Cognitive linguistics and data analysis

Data driven linguistics is a very active and lively research field, but there exists a blind spot regarding the findings of Cognitive Linguistics [1, 2] (CL for short). CL is an actively developing branch of linguistics without an established closed canon. It depends on several widely accepted premises. The most important one is a unity of language and thought. This implies that language is not produced in a separate linguistic module of the brain, but is instead processed and created by general cognitive mechanisms that are responsible for diverse cognitive tasks [3]. This has the important consequence that an understanding of linguistic structure offers a window into general principles of cognition.

Considering language in this way automatically links language research to many other areas of research. For example, with evolutionary anthropology that studies the development and role of human cognition and communication [4]. More importantly for the focus of this work, however, it connects language to human embodiment, since meaning emerges in connection with and from the physical reality of humans, rather than in abstract isolation [5]. Both of these points make computational analysis particularly challenging if its aim is to enable computers to “understand meaning" by analyzing the internal structures of language. The absence of the evolutionary functions of language and the absence of a body as well as of basal human experience mean that, from the perspective of Cognitive Linguistics, a computer can at best inadequately replicate the process of “understanding meaning". This is especially true for the most common computational linguistic method of natural language processing and large language models, where meaning of words is seen in its interrelations with other words according to the distribution hypothesis [6–8]. There, language is studied in isolation as an abstract system of signs, and text corpora are seen as containing meaning entirely within themselves. It is this, why studies of Metaphor in corpus linguistics are non-trivial. They can lead to statistical statements on the occurrence of metaphor [9] but corpus linguistics does not start at the premise of metaphor as a distinguished mechanism. Also studies in linguistic corpora in principle require preliminary delimitation of the analysis a priori, for example, restricting to certain source domains. An investigation across the entire language range, as it is aimed at here, is thus excluded. The approach presented here, on the other side can investigate metaphor across the entire English language and starts at the understanding of metaphor as a mechanism between two domains and from there develops it’s further understanding.

Also, we keep our analysis within the framework of Conceptual Metaphor Theory without trying to simulate understanding through formalisation. The mathematical structures we use do not have the purpose of imitating the structure of meaning, but only of enabling a corpus-based statistical analysis within the framework of an existing model.

Conceptual metaphor theory

A central component of Cognitive Linguistics is Conceptual Metaphor Theory. This theory does not consider metaphor as a rhetorical device in the sense of garnishing meaning packaging. Rather, metaphor is the central cognitive linking mechanism between concrete embodied experiences and abstract linguistic domains. Metaphor is not a simile, but a new transfer— hidden in its Greek root— of experiential structures into abstract topics. Because of the linguistic nature of metaphor, new abstraction can be achieved in linguistic forms, and metaphor, as the instance of transfer, can be seen as the original point of new meaning-creation.

Conceptual Metaphor Theory (CMT) emerged as a distinct field of research in linguistics with the publication of the book “Metaphors We Live By" by Lakoff and Johnson in 1980 [10]. In that book the authors put forward several main theses and developed a terminology. Essentially, the claim is that metaphors are not only a phenomenon of poetic language, but are ubiquitous in everyday language. Metaphoric transfers do not reside in individual rhetorical expressions, but systematically transfer structures between domains. With time, these mappings become part of our cognitive systems of thought and can give rise to a variety of individual metaphoric expressions. These systematic transfers are not comparisons of pre-existing structures, but constitute a model of the abstract domain in which it acquires its meaning. Since its establishment, CMT has undergone a rich development and has found application in numerous fields such as politics, literature, pedagogy, and transcultural studies [11]. However, a constant criticism concerns the lack of systematic research on the theory. The common research method of CMT studies particular language examples, supporting a separate thesis about the nature of metaphoric attributions. The selection of these language examples is based entirely on the researcher’s linguistic intuition along with their specific goal, which the example is intended to illustrate. This methodological issue is referred to as the first major problem of CMT [12]. To date, there is no empirical study that reconsiders the theoretical foundations of CMT based on real linguistic data. This is particularly problematic because of the high flexibility of languages. A language forms a complex system with rules and tendencies on one side, but also expectations and competition between rules on the other side. Many phenomena can be discovered in such a system, but the appropriate tool to detect the guiding rules and main tendencies is statistical analysis.

There are attempts to perform corpus-based metaphor analysis with automated metaphor detection. Stefanowitsch and Gries [13] reviewed these attempts, but noted that in principle all attempts at automatic metaphor detection rely heavily on a presupposed metaphor theory. For example, one could search for typical metaphor source or target vocabulary or particular sentence structures, but this requires prior knowledge of the source and target distribution or of triggers indicating metaphors. Consequently, these methods are not immune to yielding highly biased results.

For this reason, this study takes a different approach to analyze a statistically significant set of randomly selected real-world metaphors. We chose to examine metaphors based on diachronic changes over the history of the English language.

Metaphor research through diachronic changes

Some frequently used metaphors experience an increasing conventionalization. They become so-called passive metaphors, where the original metaphoric use of the word is hardly recognizable as such [14]. Typical examples of this stage are “Time is running" or “Looking up to somebody". Continuation of this process can then lead to a new acquisition of word meaning or a transfer of meaning of a word from its source domain to its target domain. This type of exploration of metaphors has been successfully pursued by [15], for example. The examination of metaphors through etymological mappings assumes a homogeneous process of meaning transfer and conventionalization across all relevant source and target domains, contingent solely upon the level of metaphorical language usage. If this is the case, no bias arises from the chosen method, and etymology, like archaeology for a historian, grants access to the study of active language.

Computational methods vs. computational meanings

The metaphor of archaeology further allows us to illustrate an important difference that characterizes the method of this study. The sharp elaboration of this difference is not only central to metaphor research but a generally significant challenge for quantitative methods, especially in the field of digital humanities. It lies in the fact that the research method does not necessarily correspond to the nature of the subject matter and makes no direct statement about it.

It is important to avoid the misconclusion: If semantic novelty can be explored computationally, it is a computational process. If we take the view of metaphor that has been expressed in numerous philosophical contributions [16–19], as well as being influential in the debate on metaphor in cognitive linguistics, then metaphor is a point of semantic innovation. It is, therefore, not a process internal to language but a moment that transcends language. In [18] Paul Ricœ ur draws a very clear distinction between the living metaphor and the linguistic code. There, the living metaphor is primary and cannot be conceived without the language-transcending function of reference.

The resulting linguistic code consists of the archaeological artifacts of these linguistic events. The investigation of these artifacts, which we can undertake here, approaches the events themselves, i.e., the metaphors, not directly but indirectly. The results of this investigation are thus not an extension of metaphor theory itself, but an analysis of the artifacts that the metaphorical events leave behind in language, and against which the explanatory power of any actual metaphor theory can now be measured.

Similar is the relationship between this study and contributions to the debate on metaphor in cognitive linguistics. Here, the language-transcending moment, i.e., the moment in which something extra-linguistic has a formative influence on language, is often localized in the relationship between metaphor and motor-sensoric experiences. There are different opinions about the exact role of motor-sensoric experiences in the formation of linguistic forms. The debate ranges from perceptuo-motor simulation [20] to a continuous process of multi-modal experiences leading to cognitive representations that can then enrich language [21, 22]. In an investigation of language itself, we can only speak of an indirect approach to this process. For this reason, this study does not take its own position in this debate and does not provide a view of what embodied metaphor means. It does, however, analyze the linguistic artifacts against which any such theory can be measured.

Structure of the paper

The paper presents various methodologically distinct analyses of the data from the Mapping Metaphor project [23] exploring different facets of metaphor formation. By adopting the Mapping Metaphor project, we thereby necessarily accept and use the classification of metaphorical domains that is employed in that data set. Metaphor formation is conceptualized as a network, where topics serve as nodes and edges represent metaphorical connections from sources to targets. First, we investigate the source and target topics of metaphors. Analyzing the distribution of connections among topics provides a global understanding of the dynamics of the metaphor network. A second set of experiments examines the network’s division into two anti-communities of topics. This is further refined by a motif and a connection type number analysis. In the third part, we explore a characteristic feature of the network, namely, preferential attachment on edges. The fourth part focuses on the connection between metaphor and semantics by identifying roles within the metaphoric network. A final, fifth part, entails a discrete curvature analysis, which also leads to a critical role analysis of spatial metaphors.

Non-randomness of metaphorical data

We first look at the in- and out-degree distribution of the directed graph representation of the Mapping Metaphor data Fig 1. The plots already include a comparison with the mean of one thousand equally sized Erdős-Rényi graphs (Materials and methods) and one standard deviation up and down. Right away, we find strong deviations of the in-degree and even stronger ones for the out-degree distribution of the metaphoric network from the random case. This already points to the interesting potential of the data for metaphor research, as non-randomness implies structure.

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Fig 1. Top: Comparison of out-degree (left) and in-degree (right) distributions of categories in the metaphor network and configuration random model.

Bottom: Absolute (left) and normalized (right) in- vs outdegree of categories in the metaphor network. Categories of the two anti-communities are distinguished by colour (categories not assignable to an anti-communities are black).

https://doi.org/10.1371/journal.pcsy.0000058.g001

For the Erdős–Rényi model the degree distribution concentrates around the mean connection number of slightly above 30 but for the metaphor data, this scale does not play a particular role. Instead, most categories cluster around low degree numbers and play only a minor role in the metaphor network. In contrast, there are a number of highly active categories, both for in- and out-degree, forming a so called heavy tail of the distribution. They show degree numbers several times higher than the degree maximally found in average random graphs.

Concrete and abstract categories separated in anti-communities

When we compare our metaphor graph with random graphs of the same size and average degree, we see striking differences and can identify important structural properties of the metaphor network. The latter’s most marked structural property is that the majority of metaphors can be assigned to one of two classes that can be characterized by connectivity properties. The first class contains mainly spatial, mechanical, concrete and physically perceptible categories while the categories in the second class are mainly temporal, emotional and social. The two classes form anti-communities, meaning that the majority of connections is between instead of inside the classes, although the first class also contains a significant number of internal connections. Fig 2 shows that in comparison to configuration random graphs, there are two systematically dominating metaphor groups, one mapping from the first, more concrete anti-community to the second, more abstract one and another with mappings within the second anti-community, which shows the continuing re-determination and re-linking of concrete themes. The first group is the one most effectively described by CMT, while there is only little research and understand for the second group and it’s role in conceptualization. The degree distribution in Fig 1 is heavy tailed. In fact, most categories have low degrees and do not contribute much to the network, while some few highly active categories have in- or out-degrees that are significantly higher than the maximal degrees found in random networks. The most active sources and targets do not match, however, as seen in the plot. The vertices at the bottom right of the graph have a high target probability, but low or at most average source activity. In contrast, the strongest sources also have a high target activity. These two different activity groups – one concentrated at the bottom right, the other on the diagonal – in fact coincide with the two anti-communities identified above. Almost all categories on the diagonal, regardless of their degree, belong to the concrete anti-community. The correlation coefficient for in- and out-degree of this set is 0.709. In contrast, for the abstract anti-community it is only 0.273, evidencing again the difference between the two classes.

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Fig 2. Anticommunities and metaphor persistence.

Left: Connectivity patterns in the metaphoric mapping network distinguish two classes of categories that coincide with concrete categories (Con) and abstract categories (Abs). These classes form anti-communities, that is, most of the connections are between the classes and not within them. The bar chart compares the number of connections between classes observed in the metaphorical network (green bars) and the average number of connections observed in an ensemble of 1000 configuration random graphs (red bars with error bars). It shows that there are two systematically dominating metaphor groups, mappings from the concrete to the abstract anti-communities, and another with mappings within the concrete anti-community, which shows the continuing re-determination and re-linking of concrete themes. Right: Comparison of multiplicity values of edges between configuration random model and the metaphor network.

https://doi.org/10.1371/journal.pcsy.0000058.g002

The identification of anti-communities and the two particularly common metaphorical processes in relation to them represents an interesting discovery in the context of the embodiment of metaphors. The vast majority of bodily experienceable domains: “tangible,” “visible,” etc., are found in the first anti-community. One of the statistically relevant processes thus describes metaphors that move from bodily-related domains to less so. The second process, however, repeatedly describes new metaphors between bodily-related domains. In literature, there exists no unified theory of metaphoric embodiment. The concept itself describes the participation of the body and the experience of corporeality in the formation of metaphorical meaning. It roughly contradicts a view where metaphors are pure transfers of abstract cognitive representations [24]. However, there are unanswered questions as to whether metaphor processing involves a one-to-one reactivation of physical sensory perception [20] and, if not, how exactly the continuum from integrated multimodal bodily experience to the imagination and the experience of oneself as mind can be understood [21, 22]. The frequency of metaphors within the first anti-community can contribute to further expanding the understanding of embodiment. Here, existing bodily experience is not simply integrated into an abstract domain. Rather, it could point to a multidirectional interweaving of imagination and experience. How bodily perception itself takes place does not remain unchanged but is constantly being linguistically reconnected in relation to other body-related domains. Possibly, the creation of meaning is not so much or not only reached by connecting abstract spaces to experience, but rather a re-experience of the concrete. In this way, metaphor would not only be embodied, but would also be an instrument of integration for our constantly renovating embodiment.

Word transfer density

In order to examine the word transfer density, we now consider the multigraph where each transferred word creates an edge between the corresponding categories. We then normalize the degrees by dividing by the total number of words in each category. Note, however, that this will in general underestimate the word transfer probability, in particular for very active mappings.

We use the same colors for the anti-communities as above in Fig 1. With the expectation of the category “light" all topics with high in-degree density probability belong to the abstract anti-community and accumulate in the lower right corner, as before. The words in the other class, the concrete categories, the predominant sources, however, do not typically occupy a position on the diagonal, but are just active sources, but not preferred targets. So instead of a diagonal slope, we find that the probability of words in a category being involved in metaphorical mappings may be high either exclusively as a source or as target depending if the category belongs to the concrete or the abstract anti-community.

Motifs and the metaphorical mechanism

The motif count reveals further structure (Table 1). We find five subgraphs of size n = 3 with Z-values either above 2 or below –2, and in four cases, the absolute Z-value is even higher than 6. The three motifs of size 3 with one missing edge are clearly more frequent. In contrast, closed triangles are almost absent in the data. Both effects, however, disappear when we restrict the analysis to the internal connections of the anti-communities.

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Table 1. Non-symmetric three-Vertices motifs with corresponding z-value. Two-vertex motifs with corresponding z-value. For complete network, only connections within concrete topics block and only connections within abstract topic block.

https://doi.org/10.1371/journal.pcsy.0000058.t001

For motifs of size 2, the deviation from randomness is even more pronounced, as symmetric connections dominate. This is true within both anti-communities, however to a much smaller extent within the abstract one. Accordingly, any metaphorical mapping from source to target makes the reverse mapping from target to source significantly more likely.

Persistence of metaphorical mappings

When looking at the multiplicity of edges, that is, the number of words transferred between categories, another deviation from a random model emerges (the black lines show the distribution for one thousand random configuration graphs with one standard deviation up and down for each data point.) As can be seen in Fig 2 in the right plot, there is a significantly higher probability of transferring words along already established mappings, so that vertex pairs with a high number of transferred words become significantly more frequent than in random models.

Such a deviation can always be a property of the methodology of the Mapping Metaphor data collection. It may also ultimately be the uneven distribution of a natural connection affinity between pairs of topics. The systematics of the deviation, however, primarily carries the properties of a preferential attachment mechanism on mappings, meaning that an already established metaphorical relation facilitates further word transfer along this relation.

Metaphor and semantics

We now turn our attention to the role that categories play in the metaphor network, which is given by the connections they form. In social network analysis, objects are considered to play the same role if they have the same incoming and outgoing neighbors. To examine the roles within the metaphor network, we employed a Hierarchical Cluster Analysis (HCA) algorithm on the collection of categories and their incoming/outgoing neighbors. The distance between categories was measured by the number of non-common connections they possessed. The output of the HCA is a binary tree called dendrogram, whose clusters (branches) represent roles in the metaphor network.

Real-world data sets often feature multiple equidistant objects, leading to numerous clustering solutions and consequently, a multitude of role schemes [25, 26]. However, the HCA algorithm yielded only four dendrograms, one of which is illustrated in Fig 3.

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Fig 3. Hierarchical representation of semantic roles in the metaphor network.

The figure is one of the four dendrograms obtained from running a Hierarchical Cluster Analysis algorithm on the set of categories from the metaphor network. Categories were represented by their incoming and outgoing neighborhoods, the distance between categories c and was computed by counting the number of non-common connections: , and Ward’s method was used as a grouping methodology.

https://doi.org/10.1371/journal.pcsy.0000058.g003

These four dendrograms encompass a total of 421 distinct graph clusters, defined as all the different subtrees that can be extracted from the four dendrograms. In S2 Fig, it can be observed that 401 graph clusters, the majority of them (95.3%), consistently appear as subtrees in all four dendrograms. The remaining twenty graph clusters, which are not present as subtrees in all four dendrograms, can be categorized as follows (S2 Fig): the four entire dendrograms (cluster size equal to 415), eight relatively small-sized subtrees (approximately 20 leaves) appearing in two out of the four dendrograms, and twelve subtrees ranging from 8 to 18 leaves, which appear in only one dendrogram.

Additionally, considering clusters as sets of leaves rather than graph structures provides an alternative perspective [25]. For instance, although the binary trees ((a, b),c) and (a,(b, c)) are distinct graph structures, they share the same set of leaves, namely, a, b, c. Similarly, the four dendrograms obtained share the same set of leaves, making them stable clusters as sets but not as graphs. In this context, we computed the clusters based on their sets of leaves, disregarding the hierarchical graph structure, and determined that there are 410 set clusters, which is 11 fewer than the number of graph clusters. Furthermore, 406 set clusters (99.0%) consistently appear as subclusters in all four dendrograms, while the remaining four set clusters appear in two dendrograms each.

In summary, the results obtained from both graph and set clusters indicate that despite being merged differently during the grouping process, leading to distinct graph structures, the semantic roles exhibit remarkable stability. The similarity between graph and set structure additionally points to the robustness also of the more defined graph structure that was obtained by the HCA.

Through evaluating the metaphor distance as a similarity measure, we aim to assess its ability to provide insights into the semantics of categories. Comparing the metaphor-based similarity distance of all categories with cosine similarity and the euclidean distance derived from fasttext word embeddings reveals a low correlation. The cosine similarity yields a value of –0.24, while the euclidean distance results in 0.23. Similarly, mutual information is present but not dominant, with values of 6.24 for both cosine similarity and euclidean distance. However, the stability of the resulting dendrograms and the consistent ordering of categories vividly demonstrate that the metaphor-based role of a category encodes a semantic position, contributing to the formation of a semantic structure. This structure predominantly goes beyond the semantic information contained in word embeddings, encompassing figurative and conceptual semantic similarities. Consequently, while the acquired semantic structure does not completely diverge from word embeddings, it exhibits semantic aspects that surpass them, underscoring the limitations of word embeddings from a broader semantic perspective.

The role of space and the global network structure

A common and widely accepted assumption in the CMT literature is the specific role of spatial and directional domains for metaphor mappings. In the original set-up of the theory, Lakoff and Johnson elaborated on the so-called orientation metaphors. These do not structure or explain a single abstract concept, but rather perform the higher-level task of organizing the relations between different abstract concepts in terms of the spatial domain. Later, space was increasingly ascribed a special cognitive role in language. Spatial thinking was seen as the platform for schema formation—the universal building blocks of language. Thus, the spatial domain was considered to be the fundamental source of all concrete structures [27]. Such linguistic considerations went along with neuroscientific research about the role of spatial thinking in the hippocampus for memory and structural thought [28] culminating in an actively pursued research program on cognitive spaces [29, 30].

However, none of the analyses conducted on the mapping metaphor data showed a central, highlighted role of the spatial and directional categories in the metaphor network. According to the theory of orientational metaphors, one might expect that mappings starting from spatial categories would influence their neighboring mappings, resulting in higher outward transitivity. In fact, the outwards transitivity of relative position and movement in a specific direction, is relatively high, but it does not exceed a general tendency found in the graph. This general tendency can be observed in Fig 4, where one sees a clear correlation between the out-degree and the outwards transitivity of the vertices. Apart from being an interesting fact about the metaphorical data in itself, this correlation also shows that spatial regions follow this relation but do not exceed it. They do not play a special role in terms of their transitivity and thus their direct influence on their neighborhood in the network.

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Fig 4. Left: Distribution of Forman Ricci vs Ollivier Ricci curvature of edges in the metaphoric network.

Right: Out-degree vs transitivity in the metaphor network.

https://doi.org/10.1371/journal.pcsy.0000058.g004

In the outlook of further possible special positions of the spatial domains and also with the aim of better understanding the general structure of the network, we consider two types of discrete curvature in the multigraph representation. The statistics of local graph curvatures are known to be good indicators of global network properties. The Forman-Ricci curvature captures the divergence or spreading in a network. Again, our analysis of the metaphor data show systematic differences from random graphs or other real world data sets. In particular, we see a rapid decrease at negative values, meaning that there are only very few edges with strong branching at their starting and/or terminal vertices (Fig 4). Such edges would provide rapid flow access to many different metaphors, but this is definitely not the structure of the metaphor network.

The Ollivier-Ricci curvature, in contrast, captures the presence of triangles and quadrangles in the network and thus the density of local neighborhoods. The existence of several seperated regimes of edges in the distribution indicates density clustering within the network on different scales. In the metaphor data, we do not find such dense local clusters separated from others as seen in (Fig 5).

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Fig 5. Distribution of Forman Ricci (left) and Ollivier Ricci (right) edge curvature in the metaphoric network.

https://doi.org/10.1371/journal.pcsy.0000058.g005

Mapping metaphor data

We conducted our research based on the data of the Mapping Metaphor project [23, 33], a collection of etymological transfers caused by metaphorical mappings. The collection, conducted by the School of Critical Studies of Glasgow University, is based on the Historical Thesaurus [34], where each entry is assigned to one of the 415 thematic categories (e.g. movement, visual arts, weapons and armour, etc.). The code used to carry out the computational methods is made available at [35].

To detect metaphorical mappings, a computer program searches for words that appear in two or more categories. It is then manually inspected if these lexical overlaps can be seen as metaphorical usage. If this is the case, an entry is created with the transferred word, the source and the target category. The entry also includes the year in which the transferred meaning was detected for the first time as well as the last year of appearance if the usage has not been preserved until today. For our purposes it is important to note that the search for metaphorical mappings was systematically performed, and that not all existing etymological mappings have been included. In particular, since the project’s main goal was to detect all possible connections between pairs of categories rather than each possible word transfer, less emphasis was placed on further findings along the same connection if several words had already been registered between a particular pair of categories. Another relevant point is that the categorisation of the lexical items does not coincide with the notion of domain from Conceptual Metaphor Theory. Rather, most categories are broader and can be considered as associations of multiple, thematically close domains. Using the Mapping Metaphor data forces us to adopt its categorisation. Therefore we will mostly use categories in the sense of the Mapping Metaphor project, which allows only for a restricted transfer of the results into the CMT framework.

The etymological circumvention is a methodological choice with several consequences to consider. Since the data, and specifically the categorization, are primarily based on the historical thesaurus and not on a project that aimed to create a metaphor database from the outset, the data is not fundamentally based on a metaphor theory. Only the Mapping Metaphor project requires a metaphor theory to be carried through; however, the influence of this metaphor theory is limited by partial automation in the data annotation process. Still, the criteria used by the MappingMetaphor project to check automatically found words for historical metaphorical use have an uncertain influence on the results of this study. These criteria are not clearly identifiable in the Mapping Metaphor project. Since we question the fundamental assumptions of any metaphor theory and aim to test them empirically, uncertainty regarding the criteria under which words appear as metaphors in the dataset is a significant potential source of error. In this way, the systematic characteristics of metaphorical transfers that are being identified may in part be artifacts of the selection process used in the Mapping Metaphors project.

More generally, the present study is highly dependent on the structure in which the limited Mapping Metaphor data is available. This has the general disadvantage that the results of other studies based on different data cannot be directly compared, tested, and combined with the present results without interpretative steps. A major such limitation is that the data in this structure is only available in English. This means that the large number of important questions that could be asked in a cross-linguistic comparison cannot be investigated.

Ultimately, this data structure only allows a limited view of the metaphorical process. Many aspects which are of interest for metaphor study are not captured in this data set and can thus not directly be inferred about by its analysis: contexts and sentence structures in which the transfer of words occurs; peculiarities of oral metaphors that do not play a role in the dataset; different conceptual mappings between the same domains and the dynamics between these mappings; transfers of partial aspects of domains and combinations of mappings in sentences and speech—to name just a few of the aspects of metaphor that cannot be examined here.

Mapping metaphor network

As the data contains pairwise relations between source and target categories, it very naturally qualifies for a graph representation. A graph in fact represents a set of elements that can be pairwise related to each other. The crucial gain here is that graphs are mathematically very well studied objects for which a richly developed research toolbox already exists. Moreover, this toolbox can be adapted and extended to particular problems in the existing formalism [36].

In the present paper we use two different graph representations of the data, as directed graphs or multigraphs, which will allow us to tackle different research questions.

In- and out-degree

The degree is a basic concept of graph and network theory. It is defined for each vertex as the number of edges it participates in. For the directed graph and multigraph we distinguish between in- and out-degree. The out-degree, as the number of outgoing connections, counts all edges in which a vertex appears as a source. Similarly, the number of edges which include a vertex as a target is the number of incoming connections to this vertex.

These simple notions allow us to identify local characteristic properties of the individual categories. On the other hand, the statistics of their distributions also provide global information about the structure of the network, by comparing them with the corresponding distributions of random graph models. That is, we can infer to which extent our networks show, or not, specific structures and features present in random models. This will be a general principle in our graph analysis.

In the multigraph case, where the categories are not viewed as uniform entities, but as word sets of different sizes, it is also interesting to look not only at the number of transferred words, which corresponds to the degree, but also at the transferred word density. This can be achieved by normalizing the in- and out-degree by the category size.

Random graph models

Directed configuration model.

The configuration model describes random graph models with a prescribed degree distribution. In the directed case, not only the in- and out-degree distribution are fixed but also the degree sequences and by this way, the in- and out-degree combination for each vertex. The outgoing and incoming edge stubs then are randomly fused [36].

Definition 4. (Directed configuration model). Let and be a fixed number of vertices and edges. We denote by and the in- and out-degree sequences of the n nodes. It holds that . To each vertex v, we attach in-going and outgoing edge connections such that each incoming stub is matched with equal probability to each outgoing stub. This way the probability for an edge to exist going from some vertex to is proportional to

(1)

The following algorithm implements a configuration model starting from a real world data model and keeping its degree sequences.

Two random connected pairs are cut and the connections are swapped, becoming . This step is repeated a given number of times k. The output network approaches the directed configuration random multigraph. One can also prevent the swapping in case one of the edges already exists. This modification of the algorithm makes the resulting network a directed configuration random graph instead of a multigraph. The drawback is that one is no longer able to analytically compute the edge probability, but this modified algorithm will be important for some of our experiments presented to create comparable random graphs [41].

Anti-community detection

The principle of anti-communities is exactly the opposite of clusters, to minimize the internal connectivity and maximize the connectivity across in- and outside of the group. Accordingly, the search can be performed using the negative graph in which disconnected nodes are connected to each other and vice versa. On this one of the well developed cluster detection algorithms can be applied. We use the Louvain cluster detection method based on greedy modularity maximization:

(2)

where A_ij is the Adjacency matrix entry of the graph, k_i, the degree of node i, m the total number of connections and c_i the cluster of node i while the Kronecker-delta function equals one only if and 0 otherwise [38, 39]. For each iteration, the algorithm produces slightly different results. In our case, the vast majority of categories are stably assigned to the same cluster. We run the assignment 10 times and keep only the overlapping sets as anti-communities while categories that change their assignment at least once are placed in the remainder set.

Edge preferential attachment

In the directed multigraph case, we can not only count the in- and out-degree for each vertex, but we can also count the edges between each two connected categories in each direction. For our data, this represents the number of words that were found to be mapped from a specific source category to some specific target.

The purpose of counting words along connections is to test whether metaphorical mappings do form persistent structures in language and mind, which is one of the main claims of Conceptual Metaphor Theory. In the Aristotelian view of metaphor as an individual rhetoric figure, individually mapped words should not influence the probability of other words being mapped along the same connection. In CMT, on the other hand, the phenomenon of metaphor is rooted not in the individual words and expressions, but in the conceptual transfer between two domains. Further, this transfer structure does not arise in the short term and temporally for production or processing of metaphoric expressions. Instead, it forms a persistent cognitive structure, which makes the transfer of many different partial aspects possible and becomes activated again and again. Once such a transfer structure gets established, the probability of further metaphoric expressions making use of this connection should increase.

The case of the random configuration model thus corresponds to the Aristotelian case. Each new formed edge of the multigraph is equally likely to form, given the degree sequences, and does not depend on previously existing identical edges. On the other hand, in the case of persistent mappings, the probability to form equal-directional edges between already connected vertices should increase.

It is possible to formulate exact models for which the probability of a new edge from u to v has a term proportional to the already existing number of edges along this possible connection:

(3)

and by comparison with the data, measure the influence of this term. However the collection method of the Mapping Metaphor data does not allow for this, as exploring further words along already detected connections has been systematically reduced. The aim of the project has not been to find all possible metaphorically transferred words but to focus on finding the mappings themselves. Thus, a quantitative examination of persistence is not possible. But we can test whether, despite the systematically lower word count, a confirmation of persistent metaphorical mappings can be found. This is done by comparing the distribution number of edges along a potential connection of the data against configuration random graph models (see).

Motif analysis

The analysis of graph motifs is a method for investigating the nature of processes behind the connections of a graph [40]. Graph motifs are defined in terms of small subgraphs that appear much more or much less often than expected from comparable configuration graphs. They reveal the logical properties according to which connections in a network emerge, and also those that do not play a role. We can then identify those different processes or mechanisms behind connection formation that possess or lack these logical properties.

We will conduct a motif analysis on the network to test if the two such logical properties drive the formation of the metaphoric network, namely, transitivity and symmetry.

Transitivity.

Transitivity can be tested by looking at motifs of size n = 3 that hold the following property:

(4)

where u, v and w are categories of the metaphoric network and represents an edge between two categories u and v. Transitivity is a logical property of every equivalence relation, for instance of analogy.

We computed the outward transitivity of a vertex u as the fraction of transitively closed directed triangles starting and ending at u, e.g., (, , ). In other words, starting at one vertex, we consider the fraction of triangles that are transitively closed.

Definition 5. (Outward transitivity). Let be a graph, the set of all vertices with an outgoing edge from v, and let K be the set of paths of length two, with . The outward transitivity T(v) of v in G is given by:

(5)

Symmetry.

In a directed graph we can distinguish between symmetric and non-symmetric relations. A relation is non-symmetric if is an edge of the graph, but is not, while a relation is symmetric if both and are in E. This property can be expressed as:

(6)

Consequently, symmetry can be quantified by computing the fraction of motifs of size n = 2 that hold 6. To find all motifs of size n = 2 and n = 3 in order to check the metaphorical mechanism for symmetry and transitivity, we make use of the motif detection software by [41].

Counting motif appearance.

Of course, when looking at our metaphorical mappings data, we cannot expect that transitivity and symmetry always hold, in particular since the topic categories are rather broad and most probably combine several semantic domains so that different mappings can involve also quite distant aspects of one and the same category. Thus, we rather study tendencies, that is, a significant over- or under-representation of particular subgraph motifs when compared to random models.

To find out which subgraphs are frequent or rare motifs of a network, we need the following definitions [42]:

Definition 6. A subgraph of is a graph that consist of a subset of V and a connected subset of edges whose nodes belong to . Furthermore, is called an induced subgraph of G if all edges such that , are also in .

Definition 7. Two graphs and are called isomorphic if there exists a bijection f between V and that preserves adjacency, that is, the bijection f is an isomorphism of G and if implies that .

Isomorphic subgraphs represent of course the same motif. To count how many times a motif of n vertices appear, one checks all non-identical connected combinations of n vertices, including the overlapped ones. The exact counting algorithm can be viewed in [41]. To assess the count and decide if a subgraph Ω is significantly over- or underrepresented in the metaphor network, it is compared with the count of the same subgraph in corresponding configuration graph models . The difference between the empirical network count and the mean random model count, , is measured in units of standard deviation :

(7)

A subgraph is considered a significantly over- or underrepresented motif if .

Similarity structure underlying the metaphoric network

Two categories are structurally equivalent if they have the same incoming and outgoing neighbors. Sets of structural equivalent categories are said to form a position in the network, since they play the same role. In practice, though, sets of structurally equivalent vertices are rare and it is therefore necessary to relax this notion and to quantify how similar vertices are regarding their connections [43]. In order to identify roles within the metaphor network, we utilized Hierarchical Cluster Analysis (HCA), which is an unsupervised method. HCA takes a set of categories, along with their incoming and outgoing neighbors, as input, and generates a classification scheme. This scheme, called dendrogram, is a binary tree that visually represents the underlying role structure of the categories in the metaphor network. HCA leverages the given metric structure of a dataset to construct the cluster hierarchy represented by the dendrogram. Agglomerative algorithms, in particular, follow an ascending grouping methodology. This process initiates with clusters consisting of single elements, referred to as leaves. At each subsequent step, the algorithm identifies the pair of closest clusters and merges them into a new cluster. The distances between this merged cluster and other clusters are then recalculated based on pairwise distances between the leaves, thereby progressively building the cluster hierarchy depicted in the dendrogram.

Clustering large data sets with, for example, distances limited to narrow ranges, may produce several different dendrograms if two or more pairs of clusters are equidistant [26]. Therefore, it is necessary to consider these ties in proximity and look for clusters, if any, that appear in a large fraction of the resulting dendrograms [25].

We run an HCA algorithm to classify categories by structural similarity in the network based on the method introduced in [43]. The set of categories is endowed with a metric structure by considering the following distance:

where c, are categories, and are the incoming and outgoing neighbors of category c, and is their symmetric difference.

Then, the Ward grouping methodology is used to produce all dendrograms resulting from ties in proximity. Cluster distances are updated at each step according to the recursive Lance–Williams formula:

where , and C_k are disjoint clusters with size and n_k, respectively, and the coefficients are given by

Then, we compute the fraction of dendrograms in which each of the resulting clusters appears, using the four contrast functions introduced in [25].

Fasttext word embedding

We want to investigate if the distances defined through the network neighborhood contain information similar or different to the semantic distance of a typical word embedding. For this we look for correlations and mutual information between this network role distance and the embedding distances of words, using fastText word embedding in 300 dimensions. The word embedding is trained on Wikipedia and based on the skip-gram model described in [44].

Typically, two different measures of semantic similarity and distance are used in word embeddings: the Euclidean distance between two words in space, or the cosine similarity between the two vectors from the origin to the words of interest. The Euclidean distance between points and in 300 dimensions is given by

(8)

and the semantic dissimilarity according to the cosine similarity is computed as

(9)

To test whether these two semantic measures are related to the metaphor network role similarity we not only look for possible correlations but also for possible mutual information [45]. The mutual information between two different measurements U and V is given by

(10)

where and are the number of samples. Compared to the linear correlation coefficient, the mutual information coefficient reveals in a much more general way the extent to which knowing the result of a measure on a data point informs the value of the second measure.

Graph curvature

To understand the structure of the metaphor network in a more complete and general way, it is useful to learn about its “shape" just as we do to investigate geometric objects. In this section we design experiments to reveal the overall “shape" of the metaphor network. The local geometry of an object is primarily described by its curvature and the the distribution of those local observations informs about the global shape as a whole. This principle of local attributes that together allow inference on global properties was already applied when we looked at the degree distribution. But unlike the degree, discrete curvature is defined not for a vertex but for an edge [46], and therefore it directly informs about the structure of relations. There exist several different translations of the continuous geometrical notion of curvature into discrete graph spaces. We shall look here into those two that have so far proved most useful in empirical data analysis, the Forman Ricci curvature and the Ollivier Ricci curvature [47, 48]. Both concepts have their origin in Riemannian geometry, but allow us to study also structural properties in graphs that have no direct analogue in Riemannian geometry [47].

Forman Ricci curvature.

The Forman Ricci curvature of a directed multigraph measures the prominence of flow structures in a network. For each connected pair of vertices it takes all uni-directed edges e.g. and counts the number of incoming edges to its source and outgoing edges of its target thus quantifying to which degree this edges contributes to a flow structure. Following [49], we define:

Definition 8. (Forman Ricci curvature). Let e be the non-empty set of edges from vertex to vertex . Let further be the set of incoming edges to and similarly the set of outgoing edges from . Then the Forman Ricci Curvature of e is defined as

(11)

A more negative Forman Curvature indicates strong flow structure, while values around zero show the absence of flow.

Ollivier Ricci curvature.

The Ollivier Ricci Curvature of an arc (u, v) quantifies how close the incoming neighbors of u are from the outgoing neighbors of v [47, 48, 50].

To define the Ollivier Ricci Curvature of an edge e from u to v, we compute an optimal transport plan between a probability measure , defined on a neighborhood of u called masses, and a probability measure , defined on a neighborhood of v called holes. The sets of masses and holes are defined as follows [46, 49, 51]:

Definition 9. (Masses and Holes of an Arc). Given an arc of a directed graph , we define the set of masses of e to be the singleton {u} if u does not have incoming neighbors. Otherwise, is the set of incoming neighbors of u. Similarly, the set of holes of e is the singleton if v does not have outgoing neighbors. Otherwise, is the set of outgoing neighbors of v.

The probability measures are the following: If the tail u of e has no incoming neighbors, we set . Otherwise, for each mass , we define , where is the number of incoming neighbors of u. Similarly, if the head v of e has no outgoing neighbors, we set . Otherwise, for each hole , we define , where is the number of outgoing neighbors from v.

We can now define the Ollivier Ricci curvature of an arc .

Definition 10. (Ollivier Ricci curvature). Given an arc of a directed graph , we define the Ollivier Ricci curvature as

where is the transportation distance defined as

Here, represents the directed distance between e₁, an incoming edge to u, and e₂, an outgoing edge from v. It is defined as the minimal number of edges needed to travel from the tail of e₁ to the head of e₂. represents the set of measures on that project to and .

In [51], it is proven that the Ollivier Ricci curvature of (u, v) can be expressed as

where quantifies the fraction of incoming neighbors of u that are at distance from v in an optimal transport plan. Positively curved edges imply that a large fraction of incoming neighbors of u are also outgoing neighbors of v, therefore they are involved in directed triangles (x, u)(u, v)(v, x). Flat edges (u, v) are, for instance, those whose incoming neighbors of u are at distance one from the outgoing neighbors of v, forming directed quadrangles (x, u)(u, v)(v, y)(y, x). Negatively curved arcs are those for which the incoming neighbors of u are at distance larger than one from the outgoing neighbors of v.