# Who is driving the conversation? Analysing the nodality of British MPs and
journalists on Twitter

Leonardo Castro-Gonzalez  Public Policy Programme, The Alan Turing Institute,
British Library, 96 Euston Rd, London, NW1 2DB, UK.  These authors contributed
equally  Sukankana Chakraborty  Public Policy Programme, The Alan Turing
Institute, British Library, 96 Euston Rd, London, NW1 2DB, UK.  Centre for
Advanced Spatial Analysis, University College London, Gower Street, London,
WC1E 6BT, UK  These authors contributed equally  Helen Margetts  Public Policy
Programme, The Alan Turing Institute, British Library, 96 Euston Rd, London,
NW1 2DB, UK.  Oxford Internet Institute, University of Oxford, 1 St Giles,
Oxford, OX1 3JS, UK  Hardik Rajpal  Public Policy Programme, The Alan Turing
Institute, British Library, 96 Euston Rd, London, NW1 2DB, UK.  Department of
Mathematics, Imperial College London, South Kensington Campus, London, SW7
2AZ, UK  Daniele Guariso  Public Policy Programme, The Alan Turing Institute,
British Library, 96 Euston Rd, London, NW1 2DB, UK.  Jonathan Bright  Public
Policy Programme, The Alan Turing Institute, British Library, 96 Euston Rd,
London, NW1 2DB, UK.

###### Abstract

Who sets the policy agenda? Much political science research has been dedicated
to investigating the influences at work in policy processes. This paper looks
at the actors influencing the policy agenda through the lens of “nodality”
—the capacity to share information and to be at the centre of social and
information networks. Nodality has traditionally been seen as a capacity or
policy tool relating to government as a set of institutions. But in a digital
world, there is greater possibility for individual actors using social media
to develop their own nodality and to shape the policy conversation. We analyse
information networks on Twitter to quantify the nodality of two sets of actors
in the UK political system - Members of Parliament (MPs) and accredited
journalists – in conversations on four policy topics: The Russia-Ukraine War,
the Cost-of-Living Crisis, Brexit and COVID-19. We perform a Transfer Entropy
analysis and a Principle Component Analysis (PCA) using different network
centralities while ensuring reproducibility and interpretability. We define
and measure two dimensions of nodality; inherent nodality, associated with an
actor’s institutional position, transferable between topics, and individual
nodality, measuring active engagement in a policy conversation on a specific
topic. These two dimensions are validated through a linear model showing how
the influence on a particular topic exercised by an actor stems from the sum
of nodality scores across these two dimensions. We thereby show the relative
influence of different types of actor on policy conversations. More generally,
we provide a methodology for measuring individual and institutional drivers of
actor influence, and understanding where influence relates to individual or
institutional characteristics.

##  1 Introduction

Much political science research has been dedicated to understanding the
relative influences of policy actors on the public policy agenda. Some
scholars focus on the media [McCombs and Shaw, 2005], some on policymakers
[Berkowitz, 1992] and some on the complex dynamics between the two [Wilson and
Kingdon, 1993, Baumgartner et al., 2006, Baumgartner et al., 2013]. This paper
tackles the long-standing question of ‘who sets the political agenda?’ through
the lens of nodality – the capacity to exchange information – which was
identified by Hood as one of the four key ‘tools of government’ used to make
policy [Hood, 1983] and further developed by Hood and Margetts [Hood and
Margetts, 2008] and Margetts and John [Margetts and John, 2023]. Nodality used
to be something that government – as a set of organisations – possessed almost
by virtue of being government, as the “water mill” at the centre of society’s
information networks [Margetts and John, 2023]. But in a digital era, where
policy debates take place in more open and diffuse information environments,
governmental actors face greater competition for nodality [Castelnovo and
Sorrentino, 2021]. Widespread use of the internet means that non-governmental
organisations or individuals such as journalists, politicians, public figures
and even citizens themselves can acquire nodality, which can challenge the
government’s capacity to capture public attention and drive policy and media
agendas [Margetts et al., 2019, Howlett, 2009, Howlett, 2022]. Therefore, as
policy-related conversations are increasingly taking place outside
institutional boundaries, there is a constant jostling for position between
both governmental and non-governmental actors [Shearing and Wood, 2003].

Framed within the context of policy discourse in the UK, we seek to analyse
the nodality of different actors on online platforms to determine how they
drive the competition to shape policy conversations. While our ultimate aim is
to analyse the relative nodality of a more diverse set of actors, including
governmental departments, NGOs, local councils and public figures, in this
paper we first test our methodology on a smaller subset of elite actors in the
UK: Members of Parliament (MPs) and accredited journalists. We propose a new
method to dynamically measure nodality on social media, where information
spreads widely and interactions are instantaneous. Although our methodology
here is applied to Twitter (now X), it can be extended to other platforms and
across a wide range of actors. Similar works include the study of exposure to
misinformation from political elites in the USA [Mosleh and Rand, 2022], or
the spread of news of COVID-19 from sources with different reputations
[Gravino et al., 2022].

Influence on social media has been researched extensively in the field of
communications (see for example, [Dubois and Gaffney, 2014, Anger and Kittl,
2011, Bakshy et al., 2011, Mei et al., 2015, Riquelme and González-
Cantergiani, 2016, Esteve Del Valle and Borge Bravo, 2018, Rezaie et al.,
2020, Zhang et al., 2023]). On X (previously Twitter), as with other social
media sites, we assume that an actor has an influence on a particular topic
when their activity generates activity by others (tweeting, retweeting,
liking, replying or quoting) on the same topic. Understanding actors’
influence on the policy and media agenda will in part relate to these dynamics
in policy conversations, which can be studied using transfer entropy, thus
allowing for temporal and predictive analysis [Schreiber, 2000, James et al.,
2016, Shorten et al., 2021].

In this paper, we investigate the nodality of two types of actors – MPs and
journalists – that both play key roles in information exchange and influence
in a policy system. Political representatives have an obligation to feed the
concerns of their constituents into policy debates, while journalists
disseminate information from policy discussions out to the public and wider
information environment. We are able to obtain data for a substantial
proportion of both sets of actors, which includes 581 out of 650 MPs and 8606
out of 9575 journalists111The missing users do not have a Twitter account or
the handles given could not be found. The dataset was also limited by the
Twitter API policy which limits the number of tweets that can be retrieved per
user to 3200.. This approach helps to address a methodological challenge faced
by previous work on policy actors and social media, where the differential
attention that actors receive relate to both institutional and individual-
level characteristics. For institutional differences, MPs can hold ministerial
or shadow ministerial positions, which means that they are more likely than
backbench MPs to receive attention (through more followers, for example),
while journalists working for mainstream news channels or newspapers are
likely to receive more attention by virtue of their institutional homes, while
independent journalists may struggle to receive attention. There are
institutional constraints for both sets of actors on social media: ministers
often need to follow a Government or party line, and journalists may need to
follow guidelines of their publishers. For individual characteristics, some
individual MPs may become active and influential on a certain topic, possibly
by virtue of their constituency being differently affected by an issue.
Individual journalists may acquire a social media presence over and above the
reputation of the publications for which they work. Our approach allows us to
develop measures of nodality across these ’institutional’ and ’individual’
dimensions, thereby understanding where influence relates to an actor’s
institutional position, and where it relates to individual-level activity,
reputation or characteristics. Developing this methodology may help in the
analysis of social media activity more generally, such as understanding widely
varying levels of hate speech and abuse directed at representatives [Vidgen et
al., 2020].

Nodality has been studied theoretically [Howlett, 2009, Howlett, 2022,
Castelnovo and Sorrentino, 2021] and quantitatively, using network science
-centrality measures [Margetts and Escher, 2006, Petricek et al., 2006], but
we believe that this is the first attempt to quantify nodality as a way to
assess actors’ communication capacity across different institutions and policy
conversations, and over time, going beyond mere information transmission. To
do so, we decompose nodality into two key components: (i) inherent nodality,
which is earned by virtue of one’s institutional role (such as a frontbench
position) and (ii) active or “topic specific” nodality, which is earned
through high levels of engagement on a particular issue. We do this using
different centrality measures and a transfer entropy analysis to establish a
connection between the two dimensions of nodality and an agent’s influence.

We test the hypothesis that that influence will vary across different types of
actor and policy topics on social media platforms [Bright, 2016, Bright et
al., 2020, Sprejer et al., 2022, Parmelee, 2013, Usherwood and Wright, 2017].
We do this by using data from social media conversations around four policy
topics in the year 2022: (1) the Russia-Ukraine war (2) the challenging
economic situation facing ordinary citizens (cost-of-living crisis) (3)
Britain’s departure from the EU (Brexit) and (4) the aftermath of the COVID-19
pandemic. All four issues were contested in policy terms and related to an
increasingly challenging economic situation, driven by Brexit [Diamond, 2023,
Dhingra and Sampson, 2022], the pandemic [Blundell et al., 2022], the Russia-
Ukraine war [Mbah and Wasum, 2022, Liadze et al., 2023] and the resulting
energy crisis [Harari et al., 2022, Francis-Devine et al., 2022], as well as a
“mini-budget” in the autumn which led to sharp and sustained rises in interest
rates [Chadha, 2022, Tosun and Lucey, 2023, Li, 2023]. We analyse Twitter data
obtained for 581 MPs and 8,606 journalists registered in the UK between
January 14, 2022 and January 13, 2023. We classify tweets using a weak-
supervision classifier [Ratner et al., 2019, Castro-Gonzalez et al., 2024] and
construct a weighted directed graph for each topic, where actors are nodes,
and any interaction is captured as an information edge. To obtain group-level
influence measures, we employ Transfer Entropy analysis on their activity time
series, while inherent and active nodalities as defined above are derived
through a Principal Component Analysis (PCA). We hypothesise that we will be
able to distinguish institutional nodality as transferable across various
topics, while active nodality is specific to a particular topic.

In this way, we address our central question: Which sets of actors drive the
policy conversation on a given topic – which are the most influential nodes?
In order to answer this question, we additionally tackle a further
methodological issue: can we separate the relative effects of each actor’s
’institutional’ and ’individual’ nodality, and how can we measure nodality on
online conversational networks across these two dimensions? In Section 2, we
present our primary findings, leading us to subsequent discussions and
conclusions in Section 3. Methods are presented in Section 4. For more
detailed information about our methods, please refer to Appendix 5.

##  2 Results

###  2.1 Granger causality analysis and qualitative analysis of conversation
networks

Figure 1: First insights for the Cost of Living Crisis conversation. (a)
Activity time series for both the journalists and the MPs groups. (b) The
giant component of the interaction network between journalists and MPs. The
size of a node corresponds to the weighted out-degree which reflects the
number of times a user has been retweeted (replied to or mentioned) by others
in the network. Nodes representing journalists are shown in grey, while MPs
are represented using coloured nodes where the colour signifies the political
party they are affiliated with (Conservative, Labour, other parties). Edges
here take the colour of the target node. Prominent nodes with high out-degrees
have been labelled to highlight actors who are integral to the discussion (as
a result of being heavily retweeted, replied to and mentioned). (c) The
activity time series of Zarah Sultana, MP.

We begin by applying a Granger causality test on the activity time-series of
MPs and journalists (shown in Figure 1a for the Cost of Living Crisis topic).
This is a preliminary step in obtaining a general understanding of whether MPs
or journalists dominate and lead the conversation on any of the four topics.
We consider multiple time lags in our analysis (1 to 7 days lag) and find that
in the cases of COVID-19 and Brexit, activity from journalists consistently
“Granger-cause” the activity of MPs, with a statistical significance of
$p<0.01$ for all considered time-lags, whereas for the Cost of Living Crisis
and the Russo-Ukrainian war discussions, result are less clear and not
statistically significant222Details of the analyses for all topics can be
found in Appendix 5..

We now proceed with a more comprehensive approach where we analyse the
conversation network qualitatively for each topic. To do this, we build a
directed network based on interactions between actors through retweets,
replies and mentions, i.e. links represent the passage of information from a
source actor who is retweeted (replied to or mentioned) by a target actor.
Figure 1b shows the conversation network for the Cost of Living Crisis topic.
We rely on network-based metrics333Centrality measures have been used
extensively in the past to determine the influence of actors on online
platforms [Dubois and Gaffney, 2014]. to determine how central the actors are
to each conversation. In our analysis, we find that nodes with the highest
outdegrees are MPs with important parliamentary positions444Prominent nodes
include Cabinet and Shadow Cabinet MPs during 2022-2023, such as Boris
Johnson, Rishi Sunak, Liz Truss, Keir Starmer, Rachel Reeves and Ed Milliband,
or political editors at influential UK newspapers, such as Pippa Crerar (from
The Guardian), Steven Swindon (The Times), and Jim Pickard (Financial Times)..
Rishi Sunak has the highest outdegree in the network which can be attributed
to the two key roles he held during the year 2022-23 —the Chancellor of the
Exchequer, which is a pivotal role in public spending, and eventually that of
the Prime Minister of the UK. We find that Rachel Reeves, the Shadow
Chancellor of the Exchequer, and Nadhim Zahawi, who was the Chancellor under
Liz Truss’ premiership in 2022, are also among the prominent nodes in the Cost
of Living Crisis network. At the same time, we observe an anomaly here. One of
the other actors central to the conversation is Zarah Sultana, a Backbench
Labour MP, who despite her less significant role in the parliament has a high
outdegree in the cost of living crisis network —an effect of a strong
engagement with journalists who interact with her frequently on the topic. We
further note that her nodality on the cost of living crisis discussion is
topic-specific, and her activity, as well as nodality, is limited on the other
topics (Figure 1c). In contrast, actors who have senior positions within the
Cabinet or shadow frontbench such as Boris Johnson, Rishi Sunak, Liz Truss and
Keir Starmer consistently appear central in all four conversations. These
findings align well with the two dimensions of nodality outlined in Section 1—
_inherent_ or institutional nodality which can be attributed to one’s
institutional role, and _active_ nodality which can be acquired at an
individual-level by being active on a specific topic (such as in the case of
Zarah Sultana).

###  2.2 Quantifying the nodality of MPs and journalists

We now attempt to quantify the inherent and active nodalities of actors in
each of the studied conversations. The details of the approach are outlined in
Section 4.3, but principally, we construct two networks for each topic: the
conversation network (i.e. topic-specific network) and the null network
capturing user interactions outside of the topic of focus. We then obtain the
centrality measures of actors on both networks to perform a principal
component analysis (PCA) for each topic, to quantify the inherent and the
active nodalities of the actors. Guided by past work [Petricek et al., 2006,
Margetts and Escher, 2006], we use six standard network-metrics for the PCA.
In addition, we introduce two new metrics: the funnel bandwidth and the
amplification bandwidth of an actor. The funnel bandwidth of a node measures
its interaction activity, weighted by its own follower count, i.e. a measure
of the extent to which an actor facilitates the passage of information in the
network. Amplification bandwidth, on the other hand, measures the exposure of
information from a user in two hops. We measure this as the number of times a
user $i$ interacts with others, weighted by the number of followers of each
target user $j$. These new measures are based on the follower count of the
actors which has been extensively used in the literature to examine the role
of users in online discussions [Riquelme and González-Cantergiani, 2016]. We
thus finalise _eight_ network-metrics —shown in Table 4.5 —, which on their
own and combined with each other capture different aspects of nodality. We now
employ the metrics in a PCA to quantify the two dimensions of nodality.

Figure 2: Details of the PCA and the $k$-means clustering for the Cost of
Living Crisis conversation. (a) Figure showing the PCA results. (b) The three
clusters represented in the interaction network. Nodes are coloured based on
the cluster they belong to in the PCA (representing their nodalities in
comparison to the mean nodality of the population). The size of the nodes
reflects their weighted outdegrees. (c) Schematic of the general PCA results
for the four topics. Observe that it has a notable triangular structure.
Regions of sparse data are characterised by low active nodality (and high or
low inherent nodality). Actors with low inherent nodality and limited
participation in the conversation have low centrality measures and thus are
excluded as outliers while fitting the data into a normal distribution prior
to the PCA transformation. This is also true for actors with high inherent
nodality and low active nodality. However nodes with high inherent nodalities
are excluded less because they are retweeted by others in the network (given
their prominent institutional roles).

We begin by generating all possible combinations of _three_ or more measures
—which gives us 255 combinations —of which we retain only those combinations
that capture the inherent and the active nodalities of an actor along the
first and second principal components. We verify this by employing an
eigenvector test555Details can be found in 4.3., where we ensure that the
features from both networks either all positively (or negatively) correlate
with the first principal component666The weights of all features on the first
principal component are either all positive or all negative.. Given that the
inherent nodality of an actor is a reflection of their institutional role in
the real-world and is independent of their activity on any topic, we
hypothesise that the correlations in node measures on both the topic and the
null networks as observed along the first principal component should capture
the inherent nodality of an actor. We further assume that the residual
nodality captured along the second principal component is the topic-specific
active nodality. We verify this by ensuring that the features related to the
topic network are positively correlated with the second principal component
and orthogonal to the features obtained on the null network. In this way, we
use the two principal components to distinguish between the inherent and
active nodalities of actors. We only retain combinations of measures that
satisfy the above form and we move into the next phase of our analysis where
we identify the set of measures that maximises the number of common inherently
nodal actors across all four topics. We expect actors with high inherent
nodalities to consistently appear nodal, irrespective of the topic of focus.
Therefore for each of the remaining set of features, we apply a $k$-means++
clustering, where $k=3$. The clustering approach groups actors based on their
nodalities and we obtain three classes of policy-actors, those with inherent
nodality: (i) higher than the mean of the studied population (nodal actors),
(ii) same as the mean of the population and (iii) lower than the mean of the
population (non-nodal actors). We then compare the nodal clusters across all
four topics. The combination of metrics that maximises the number of shared
inherently nodal actors across all topics includes: the strength of a node,
their degree centrality, and the funnel bandwidth of the node.

Figure 3: Results of the PCA and $k$-means clustering for the four
conversations. (a) The PCA results for the four topics. In a clockwise sense,
starting from the top left, we find the results for the Russia-Ukraine War,
Covid, Brexit and Cost of Living Crisis. (b) The result of the Transfer
Entropy Analysis done for the resulting three clusters of the four
conversations.

We now combine the clustering results from the PCA with the conversation
network for the cost of living crisis discussion in Figure 2b. We find that
the clusters form three tiers of the conversation network. Nodes in the
inherently nodal cluster777These are nodes with nodality score higher than the
mean of the studied population. assume central positions in the conversation,
i.e. we find that most of the interactions in the network are centred around
these nodes and they exhibit high out-degree to in-degree ratios. We term
these nodes _leaders_. Nodes with less than average inherent nodality (or the
non-nodal actors) largely appear on the periphery of the network with limited
connections to the rest of the network (average degree of 2). As these nodes
typically appear to be at the periphery or the receiving end of the
information propagation process, we term them _receivers_. Finally, we observe
that nodes with average nodality form the layer between the central nodal
cluster and the peripheral non-nodal cluster. We find that these nodes appear
to bridge the interaction between the _leaders_ and the _receivers_ , by
receiving information from the _leaders_ (through retweets/mentions and
replies) and passing it onto the _receivers_ , and hence we term them
_funnelers_.

To validate the PCA results and our hypothesis related to the role of each
cluster in the topic-related discussion, we use a transfer entropy (TE)
analysis. The TE analysis helps us measure the extent to which a group’s
activity can predict the activity of others, measured as the reduction in the
entropy of a group’s activity that can be estimated from the activity time
series of the complement group888Details of the method can be found in Section
4.4.. The difference in the share of entropy of Y predicted by the time series
of X and the share of entropy of X predicted by the time series of Y,
determines the direction of the flow of information. If this quantity is
positive, there is an information flow from $X\rightarrow Y$, whereas if this
is negative, information flows from $Y\rightarrow X$.

The results of the TE analysis aggregated across all four topics are shown in
Figure 3. We find that _leaders_ exhibit the highest share of influence among
the three clusters. This supports our hypothesis that information in the
network is typically disseminated from these nodes. _Receivers_ , on the other
hand, maintain a negative share of influence, which indicates that they mainly
respond to the ongoing activity in the network. Whereas _funnelers_
demonstrate low positive values suggesting that while their activity may
influence the activity of the _receivers_ , their own activity is influenced
by the activity of the _leaders_. This means that information in the network
flows from leaders to funnelers to receivers, as also concluded from our study
of the topology of the network and the PCA. It is worth noting that despite of
using two different data sets —conversation networks for the PCA and time
series data for the TE analysis —our results from both analyses complement
each other well.

###  2.3 Quantifying nodality at group level

Figure 4:  Boxplots illustrating the inherent and active nodality scores of
the different groups of MPs and journalists across all four topics, as
obtained from the PCA. (a) shows the inherent nodality scores of each group
while (b) shows the active nodality scores.

We now use the approach proposed above to answer our original research
question —who among the MPs and journalists leads the conversation on the
selected policy-related topics? So far we have shown that not all MPs (or
journalists) have the same nodality, and that an actor’s capacity to influence
online conversations often relies on their institutional roles. We therefore
split MPs and journalists into sub-groups that reflect their organisational
roles in the real world999For the MPs, we exploit the structure of the British
parliament to classify them into four groups: the Cabinet, which forms the
executive power; the Shadow Cabinet, formed by the Opposition party that
oversees the activities of the Cabinet; the Backbench, formed by MPs of the
ruling party who do not have a place in the Cabinet; and its analogous for the
opposition party, simply called the Opposition. We also divide journalists
into two groups based on their follower counts. We find that the ratio of the
Cabinet to the Backbench MPs and of the Shadow Cabinet to the Opposition
(backbench) MPs is roughly 10%. Therefore we split the journalists by
separating the 10% most-followed journalists from the rest of the journalists
(and term them prominent journalists).. The groups are listed in Table 1,
along with their sizes.

As before, we perform a PCA on the conversation networks for all four topics
and we determine the mean inherent and active nodality score of each group.
Results of the PCA are shown in Figure 4. We find that the Cabinet and the
Shadow Cabinet MPs exhibit the highest inherent nodalities across all topics,
typically followed by the prominent journalists, the backbench MPs (both
government and opposition) and finally, by the rest of the journalists who
exhibit the lowest inherent nodality scores. On examining the active
nodalities of each group, we find that overall the median active nodality of
all six groups of actors is similar to the mean active nodality of the studied
population across all topics. Only the Shadow Cabinet distinguishes itself
from the rest, by exhibiting an above average active nodality for the topic of
Cost of Living crisis and a less than average active nodality for the Brexit-
related conversation.

We simultaneously perform the TE analysis for each group and each topic over
time. Results are shown in Table 1. For each topic, we highlight the top two
groups with the highest shares of influence on the rest of the
population101010A schematic of how this table is computed is shown in Figure 6
of Section 4.4.. We find that while the Cabinet MPs and the prominent
journalists lead the activity on the Russo-Ukrainian conflict over time, the
Shadow Cabinet and the Opposition backbench drive the discussion on the cost
of living crisis. Furthermore, we observe that the Opposition backbench also
leads the conversation on Brexit and COVID-19, along with prominent
journalists.

Table 1: Temporal average influence score of each one of the groups for the
different studied topics. Highlighted cells are those having a positive
influence within the 95% confidence interval

Group |  Russia-Ukraine War | COVID |  Cost of Living Crisis | Brexit |  Number of nodes
---|---|---|---|---|---
Cabinet MP | $6.53\%\pm 2.81\%$ | $0.91\%\pm 2.45\%$ | $0.02\%\pm 3.14\%$ | $-1.41\%\pm 2.81\%$ | 58
Backbench MP | $1.11\%\pm 1.37\%$ | $0.33\%\pm 1.10\%$ | $0.10\%\pm 1.26\%$ | $-1.37\%\pm 1.51\%$ | 237
Shadow Cabinet MP | $-0.1\%\pm 4.84\%$ | $0.71\%\pm 4.25\%$ | $9.41\%\pm 5.04\%$ | $-1.08\%\pm 5.78\%$ | 27
Opposition backbench MP | $1.26\%\pm 0.99\%$ | $1.11\%\pm 0.76\%$ | $3.22\%\pm 1.41\%$ | $1.85\%\pm 0.90\%$ | 246
Prominent journalists (10% most followed) | $2.75\%\pm 1.01\%$ | $0.95\%\pm 0.79\%$ | $0.38\%\pm 1.20\%$ | $1.44\%\pm 1.16\%$ | 657
Rest of journalists | $-0.34\%\pm 0.94\%$ | $-0.17\%\pm 0.77\%$ | $-1.18\%\pm 1.48\%$ | $-1.02\%\pm 1.63\%$ | 3724

These results suggest that the ability of an actor to influence the discussion
on a topic does not entirely depend on their institutional status but can also
depend on other factors (such as their activity on the topic). To examine this
further, we perform a regression analysis that determines the relationship
between the capacity to influence the conversation on a topic and the
different dimensions of nodality (i.e. inherent or institutional and active or
individual nodality). To do this, we employ a mixed-effects model where the
dependent variable is the influence exerted by a group on the activity of a
topic, and the independent variables include the collective inherent and
active nodalities of the groups and a related crossed term (inherent-active).
Given that the influence of a group on the remaining population is measured
using the time-series of their activities, we also include time as a
independent variable. A _two_ -week time period is used to compute the
influence of each group (the dependent variable) and the PCA-related terms
(the independent variables). Details of the model are given in Section 4.5.

Table 2: Results of the Linear regression for the model relating the group
influence and the Nodality of its integrants. $N=1045,\ R^{2}=0.6137$
---
Variable | Coefficient |  $p$-value
Inherent** | $0.010\pm 0.001$ | 0.000
Active** | $0.008\pm 0.004$ | 0.025
Inherent-Active interaction | $-0.001\pm 0.001$ | 0.111
Time** | $-0.002\pm 0.000$ | 0.000
Intercept** | $0.088\pm 0.029$ | 0.002

Results of the regression analysis are presented in Table 2. We make four key
observations: (i) the capacity of a group to influence the discussion on a
specific topic depends not only on their institutional status but also on how
active they are on the topic; (ii) the capacity of a group to influence the
topic-related activity is positively associated with the mean inherent and
active nodalities of the group; (iii) we also verify the independence of the
_two_ dimensions of nodality (inherent and active), given that the crossed
term (inherent-active) does not significantly contribute to overall capacity
of the group to influence the activity related to a conversation; and finally,
(iv) assuming that the mean nodality of a group is fixed, we observe that the
influence exerted by the group on the activity of a topic decreases with time.

##  3 Discussion

To date, the concept of nodality has been studied largely from a theoretical
perspective [Howlett, 2009, Castelnovo and Sorrentino, 2021, Howlett, 2022,
Margetts and John, 2023], with very few attempts to quantify nodality in an
empirical setting [Petricek et al., 2006, Margetts and Escher, 2006]. Here for
the first time, we focus on computing nodality scores for individual actors in
policy conversations. More specifically, for our two sets of elite actors (MPs
and journalists), we distinguish and measure two dimensions of nodality; (i)
topic-specific active nodality at the individual level, relating to the
activity of an actor on a specific topic, and (ii) inherent nodality relating
to the actor’s institutional role and status. Distinguishing between the two
types of nodalities provides a useful way to determine the _nodal advantage_
of actors in prominent roles, such as in the case of Ministers and political
editors, who by virtue of their institutional status possess nodality which
can be transferred across topics. Our empirical results show that while the
inherent nodality of institutional groups (such as Cabinet Ministers and
backbench MPs) is consistent across topics, the individual (active) nodality
of actors vary with the topic of focus. The methodology described in the paper
can be generalised to any set of topics. For the four topics studied here, we
find that node strength, degree centrality and funnel bandwidth (which is a
measure reflecting the extent to which a node can amplify existing information
in the network) is the best combination of measures to decompose nodality into
individual (active) and institutional (inherent) nodalities.

Our results are further consolidated using a linear model which validates the
independence of the two dimensions of nodality and establishes a relationship
between the collective (mean) nodality of a group of actors and their capacity
to influence the conversation on any given topic. A transfer entropy analysis
is used to measure the influence of a group, in relation to the rest of the
population, on any topic. We find that the Cabinet MPs and the prominent
journalists hold the highest shares of influence in the discussion relating to
the Russia-Ukraine War. Whereas the discussion on the COVID-19 pandemic is led
by the opposition backbench MPs and the prominent journalists (there is
however notably less activity on this topic as compared to other topics
through 2022). This pattern continues in the discussion relating to Brexit
(also with limited activity during 2022), where the conversation is again
dominated by prominent journalists and opposition backbench MPs, with
restrained activity on the topic from Labour and Conservative MPs in prominent
roles. Finally, for the Cost-of-Living Crisis, we observe that the opposition
MPs (both the shadow cabinet as well as the back-benchers) lead the
conversation, highlighting reluctance of the ruling party to engage on this
topic.

The main contribution of this work is to propose a method that measures the
nodality of individual actors and their collective capacity to drive a
conversation. For any given topic, our method measures the capacity of a group
to influence the rest of the studied population, at any given point in time.
In addition, by determining the scores for both individual and institutional
nodality for each member of the group it also offers insights into the
dynamics of influence within the group —enabling an understanding of the
respective contributions of members (and of their role and activity) within
the larger context. This work also enriches our understanding of the role of
policy-actors in policy-related conversations, as it goes beyond studies which
look at the influence of individuals in a conversation only in terms of their
network attributes or positions in the network (i.e. centrality metrics)
[Esteve Del Valle and Borge Bravo, 2018, Dubois and Gaffney, 2014, Riquelme
and González-Cantergiani, 2016, Zhang et al., 2023]. Our study expands on the
literature of politics and journalists on Twitter [Parmelee, 2013, Jungherr,
2014, Mosleh and Rand, 2022, Kim and Park, 2012, Graham et al., 2013, Graham
et al., 2015, Bovet and Makse, 2019, Nożewski and Baranowski, 2021, Shapiro
and Hemphill, 2017, Bright, 2018], introducing the actor’s role in their
institutions and their individual activity on a topic as key elements in
understanding the influence they exert.

Going forward, our methodology could support further research investigating
the activities and influence of politicians on online platforms. For instance,
here we classify _inherent_ nodality as pertaining to institutional status,
but there are other potential drivers of inherent nodality that our approach
might help to identify, such as educational background, based on the
hypothesis that elite actors attending the same private school or university
are likely to be more connected in an online social network (as compared to a
null network) —which might then drive their nodality on the platform,
independent of individual levels of activity. Alternatively, our methodology
might assist the study of different, more negative kinds of attention, such as
hate speech directed at political representatives, where our approach might
help assess which element of the hate directed at ministers relates to their
institutional role, and which to their individual characteristics. This has
been a challenge to examine in the past [Gorrell et al., 2020], given that
ministers are more well-known and hence get more hate, regardless of race or
gender [Agarwal et al., 2021]. Likewise, for journalists, who are increasingly
suffering threats of violence online, such an approach might help determine
whether the hate they receive relates to the publications they work for, or to
their historic activity on policy-topics such as gender.

Finally, there are some limitations to this work. First, our elite networks
include only British journalists and MPs. This is a limited set of individual
actors, albeit an important one in terms of nodes that have both individual
and institutional nodality, and who play an important role in setting the
policy agenda. Second, we have covered only four policy topics, and future
work should expand the range to further validate the analysis. In this way we
might identify larger domains, such as defence and security (rather than the
Ukraine war as focused on here), which could allow us to say something
generalisable about elite networks in particular domains. Third, our results
represent an analysis of the volume of activity – not the content. To tackle
some of the methodological extensions noted above, it could be important to
focus on content, such as different types of hate speech by targeted group,
which have been shown to offer distinct challenges for classification and
detection. Fourth, we have looked at only one specific social media platform -
Twitter, now X, a platform currently experiencing a
decline111111https://variety.com/2023/digital/news/musk-twitter-x-acquisition-
one-year-user-revenue-decline-1235770297/, which may not be possible to
analyse in this way in the future, due to the lack of data and also lack of
universality (over the last years, virtually all politicians and journalists
have maintained official Twitter accounts, but with the increasing uncertainty
over X, they have started to transfer their activity to Instagram or Threads).
It will be increasingly important therefore, to conduct such research on
different platforms. Each one of the four limitations above could be tackled
in future work: the expansion of types of actors, topics and online platforms.

##  4 Methods

###  4.1 Data set

Our database comprises of tweets published by British MPs and journalists
between January 14, 2022 and January 14, 2023, collected using the Twitter
API. Of the 650 House of Commons MPs, we find that 581 MPs have active Twitter
accounts and a total of 580,312 tweets were collected from these users for the
above time frame. We obtain a list of 9594 journalists from journalism.co.uk
which lists journalists in the UK along with their Twitter accounts. Of these,
we were able to retain information on 8606 journalists, and a total of about
5M tweets were collected over the given time period.

###  4.2 Dimensions of nodality

In [Petricek et al., 2006, Margetts and Escher, 2006], the authors present a
qualitative analysis of multiple dimensions of nodality (for instance,
extroversion, visibility of government websites in different countries) using
network metrics such as centrality measures. Here we extend this analysis by
proposing a methodology to quantify nodality on information networks, in
addition to extending the dimensions of nodality studied in [Margetts and
John, 2023]. We consider not only the extroversion and visibility of nodes in
the information network, but also the within-node importance, the engagement a
node generates, its authority, its capacity to transmit information, and its
ability to amplify messages in the network. These additional dimensions are
each paired with centrality measures, such as degree centrality, eigenvalue
centrality, betweenness centrality, strength, and the authority and hubs
measurements [Kleinberg, 1999]. These measures have been extensively used in
the literature to gauge node influence in networks [Anger and Kittl, 2011,
Bakshy et al., 2011, Mei et al., 2015, Riquelme and González-Cantergiani,
2016, Essaidi et al., 2020, Rezaie et al., 2020, Zhang et al., 2023]. A
description of each dimension of nodality and how it applies to our research
can be read in Table 4.5.

We also introduce two new measures which focus on the broad audience reached
by individual actors in the network. These measures incorporate the number of
followers of each node, also referred to as a measure of popularity of a node
[Riquelme and González-Cantergiani, 2016], allowing us to capture the reach of
a node in the network. These are:

$\displaystyle\nu^{i}=\sum_{h}\frac{f^{i}w_{hi}}{\langle
k_{\mathrm{in}}\rangle^{*}}\hskip 10.0pt\mathrm{and}\hskip
10.0pt\mu^{i}=\sum_{j}\frac{f^{j}w_{ij}}{\langle
k_{\mathrm{out}}\rangle^{*}}.$ (1)

$f^{i}$ represents the number of followers of node $i$, $w_{ij}$ is the weight
of link $i\to j$, thus the number of times node $j$ interacts with information
from $i$, and $\langle k_{\mathrm{in/out}}\rangle^{*}$ is the weighted in/out-
degree average of the network.

While $\nu^{i}$ is interpreted as the extensive exposure that a node $i$ gives
to other nodes through interactions, $\mu^{i}$ represents the exposure that
nodes interacting with node $i$ provide by virtue of their own follower count.
We call these two measures funnel bandwidth and amplification bandwidth
respectively. Unlike the initial six measures that solely integrate structural
information from the network, these two additional metrics incorporate
Twitter-specific properties into the network analysis.

###  4.3 Active and inherent nodality through Principal Component Analysis

We perform a PCA [Jolliffe and Cadima, 2016] with different tests and checks
to ensure that the resulting first two components directly correspond to the
inherent and the active nodalities. At a higher level, this series of steps to
obtain individual scores related to the active and inherent nodalities of
users can be described as follows:

  1. 1.

The PCA is performed for each topic using the measures presented in Table 4.5
for the topic-specific network and the null network. By definition, users have
inherent nodality across both networks (as it should be independent of
topics), while their active nodality exists only on the topic-specific
network.

  2. 2.

However, given the measures presented in Table 4.5, we understand that we are
only studying a subset of conversations. Thus, to find the best combination of
measures for our subset of conversations, we perform different PCAs with
different offsets for each one of the features. We analyse all different
combinations for the 8 measures, thus performing
$\sum_{k=0}^{7}\binom{8}{k}=255$ PCAs per topic network, where $k$ is the
number of offsets and 8 refers to the total number of features we have.

  3. 3.

We filter the combinations of features that allow us to capture the inherent
and active nodalities along the principal components. To achieve this, we
perform a test on the first two eigenvectors of each PCA transformation. We
only retain those feature combinations where (1) the first eigenvector from
the PCA transformation has the same sign for all features across both
networks, and (2) the signs of the entries for the null network in the second
eigenvector are opposite to those of the topic-specific network. As an
illustrative example, if suppose, we perform a PCA with any 3 features of the
ones listed in Table 4.5, we obtain the related eigenvectors from the PCA
transformation as
$\boldsymbol{e}=(f_{1}^{t},f_{2}^{t},f_{3}^{t},f_{1}^{0},f_{2}^{0},f_{3}^{0})$,
where $f_{n}$ refers to any of the three features from Table 4.5 and the
superscript $t/0$ refers to the network it belongs to (topic-specific/null).
Representing the signs of the entries of the eigenvectors as 1 and -1, two
eigenvectors passing our test would look like
$\boldsymbol{e_{1}}=(1,1,1,1,1,1)$ and $\boldsymbol{e_{2}}=(-1,-1,-1,1,1,1)$.
For this particular example, $\boldsymbol{e_{1}}$ captures correlated features
across the topic and the null network along the first principal component,
while $\boldsymbol{e_{2}}$ extracts the features that are related to the
topic-specific network.

  4. 4.

Following our hypothesis of two types of nodality, from the results of our PCA
we can distinguish between three types of users: those having high inherent
nodality, those having average inherent nodality, and having less than average
inherent nodality. Following this assumption, we perform a $k$-means++ [Arthur
and Vassilvitskii, 2007] algorithm with $k=3$, each cluster containing one of
the three types of users just described. The robustness of the clusters is
then proven by a number ($M=50$) of repetition to then form stable clusters.
This enables us to eliminate the bias given by the initial position of the
centroids [Kanungo et al., 2004, Nielsen and Nock, 2015, Bahmani et al.,
2012].

  5. 5.

Once having the three clusters for a given PCA, we are now in search of the
best combination for the four different topics. From out of these three
clusters, we could expect, following our definition, that the high-inherent
cluster and the low-inherent nodal cluster share a high number of nodes
throughout the four topics. Thus, as our last check to pick the best
combination of features for our set of four conversations, we select the one
that maximises the number of nodes shared in each respective cluster between
the four topics.

###  4.4 Influence score and Transfer Entropy

Transfer entropy [Schreiber, 2000] is typically interpreted as the shared
information between X’s past and Y’s present, while also taking into account
Y’s past. In [James et al., 2016], the authors highlight instances where
Transfer Entropy can overestimate information flow and underestimate influence
between two timeseries. Keeping this in mind, and to avoid any such errors we
first determine the transfer entropy from $X$ to $Y$, $T_{X\to Y}$ as a share
of the entropy of $Y$ at time $t$ and time delay $k=1$ and then compute the
difference between the transfer entropy from $X$ to $Y$ and the transfer
entropy from $Y$ to $X$, to estimate the net influence score such as in:

$\varphi(X,Y,t,k=1)=\frac{T_{X\to Y}(t,k=1)}{H_{Y}(t)}-\frac{T_{Y\to
X}(t,k=1)}{H_{X}(t)},$ (2)

where $H_{X}(t)$ is the entropy of $X$ at time $t$. The influence score
$\varphi$ at time $t$ gives the net share of information between $X$ and $Y$
at time $t$. $\varphi$ lies on the interval [-1, 1], where -1 refers to $Y$
causing the total entropy of $X$ at time $t$ and 1 represents the opposite.
The transfer entropy $T_{X\to Y}$ is computed in the same way as originally
presented in [Schreiber, 2000].

In our study, we measure the influence score of actors using their activity
time series. The “shared information” here is related to the frequency of user
activity on Twitter around a particular topic and therefore, the TE analysis
it helps us measure the influence that a user has on someone else’s activity
on the given topic. More details on how the influence score can be read in
Appendix 5.3.

###  4.5 Relation between influence score and active and inherent nodality

We are interested in determining the relationship between the capacity of
actors to influence conversations and the inherent and active nodalities of
these actors. We are inspired by [Margetts and John, 2023], who define “nodal
effectors” as a type of nodal actors who assumes a central position in an
information network and enable the dissemination of knowledge in the network,
while also influencing or nudging the behaviours of listeners. Here we think
of MPs and journalists as nodal effectors who can similarly influence the
extent to which other agents in the network engage in a conversation related
to a particular topic.

We use the methodology described in 4.4 to determine the influence scores of
actors at the group level (such as the Cabinet and the Shadow Cabinet) and we
relate it to the individual nodality scores obtained using the PCA in the
following way:

$\varphi(G,\neg G,t)=\mathrm{constant}+\gamma t+\sum_{i\in
G}\alpha_{\mathrm{inherent}}\mathrm{PC}1_{i}^{t}+\alpha_{\mathrm{active}}\mathrm{PC}2_{i}^{t}+\beta\mathrm{PC}1_{i}^{t}\times\mathrm{PC}2_{i}^{t},$
(3)

where $G$ refers to a given group, which in our case can be the Cabinet, the
Shadow Cabinet, etc. $\neg G$ refers to the complement of $G$, thus all nodes
not belonging to $G$. PC1 and PC2 refer to the first two components of the
resulting PCA analysis outlined in Section 4.3. $\alpha_{\mathrm{inherent}}$,
$\alpha_{\mathrm{active}}$ and $\beta$ are three independent coefficients. The
term PC1$\times$PC2 is included to capture any correlations between PC1 and
PC2. We also include a temporal term $\gamma t$ to capture any correlation
between the influence and time.

We fit Eq. (3) with the values of $\varphi$ for different groups and times. To
obtain temporal values from the PCA, we create snapshots of networks
constructed over shorter time-frames that use the same time period as the TE
to compute the influence scores. Here TE computes the measure of influence the
activity of a group (over a time-period of 2 weeks [t-1,t]) exerts on the
activity of the rest of the population at (t+1) week. The networks for the PCA
analysis are similarly generated using tweet data during the weeks (t-1) and
t, to obtain the nodality scores.

The centrality measures used to study different aspects of nodality and their interpretation. |  |
---|---|---
Network measures |  Interpretation | Indicator of  c c c c c c c c c c c c c c c Degree centralityc c | A high degree centrality is obtained when a node interacts with a high number of actors within the conversation network. This contrasts with a low degree centrality related to a “shy” actor not interacting with many others. | Extroversion
Eigenvalue centrality |  A high eigenvalue centrality is related to the probability that information will pass through a particular node. In this sense, a node with a respective high eigenvalue centrality is interpreted as a node whose activity will most probably be seen in the conversation network by the other actors. | Visibility
Betweenness centrality |  Betweenness centrality is related to the likelihood that information will pass through a particular node. Thus, an actor who is important to deliver a message in the conversation network is interpreted to be one with a high betweenness centrality. |  Within-nodes importance
Strength |  The strength of a node takes into account the number of links with other nodes and the weights associated with them. Thus, a node with a small number of ties but with high weights can be as “strong” as one with a high number of ties with low weights. |  Engagement
Authority (HITS algorithm) |  A high authority score reflects, in our case, a node that has a high number of out-degree links, thus having a high number of other nodes interacting with it. The algorithm reflects in this sense the authority of the node in the network [Kleinberg, 1999]. |  Authority
Hubs (HITS algorithm) |  A high hubs score reflects, in contrast with the authority score, a node that has a high number of in-degree links, thus reflecting the node interacting with a high number of other nodes. The algorithm reflects in this sense how much the node acts as a hub of information within the network [Kleinberg, 1999]. |  Information transmission
Funnel bandwidth $\nu$ |  This term reflects how much amplification can a node give to other’s messages when interacting with them. By having an important number of followers, a node can give others visibility by interacting with them. The more interactions, the more visibility a node gives to others. The former is then acting as a funnel of information |  Information transmission
Amplification bandwidth $\mu$ |  The amplification bandwidth relates to who interacts with a given node. The more nodes interacting with you, the more frequent those interactions are, and the more followers your interlocutors have, the more amplified your activity will be. | Amplification

##### Acknowledgments.

The authors would like to thank Angus R. Williams for providing the first list
of Twitter handles associated with MPs. The authors would also like to thank
Eirini Koutsouroupa for her support and effort to get this work published.

##### Funding Statement.

This work was supported by the Public Policy Programme of The Alan Turing
Institute under the EPSRC grant EP/N510129/1. This work was supported by the
Ecosystem Leadership Award under the EPSRC Grant EP/X03870X/1 & The Alan
Turing Institute.

##### Data Availability Statement.

Our code and experimental results are available in our GitHub repository:
https://github.com/LeonardoCastro/Nodality.

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##  5 Appendix

###  5.1 Weak supervision classifier

We build a classifier using a weak-supervision approach. This allows us to go
beyond the keyword search and build heuristic rules related to a particular
topic. The basis of the classifier consists of a set of labelling functions
(LFs) and a probabilistic model that works on the labelled data obtained from
applying the prior functions to the unlabelled data set. LFs are rule-based
functions that assign labels based on specific criteria. These criteria
encompass diverse elements such as keywords, regular expressions, Named-Entity
Recognition (NER) systems, and NLP features like polarity and subjectivity.
This diverse set of criteria results in topic-specific LFs tailored to the
nuances of each topic, surpassing simple keyword searches.

Weak supervision has found application in various contexts such as medicine
and health sciences [Fries et al., 2021, Silva et al., 2022], chemistry
[Mallory et al., 2020], detection of fake news [Shu et al., 2021], and more
generally to study sentiment analysis [Jain, 2021], and ontology and
computational linguistics problems [Maresca et al., 2021, Berger and
Goldstein, 2021]. In the present section we present the two main elements of
this classifier: the Labelling Functions -or heuristic rules- used to build
the classifier (Table 4) and the confusion matrix (Table 3) obtained when
testing it.

If for a particular classification exercise, there are $m$ labelling functions
and $n$ unlabeled data points, then the process of applying all labelling
functions to the entire data set will result in an $n\times m$ matrix
$\boldsymbol{M}$ where each entry reflects how a given labelling function
acted on a data point. By having a training data set with golden labels
$\boldsymbol{Y}$, the probabilistic LabelModel [Ratner et al., 2019, Castro-
Gonzalez et al., 2024] computes the conditional probability of labeling
functions given the matrix $\boldsymbol{M}$,
$P(LF|\boldsymbol{M},\boldsymbol{Y})$. The latter probabilities are used as
parameters to then obtain a final label for each data point.

In our case, we build a collection of labelling functions for each one of the
four studied topics. A detailed list of them is found in the Appendix 5 and in
our GitHub repository121212https://github.com/LeonardoCastro/Nodality. With
the purpose of illustrating how labelling functions are created, we give two
examples of pseudo LFs for the topic of the Cost of Living Crisis:

$\displaystyle\mathrm{LF1:}$ $\displaystyle\ \texttt{label CRISIS if any
keyword in keywords in datapoint else ABSTAIN}.$ $\displaystyle\mathrm{LF2:}$
label CRISIS if ‘(rising|inflation|rise)\W+(?:\w+\W+)0,6?(inflation|is
rising)’ in datapoint else ABSTAIN

While LF1 refers to a normal keyword search, LF2 refers to a more complex
regular expression regarding the rise of inflation. LF2 allows us to look for
different forms of the same sentence “The inflation is rising”. This
flexibility goes beyond the keyword search to label tweets that, without
mentioning specific keywords or explicitly talking about the topic, lay down
within the context of the topic itself.

#### Confusion matrix of the classifier

Table 3: Confusion matrix of the classifier built. The size of the test data
sample is $N=598$. Actual Topic
---
Predicted Topic |  | Other | Ukraine | COVID-19 | Cost of Living Crisis | Brexit
Other | 8.99e-1 | 1.65e-2 | 2.27e-2 | 5.98e-2 | 2.00e-3
Ukraine | 0.00 | 1.00 | 0.00 | 0.00 | 0.00
COVID-19 | 0.00 | 0.00 | 1.00 | 0.00 | 0.00
Cost of Living Crisis | 0.00 | 0.04 | 0.00 | 0.96 | 0.00
Brexit | 0.00 | 0.00 | 0.25 | 0.00 | 0.75

#### Labeling Functions

Table 4: A description of the labelling functions used for all classification tasks. The full list of keywords and regular expressions can be found in our GitHub repository. Topic | Type |  Explanation
---|---|---
Ukraine | Keyword search |  If any keywords belonging to the “Russia-Ukraine War” topic, the classifier labels the text as talking about the particular topic. The keywords include “ukraine”, “russia”, “putin”, “zelensk”, etc.
COVID | Keyword search |  If any keywords belonging to the “COVID-19” topic, the classifier labels the text as talking about the particular topic. The keywords include “covid”, “vaccine”, “sars-cov-2”, “pandemic”, etc.
COVID | Regex |  Regular expressions related to the “COVID-19” topic are searched. If any is found, then the classifier labels the text as talking about the topic. The only regular expression in this case is “vaccin$\backslash$W+ rollout’.
Cost of Living Crisis | Keyword search |  If any keywords belonging to the “Cost of Living Crisis” topic, the classifier labels the text as talking about the particular topic. The keywords include “cost-of-living”, “inflation”, “interest rate”, “windfall”, etc.
Cost of Living Crisis | Regex |  Regular expressions related to the “Cost of Living Crisis” topic are searched. If any is found, then the classifier labels the text as talking about the topic. Regular expressions include “cost$\backslash$W+of$\backslash$W+living”, “(rising|increasing|historic|high) inflation”.
Brexit | Keyword search |  If any keywords belonging to the “Brexit” topic, the classifier labels the text as talking about the particular topic. The keywords include “brexit”, “article 50”, “northern ireland protocol”, “european union”, etc.

###  5.2 Results

####  5.2.1 Granger causality analysis

To have an overall perspective of the daily activities of the MPs and the
journalists, in Figure 5 we plot the daily activities for the four studied
topics in a logarithmic scale. Notice how, in the case of Ukraine (Figure
5(a)), the daily activity increases an entire order of magnitude. The peak of
activity in this case (in day 43), coincides with the start of the Russo-
Ukrainian conflict.

(a)

(b)

(c)

(d)

Figure 5: Activity time series of journalists and MPs on Twitter around the
four studied topics.

Given the apparent synchronicity between the activities of journalists and MPs
in the Ukraine conversation, we perform a Granger causality test to have a
first idea of which group is driving the conversation in each one of the four
topics. We test for lags going from 1 to 7, thus taking into account the
activity of one day before, two days before, …, and until an entire week’s
activity is considered in the test. Results are as follows:

  * •

Ukraine: The activity of journalists is statistically significant with a
p-value $<0.01$ (s.s.) from a lag of 1 day to a lag of 7 days to Granger-cause
MPs’ activities. MPs activity of 6 and 7 days before are s.s. to Granger-cause
journalist’s activity.

  * •

COVID: The activity of journalists is s.s. from a lag of 1 day to a lag of 7
days to Granger-cause MPs’ activities. The activity of MPs is not s.s. to
Granger cause journalist’s activity.

  * •

Cost of Living: Journalist’s activities during the prior first 4 days are s.s.
with p $<0.05$ to Granger cause MP’s activity. The latter’s activities from
day 2 to day 7 are s.s. with p $<0.01$.

  * •

Brexit: The activity of journalists is s.s. from a lag of 1 day to a lag of 7
days to Granger-cause MPs’ activities. The activity of MPs on the 7th day
before is s.s. with p-value $<0.05$ to Granger cause journalist’s activity.

Only for the COVID conversation we obtain a clear result of journalists’
Granger causing MPs’ activity on Twitter. This is an interesting result given
that during the pandemic, governments around the world were the main source of
information. For the other three conversations (Ukraine, Cost of Living and
Brexit), we obtain feedback between the two groups, thus not allowing us to
have a clear view of who is Granger causing the other one.

Given the above results, we can move into a deeper level of analysis to
understand if there are particular actors who are more influential than
others. By doing so we can stop talking about MPs and journalists as two big
groups and focus on which are the most important actors within these two
categories.

###  5.3 Transfer entropy and influence score

Since its first introduction in [Schreiber, 2000], transfer entropy has been
used to measure the transfer of information from one event to another.
Following the Python-like notation presented in [James et al., 2016], where
$x_{t-k:t}=(x_{t-k},\ldots,x_{t-1})$, for any two time series $\\{x_{t}\\}$
and $\\{y_{t}\\}$, we can compute the information flow from one to another by
computing

$T_{X\to Y}(t,k)=\sum
p(y_{t},y_{t-k:t},x_{t-k:t})\log\Big{(}\frac{p(y_{t}|x_{t-k:t},y_{t-k:t})}{p(y_{t}|y_{t-k:t})}\Big{)}.$
(4)

In this case, we are using the original expression as when first introduced in
[Schreiber, 2000].

As explained in Section 4.4, we compute the influence score from $X$ to $Y$ at
time $t$ and time delay $k=1$ as

$\varphi(X,Y,t,k=1)=\frac{T_{X\to Y}(t,k=1)}{H_{Y}(t)}-\frac{T_{Y\to
X}(t,k=1)}{H_{X}(t)},$ (5)

with $H_{X}(t)$ the entropy of $X$ at time $t$.

We computationally measure the entropy of time series $\\{x_{t}\\}$ and the
transfer entropy of $\\{x_{t}\\}$ on $\\{y_{t}\\}$ inspired on the work of
[Shorten et al., 2021], sampling on the activity time series of the two given
groups to then compute an average transfer entropy. For more details, please
refer to our Github
repository131313https://github.com/LeonardoCastro/Nodality/blob/main/notebooks/helper_functions.py.

A schematic of how the latter influence score is computed can be seen in
Figure 6.

Figure 6: The different steps to obtain the influence score for a given group.
(a)-(b) The share of entropy time series obtained from our Transfer Entropy
analysis for the Top 10% most followed journalists and the Cabinet, for the
Russia-Ukraine War. (c)-(d) The resulting influence score over time obtained
for the same two groups and for the same topic.