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Citation: Tantos, A.; Kosmidis, K.
From Discourse Relations to Network
Edges: A Network Theory Approach
to Discourse Analysis. Appl. Sci. 2023,
13, 6902. https://doi.org/10.3390/
app13126902
Academic Editor: Kuei-Hu Chang
Received: 13 April 2023
Revised: 24 May 2023
Accepted: 25 May 2023
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applied
sciences
Article
From Discourse Relations to Network Edges: A Network
Theory Approach to Discourse Analysis
Alexandros Tantos 1,*,† and Kosmas Kosmidis 2,†
1Department of Philology, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
2Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece; kosmask@auth.gr
*Correspondence: alextantos@lit.auth.gr
† These authors contributed equally to this work.
Abstract: In this paper, we argue that discourse representations can be mapped to networks and
analyzed by tools provided in network theory so that deep properties of discourse structure are re-
vealed. Two discourse-annotated corpora, C58 and STAC, that belong to different discourse types
and languages were compared and analyzed. Various key network indices were used for the dis-
course representations of both corpora and show the different network profiles of the two discourse
types. Moreover, both network motifs and antimotifs were discovered for the discourse networks in
the two corpora that shed light on strong tendencies in building or avoiding to build discourse re-
lations between utterances for permissible three-node discourse subgraphs. These results may lead
to new types of discourse structure rules that draw on the properties of the networks that lie behind
discourse representation. Another important aspect is that the second version of the STAC corpus,
which includes nonlinguistic discourse units and their relations, exhibits similar trends in terms of
network subgraphs compared to its first version. This suggests that the nonlinguistic context has a
significant impact on discourse structure.
Keywords: discourse structure; discourse relations; network analysis; network motifs; network
edges
1. Introduction
One of the main research goals in computational theories of discourse structure has
been to reveal whether there are specific discourse coherence patterns followed by speak-
ers in single-author texts, monologues and dialogues that could be exploited for building
NLP applications ([1–6]). Being able to answer this question allows to also address another
question of bigger theoretical depth and importance, namely how coherence and cohesion
are realized in the discourse. Exploring discourse relations remains a topic of growing
interest, and several rather sophisticated approaches have recently been proposed [7–12].
The current trend in attempting to answer this and similar questions in NLP in gen-
eral has been to train Transformer models ([13–15]) and construct the so-called probing
tasks to indirectly obtain a deeper understanding of discourse coherence and cohesion.
However, it still does not seem feasible to extract interpretable semantic properties of the
discourse out of these deep learning models, and one can still use some valuable lessons
from the formal discourse semantic tradition in order to achieve this ([16–25]).
In this paper, we intend to focus on three different types of recorded
discourse-annotated data spanning two languages, English and Greek: (a) the single-
author written texts of two daily newspapers with large circulation in Greece that were
included in the corpus C58 ([26]) and (b) the multiparty dialogue texts of the STAC cor-
pus annotated in two stages, resulting in two discourse-annotated versions; (i) the first
version includes chat logs (or chat moves) recorded in a virtual environment during an
online game session ([27–29], and (ii) the second version includes the same multiparty
Appl. Sci. 2023,13, 6902. https://doi.org/10.3390/app13126902 https://www.mdpi.com/journal/applsci
Appl. Sci. 2023,13, 6902 2 of 23
dialogues accompanied by the messages automatically generated by the game software.
These messages describe the nonlinguistic events that take place during a game session
and thus situate the linguistic utterances in the broader nonlinguistic context of usage.
The availability of discourse-annotated corpora allows us to use quantitative meth-
ods to pin down the essence of discourse coherence and cohesion but also to profit from
the numerous tools provided by the mathematical field of network analysis ([30–32]). Ex-
isting formal discourse theories have been providing the formal means to construct dis-
course representations that can be mapped to networks that reflect the information flow
of discourse and reveal hidden properties of the discourse. Network edges play a cru-
cial role in a discourse dialogue network. They represent the connections or relationships
between different discourse units, such as utterances or events, within the network. The
importance of network edges lies in their ability to capture the flow and dynamics of dis-
course interactions.
Most notably, network edges provide a pathway for the exchange of information and
meaning between discourse units. They indicate how different units are connected and
how information is transmitted within the discourse. By following the network edges, one
can trace the progression of ideas, arguments, or conversations in the discourse dialogue.
The discovery of significant network patterns in the discourse representations of the
corpora would suggest that the speakers engage in discourse by following implicit strate-
gies, and this might mean that speech acts are related in predictable ways. Moreover,
traceable differences and commonalities in the network patterns of single-author written
texts and multiparty dialogue texts, as well as between different languages, would help
formulate theoretically driven corollaries related to different discourse types.
Based on two corpora and the three datasets that accompany them, the network anal-
ysis that we provide gives important and counter-intuitive insights to speakers’ prefer-
ences in constructing and interpreting discourse structure. Section 2describes the prin-
ciples behind the types of discourse units and discourse relations adopted in the two
corpora, C58 and STAC. Section 3offers a brief overview of the compilation and anno-
tation process of C58 and STAC. Section 4provides a detailed analysis of (a) the mapping
between discourse representations and networks, (b) the profile of discourse representa-
tions through key network indices and (c) the presence of and antimotifs for three-node
subgraphs for the discourse networks of all three datasets. The last section, Section 6,
sums up the findings of this study and offers a series of theoretically driven reflections
related to the types of restrictions on discourse inference and interpretation imposed by
the discourse structure.
2. Discourse Relations and Discourse Units
There are numerous formal theories of discourse representation in the literature, and
corpora have been annotated for discourse structure based on the principles of these the-
ories (Segmented Discourse Representation Theory (SDRT; [21,23], Rhetorical Structure
Theory (RST; [33]), the Linguistic Discourse Model (LDM; [34]), the Discourse Graphbank
model ([35]) and Discourse Lexicalized Tree Adjoining Grammar (DLTAG; [36]))). Briefly,
Segmented Discourse Representation Theory (SDRT) is a dynamic semantic theory of dis-
course interpretation that focuses on the interplay between discourse interpretation and
discourse coherence. It is an extension of Discourse Representation Theory (DRT), which
is a framework for exploring meaning under a formal semantics approach. In SDRT, a
text is segmented into constituents related to each other by means of rhetorical relations,
resulting in a structure known as a segmented discourse representation structure (SDRS).
Among the above approaches, SDRT offers a well-principled model of discourse in-
ference and interpretation and has also been adopted as the annotation model for the
annotation of the STAC corpus as well as the C58 corpus. The availability of both corpora
serves the process of stressing the differences and similarities between the three different
types of recorded discourse data: single-author written texts, nonsituated and situated di-
alogues. Additionally, SDRSs, the discourse representations that SDRT adopts and that is
Appl. Sci. 2023,13, 6902 3 of 23
analyzed in the rest of the paper, can be mapped to non-tree-like graphs, contrary to the
above-mentioned discourse theories.
Although SDRT has been chosen as the basis of the current work, all discourse the-
ories are more or less aligned as to how to proceed in annotating linguistic utterances.
The basic premise is that discourse structure is hierarchically organized around discourse
units, the elementary discourse units or EDUs in SDRT parlance. However, SDRT includes
also Complex Discourse Units or CDUs to cope with semantic groupings of EDUs in the
discourse and, recently, refs. [28,29,37] extended SDRT’s classic definition of a discourse
graph with a third type of node, the elementary event units or EEUs, so that nonlinguistic
events are considered as nodes during the construction of the discourse graph. Here is the
definition of a discourse graph by [28]:
Definition 1. A discourse graph Gis a tuple (V, E1, E2, ℓ, Last), where V is a set of nodes
(EDUs, EEUs, and CDUs); E1, a set of edges representing discourse relations; E2, a set of edges re-
lating each Complex Discourse Unit (CDU) to its sub-units; ℓ, a labelling function from elements
of E1 to discourse relation types; and Last, a label for the last unit in Vrelative to textual order.
2.1. Elementary Discourse Units
The EDUs are linguistic utterance nodes on the discourse graph and are considered
the basic building blocks of a discourse representation. The intricate semantic and prag-
matic manner that the EDUs are related is expressed through a number of discourse rela-
tions, such as Narration,Elaboration,Explanation,Result and Narration.
All the above-mentioned discourse theories have a different theoretical standpoint re-
garding both the definition criteria of discourse relations and the set of discourse relations
that characterize a coherent text. However, for the purposes of this paper, these heated
theoretically driven debates can be safely ignored, since the main target is to unveil deep
properties of the discourse structure viewed as a network and not on the type of discourse
relations that create them.
Moreover, all discourse theories acknowledge the need of a discourse segmentation
algorithm that tokenizes the discourse into EDUs, and they are more or less aligned re-
garding the segmentation criteria.
The above-mentioned theoretical differences aside, another important classification
axis of discourse relations pertains to whether a discourse relation should be considered
as coordinating or subordinating ([38]). Briefly, coordinating relations, such as Narration
and Continuation, relate EDUs that share a general common topic and can be thought of as
providing information on the same level of detail on that topic, while subordinating rela-
tions, such as Elaboration and Explanation between two EDUs are asymmetrically related,
since one of the two plays a subordinate role relative to the other. Ref. [38] argues that it
is not an easy task to classify discourse relations as either coordinating or subordinating
and presents linguistic tests that help decide which discourse relations are subordinating
and which are coordinating in a given context.
The relevant part of the distinction between coordinating and subordinating relations
for the construction of the discourse representation is that the first type of relation is in-
dicated through horizontal connections, as in the simple graph in Figure 1(1), for the
coordinating relation Narration between π1and π2and the second type through vertical
ones, as in Figure 2for the subordinating relation Explanation between π1and π2in (1).
1. π1Yesterday evening, John had a great meal and
π2won a dancing competition.
π1Continuation
//π2
Figure 1. Coordinating connection between π1and π2.
2. π1John fell.
π2Max pushed him.
Appl. Sci. 2023,13, 6902 4 of 23
π1
Ex pl anation
π2
Figure 2. Subordinating connection between π1and π2.
Subordinating and Coordinating Relations
Although the classification of discourse relations as subordinating or coordinating
has important semantic effects in discourse interpretation, reflected in discourse anaphora
and attachment availability ([39]), it is highly context-dependent, as pointed out by [38,40],
which suggests that the contexts that they appear in and the conditions that dictate their
classification as either coordinating or subordinating should be further investigated.
As [38] mentions, a discourse relation does not share common underlying content with
a homogeneous set of other discourse relations that can be assigned to either the coordi-
nating or the subordinating type. Essentially, the two types of discourse relations reveal
an important aspect of the core of discourse structure related to the way speakers interpret
the information in the discourse; this aspect is often called information packaging in the
SDRT tradition.
However, beyond the linguistic contexts that need to be studied further so that the
linguistic classification tests suggested by [38] can be confirmed and/or refined, this paper
aims to abstract away from the distinction between subordinating and coordinating rela-
tions and to inspect deeper properties of discourse evolvement that would probably play
a role in the classification of discourse relations. We claim that the study of network motifs
and other key network indices is the formal means to reveal these properties and, then, the
distinction between subordinating and coordinating relations may be examined further in
light of the findings in this paper. The most important feature of discourse representations
that can be exploited to identify network properties is the presence of directionality in the
discourse edges, as is made clear in Section 4.
SDRT’s formalism supports directed acyclic graphs (DAGs), since, following SDRT’s
principled method of building graph representations of discourse, every EDU contributes
to the information flow of discourse, and its advancement implies that there can be no
loop constructed that could ultimately allow self-relational connections.
2.2. Complex Discourse Units
The second type of node on a discourse graph is the CDU. Following [41], a CDU
represents a semantically coherent group of EDUs that collectively serve as an argument
to a discourse relation between this group and another EDU. Accordingly, SDRT assigns
semantic content and internal structure to CDUs and, moreover, since CDUs are first-class
citizens on the discourse representation level, they can be thought of as complex speech
acts that participate in discourse inference as arguments of discourse relations. Admitting
a second kind of node that has subconstituent nodes inevitably results in acknowledging a
second kind of relation apart from discourse relations, and this is reflected in the definition
of CDUs in [41]:
—Undirected unlabeled edges connect a Complex Constituent to its subconstituents,
introducing recursivity in the structure.
Hence, a Complex Discourse Unit or Complex Constituent is a node of the graph that
has some subconstituents identified by the second kind of edge. We may write α∈πas
shortcut for αis a subconstituent of π.
The classic example introduced in [21], and repeated in Figure 3, may serve as the
basis for distinguishing the two types of relations displayed in the discourse graph of
Figure 3(you can safely ignore the top-most label, π0, which signifies the content that is
there at the beginning of the discourse). The directed edges indicate discourse relations,
whereas the undirected ones represent the relation between a CDU and its subconstituents.
Appl. Sci. 2023,13, 6902 5 of 23
The CDUs in the graph are the primed πs; π′groups the content of π2and π3, and π′ ′ has
π3and π4as its subconstituents.
3. π1John had a great evening last night.
π2He had a great meal.
π3He ate salmon.
π4He devoured lots of cheese.
π5He won a dancing competition.
For the purposes of this paper, we focus on the discourse relations of the graphs
and not on the secondary subconstituency relationship between a CDU and its subcon-
stituents. The way that groupings of EDUs in terms of CDUs may indirectly affect the
discourse structure may, however, be the subject of a separate future study.
π0
π1
Elaboration
π′
r
r
r
r
r
r
r
r
r
r
L
L
L
L
L
L
L
L
L
L
π2Narration
//
Elaboration
π5
π′′
r
r
r
r
r
r
r
r
r
r
L
L
L
L
L
L
L
L
L
L
π3Narration
//π4
Figure 3. The discourse graph of (3).
2.3. Elementary Event Units
The third kind of node admitted in a discourse representation graph with equal status
to EDUs is the EEU. EEUs represent nonlinguistic events in the discourse, especially in
the context of situated dialogues, with little to no descriptive content. By nonlinguistic
events, in the context of situated dialogues, we mean the different moves in the recorded
game. For instance, π2and π3in ref. [28] refer to two moves by the server and the UI
that represent two nonlinguistic events. On the other hand, EDUs represent linguistically
uttered events, namely events that have been explicit realizations, e.g., through a verb
or an event nominal. EEUs can be either related with other EEUs or with other EDUs,
opening the window for relating linguistic with nonlinguistic events.
Although the use of nonlinguistic events in discourse is pervasive, for good reasons,
pragmatically driven theories of discourse have focused mainly on the way that reference
to nonlinguistic events is realized through the use of indexical and demonstrative expres-
sions, ignoring the complex ways that nonlinguistic events affect discourse structure. Con-
trary to linguistic events, due to EEUs’ lack of propositional content, it is challenging for
the interpreters to individuate and conceptualize them. Different interpreters may assign
different content to a nonlinguistic event. However, speakers engage in situated dialogues
on a daily basis, and they fully integrate nonlinguistic events throughout discourse infer-
ence and interpretation. Allowing EEUs to interact with EDUs as well as with CDUs
opens up a way to enrich the discourse graph and study in depth the interaction between
linguistic and nonlinguistic events. Refs. [28,37] offer a formal model of situated dialogue
to capture these effects and the study of STAC. The first discourse-annotated corpus that
includes EEUs by [29] has already pointed out interesting ways that discourse is affected
by the presence of EEUs. For instance, Figure 4, which displays the discourse graph of (4),
taken from [29], shows the direct relation between the CDUs. The labels π1,π2and π3
Appl. Sci. 2023,13, 6902 6 of 23
represent three EEUs generated during an online game session that was recorded during
the compilation of the STAC corpus (for more on the description and the basic features of
the STAC corpus, go to Section 3.1). π4is clearly related to π3with commentary, and the
graph encloses a rich repository of linguistic and nonlinguistic information that explains
in a more complete fashion the current discourse.
Additionally, π2and π3are subconstituents of π′. As [29] points out, CDUs that
have EEUs as their subconstituents, such as π′, serve very often as arguments to discourse
relations and are essential in a situated discourse, since speakers frequently refer back to
grouped nonlinguistic events in various ways.
4. π1william rolled a 6 and a 1 [Server].
π2william will move the robber [UI].
π3william stole a resource from GWFS [Server].
π4oucho [GWFS]
π5you can have it back for some ore. [william]
π1Result
//π′
r
r
r
r
r
r
r
r
r
r
L
L
L
L
L
L
L
L
L
L
π2Result
//π3
Commentary
π4Continuation
//π5
Figure 4. The discourse graph of (4).
Lastly, according to [28], EEUs differ from EDUs in terms of their semantics, due to
the fact that they always denote actual events, and their order matches the order of the
events they describe. However, although one would expect that EEUs always describe a
sequence of events, there is a wide range of relations between EEUs and between EEUs
and EDUs. Ref. [29] describes in detail the frequency distribution of the discourse re-
lations that EEUs are assigned as their arguments, and they include Result,Elaboration,
Correction,Background,Sequence,Continuation and Question–Answer Pair.
3. The Datasets
As mentioned above, we aimed to examine as broad a spectrum of discourses as
possible given the availability of the discourse-annotated corpora (single-author written
texts, chat-only and situated multiparty dialogue texts) spanning two languages, English
and Greek. The following two sections briefly describe the main features, the collection
and the annotation process that have been followed during the compilation of the two
datasets, the STAC corpus and the C58 corpus.
3.1. The STAC Corpus
There are several discourse-annotated corpora with different theoretical outset but
similar concerns and targets, including the Penn Discourse Treebank (PDTB; [42]), the RST
Discourse Treebank (RST- DT; [43]), DISCOR [44], ANNODIS [45] and, more recently, the
STAC corpus ([27]). The largest of them is RST-DT, which is compiled and annotated based
on RST’s annotation principles. However, the STAC corpus ([29]) is the first corpus that
provides discourse structures for multiparty dialogues situated in a virtual environment,
which is the reason why we chose to use it for the comparison of different discourse types.
The corpus was annotated in two stages leading to two subcorpora: the first sub-
corpus includes the annotated chat moves of the players during a game session, and the
second extends the annotated game sessions of the first subcorpus by adding the annota-
tion of nonlinguistic events that were automatically generated by the same game session.
The two subcorpora are offered for direct comparison between situated and nonsituated
discourse analysis, as well as for exploring the interaction between the linguistic and the
Appl. Sci. 2023,13, 6902 7 of 23
nonlinguistic events in the situated subcorpus. In this paper, we follow the convention
of [29], and we refer to the annotations of the first subcorpus as the chat-only annotations
and the annotations of the second subcorpus as the situated annotations.
Each game is divided into dialogues, or subsections, of the game that encompass one
or more bargaining sessions. As mentioned above, EEUs lack propositional content and,
thus, it is already challenging for the interpreters that participate in the dialogue to in-
dividuate and conceptualize them, rendering it almost impossible for the annotators to
construct and include them in the situated corpus. However, the messages that are auto-
matically generated by the game software and shared between the players in a common
virtual environment set the stage for including nonlinguistic events that occur in a con-
trolled and common virtual environment.
As expected, the subcorpus that includes the situated annotations consists of many
more EEUs (31,811) than the chat-only annotated subcorpus (12,588). (All the STAC-
annotated data are stored in tabular format—converted as pickle files—and are available
upon request from the corresponding author.)
3.2. The C58 Corpus
C58 is the first corpus annotated with discourse relations for Greek, based on SDRTs
theoretical framework for the implementation of its annotation scheme, and serves as one
of the first resources for Greek discourse parsing (since one of the main goals of creating
the C58 corpus was to quantitatively approach the interface between intra- and intersen-
tential semantics, it has also been annotated for the allowing verbal aspect and thematic
roles). C58 consists of 58 journalistic texts (more than 1000 annotated DUs), sampled from
two popular Greek newspapers in northern and southern Greece, “Makedonia” and “Ta
nea”, respectively, which belong to the Corpus of Modern Greek Texts, a reference corpus
of Greek compiled by the Center of Greek Language.
The Corpus of Modern Greek Texts includes approximately 7000 and 4500 articles
from each newspaper, accordingly. Ref. [26] decided to collect texts proportionally from
both newspapers in order to neutralize any possible dialectical factors. The initial corpus
of the Center of Greek Language is divided into 56 text genres, essentially covering the
whole extent of the journalistic genre type, ranging from book review to economic news.
The scope of C58 texts was narrowed down to 27 genres, based on the mean size of their
texts, in an attempt to meet the balance and representativeness criteria in the sample.
Ref. [26] describes in detail the compilation process, including the data collection
criteria and the annotation cycle of the corpus undertaken by the team of five linguistics
students, as well as the various challenges that they had to face throughout the process
(all C58 texts have been annotated with the brat annotation tool, and the resulting brat-
exported annotated data are available at http://github.com/atantos/C58).
4. Networks and Connectivity
Networks or graphs are sets of nodes and edges ([46–49]). The nodes of the network
are representations of objects, and the edges represent relations between these objects. In
a social network, for example, nodes represent persons, and two nodes are connected if
these persons know each other. In a biological protein interaction network, nodes are
proteins that are connected if they interact with each other through some biochemical
reaction ([50]). In our case, a discourse representation can be thought of as a network,
since it can be directly mapped to a graph with nodes and edges. The nodes in our net-
works are utterance labels, and these labels are connected if a discourse relation exists
between them. This is markedly different from other linguistic networks studied in the
past ([51–53]), where usually the nodes represent words and the aim is to study relations
between words in a sentence. Most importantly, as mentioned earlier, our networks are di-
rected networks. Notice that while constructing and analyzing the discourse networks of
the two discourse-annotated corpora, we did not include the relation between EDUs and
Appl. Sci. 2023,13, 6902 8 of 23
their subconstituents, since these relations are not considered as relations with the same
impact and role throughout discourse inference and interpretation ([41]).
Networks have become an invaluable tool of applied mathematics due to their versa-
tility in describing so many diverse physical systems using a common framework ([54–61]).
A brief search of the Scopus bibliographic database returns more than 750 articles pub-
lished in 2021 that contain the phrase “Complex Network” in their title.
4.1. Key Network Indices and Their Relevance to Discourse
Graphs can be either undirected or directed. A directed graph, also called a digraph,
is a graph in which the edges have a direction. Pictorially, this is indicated with an arrow
on the edge. If vand ware vertices, an edge is an unordered pair [v,w], while a directed
edge is an ordered pair, (v,w)or (w,v). The directed edge (v,w)is drawn as an arrow
from vto w. Of course, it is possible to treat a digraph as a nondirected network and
still obtain valuable information. Our graphs are unweighted, i.e., all edges are treated as
equal, and contain no multilinks or self-loops.
There are several quantities of interest in the study of networks. Some of them are
meaningful for directed as well as for undirected networks, while others are defined for
directed graphs only. Thus, typically, the study of a directed network starts by treating it
as undirected. When the basic properties of the undirected graph are calculated, then one
proceeds by examining the quantities of interest that are associated with the directionality
of the graph links. This is the approach that we follow here. When we treat discourse
networks as undirected graphs, we are interested in the following quantities:
(a) The number of nodes nsize and the number of edges nedge of the network. In terms
of discourse networks, these two indices indicate the size of the network.
(b) The fraction of edges fe, which is the ratio of the number of edges nedge to the
maximum possible number of edges nemax , i.e.,
fe=nedge
nema x
=nedge
(nsize)(nsize−1)
2
(1)
where we use the fact that the max number of undirected edges in a network of
nsize nodes is nem ax =(nsize)(nsize−1)
2. The fraction feis a number between zero and
one. In terms of discourse graphs, if feis closer to 0, it can be interpreted as an
indicator of low connectivity, while in the opposite case, the discourse graph can
be considered to have a highly connected structure.
(c) The degree centrality of a node is a non-negative integer denoting the number of
edges emanating from a node. In our case, utterance labels with a high node degree
can be interpreted as having high significance for the discourse evolvement. Here,
we study the maximum, minimum and average degree of a network’s nodes kmax ,
kmin and kav , respectively, as well as the standard deviation of the degree distribu-
tion kstd .
(d) The mean clustering coefficient C. The (local) clustering coefficient of a node iin a
network is defined as
C(ki) = ti
[ki(ki−1)/2](2)
where tiis the number of triangles (loops of length 3) attached to this node divided
by the maximum possible number of such loops ([49]). Here, we compute the av-
erage of the local clustering coefficients. It is a number in the range 0 ≤C≤1.
Clustering coefficient values C≃1 indicate that two nodes connected to a third
common node have a high probability of also being connected to each other. So-
cial networks are typically networks with a high clustering coefficient, since it is
probable that individuals that have a common acquaintance know each other as
well ([48,59,62]).
Appl. Sci. 2023,13, 6902 9 of 23
(e) The degree assortativity coefficient r([63]). A network is assortative when nodes
of high degree tend to connect to nodes with high degree. It is disassortative when
nodes of high degree tend to connect to nodes with low degree. The assortativity
coefficient rlies in the range −1≤r≤1, with r=1 indicating perfect assortativity
and r=−1 indicating perfect disassortativity. To be more formal, the assortativity
coefficient ris defined as
r=∑jk jk(ejk −qjqk)
σ2
q
(3)
The term qkis the distribution of the remaining degree, i.e., the number of edges
leaving the node, other than the one that connects the pair. This distribution can
be derived from the degree distribution pkas qk=(k+1)pk+1
∑j≥1jp j([63]). The quantity
ejk represents the joint probability distribution of the remaining degrees of the two
vertices. This quantity is symmetric on a undirected graph and follows the sum
rule ∑jk ejk =1 and ∑jejk =qk.
In Table A1 of the Appendix A.1, we present a table with basic statistics for a number
of real-world networks. It provides, for several real-world networks, numerical estimates
for the important properties of clustering and assortativity. As one can see from Table A1
the clustering coefficient for social networks tends to be of higher value. For example,
the network of film actor collaborations has a C=0.78, while a network of collabora-
tions between physicists has been found to have a C=0.56. These are typical values for
social networks. Technological and biological networks by contrast tend to have some-
what lower values. The power grid network, for instance, has a clustering coefficient of
only about 0.08. A high clustering coefficient for the study of discourse networks may
serve as an important proxy for pinning down discourse coherence and the way that it is
maximized, since it appeals to discourse structures with a higher quality of discourse con-
nectivity. However, to establish such a conclusion would require cross-checking whether
discourse networks with a high clustering coefficient are also ranked as highly coherent
by speakers.
Concerning the assortativity property, one can observe that social networks are in gen-
eral assortative networks (r>0). This is quite well known to sociologists, as people have,
it appears, a strong tendency to associate with others whom they perceive as being similar
to themselves in some way. In sociology, this tendency is called homophily, or assortative
mixing. More rarely, one also encounters disassortative mixing, the tendency for people to
associate with others who are unlike them. Assortative (or disassortative) mixing is also
seen in some nonsocial networks. Papers in a citation network, for instance, tend to cite
other papers in the same field more than they do papers in different fields. Web pages
written in a particular language tend to link to others in the same language. The assorta-
tivity coefficient rthat we study here is a measure of a particular type of homophily, i.e., it
measures the degree of assortativity. One may consider the nodes of a graph as belonging
to two groups. A group of highly connected nodes and a group of poorly connected ones.
The assortativity coefficient rmeasures the tendency of nodes to connect (or not) within
their respective group.
In terms of discourse representation, a highly assortative discourse network may sig-
nal the connectedness between highly important nodes. In the case of a highly assortative
discourse, one may also claim that a few or even a single common topic or line of argu-
mentation is promoted that is supported by several influential utterances in the discourse.
On the other hand, a disassortative discourse network may signal a lack of connected-
ness between highly important nodes that can be further interpreted as a type of mutual
avoidance between nodes with a high degree centrality.
4.2. Network Motifs and Discourse Structure
The most famous nontrivial quantity of interest characterizing directed networks is
the existence or absence of network motifs ([50,64]). The basic premise behind network
Appl. Sci. 2023,13, 6902 10 of 23
motifs is that networks are made of repeating occurrences of simple patterns. Thus, a net-
work is scanned for specific n-node subgraphs (here n=3). We refer to these three-node
subgraphs as patterns and denote them with the letter S, followed by an index number;
see Figure 5. The number of occurrences of these patterns in a graph is compared with the
number of subgraphs found in appropriately randomized networks, and z-scores are cal-
culated so that one can estimate how likely a particular subgraph is to appear by chance.
Z-scores higher than 2 indicate that a particular pattern is frequently present in the actual
network, and it is unlikely that this is due to pure chance, since it is infrequent in the ran-
domized graphs. Such a pattern is then termed a network motif. It is equally important
to locate patterns that are missing from the original network while they appear in the ran-
domized graphs. Such an absence would indicate that there is a mechanism prohibiting
their appearance. We identify them when their associated z-score is less than −2. Such
patterns are termed antimotifs, i.e., patterns that are rarer than at random, and may be
equally important for revealing aspects of the discourse structure. Thus, we must empha-
size the difference between a pattern and a motif. For a pattern to be termed a network
motif, it is required that it appears much more often or rarely than it would, on average,
appear in random networks.
S1: Linear Chain S2: Feed Forward Loop
S3: Cycle S4: Dual Cause S5: Common Cause
Figure 5. The basic patterns for three-node network motifs.
Higher-order patterns (i.e., n≥4) can in principle be studied, but such a task be-
comes prohibitively difficult in practical cases. There are 199 possible 4-node subgraphs,
9364 5-node subgraphs and so on ([65]). Moreover, to count the number of a specific
4-node subgraph occurrences in a graph of 1600 nodes, the (1600
4)combinations of nodes
should be checked. This is a number on the order of 1011, i.e., the computational task
quickly becomes unfeasible. Thus, the vast majority of research is conducted consider-
Appl. Sci. 2023,13, 6902 11 of 23
ing the existence of three-node motifs and has been proven a rather fruitful, especially in
biological networks ([64,66–71]).
Figure 5shows the five three-node subgraphs (patterns) that are possible with two
of three directed edges, while no bidirectional edges are allowed. It was a celebrated
result in biological networks that the feed-forward loop (S2 in Figure 5) is a network motif
of the Transcriptional Regulatory Network (TRN) of the bacterium E. coli ([65]). These
subgraphs cannot not be directly mapped to the discourse graphs based on the annotation
principles set by both C58 and STAC. Recall that discourse graphs admit two different
types of directional edges depending on the type of discourse relation between the two
related nodes, namely coordinating vs subordinating ones, and are drawn horizontally
or vertically, respectively. Moreover, as mentioned in Section 2.1, our network analysis
aims to offer a deep insight to principles of discourse evolvement and to contribute to
the discussion as to what dictates the distinction between coordinating and subordinating
relations.
Between the five possible three-node subgraphs in Figure 5,S3 is excluded, since dis-
course structure graphs are not allowed to include loops that could relate a node back to
itself. Naturally, discourse advancement forbids this type of subgraph and, thus, S3 is
excluded from our analysis (although it is not analyzed, we included it in the graphs in
Section 5and in Table A2 with the mean pattern values; see the discussion below). Before
responding to the question of whether discourse graphs are characterized by one or more
network motifs or antimotifs in Section 5, it is important to realize the types of discourse
subgraphs that are mapped to the different S∗network subgraphs. Notice that Figures 6–9
include only the corresponding discourse graphs that respect the Right Frontier Constraint
(RFC) and not all possible three-node subgraphs that may satisfy the definition of the net-
work subgraph patterns. However, for our purposes, the distinction between coordinating
and subordinating discourse relations does not affect the results of our study (however,
see [72] for certain types of RFC violations related to this distinction).
y
xz
π1Coord
//π2Coord
//π3
π1
Subord
π2
Subord
π3
π1Coord
//π2
Subord
π3
Figure 6. S1: The linear chain pattern and its correspondence to discourse graphs.
S1 represents the uninterrupted sequence of related utterances, and this can be trans-
lated in three different discourse graphs. Figure 6displays the possible three-node sub-
graphs included in a discourse graph, with either subordinating, coordinating or both
types of discourse relations corresponding to the linear chain pattern S1. All three sub-
graphs are expected to be found frequently in single-author written texts, due to coherence
realization in these texts that we expect to promote uninterrupted sequences of related ut-
terance, while the opposite may be expected to occur for the multiparty dialogue texts
of the STAC corpus, namely that the participants frequently interrupt the discourse and
introduce new topics and/or lines of argumentation. All of these conjectures remain to be
confirmed or rejected in Section 5.
Appl. Sci. 2023,13, 6902 12 of 23
y
xz
π1
π2
π3
Figure 7. S2: The feed-forward loop pattern and its correspondence to discourse graphs.
z
xy
π1
π2π3
π1
π2
π3
Figure 8. S4: The dual-cause pattern and its correspondence to discourse graphs.
z
xy
π1
π2
π3
Figure 9. S5: The common-cause pattern and its correspondence to discourse graphs.
S2 corresponds to fully connected discourse subgraphs of three nodes, whereby all
three nodes are related to the other two nodes. In terms of subgraph connectivity, S2 refers
to maximally connected discourse graphs, which in turn would mean that in these three-
node subgraphs, the maximum quality of discourse coherence is served if one admits that
network connectivity is considered a direct and strong indication of discourse coherence.
Note that all discourse semantic theories either tacitly agree with the statement that the
more edges or connections a discourse representation graph has, the more probable it is
that the discourse is coherent, or they integrate it as a principle in their theory, e.g., the
Maximize Discourse Coherence principle (MDC) in SDRT ([73–75]).
The dual-cause pattern in S4 maps to discourse subgraphs in which π3plays a cen-
tral role and could be perceived as labeling an utterance that strengthens the quality of
discourse coherence. Note that the central role of π3we mention here is not identified
with RST’s nucleus–satellite distinction by [33] but is attributed to the fact that π3is re-
lated to both of the following two nodes in the discourse subgraph. Although π3of the
discourse subgraphs that corresponds to S3 is related to the previous two πs, as in S2,
there is no indirect relation to π1through π2in which the distance d(π1,π3)equals 2. The
distance between two nodes A and B is the length of the shortest path between A and B.
The absence of an edge between π1and π2does not mean that they are not related indi-
Appl. Sci. 2023,13, 6902 13 of 23
rectly through an intermediate, though, πbut that the number of intermediate nodes, as
well as the type of discourse relations that intervenes between them is not specified.
Regarding the permissible discourse graph that corresponds to the common cause
pattern S5, π1with two outgoing edges plays a prominent role too, since it is directly
related to both π2and π3, while π2and π3are only indirectly related to each other in the
sense described for S4; namely there is no intermediate node between π2and π3.
Notice that the edges of the discourse networks created for both corpora, C58 and
STAC, are unweighted. In terms of the relevant distinction made by [29] between quasi-
tree-like and truly non-tree-like discourse units, the extracted three-node subgraphs are
not classified based on whether they include quasi-tree-like discourse units or not, since
our network analysis does not require distinguishing subgraphs that include one or more
quasi-tree-like nodes from the truly non-tree-like nodes (i.e., subgraphs that have node(s)
with a degree of 2 or higher). Essentially, what this means is that 3-node subgraphs that
consist of entirely non-tree-like nodes include only one discourse relation between any
two nodes.
Therefore, there is no weight assigned to three-node subgraphs that are more strongly
connected in terms of the number of discourse relations that relate any pair of the three
nodes or in terms of the degree of the nodes.
5. Interrogating the C58 and the STAC Networks
Figure 10 shows, in the form of histograms, the distribution of the basic properties of
55 discourse networks derived from Greek newspaper articles that were included in C58.
The largest of these networks contains 69 nodes, while the smallest contains only 4 nodes
(we left out 3 of the original 58 texts included in C58, since they consisted of less than
3 utterances). Similarly, concerning the edges, the largest graph contains 87 edges, while
the smallest only 3. Most of the networks are sparse, as the mean value of the fraction of
edges is equal to 0.12. There are, however, some exceptions, as the maximum value of the
fraction of edges among the 55 networks equals 0.67. Most of the networks have a low
clustering coefficient (mean value of Cis 0.22). Lastly, the vast majority of the networks
are disassortative, since their assortativity coefficients rare negative.
Figure 11 shows the distribution of the z-scores for the 5 basic network patterns de-
picted in Figure 5. To be specific, for each of the 55 C58 discourse networks, we measure
the number of Si,i=1 . . . 5 patterns. Then, we create 100 random networks with the
same number of nodes and edges as the actual network and measure the occurrence of the
same Sipatterns in each of the random networks. For each pattern, and for each actual
network, we calculate a z-score as follows. Let nidenote the number of Sipattern occur-
rences in the actual network. Let µand σdenote the mean number and standard deviation
of occurrences of pattern Siin the randomized networks. Then, the z-score is calculated as
z=ni−µ
σ. Figure 11 shows that the feed-forward loop S2 is clearly a network motif for a lot
of the 55 networks. There are several cases with very high z-scores. (More than 10 in some
cases!) The fact that the feed-forward loop, which is the most famous motif in biological
networks, is found to appear in linguistic networks as well is certainly intriguing. It is also
intriguing that S4, i.e., the dual-cause pattern shown in Figure 5, is clearly an antimotif,
as suggested by the large negative z-scores of its distribution. Furthermore, interestingly,
the linear chain pattern S1 that represents a sequence of uninterrupted related utterances
is neither a motif or an antimotif for the discourse-annotated single-author written texts
in C58, since it does not appear more frequently or rarely than it would at random for the
vast majority of the discourse graphs. In particular, we compared the observed frequency
of the S1 pattern with what would be expected by chance, given the size and composi-
tion of the texts in the dataset. Our findings suggest that the usage of this particular linear
chain pattern is not a common strategy of the authors of the texts in the C58 dataset. Lastly,
the common-cause pattern S5 does not appear to be a network motif or antimotif.
Appl. Sci. 2023,13, 6902 14 of 23
Figure 10. C58 networks—histograms of basic undirected network properties: number of nodes
nsize, number of edges nedge, fraction of edges, i.e., the ratio of nedge to the maximum possible
number of edges, mean degree kav, degree standard deviation kstd, clustering coefficient Cand as-
sortativity coefficient r.
Figure 11. C58 networks—histograms of network motif z-scores. Three-node subgraph patterns are
denoted as follows: S1—linear chain, S2—the feed-forward loop, S3—a cycle, S4—dual-cause and
S5—common-cause.
Appl. Sci. 2023,13, 6902 15 of 23
Figure 12 shows, in the form of histograms, the distribution of the basic properties
of 299 STAC (chat-only) networks derived from dialogues in the English language. These
networks are quite heterogeneous, as the largest of them contains 322 nodes, while the
smallest contains only 2 nodes. Similarly, concerning the edges, the largest graph contains
166 edges, while the smallest only 1. The STAC (chat-only) networks are much sparser
than the C58 networks, as the mean value of the fraction of edges is equal to 0.02. It should
be noted that there is a complete absence of clustering. All STAC (chat-only) networks
have a clustering coefficient C=0. Moreover, similarly to the C58 networks, the STAC
(chat-only) networks are also disassortative (i.e., assortativity coefficients rare negative
for all the networks.)
Figure 12. STAC (chat-only) networks—histograms of basic undirected network properties: number
of nodes nsize, number of edges nedge, fraction of edges, i.e., the ratio of nedge to the maximum
possible number of edges, mean degree kav, degree standard deviation kstd, clustering coefficient C
and assortativity coefficient r.
Next, we scanned the STAC (chat-only) networks for the 5 basic network patterns
depicted in Figure 5. As shown in Table A2 in the Appendix A.2, the actual counts of all
the S∗patterns are zero, except for the S4 pattern, which indeed has nonzero values. This
is a rather intriguing fact and is in marked contrast to our observations for the C58 corpus
networks. As mentioned above, the presence or absence of a specific pattern, although
interesting in itself, does not suffice to term a pattern as motif or antimotif.
Appl. Sci. 2023,13, 6902 16 of 23
Thus, Figure 13 shows, for the STAC (chat-only) networks, the z-scores’ distribution
for the 5 basic network patterns depicted in Figure 5. The plot shows that, unlike in C58,
the feed-forward loop S2 is not a network motif in the case of the STAC (chat-only) net-
works. The z-scores of the S2 distribution are negative, but since all of them are small in
absolute values (less than 0.6), we cannot claim that they consist of an antimotif.
Figure 13. STAC (chat-only) networks—histograms of network motif z-scores. Three-node subgraph
patterns are denoted as follows: S1—linear chain, S2—the feed-forward loop, S3—a cycle, S4—dual-
cause and S5—common-cause.
It is, however, intriguing that S4, i.e., the dual-cause pattern shown in Figure 5, is still
an antimotif, as suggested by the large negative z-scores of its distribution. Moreover, the
z-scores’ distribution suggests that the linear chain S1 and the common-cause S5 patterns
are clearly antimotifs of the STAC (chat-only) networks.
Figure 14 shows the distribution of the basic properties of 299 English discourse net-
works derived from the situated discourse corpus described above. While the text files of
the corpus are the same as those presented in Figures 12 and 13, additional information
of the nonverbal communication of the participants is included and, thus, the resulting
networks are much larger than before. The largest of them contains 1628 nodes, while
the smallest contains 29 nodes. Similarly, concerning the edges, the largest graph contains
818 edges, while the smallest only 15. These networks are much more sparse than be-
fore, i.e., the mean value of the fraction of edges is equal to 0.005. It should be noted that
there is a complete absence of clustering. Again, all networks have a clustering coefficient
C=0. Additionally, similar to the previous cases, the networks are still disassortative
(assortativity coefficients rremain negative for all the networks).
Appl. Sci. 2023,13, 6902 17 of 23
Figure 14. STAC (situated) networks—histograms of basic undirected network properties: number
of nodes nsize, number of edges nedge, fraction of edges, i.e., the ratio of nedge to the maximum
possible number of edges, mean degree kav, degree standard deviation kstd, clustering coefficient C
and assortativity coefficient r.
Finally, focusing on the situated STAC networks, Figure 15 displays the z-scores’ dis-
tribution for the 5 basic network patterns depicted in Figure 5. Despite the increased
nonverbal information included in the situated dialogues, the results remain consistent
with those of the nonsituated networks, namely that S1, S4 and S5 are antimotifs, since
they appear with considerably less frequency than expected by chance, while S2 is neither
a motif nor an antimotif.
All numerical simulations and data analyses were performed on a workstation
equipped with 2 Intel Xeon Gold 6140 Processors (72 cpu cores in total) provided by
the MSc program “Computational Physics” of the Physics Department, Aristotle Univer-
sity of Thessaloniki. The estimation of network motifs was performed through custom
code written in python. Basic network properties were studied with python modules
NetworkX [76] and retworkx [77]. The code was parallelized using the python module
dask [78]. The parallelization step is essential to the project, since the search for network
motifs is a computationally intensive task, even for networks with moderate size. For a
network of 103nodes, the process requires approximately 64 h in a Xeon Gold 6140 Proces-
sor. Thus, without parallelization, the processing of the 299 STAC networks is practically
prohibited. Lastly, the data preprocessing was conducted in Python and R on a PC with
an i9 Intel processor (8 cores in total).
Appl. Sci. 2023,13, 6902 18 of 23
Figure 15. STAC (situated) networks—histograms of network motif z-scores. Three-node subgraph
patterns are denoted as follows: S1—linear chain, S2—the feed-forward loop, S3—a cycle, S4—dual-
cause and S5—common-cause.
6. Discussion
We applied network analysis to two discourse-annotated corpora, STAC and C58, to
uncover deep properties of discourse representations. The corpora contain English and
Greek texts from two discourse types: multiparty dialogues and single-author written
texts. We also analyzed the chat-only and situated versions of the multiparty dialogues
in STAC. This helps us understand and compare how discourse structure varies across
discourse types and languages.
Initially, we presented key network indices for all discourse networks. C58 has a
higher fecompared with both STAC versions, though not significantly high. ferepresents
network connectivity. The networks differ in size, measured by number of nodes (nsize)
and edges (nedge), but all are sufficiently large for reliable analyses. Despite being sparse,
C58’s discourse network exhibits a significantly higher mean fraction of edges than the
STAC networks, indicating greater connectivity in single-author written texts, as expected.
Furthermore, the clustering coefficients of C58 networks are noteworthy, comparable
to those found in social networks (see Figure 10 and Table A1). However, they exhibit a
tendency towards disassortativity, a characteristic commonly observed in technological or
biological networks but rarely seen in social networks.
Our study revealed remarkable differences and similarities between two discourse
types: single-author written texts and dialogue texts in their chat-only and situated ver-
sions. Specifically, we examined the presence of network motifs and antimotifs:
1. The linear chain pattern (S1) is not frequently observed as a motif in C58, despite
the expectation that journalistic discourse would involve uninterrupted related se-
Appl. Sci. 2023,13, 6902 19 of 23
quences. In contrast, both versions of STAC systematically avoided the S1 subgraph
pattern, as evidenced by the absence of occurrences (see Table A2). These findings
suggest that multiparty dialogue networks tend to avoid three uninterrupted related
utterances, indicating that speakers do not participate in or contribute to a continu-
ous line or chain of utterances. In C58, authors neither avoid nor prefer to establish
connections between utterances in the form of a linear chain pattern.
2. The feed-forward pattern (S2) emerges as a motif in C58, indicating a statistically
significant preference for constructing fully connected three-node subgraphs in most
single-author written texts. This finding highlights the strong preference for fully
connected three-node subgraphs in the discourse structures of single-author written
texts. On the other hand, the two STAC discourse networks do not exhibit a net-
work motif or antimotif, although it should be noted that no occurrences of the S2
subgraph pattern were recorded in these networks.
3. Among all three discourse networks, the dual-cause pattern (S4) stands out as the
only subgraph pattern that acts as an antimotif. In the three-node subgraphs of S4,
there is a node π3with two incoming edges originating from two utterance labels, π1
and π2, which are not directly related to each other but precede in the discourse.
4. The common-cause pattern (S5) serves as an antimotif for the STAC corpora, while
it does not exhibit characteristics of either a motif or an antimotif in the C58 corpus.
This suggests that a commonly observed discourse strategy does not favor the scat-
tered presentation of various aspects of an event described by an utterance. More
typically, we witness a sequence of utterances that play a subordinate role to an ini-
tial utterance. It is generally avoided to circle back later in the discourse to add more
aspects to that initial utterance.
The above findings can be summarized and as follows:
• Aw Table 1illustrates, both versions of the STAC corpus exhibit similar behavior
regarding the four three-node subgraph patterns. This similarity suggests that the
additional structures introduced by annotating EEUs (Embedded Event Units) and
their interactions with other discourse units in the situated STAC corpus strongly
avoid the same three subgraph patterns (S1, S4 and S5) observed in the discourse-
annotated structures of the chat-only STAC corpus. This parallelism between the
two STAC versions supports the argument made by [28,29,37] that EEUs introduce
higher-level structures with their own complexity and idiosyncrasies. In other words,
if the presence of EEUs were random and lacked systematic structure, the distribu-
tion of the corresponding S∗subgraph patterns in the situated STAC corpus would
resemble that of a randomly generated network, rendering it inconclusive for the
existence of motifs or antimotifs (see Table A1).
• This study’s findings indicate that the presence of the antimotifs S4 and S5 in the
STAC discourse networks suggests a strong restriction on establishing discourse rela-
tions between π1and π3when there is a distant discourse relation between them, and
π2does not serve as the bridge connecting π1and π3. This restriction imposed by
the discourse structure may be influenced by the distance between the two utterance
labels, π1and π3, as the distance in terms of utterances is expected to impact the de-
velopment of discourse in multiparty dialogue texts. However, it is important to note
that the baseline approach of attaching an utterance label πto the last available in the
discourse, as noted by [72], fails to capture 40% of attachments in the ANNODIS cor-
pus. This empirical fact emphasizes that there are numerous attachment candidates
for a given πthat extend beyond its immediate vicinity, further supporting the argu-
ment for the strong restriction imposed by the discourse structure.
• A general observation related to S4 and S5 is that although both are considered anti-
motifs for the two STAC discourse networks, there is a noticeable difference between
the two types, since occurrences of the S4 pattern have been recorded in both corpora,
while no occurrence has been observed for the S5 pattern. However, as mentioned
above, the presence or absence of a specific pattern, although interesting in itself,
Appl. Sci. 2023,13, 6902 20 of 23
does not suffice to term a pattern as motif or antimotif. S4 is an antimotif for the C58
discourse networks, too, as mentioned above, but since S5 is neither a motif nor an
antimotif for these discourse networks, our network analysis suggests that the above-
mentioned restriction holds for both corpora but only for the dual-cause pattern, S4.
Table 1. Network motifs and antimotifs of the two corpora, C58 and STAC, for three-node subgraph
patterns.
Corpora S1S2S4S5
C58 neither motif antimotif neither
STAC
(chat-only) antimotif neither antimotif antimotif
STAC (situated) antimotif neither antimotif antimotif
Author Contributions: Conceptualization, A.T. and K.K.; methodology, A.T. and K.K.; software,
A.T. and K.K.; writing—original draft preparation, A.T. and K.K. All authors have read and agreed
to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: All datasets are available upon request.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Appendix A.1. Basic Statistics for a Number of Real-World Networks
Table A1. Basic statistics for a number of real networks: The properties presented here are Number
of nodes nsize, Number of edges nedge, clustering coefficient Cand assortativity coefficient r. Data
taken from [31].
Network Nsize Nedge C r
Film actors 449,913 25,516,482 0.78 0.208
Company Directors 7673 55,392 0.88 0.276
Math coauthorship 253,339 496,489 0.34 0.120
Physics coauthorship 52,909 245,300 0.56 0.363
Biology coauthorship 1,520,251 11,803,064 0.60 0.127
Email address books 16,881 57,029 0.13 0.092
Student dating 573 477 0.001 −0.029
WWW nd.edu 269,504 1,497,135 0.29 −0.067
Roget’s thesaurus 1022 5103 0.15 0.157
Internet 10,697 31,992 0.039 −0.189
Power grid 4941 6594 0.080 −0.003
Train routes 587 19,603 0.69 −0.033
Software packages 1439 1723 0.082 −0.016
Software classes 1376 213 0.012 −0.119
Electronic circuits 24,097 53,248 0.030 −0.154
Peer-to-Peer network 880 1269 0.011 −0.366
Metabolic network 765 3686 0.67 −0.240
Protein interactions 2115 2240 0.071 −0.156
Marine food web 134 598 0.23 −0.263
Freshwater food web 92 997 0.087 −0.326
Neural network 307 2359 0.28 −0.226
Appl. Sci. 2023,13, 6902 21 of 23
Network theory allows for the comparison of several different complex systems that
would otherwise be considered completely isolated and beyond comparison. A recent
and rather interesting case of the application of network theory in a novel class of social
networks, i.e., the follower–following graph, can be found in [79].
Appendix A.2. Mean Values of the S∗Patterns in the Three Datasets
Table A2 below presents the mean number of the occurrences of each S∗pattern in
the three datasets we examined. The numbers represent the ratio of the total number of
occurrences of each pattern in all sets, i.e., the sum of the occurrences in each network,
divided by the number of networks in each dataset, i.e., 55 for the C58 corpus and 299
for STAC.
Table A2. Mean Pattern values.
Dataset S1S2S3S4S5
C58 24.06 4.24 0 1.33 17.69
STAC (chat-only) 0 0 0 3.64 0
STAC (situated) 0 0 0 4.64 0
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