# Measuring the influence of beliefs in belief networks

Aleksandar Tomašević
Department of Sociology
University of Novi Sad
Novi Sad, Republic of Serbia
<EMAIL_ADDRESS>
###### Abstract

Influential beliefs are crucial for our understanding of how people reason
about political issues and make political decisions. This research proposes a
new method for measuring the influence of political beliefs within larger
context of belief system networks, based on the advances in psychometric
network methods and network influence research. Using the latest round of the
European Social Survey data, we demonstrate this approach on a belief network
expressing support for the regime in 29 European countries and capturing
beliefs related to support for regime performance, principles, institutions,
and political actors. Our results show that the average influence of beliefs
can be related to the consistency and connectivity of the belief network and
that the influence of specific beliefs (e.g. Satisfaction with Democracy) on a
country level has a significant negative correlation with external indicators
from the same domain (e.g. Liberal Democracy index), which suggests that
highly influential beliefs are related to pressing political issues. These
findings suggest that network-based belief influence metrics estimated from
large-scale survey data can be used a new type of indicator in comparative
political research, which opens new avenues for integrating psychometric
network analysis methods into political science methodology.

##  1 Introduction

Belief systems are one of the central topics in scientific study of politics.
In the most broader sense, belief systems describe interrelationships of
various beliefs related to politics and classical definitions emphasize that
those beliefs are bound together by some form of functional interdependence
[18]. Describing the structure of belief systems, their formal properties and
the influential elements of that structure is important for understanding how
people draw on different elements of a belief system when evaluating
politicians, policies and governments [10]. As belief systems may encompass a
large number of connected elements, studies seek to identify the most
influential or central elements of belief systems [9] as those beliefs may be
crucial for our understanding of how people reason about political issues,
make political decisions and form their knowledge about "what goes with that"
in world of politics [41, 48, 35, 44].

Recently, a new approach to the study of belief systems emerged, inspired by
various methods and tools from network science [22, 7, 9, 9]. From network
perspective, measuring influential beliefs is identical to measuring node
influence in a network. Influential nodes have a critical role in the spread
and propagation of influence in many types of social and physical networks and
identification of such nodes has been widely studied by scientists from
different disciplines [49, 87, 92, 73, 59, 53, 69, 91]. One of the possible
approaches to detect influential nodes in social networks is to simulate
spreading dynamics and then optimize the spreading process. The optimal
spreading solution would reveal the most important set of nodes which are are
crucial to the spreading process, but this optimization problem is NP hard
[47]. The alternative approach is to quantify the influence of the nodes using
structural network information [60]

The aim of the current study is to develop a rigorous approach for measuring
the influence of beliefs in belief networks based on extracted structural
information. Previous studies relied on several classic centrality measures
which have been proven either unstable, unreplicable, or not suitable for
cross-country comparisons in different political contexts [11, 9, 8]. If the
goal of the analysis is to quantify influence, then simple, classic centrality
measures, such as strength, closeness or betweenness centrality, can at best
can identify rare, highly influential nodes, but they perform poorly at either
ranking all nodes by their influence or providing precise and reliable
quantitative measurement of influence of all nodes in the network [52]. Since
our aim is not just the identification of the most influential beliefs, the
present study is oriented towards more recent and advanced centrality
measures.

In order to apply these measures, we will estimate, analyze, and discuss
belief networks using the attitudinal entropy (AE) framework [21, 24, 81],
also known as network theory of attitudes [6]. This approach is based on
models analogous to Ising models for ferromagnetism in statistical physics and
represents a rare formal model of attitudes. The central idea is that
attitudes are high-level properties emerging from configuration lower-level
elements such as beliefs, feelings and behaviours. In order to simplify the
terminology, we will follow recent work [37] and define belief networks as
configurations of attitudinal elements which underpin a general attitude.
Naturally, the basic elements of belief networks are beliefs and not feelings
or behaviors. Beliefs encompass a variety of phenomena including assumptions
about the state of the world (including future states of the world), views on
moral and political issues, personal preferences, and evaluations.

More formally, belief networks are networks in which nodes represent belief
variables and edges represent couplings or interactions between two beliefs at
a functional level. Strength of the couplings determines the probability that
beliefs are in consistent states [23]. A positive coupling between two beliefs
indicates that on a population level, if we find a person holding one belief,
the likelihood that the same person will hold a second belief is increased.
Therefore, we can say that two beliefs have influence over one another, but
also remain agnostic about the mechanism of causal effects driving this
influence [10]. Belief networks approach is consistent with the idea that
individuals have some level of cognitive organization of various beliefs and
values [7, 22] and with the classical social constraint approaches [88, 18].
Belief network methods do not start with a priori conceptions of how beliefs
ought to be organized and network models allow multiple organizational schemas
to emerge in different subgroups of the investigated samples.

Using AE framework to model belief networks gives us two important formal
properties: energy and temperature. Energy of the network can be understood as
formalization of the concept of potential dissonance. Potential dissonance
refers to the actual inconsistency between beliefs which is translated into
felt dissonance if enough attention is given [23]. Stable, consistent belief
networks have lower energy. Probability that the state of any given belief
will be in accordance with other beliefs in the network depends on how much
attention is focused on the attitude object. Attention therefore represents
the strength of the interdependence between beliefs and it has an analogous
effect on the network as inverse temperature has on thermodynamic systems.
Lower temperature corresponds to a state in which higher attention is given to
the attitude object. Therefore, if temperature is high, beliefs are randomly
associated and if temperature is low, beliefs are systematically organized in
a belief network [21, 81, 23].

These formal properties of belief networks are important as we do not expect
to find the same distribution of node influence in belief networks with
different levels of consistency and different temperatures. To test this
assumption, we will estimate multiple belief networks for different social
groups for which we expect different levels of energy and temperature and
examine if the average belief influence is associated with these formal
properties of belief networks between groups. This association is crucial for
the validity of influence measurement, because once it is confirmed, further
attempts to explain why certain beliefs are influential (or not) in a specific
belief system may be based on the notions of dissonance, consistency, and
attention of that system. This is especially relevant to the issue of belief
or attitude change, where influential beliefs may be targets of interventions,
and recent work has established that change is more likely in high-energy
networks [23].

Similarly, we aim to prove the external validity of influence measurement by
comparing the influence of a specific belief in a belief network with an
external measurement of a social issue related to that belief. For example, we
will estimate belief networks on subsamples of respondents from different
countries, measure the influence of belief related to the satisfaction with
democracy in each country and compare these measures with external measures of
quality of democracy. We expect a significant correlation between these
measures, which would show that belief influence is not a result of some
arbitrary network centrality measurement process.

To summarize, our main aim is to develop an approach of measuring the
influence of beliefs in belief networks which enables us to associate the
influence of a belief with both formal attributes of belief networks and
external measurements related to the social issues belonging to the same
domain as beliefs whose influence was measured. We will present this approach
using an example based on recent European Social Survey data concerning the
support of European citizens for regimes in their countries. After we present
the motivating example in the next section, we will describe the general
analytical strategy for measuring node influence, give a formal description of
the methods and show the results obtained by applying our approach to the
aforementioned example.

##  2 Motivating example

Our motivating example refers to the citizens‘ attitude towards the regime in
their countries. This type of complex attitude has been regarded as crucial
for the political legitimacy of liberal democracy through public evaluations
of regime performance [67] and it can be shaped by various dispositions and
beliefs [82]. We rely on a multidimensional concept of political support [65,
56] which encompasses five levels or objects of support, four of which can be
related to the support for a regime or government and one (political
community) which is more general than any specific regime and will not be
included in our example. Four regime-related levels are: regime performance,
regime institutions, regime principles, and political actors. Therefore, the
macrostate of the belief network we are investigating is the global evaluation
of support for the regime which reflects satisfaction with governmental
political institutions, principles and specific actors. As such, the
macrostate can be neutral, positive, or negative. However, we are not
interested in the distribution of the macrostates as our focus is on the
microstates and the structure of the belief network and its influential
elements. Elements of such belief network correspond to different levels of
political support.

Support for regime performance is reflected in satisfaction with various areas
of governmental activity (economy, healthcare, democracy) and can be viewed as
the belief of the citizens that the regime has fulfilled their expectations
and wishes [82, 85]. These beliefs emerge as summative satisfaction judgments
based on different individual performance perceptions (of various institutions
or actors) [82].

Support for regime institutions is very closely related to political trust in
those institutions. This relationship between trust and support is an
important issue in political science as both constructs have implications for
the political health of a country [84, 64]. Political trust is a type of
future-based belief that politicians will serve the citizen’s interests and
implement their campaign promises [85, 77]. Unlike social trust, which is
moralistic in nature, political trust has a rationalistic nature which is
rooted in performance of regime’s institutions and officials [79]. Recent
approaches to political trust from a belief network perspective [77, 90] have
unveiled a complex structure of different trust beliefs and different studies
had confirmed the association between political trust and beliefs related to
satisfaction with democracy and other political institutions [42, 25]. Based
on these findings, we include measures of political trust in our belief
network as beliefs expressing support for regime institutions.

Support for regime principles in democratic societies is primarily reflected
in satisfaction with democracy, which is often viewed as one of the most
important elements for the success of a democratic system to implement
effective public policies [12, 25] and important element of political health
of democratic society [17, 80]. Satisfaction with democracy has a close
relationship with various political performance evaluations which rely on
regime’s record of policy outputs or outcomes [67, 50]. In order to
differentiate between different principles of democratic regimes, we also
include several measures related to the intrinsic quality of democratic
governance [2] such as beliefs related to issue of giving voice to citizens,
which is a measure of government responsiveness [74], and beliefs related to
transparency, fairness and impartiality of political procedures.

Finally, support for political actors will be operationalised as trust in
politicians and political parties. Taken together, mentioned beliefs will be
the constitutive elements of belief networks, which will be used as an example
for measuring the influence of specific beliefs. Given the evidence that
relationship between constructs such as trust and satisfaction with government
is reciprocal and dependent on other beliefs related to the regime performance
[82, 85], belief network estimation and analysis will enable us to analyze the
structure, consistency, attention and influential beliefs which give rise to
the macrostate of the network which can be defined as a general attitude
towards the regime.

To produce a more fine-grained analysis, we will consider belief networks of
different groups of European citizens. Since we seek to establish the link
between the influence of beliefs in belief networks and the formal properties
of belief networks (temperature and energy), we will consider belief networks
of groups for which we expect different levels of these properties. Therefore,
we will divide our sample into subsamples based on: ideological position of
the respondent, interest in politics, and the average time spent following
news about politics.

Previous work has established that belief systems of individuals positioned on
the left should be characterized by openness to social change and a preference
for greater equality, and those on the right should be characterized by
preferences for stability and hierarchy [43]. Left-right division plays a
dominant role in many political systems across Europe and more politically
engaged and interested citizens are more likely use utilize left-right schema
in order to organize their preferences and attitudes and utilize ideological
messages in order gain "what goes with what" knowledge about the world of
politics [48, 44, 35]. More generally, given that ideology organizes attitudes
into thematically consistent networks, has a function of moderating different
beliefs and attitudes and given the fact that political issues are considered
in light of their ideological implications [34, 43, 46, 5], we can expect
different levels of both temperature and energy in groups which are leaning
further from the political center, as well as differences in belief influence.

In a similar fashion, we can expect that for different levels of political
interest and media use, we can expect belief networks with different
properties. Media are a crucial source of political information for citizens
and can alter citizens’ interest in politics and their evaluation of the
regime and democracy performance in their countries [12]. Previous studies
have demonstrated that there strong links between both media use and interest
in politics and various political attitudes [76, 75, 4]. It should be noted
that media use and interest in politics are not necessarily positively
correlated as some types and forms of media can enhance interest in political
affairs while others can suppress it [12]. In literature, the former is known
as virtuous cycle theory [66, 1, 76], while the latter is known as media
malaise theory [62]. In both theories, interest for politics may be
conceptualized as a mediator between media use and various evaluations of
political performance. Investigating this relationship is beyond the scope of
this paper, but the results of these studies give support to our expectation
of differences in belief networks based on the answers of respondents with
different levels of interest in politics and in subsamples of respondents with
different reported levels of media use.

Studies have shown that support towards the regime in comparative perspect is
highly sensitive to different institutional contexts [56, 25]. Therefore, by
estimating belief networks on subsamples from different countries, we can
expect different structure, energy and temperature levels as well as
distribution of important beliefs. As stated in the introduction, important
beliefs in each country should be associated with external indicator from the
same domain (economy, democracy, healthcare, etc.). When inspecting the
attitude towards the regime, we expect that the influential beliefs will be
those related to pressing social issues or the domains of weak regime
performance.

To summarize, we will estimate belief networks representing support for the
current regime. Nodes of these networks represent beliefs that belong to one
of four groups:

  1. 1.

support for regime performance,

  2. 2.

support for regime institutions,

  3. 3.

support for regime principles,

  4. 4.

support for political actors.

Belief networks with those elements will be estimated for different
subpopulations based on:

  1. 1.

ideology measured on left-right scale,

  2. 2.

interest in politics,

  3. 3.

media use,

  4. 4.

country of the respondent.

##  3 Measuring node influence

In terms of centrality metrics on social networks, the notion of influence is
generally used to represent the node’s ability to affect other nodes and is
closely connected to important concepts in social sciences such as power,
status or prestige. More generally, we can say that node centrality determines
its influence in terms of connectivity, communicability, and controllability
[83]. However, before applying the specific centrality metric it is necessary
to clarify what node centrality or influence means and represents in a
particular context of application [11]. In classical belief systems
literature, the influence or centrality of belief was referred to as the role
beliefs play in the belief system as a whole [18]. We could reformulate this
idea in network terms and conceptualize influential beliefs as super-mediators
[72] where influence comes from node’s capability to receive information from
different nodes and to propagate them to numerous others. This means that
influential beliefs are driving force of constraint and consistency of the
network as they are strongly coupled with their neighboring nodes and they
have the ability to directly or indirectly constrain the values of different
beliefs throughout the network. However, having many influential beliefs in a
network makes consistent states of the network less likely as mutual
constraints impose more narrow choice of consistent belief configurations.

Having in mind this general idea and following recent work on node influence
[73, 69, 83], we can say that the influence of node $i$ is greater if:

  1. 1.

$i$ has high a degree,

  2. 2.

the average degree of $i$’s closest neighbors is also high,

  3. 3.

$i$ is positioned in the core of the network.

All three conditions taken together enable a belief to constrain the majority
of other nodes in the network through direct or indirect couplings.

Although many new centrality metrics have developed in the last decade, not
many of them have been used in the existing applications in network science in
general [83], and particularly in belief networks literature, where strength,
betweenness and closeness centrality are almost exclusively used, including
some of the specific variations of those measures such as network loadings
[15]. In biological measurements of node influence, it was shown that single
simple centrality measure cannot identify influential nodes in a statistically
significant way [26]. Given our description of node influence in a belief
network, in order gain reliable measurement the influence measure has to
capture both local and global properties. Local centrality metrics measure the
centrality of a node based on its local neighborhood topology and usually
represent some variation of node degree centrality [83]. On the other hand,
global information in centrality measures is obtained through an iterative
process, starting from the node’s neighborhood and spreading through the
entire network.

K-shell is one of the most important node ranking methods used in social
network analysis which is based on both types of information [60]. It
describes node’s influence starting from node’s local information (degree) and
iteratively incorporates the global network information [49]. The basic idea
of this approach is that nodes which are positioned in the core of the network
have higher spreading capabilities than those with high node centrality as
measured with the classic network centrality measures such as degree
centrality, betweenness centrality and closeness centrality. The key idea of
this approach is to assign an integer value, called k-shell value, which would
be representative of node’s position within the network in relation to its
connectivity patterns.

The initial step of the k-shell algorithm is to remove all nodes in the
network whose degree $k=1$ and keep removing nodes whose degree $k\leq 1$
after accounting for the updated degrees of nodes given the previous removals.
This process continues until there are no nodes in the network with degree
$k\leq 1$. All the removed nodes at this point are being assigned with the
k-shell value 1. Then the procedure is repeated with incremented k-shell
values of 2, 3, and so on. A node with a higher k-shell value is located in a
more central position in the network.

One of the biggest strengths of k-shell approach is its robustness in the face
of incomplete network information. The k-shell ranking remains robust even
when 50 % of edges are removed from the network [49]. However, major drawback
of the classic k-shell decomposition approach is that there are many nodes
sharing the same k-shell values, so different approaches are proposed in order
to increase the monotonicity of k-shell value rankings [60, see].

One of the main routes for improving the monotonicity of k-shell decomposition
is based on the introduction neighborhood rule, which means taking into
account k-values of both the node and its neighbors when measuring node
influence [13, 3, 57, 58, 55, 63, 59]. For example, [3] extended the k-shell
method with the approach which distinguishes between neighborhood coreness,
the sum of k-shell values of all neighbors of the focal node and extended
neighborhood coreness, the sum of neighborhood coreness of all neighbors of
focal node).

The key issue in these approaches is how to weight the contribution of the
neighborhood k-shell values to node’s influence. Gravity index centrality [58]
was proposed as a solution. This measure takes inspiration from classical
gravity formula proposed by Newton and estimates the strength of influence
nodes have on each other by simultaneously considering the intrinsic
influences of the nodes themselves and the distance between them [54]. K-shell
values of two nodes are taken as their mass and the shortest path distance
between them as the distance for gravity formula. The influence of node $i$
becomes the sum of the force it exerts to nodes in its neighborhood. This way,
nodes with the same k-shell value may have different influence because they
are placed in different neighborhoods and have different distances from their
influential neighbors. This approach can be time consuming and computationally
heavy for larger networks, but it performs better than degree centrality or
classic k-shell method [60]. In the original version, gravity index centrality
was limited to unweighted networks in the same way as the original k-shell
method, but since we have an improved k-shell algorithm for weighted networks
[38], it’s trivial to adapt gravity index centrality for weighted undirected
networks. Other, more recent, improvements of gravity centrality, such as [53]
where k-shell value as mass was replaced with a combination of degree and
local clustering coefficient, are limited to unweighted networks and will not
be discussed in this paper.

There are other, more recent methods based on k-shell values and gravity
centrality. In a recent paper [86], the attraction coefficient based on the
difference of k-shell values of two nodes was introduced and incorporated into
the standard gravity centrality equation with mass of the nodes being
represented by their degree. The motivation of this approach is to use the
differences in node locations (represented by the k-shell differences as the
basis of the attraction coefficient) when calculating the mutual interaction
between two nodes. Node in a more central location in the network (higher
k-shell value) will have more attraction than the nodes located in the
periphery. However, the drawback of this approach, especially when considering
small networks, is that if two nodes belong to the same shell do not exert
influence over one another which means that connections with nodes with the
same k-shell value do not contribute to node’s influence.

Recently, Global structure model (GSM) [78] emerged as an alternative to
gravity index centrality while retaining the similar logic to the one employed
by the gravity model and k-shell as basic measure of influence. GSM
differentiates between self-influence and global influence of a node. Self-
influence is based on k-shell value of the node, while its global influence is
primarily defined by the k-shell values of its neighbors and how far are those
neighbors from the focal node. Global influence follows similar logic as the
gravity formula, but it doesn’t utilize mass product as only k-shell values of
neighbors and not the focal node are considered. Also, the distances between
nodes are not squared in the appropriate part of the equation for global
influence.

Given that GSM method works for weighted networks and that it is essentially
based on the same information as gravity index centrality (k-shell value and
shortest path distance), we will employ both measures to quantify the
influence of belief nodes in belief networks. In this way, we have two robust
influence measures which combine both local and global network information and
are capable of measuring all three aspects of belief influence.

##  4 Data

The main analysis of our work is based on European Social Survey round 9 data
[33]. Round 9 is the latest round of the ESS and introduces new rotating
module "Justice and fairness in Europe". This module contains new items
related to regime principles which are not present in the previous rounds of
the survey and therefore we limit our analysis only to this round. Round 9
data were collected in 2018 and 2019, with most countries starting fieldwork
in late 2018 and finishing early 2019. ESS items included in our study are
presented in table 1. We used unweighted data and the analysis is based on the
entire sample which encompasses 29 countries 111Information about sample sizes
for each country and socio-demographic composition of the sample can be
accessed using ESS Online analysis tool..

Name | Survey item | Group
---|---|---
Economy | On the whole how satisfied are you with the present state of the economy in [country]? | Regime performance
Education | Please say what you think overall about the state of education in [country] nowadays? | Regime performance
Health | Please say what you think overall about the state of health services in [country] nowadays? | Regime performance
Democracy | And on the whole, how satisfied are you with the way democracy works in [country]? | Regime principles
PeopleAllow | How much would you say the political system in [country] allows people like you to have a say in what the government does? | Regime principles
PeopleInf | And how much would you say that the political system in [country] allows people like you to have an influence on politics? | Regime principles
Transparent | How much would you say that decisions in [country] politics are transparent, meaning that everyone can see how they were made? | Regime principles
CitInterest | How much would you say that the government in [country] takes into account the interests of all citizens? | Regime principles
FairChance | How much would you say that the political system in [country] ensures that everyone has a fair chance to participate in politics? | Regime principles
Parliament | How much you personally trust [country]’s parliament? | Regime institutions
Police | How much you personally trust the police | Regime institutions
Legal | How much you personally trust the legal system? | Regime institutions
Politicians | How much you personally trust politicians? | Political figures
Parties | How much you personally trust political parties? | Political figures
News | On a typical day, about how much time do you spend watching, reading or listening to news about politics and current affairs? | Grouping variable
PolInt | How interested would you say you are in politics? | Grouping variable
LRScale | In politics people sometimes talk of ’left’ and ’right’. Where would you place yourself on this scale? | Grouping variable

Table 1: European Social Survey items, variable names and groups

Following previous work [23], we transformed all belief variables to 0-1
scale. For grouping variables, we kept the original scale of variable related
to interest in politics (Very interested, Quite interested, Hardly interested,
Not at all interested) as groups. For variable related to following news about
politics, we transformed the original variable and created four groups based
on quartile values in minutes (0-30, 30-60, 60-90, 90+). Finally, for the
ideological left-right scale we transformed the original variable and created
five groups (Far Left, Left,Center, Right and Far Right).

External country-level data have been gathered from public data sets from
different sources: World Bank 222All data publicly available at
https://data.worldbank.org/, V-Dem Institute [19] and Institute for Health
Metrics and Evaluation (IHME) 333Healthcare Access and Quality Index data are
published by IHME in 2017, publicly available at
http://ghdx.healthdata.org/record/ihme-data/gbd-2015-healthcare-access-and-
quality-index-1990-2015. The variables are shown in table 2. The values of all
indicators refer to reported values for year 2018 from all sources, except
Healthcare Access and Quality Index whose latest available values refer to
year 2015. Since ESS data contain survey items recorded in late 2018 and early
2019, 2018 country-level indicators should describe the quality of government
and regime performance at the time or shortly preceding the time at which the
survey responses were recorded. For additional analysis, we also used lagged
values of indicators which are taken from the same sources 5 and 10 years
before.

Variable | Source
---|---
GDP per capita (PPP), thousands of US dollars | World Bank
Government Effectiveness | GovData360, World Bank
Political Stability and Absence of Violence | GovData360, World Bank
Healthcare Access and Quality Index | Institute for Health Metrics and Evaluation (IHME)
Liberal Democracy Index | V-Dem
Electoral Democracy Index | V-Dem
Table 2: External country-level indicators and their sources

##  5 Methods

The analytical strategy of our approach contains following elements:

  1. 1.

Unique variable analysis,

  2. 2.

belief networks structure estimation

  3. 3.

estimation of formal properties of belief networks

  4. 4.

belief influence measurement

In the previous sections, we have described the motivating example for our
study and shown the ESS variables we will use for the analysis. Before, we
proceed with the estimation of networks, we will employ Unique variable
analysis (UVA) [14] in order to check if there are redundancies between the
survey items we have chosen. If such redundancies are detected we will remove
problematic items to achieve a more parsimonious representation of belief
networks.

The next step is to estimate belief networks in different subsamples. This
will be achieved using psychometric network analysis methods [32, 30, 6] with
partial correlations representing couplings between beliefs. Finally, for each
estimated belief network we will measure the influence of beliefs using
Gravity Index Centrality [58] and Global Structure Model [78].

The entire analysis was performed in R [70]. European Social Survey data was
imported using essurvey package [16]. Main analysis was based on EGAnet [39],
psychonetrics [29], qgraph [31]. We provide new R implementation of weighted
k-shell method as well as GIC and GSM metrics based on igraph [20] package.
The entirety of data, code, notebooks and results can be found in an online
repository: https://github.com/atomashevic/belief-influence.

###  5.1 Unique Variable Analysis

Unique variable analysis (UVA) is a novel approach for detecting and removing
redundant survey items from the analysis [14]. In psychometrics, redundancy of
survey items can refer to the potential for a shared substantive cause, shared
semantic reference or general response tendency. Through development of the
psychometric network analysis [32, 28, 6] a less favorable view of redundancy
has emerged. Because elements of psychological networks (including belief
networks) are assumed to be casually autonomous, redundancy of a pair of
survey items due to their shared causes problems for the interpretation of the
entire network. Nodes who are mutually redundant are strongly connected and
have an identical or quite similar global network position, thus negatively
affecting almost every approach to measure the influence of the nodes as it
would lead to inflated measurements for those nodes having one or more
redundant pairs.

Our application of UVA is based on weighted topological overlap (wTO) [89].
Weighted topological overlap is a network measure of redundancy which
quantifies the similarity of two nodes’ pattern of connections with all other
nodes in the network, including the similarities in the weights of edges they
have with common nearest neighbors. The basic idea of UVA is to estimate wTO
for each pair of variables and values above a a chosen threshold used to
evaluate which groups of nodes are considered highly redundant. Most redundant
nodes are then removed from the network.

This is a simple approach which does not require estimation of a latent factor
model to detect redundant variables, and no references are made to the factor
or latent structure of the variable set being investigated. In order to
estimate wTO, UVA method relies on estimating the network model using the
standard approach in the psychological network literature, partial correlation
network estimation with graphical lasso regularisation with EBIC model
selection and tuning parameter $\gamma=0.5$ [30, 40]. For wTO, we applied the
threshold of 0.25 which is the mean value of previously reported evidence for
wTO cutoff values [68, 14]. Once groups of redundant variables (pairs or
triads) are identified, items which relate to a broader political concept and
have higher standard deviation remain in network while others are removed.

###  5.2 Network Structure Estimation

Estimation of the belief network relies on the methods belonging to the
psychometric network analysis [32, 61, 6] or the analysis of multivariate
psychometric data using network structure estimation and network description
in order to capture complex interactions between variables. Using this
approach, the joint probability distribution of a set of variables is
described in terms of pairwise statistical interactions. This results in a
graphical model knows as pairwise Markov random field (PMRF) [51, 28, 6]. If
we view this type of a graphical model as a network than each variable is
represented as a network node and nodes are connected by edges. Edge between
two nodes represents a conditional association between two variables,
conditional on all other variables in the data. For multivariate normal data,
these associations can are represented by partial correlation coefficients who
are then directly used as edge weights in the resulting network. If data are
continuous, as in our case, resulting is PMRF is the Gaussian Graphical model
(GGM). Formally, GGM is a model for a variance covariance matrix
$\mathbf{\Sigma}$:

$\Sigma=\Delta(I-\Omega)^{-1}\Delta,$ (1)

where $\Delta$ is a diagonal scaling matrix, $I$ is the identity matrix, and
$\Omega$ is a square symmetrical model matrix with zeroes on the diagonal and
partial correlation coefficients on the off diagonal (the edge weights of the
network).

In practice, we will use an automated model selection using step-up search
procedure. This procedure starts from an empty network and iteratively add
edges between nodes in a manner which increases the network model’s fit to the
data as estimated by Bayesian Information Criterion (BIC). New edges are
partial correlations whose addition to the model has the largest value of
statistically significant modification index ($\alpha<0.0001$). This approach
is preferred for smaller networks where we do not expect sparse structure [6]
rather than regularized approaches which are more common and come with the
assumption of network sparsity. Regularized approach was used only in the
previous step (UVA), where the goal was not to approximate the network
structure but rather to explore the dyadic redundancies between beliefs.

For every analysis that includes multiple groups, we perform model selection
for different classes of models. Given the GGM model shown in equation 1, we
can constrain any model parameters to be strictly equal or free between
groups. Of special importance are the models where we constrain the network
structure itself $\Omega$ and/or and the scaling matrix $\Delta$ to be equal
between groups. The relationship of these constrained models to models without
constraints will provide information about whether networks and their formal
properties differ between groups or not.

###  5.3 Formal properties of belief networks

As we stated in the introduction, the two formal properties of belief networks
we are interested in are energy and temperature. We modify the previous
approach [21, 23] and first estimate the energy for the individual survey
respondent $k$:

$H_{k}=-\sum_{i,j}w_{ij}b_{i_{k}}b_{j_{k}},$ (2)

where $\omega_{i,j}$ is the element of $\Omega$ partial correlation
coefficient between pair of beliefs $b_{i}$ and $b_{j}$, and $b_{i_{k}}$ and
$b_{j_{k}}$ are the values of two belief variables for respondent $k$. The
average energy $\hat{H}$ of the belief network is the mean value of all
individual energies.

For temperature, we use the approach of the previous studies [21, 23] and
calculate it as the average value of the elements of scaling matrix $\Delta$.
Lower average scaling value indicates higher overall partial correlations in
the network. Correlations implied by the chosen model result from dividing the
measured partial correlations between two beliefs by the product of their
scaling values. Therefore, the average scaling value approximates the
temperature of the belief network.

###  5.4 Node influence

We applied the k-shell method for weighted networks [38] in order to retrieve
k-shell values for nodes in our networks. The core of this method is the
alternative concept of node degree, which is calculated using its unweighted
degree (number of edges) and the sum of edge weights (also known as node
strength). The weighted degree of node $i$ is:

$k_{i}^{\prime}=[k_{i}^{\alpha}(\sum_{j}^{k_{j}}w_{ij})^{\beta}]^{\frac{1}{\alpha+\beta}},$
(3)

where $k_{i}$ is unweighted degree and $\sum_{j}^{k_{j}}w_{ij}$ is the sum of
edge weights (strength centrality). In case of $\alpha=\beta=1$ the weights
and the degree are treated equally and weighted degree is a geometric mean of
degree and strength centrality. In our case, we expect dense networks with few
nodes and therefore to increase to monotonicity of this measure we set
$\alpha=1$ and $\beta=2$ to put more emphasis on edge weights.

Once we have the weighted degrees, we apply the original k-shell decomposition
algorithm [49] to obtain k-shell values $k_{s}$. Then we can proceed to
calculate Gravity Index centrality [58] for any node $i$:

$GIC(i)=\sum_{j\in\theta_{i}}\frac{ks_{i}\times ks_{j}}{dist_{ij}^{2}},$ (4)

where $\theta_{i}$ is the nearest neighborhood of node $i$, $ks_{i}$ and
$ks_{j}$ are k-shell values and $dist_{ij}^{2}$ is the squared shortest path
distance obtained using Dijkstra algorithm. Shortest path distances were
calculated using the original degrees and not weighted ones. Weighted degrees
are used only for k-shell decomposition and discarded afterwards.

Global structure model [78] has two components: self-influence and global
influence of the node. Self-influence of node $i$ is based on its k-shell
value and is calculated in the following way:

$SI(i)=e^{\frac{K_{s}(i)}{m}},$ (5)

where $K_{s}(i)$ is the k-shell value of node $i$m and $m$ is the number of
nodes in the network. Global-influence of the same node is:

$GI(i)=\sum_{j\in\theta_{i}}\frac{K_{s}(j)}{dist_{ij}}.$ (6)

Finally, the node influence is calculated as the product of two components:

$GSM(i)=SI(i)\times GI(i)$ (7)

##  6 Results

The first step of the analysis was the application of Unique Variable analysis
to remove the redundant variables. Before the first round of UVA, the highest
value of wTO and the highest redundancy was between variables capturing trust
in the legal system and trust in police (0.943). After first round of UVA, the
following variables were removed: Transparent, FairChance, PeopleInf, Police,
Parties (see Table 1 for further info on variables). This removal resulted
with significantly smaller wTO and highest redundancy after 1st round of UVA
was between trust in Parliament and trust in legal system (0.352). We
performed a second round UVA and removed following variables from redundant
pairs with wTO higher than the threshold (0.25): Legal, Politicians.

After removing the redundant variables, a set of 7 variables remained:
Economy, Education, Health, Democracy, PeopleAllow, CitInterest and
Parliament. For this set of variables we estimated a set of belief network
models for several groups of respondents. These groups are based on variables
News, PolInt and LRScale and in total we estimated belief networks for 13
groups. For each grouping variable, we constructed 8 statistical network
models. For benchmark purposes, we created 4 empty network models (no partial
correlations included in model) and 4 nonempty models using step up procedure
with different constraints between different groups:

  * •

Model 1: All model parameters free (empty network)

  * •

Model 2: Equal edges and weights (empty network)

  * •

Model 3: Equal temperature (empty network)

  * •

Model 4: Equal edges, weights and temperature (empty network)

  * •

Model 5: All model parameters free

  * •

Model 6: Equal edges and weights

  * •

Model 7: Equal temperature

  * •

Model 8: Equal edges, weights, and temperature

For all cases, model 5 had the lowest values of BIC and therefore was selected
as best fitting model. Second best model is the model with equal edges and
weights. This means that in all cases the estimated network performed better
than the null model and that there are differences in network structure and
temperature between various groups.

For groups estimated on the basis of the ideological self-identification of
the respondent, differences between belief networks can be seen in figure 1.
All 5 belief networks exhibit low dimensionality and a relatively simple
psychological structure which is expected in belief systems [27].

Figure 1: Belief networks of different ideological groups

In these belief networks, some edges (e.g. Health-Education) are constant
between groups, while others differ (e.g. Democracy-Parliament). Given the
simple network structure, these slight differences in edges and their weights
produce differences in belief network energy and temperature, which are shown
on figure 2.

Figure 2: Differences in absolute average energy and temperature for different
groups of respondents. Note: Higher values of absolute energy correspond to
lower energy (consistent belief network)

We see that groups on the extremes of ideology, interest in politics, and news
watching have consistently higher absolute energy and more consistent belief
networks. In the case of ideology, we see that respondents belonging to Far
Left have the most consistent attitude towards the regime, followed by Far
Right. It is interesting that the third case of highly consistent belief
networks is found for respondents not interested in politics, suggesting that
consistent attitude towards the regime does not require high involvement in
daily political affairs. Smallest differences between groups were found when
comparing groups based on time spent following news about politics. Similar
level of consistency was found in groups with the lowest and the highest time
spent.

Similar trend was observed for temperature, with groups on the scale extremes
having higher temperature meaning belief system has weaker connectivity and
that less attention is payed to the attitude object. This result is
interesting in the case of ideology groups as it seems that respondents on
ideological extremes pay less attention to the activities and performance of
the regime. This finding supports the idea that ideology operates as a
cognitive heuristic and reduces the attention needed for an individual to form
a consistent attitude towards the attitude object [71]. In other groups, those
moderately interested in politics pay most attention to the regime.
Unexpectedly, the group following news about politics more than 90 minutes per
day pays the lowest attention to the regime.

Table 3: Gravity Index centrality of belief nodes for different groups. Note:
Bolded values are the most central nodes in each row.

Category | Group | PeopleAllow | Parliament | Economy | Democracy | Education | Health | CitInterest
---|---|---|---|---|---|---|---|---
Ideology | Far Left | 2.95 | 5.82 | 4.57 | 6.03 | 3.02 | 2.56 | 5.06
Left | 4.73 | 7.84 | 5.56 | 7.87 | 4.57 | 4.2 | 6.54
Center | 4.8 | 6.23 | 5.59 | 7.73 | 3.85 | 3.39 | 5.63
Right | 3.81 | 7.52 | 5.07 | 7.85 | 6.02 | 5.19 | 5.78
Far Right | 1.47 | 3.12 | 4.64 | 6.43 | 4.28 | 2.83 | 3.84
Interest in politics | Very | 5.68 | 7.58 | 6.67 | 10.43 | 3.05 | 3.46 | 7.42
Quite | 3.68 | 6.92 | 6.31 | 8.25 | 4.85 | 4.00 | 5.63
Hardly | 3.27 | 6.37 | 6.07 | 8.77 | 4.71 | 3.95 | 5.66
Not | 1.81 | 3.48 | 4.87 | 6.65 | 4.45 | 3.39 | 4.08
News following (minutes) | 0-30 | 3.98 | 4.52 | 5.05 | 6.33 | 2.97 | 2.28 | 5.13
30-60 | 5.22 | 7.78 | 6.1 | 7.43 | 4.45 | 3.29 | 6.26
60-90 | 3.16 | 6.25 | 6.51 | 9.17 | 4.32 | 3.94 | 3.75
90+ | 3.85 | 7.99 | 6.06 | 9.44 | 4.96 | 4.4 | 7.49

Results of influence measurements using GIC are presented in table 3. In most
groups, satisfaction with democracy (SWD) is the most influential node. We can
also observe that all measurements have the same scale irrespective of which
subgroup of respondents is considered. This makes the notion of average or
total influence of beliefs in the network a meaningful measure. There are no
issues with low monotonicity even though small networks are considered.

Table 4: Global Structure Model centrality of belief nodes for different
groups. Note: Bolded values are the most central nodes in each row.

Category | Group | PeopleAllow | Parliament | Economy | Democracy | Education | Health | CitInterest
---|---|---|---|---|---|---|---|---
Ideology | Far Left | 169.62 | 950.94 | 1210.29 | 1181.47 | 1037.39 | 1210.29 | 902.91
Left | 2376.04 | 1932.16 | 3316.01 | 2898.24 | 422.64 | 893.31 | 1644.95
Center | 6813.62 | 9084.82 | 9652.63 | 7381.42 | 7097.52 | 9084.82 | 7665.32
Right | 2284.65 | 1864.28 | 3107.13 | 2412.59 | 1681.5 | 2229.82 | 2467.42
Far Right | 91.00 | 349.83 | 353.36 | 261.49 | 123.68 | 353.36 | 346.3
Interest in politics | Very | 403.43 | 2647.59 | 2832.97 | 1603.17 | 3655.44 | 2558.81 | 3799.05
Quite | 2571.87 | 9084.82 | 9084.82 | 7026.54 | 7097.52 | 8658.97 | 2780.75
Hardly | 2611.03 | 9056.43 | 8658.97 | 6146.45 | 6529.72 | 8517.02 | 7274.96
Not | 206.72 | 300.36 | 672.38 | 854.88 | 662.78 | 432.25 | 167.85
News following (minutes) | 0-30 | 2284.65 | 3838.22 | 3107.13 | 2467.42 | 2467.42 | 3655.44 | 2650.2
30-60 | 6813.62 | 9084.82 | 10220.43 | 7665.32 | 7381.42 | 9652.63 | 7665.32
60-90 | 141.35 | 614.75 | 864.49 | 576.33 | 576.33 | 749.22 | 353.36
90+ | 7097.52 | 9084.82 | 9652.63 | 7097.52 | 7097.52 | 5678.02 | 3690.71

On the other hand, GSM influence measurement presented in table 4 shows
different results. In most cases, satisfaction with economy is the most
influential node, but we have higher variability between groups for most
influential nodes. Also, the scale of the values seems to vary with unclear
interpretation of the order of magnitude differences between groups (e.g., the
influence of Economy belief in Far Right and Right groups). There some issues
with low monotonicity as multiple beliefs have the same influence.

As stated before, we are interested in the relationships between influence
measures and formal properties of belief networks. They are shown on figure 3.
There exists a clear inverse linear relationship between energy and
temperature. Consistent low-energy belief networks have higher temperature,
which means lower connectivity and attention. These results point towards a
trade-off in this type of belief networks, where either consistency in
attitude towards the regime is obtained (on average in a group) or higher
state of belief connectivity and higher attention to the attitude object is
obtained.

Figure 3: Scatterplots showing relationships of formal network properties and
influence measurements. First (top) row: Belief network temperature and
average energy. Second row, left: Average energy and average GIC of belief
network. Second row, right: Temperature and average GIC of belief network.
Third row, left: Average energy and average GSM of belief network. Fourth row,
left: Average energy and standard deviation of GSM. Fourth row, right:
Temperature and standard deviation of GSM.

Second row of panels on figure 3 shows the relationship of the mean value of
Gravity index centrality with the average energy and temperature of the belief
network. In low-energy highly consistent networks, we found a low average
influence of beliefs and it seems that higher consistency of beliefs is
possible in the absence of highly influential beliefs. The reverse is found in
case of temperature. As temperature refers to connectivity of the network and
as GIC (any almost every other measure of centrality) measures influence on
the basis of strength of connections (of focal node, its neighborhood, or the
entire network) this result is expected.

Similar conclusions can be obtained by inspecting the third row of panels on
figure 3, although due to the wider range of GSM, there seems to be a weaker
linear relationship between this measure of influence of energy and
temperature. The same can be said for the standard deviation of GSM, measuring
the variability of belief influence within the network.

Figure 4: Scatterplots showing the relationship between belief influence
(measured by GIC) and country-level external indicators

Finally, the relationship between the influence of beliefs as measured by GIC
and external country-level measures associated with the corresponding domains
is shown on figure 4. The influence of belief expressing satisfaction with the
national economy is negatively correlated with GDP per capita ($r=-0.352$,
$p=0.06$) and with Government Effectiveness index ($r=-0.530$, $p=0.03$) which
is related to the quality of public services and national policy formulation
and implementation, including policies related to the national economy [45].
The correlation with GDP per capita increases when we take lagged values into
consideration. Correlation with 5-year lagged GDP is $r=-0.397$ ($p=0.03$),
while with 10-year lag we have $r=-0.447$ ($p=0.02$). This is the only
relationship where lagged values are shown to correlate better with belief
influence measured by GIC.

Similar results are obtained for influence of satisfaction with democracy and
V-Dem indices of Liberal ($r=-0.52$, $p=0.004$) and Electoral democracy
($r=-0.52$, $p=0.003$). As the majority of European countries in the sample
score high on these indices, it is important that we observe in those
countries with lower score a higher influence of satisfaction with democracy.
The influence of satisfaction with healthcare system is also found to be
greater in countries with lower values of Healthcare Access and Quality Index
($r=-0.344$, $p=0.07$). HAQ gives insight into the personal health-care access
and the quality of a wide range of medical services [36]. Similar correlation
is found between the influence of trust in national parliament and Political
stability index, which indicates the likelihood that the government will be
destabilized or overthrown by unconstitutional or violent means ($r=-0.31$,
$p=0.09$).

All results shown on 4 are based on influence measured by GIC. GSM values show
close to zero correlations with all external country-level measures.

##  7 Limitations

Study has several notable limitations. First, the analysis is based on ESS
data, which means a fixed, closed set of survey items are available and form
the basis of the belief system analysis. If different survey items were
available, after redundancy analysis, the resulting belief network may have
different number of different items items. However, using UVA approach
minimizes the risk of item selection bias as we reduced the number of items
from a larger set of items present in the survey to core questions related to
basic political institutions of European countries.

Second, we use the unweighted, raw version of the survey data. Although we are
not interested in country-level means of survey items, structural post-
stratification inequalities in national subsamples may influence the results
concerning comparisons of networks estimated from those subsamples.

Third, influence measures are selected based on literature review and the
analysis of previous applications on network different than belief networks.
It is possible that there is a another subset of all proposed centrality
measures capable of measuring the influence of beliefs in this context and not
taken into consideration in our paper.

Third, reported correlations are based on 29 country-level subsamples and
reported $p$-values have to be treated with caution. Given the theoretical
distribution of Pearson correlation coefficient, it is not unlikely that some
of these reported correlations do not differ from zero on a global level.

##  8 Discussion

Main findings of our study contribute to the hardcore of theory of belief
system networks [10, see]. Our results present evidence of the relationship
between previously established properties of belief networks and new measures
of belief influence. Also, we have shown that similar relationship exists
between belief influence and external indicators related to same belief domain
(such as democracy or economy). Taken together, these results support the
claim that centrality-based influence metrics in belief networks provide
valuable information, in contrast to the studies questioning the utility of
node centrality in psychometric network analysis. [11].

Previous results in network theory of attitudes have shown that when attitude
is important, we have higher consistency and stability of the attitude,
because different attitude elements rein each other in a system with low
temperature [6]. Our results show that belief networks with higher average
influence have lower temperature. Higher average influence implies higher
connectivity and higher average centrality in the network as it is highly
probable that on the neighborhood level, nodes are capable of influencing on
another.

By focusing on differences in belief networks of various subgroups our results
demonstrated new findings related to energy and temperature of belief
networks. The inverse relationship between these two properties is especially
interesting given that recent belief network study [10] demonstrated that
higher levels of network connectivity (lower temperature) lead to
ideologically consistent (lower energy) left-wing or right-wing belief
networks. However, this conclusion comes under the assumption that every
element of the belief network has a potential to be in left-wing or right-wing
state and therefore the very nature of belief network elements is ideological,
while in our case we are concerned with beliefs about regime, its principles
and institutions, where positive or negative evaluations cannot be directly
connected with ideological content. We find higher consistency of attitude
towards the regime in groups positioned on ideological extremes, but the
connectivity of these networks is lower. In this case, ideology provides
consistency but without strong links between non-ideological beliefs. This
finding is in line with recent study [5] which demonstrates that ideology can
serve as a mechanism to validate one’s reactions to different political issues
and increase certainty in thoughts related to them. If the ideology a the
proxy mechanism which provides certainty in our belief networks, then it would
be expected that attention towards the attitude object (the regime) becomes
lower in groups positioned on ideological extremes.

We also found unexpected results in our study. First of all, we did not expect
that both groups highly interested in politics and not interested at all show
the same level of attention in their belief networks and that both have lower
attention than moderate groups. These results point towards the conclusion
that general interest or disinterest in politics does not correlate with level
of attention paid towards the actions of the government. Second we did not
expect almost no differences between groups of respondents who spent different
time following the news. Taken together, these findings are more likely to
support media malaise theory [62] as we observe no positive effect of media
use and interest in politics and on consistency and attention of belief
networks.

Finally, even though we have found both GIC and GSM measures to have
relationship with energy and temperature, GSM showed absolutely no association
with external measures used. Overall, despite its conceptual similarities with
GIC, GSM proved to be a poor measure of influence in our example of belief
networks. On the other hand, Gravity index centrality emerged as a suitable
measure of influence, especially when considering correlations with related
country-level indicators. Although the sample of countries is small and
limited to European countries, correlations between the influence of specific
beliefs and country-level indicators of regime performance have proved to be
equal of greater than previously reported correlations between country-level
average of Satisfaction with democracy and similar group of indicators [67,
e.g.].

In light of these results, we believe that the main task of future studies is
to utilize developments in belief systems network research for creating
country-level indicators based on belief influence from large-scale
comparative surveys. Our study demonstrated the potential of this approach on
one clear example based on European Social Survey round 9 data. Further
studies are needed to demonstrate that this approach is viable for different
belief networks and Gravity index centrality’s performance needs to be
evaluated in comparison with both simple and advanced centrality metrics.
Psychometric network analysis and the theory of belief system networks have a
lot to offer to comparative political science studies and network-based belief
influence measurement may be the right first step towards bringing these
fields closer through multidisciplinary collaborations.

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