present model could not mimic the real world data entirely
due to certain limitations.
Firstly, the empirical data was not accurate. Multiple time points had to be generated
via estimation, which caused the empirical data to be unreliable and outside the range
of the model state values. Consequently, a relatively high error value was generated
that rather could be attributed to the imprecision of the empirical data than to the model.
Secondly, due to the scope of this project only a limited amount of events, with the
main focus on political events, were taken into account. However, real-world processes
are more complex and cannot be summarized in twelve events, as the sum of many
small events also could generate a significant impact on a sympathy state. Therefore,
future research could expand the number of events included in the model, thus, resulting
in a model that can better represent the complex relations between country sympathies.
Thirdly, due to the time constraints, only a subset of all European countries was in-
cluded in the model. Thus, perhaps important sympathy states were lost that could have
improved the accuracy of the model. All of these factors could be improved upon in
future research, which would improve the validity of the model.
In conclusion, in spite of the limitations mentioned above, the present model offers
a first attempt in modeling sympathy state relations. The model achieved to show that
a sympathy state exhibits an influence over another sympathy state and that this inter-
action can be modulated by external factors. Therefore, these results unfold a promising
future research field for simulating population-based sympathy interactions.
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Appendix A Sympathy States Used
List of all sympathy states used:
Georgia/Azerbaijan, Russia/Azerbaijan, Russia/Belarus, Ukraine/Belarus, France/Belgium,
Netherlands/Belgium, Lithuania/Denmark, Norway/Denmark, Finland/Estonia, Estonia/Finland,
Sweden/Finland, Belgium/France, Italy/France, Armenia/Georgia, Finland/Germany, Geor-
gia/Germany, Greece/Germany, Hungary/Germany, Netherlands/Germany, Romania/Germany,
Georgia/Greece, Romania/Hungary, Ukraine/Hungary, Norway/Iceland, Belarus/Italy, Malta/It-
aly, Moldova/Italy, Romania/Italy, Belarus/Lithuania, Belgium/Netherlands, Germany/Nether-
lands, Denmark/Norway, Estonia/Norway, Finland/Norway, Iceland/Norway, Lithuania/Nor-
way, Sweden/Norway, Hungary/Romania, Italy/Romania, Moldova/Romania, Spain/Romania,
Armenia/Russia, Azerbaijan/Russia, Belarus/Russia, Estonia/Russia, Finland/Russia, Geor-
gia/Russia, Lithuania/Russia, Moldova/Russia, Ukraine/Russia, Belgium/Spain, Italy/Spain, Ro-
mania/Spain, Denmark/Sweden, Estonia/Sweden, Finland/Sweden, Ice- land/Sweden, Nor-
way/Sweden, Azerbaijan/Ukraine, Belarus/Ukraine, Georgia/Ukraine, Moldova/Ukraine, Rus-
sia/Ukraine, Greece/UK, Iceland/UK, Lithuania/UK, Malta/UK, Norway/UK, Lithuania/Ireland,
UK/Ireland, Ireland/Spain, Ireland/UK
