Defining Nodes and Edges in Other Languages in Cognitive Network Science—Moving beyond Single-Layer Networks

: Cognitive network science has increased our understanding of how the mental lexicon is structured and how that structure at the micro-, meso-, and macro-levels influences language and cognitive processes. Most of the research using this approach has used single-layer networks of English words. We consider two fundamental concepts in network science—nodes and connections (or edges)—in the context of two lesser-studied languages (American Sign Language and Kaqchikel) to see if a single-layer network can model phonological similarities among words in each of those languages. The analyses of those single-layer networks revealed several differences in network architecture that may challenge the cognitive network approach. We discuss several directions for future research using different network architectures that could address these challenges and also increase our understanding of how language processing might vary across languages. Such work would also provide a common framework for research in the language sciences, despite the variation among human languages. The methodological and theoretical tools of network science may also make it easier to integrate research of various language processes, such as typical and delayed development, acquired disorders, and the interaction of phonological and semantic information. Finally, coupling the cognitive network science approach with investigations of languages other than English might further advance our understanding of cognitive processing in general.


Introduction
Cognitive network science uses the mathematical tools of network science to examine how the structure of cognitive or linguistic representations influence processing [1][2][3][4][5].Many people might be familiar with the tools of network science from its application to social networks, where nodes represent people, and connections (sometimes called edges) are placed between people who are friends with each other.However, the theory and methodology of network science can be used to map out more than just people who know each other [6].In a psycholinguistic context, nodes may represent individual words, and connections are placed between words that are semantically [7] or, as in Figure 1, phonologically [3] related to form a network of representations in the mental lexicon, or that part of memory that stores information about the words one knows.
The cognitive network science approach should not be confused with artificial neural networks, connectionist networks, or parallel distributed processing models [8][9][10].In the latter approaches, which were inspired by the processes used by groups of neurons to process information (see [11,12] for details on the complexity of actual neurons), nodes are better described as "processing units", and connections can be modified via various learning algorithms to amplify or dampen activation passed from unit to unit.The "network" is hierarchically organized into layers of input units, (sometimes several layers of) hidden units, and output units, and is given a specific task to perform (e.g., classification, prediction, pattern recognition or completion, noise filtering, etc.).Although the "networks" used in the artificial neural network approach have led to many advances in the cognitive and language sciences [10], they are not the same networks used in the cognitive network science approach (which can be described as "maps of how cognitive representations are related to each other").
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Figure 1.
As in [3], nodes represent words, and connections are placed between words that are phonological similarity to each other as defined by a simple computational metric (add, delete, or substitute a phoneme in a word to form the other word).
The cognitive network science approach should not be confused with artificial neural networks, connectionist networks, or parallel distributed processing models [8][9][10].In the la er approaches, which were inspired by the processes used by groups of neurons to process information (see [11,12] for details on the complexity of actual neurons), nodes are be er described as "processing units", and connections can be modified via various learning algorithms to amplify or dampen activation passed from unit to unit.The "network" is hierarchically organized into layers of input units, (sometimes several layers of) hidden units, and output units, and is given a specific task to perform (e.g., classification, prediction, pa ern recognition or completion, noise filtering, etc.).Although the "networks" used in the artificial neural network approach have led to many advances in the cognitive and language sciences [10], they are not the same networks used in the cognitive network science approach (which can be described as "maps of how cognitive representations are related to each other").
An important distinction between the artificial neural network approach and the cognitive network science approach is the central tenet of network science, which states that the structure of the network influences processing in that system [13].Computer simulations have shown that a search algorithm can work very efficiently in a network structured in one way, but very inefficiently in a network with the connections structured in a different way, even when the networks have the same number of nodes and same number of connections [14,15].Thus, the way in which the nodes in a system are connected has an important influence on processing in that system (for examples of this in social networks, see [16,17]).
In the domain of psycholinguistics, the cognitive network science approach was used to examine the structure of a network of English words that were phonologically related ( [3] and see Figure 1).Subsequent behavioral experiments demonstrated that measures of the structure among individual nodes (i.e., the micro-level), among subsets of nodes (i.e., the meso-level), and of the whole network (i.e., the macro-level) were predictive of behavior in various psycholinguistic tasks.For example, at the micro-level of the phonological network, the clustering coefficient measures the extent to which phonological neighbors of a word are also neighbors with each other.This micro-level measure predicted behavior As in [3], nodes represent words, and connections are placed between words that are phonological similarity to each other as defined by a simple computational metric (add, delete, or substitute a phoneme in a word to form the other word).
An important distinction between the artificial neural network approach and the cognitive network science approach is the central tenet of network science, which states that the structure of the network influences processing in that system [13].Computer simulations have shown that a search algorithm can work very efficiently in a network structured in one way, but very inefficiently in a network with the connections structured in a different way, even when the networks have the same number of nodes and same number of connections [14,15].Thus, the way in which the nodes in a system are connected has an important influence on processing in that system (for examples of this in social networks, see [16,17]).
In the domain of psycholinguistics, the cognitive network science approach was used to examine the structure of a network of English words that were phonologically related ( [3] and see Figure 1).Subsequent behavioral experiments demonstrated that measures of the structure among individual nodes (i.e., the micro-level), among subsets of nodes (i.e., the meso-level), and of the whole network (i.e., the macro-level) were predictive of behavior in various psycholinguistic tasks.For example, at the micro-level of the phonological network, the clustering coefficient measures the extent to which phonological neighbors of a word are also neighbors with each other.This micro-level measure predicted behavior in tasks that assess speech perception [18], speech production [19], word-learning [20], short-and long-term memory [21], and perception of the speech to song illusion [22].
At the meso-level, a subset of nodes referred to as keyplayers (which maximally fracture the network when they are removed [23]) were responded to more quickly and accurately in several tasks than words that were comparable to the keyplayers on a number of psycholinguistic measures, but were not in the set of keyplayers [24].At the macrolevel, words outside of the giant component (i.e., the largest group of connected nodes in a network) were produced in a picture-naming task more quickly by people with aphasia than words that were in the giant component despite the fact that words in the giant component tend to have psycholinguistic characteristics (e.g., common, short words) that favor rapid retrieval [25].Similar processing benefits to words outside the giant component have also been observed in spoken word recognition and serial recall tasks ( [26]; see also [27] for another macro-level measure).Together, these studies demonstrate that measures of the structure of the network at the micro-, meso-, and macro-levels predict cognitive and language-related behaviors.
One important point to note about the studies described above is that most of those analyses and psycholinguistic experiments have used words from the English language.As observed by [28], studies of English have provided much insight into the mechanisms that underlie both normal and disordered language processing, and there is available a large number of databases with information about words in the English language, many of which are freely accessible (e.g., [29]).However, there are over 6000 other languages in the world found in a diverse range of language families (e.g., Indo-European, Sino-Tibetan, Afro-Asiatic, and Austronesian), making it unclear how broadly the results of studies in and analyses of English can be generalized [30,31].
Importantly, the phonological networks of words in several other languages have ben examined, including Mandarin, Hawaiian, Basque, and Spanish, which were chosen to represent some of the other language families found in the world [32].Similar network structures were observed across all of the languages they examined (see also [33]).Furthermore, an increasing number of analyses and psycholinguistic experiments have been conducted from the cognitive network science approach using languages other than English, including Mandarin [34], Hebrew [35], bilingual code-switchers [36], and in children across 10 different languages [37], with similar results typically being observed across languages.
However, more network analyses and psycholinguistic experiments in languages other than English are required for at least two reasons.One reason more research is needed in languages other than English is because languages with similar overall network structures may process certain aspects of language in different ways.One example of this is the different influence that degree (known as phonological neighborhood density in psycholinguistics [38]) has in speech perception and speech production in English and in Spanish.
In speech production in English, words with many phonological neighbors/high degree are produced more quickly and accurately than words with few phonological neighbors/low degree [39][40][41].In speech production in Spanish, however, words with many phonological neighbors/high degree are produced slower and less accurately than words with few phonological neighbors/low degree [42,43].
Turning now to speech perception, in English, words with many phonological neighbors/high degree are recognized slower and less accurately than words with few phonological neighbors/low degree [38].In speech perception in Spanish, however, words with many phonological neighbors/high degree are recognized more quickly and accurately than words with few phonological neighbors/low degree [44].
A further analysis by [45] of groups of connected words that were not part of the giant component (referred to simply as components, or as lexical islands in the context of language networks) found that lexical islands in English contained words that were only phonologically related (e.g., reckless, necklace), whereas lexical islands in Spanish contained words that tend to be phonologically and semantically related (e.g., hacendado [land owner-masculine], arrendado [rented-masculine]).It was suggested in [45] the phonological similarity among words in English, but the phonological and semantic similarity among words in Spanish might account for the differences observed in speech production and speech perception tasks across the two languages.These findings highlight one reason why more analyses and psycholinguistic experiments in languages other than English are required.
A second reason that more analyses and psycholinguistic experiments are needed in languages other than English is that there are languages in the world that differ from English on a number of linguistic characteristics, such as the inventory of sounds used to make words, the phonotactic rules governing the sequencing of sounds in words, etc.
Given that several analyses suggest that the structure of the phonological network emerges from phonotactic-like restrictions on word formation [46][47][48], it is logical to deduce that a language with different linguistic constraints may have a different network structure, and, therefore, that the different network structure might have different influences on cognitive processing.
To explore how differences in linguistic characteristics across languages might lead to different cognitive network structures, we considered two lesser-studied languages that differ in radical ways from English: American Sign Language (ASL) and Kaqchikel.In applying network analysis to any domain, it was argued in [49] that it is important to clearly define what system a network is supposed to represent, what the connections in a network represent, and what the nodes in a network represent.In what follows, we discuss, in the context of single-layer networks, how connections might be defined in ASL and how nodes might be defined in Kaqchikel.Analyses of the ASL network and of the Kaqchikel network reveal several differences in network structure compared to the English network in [3], which pose a challenge to the application of network science methods to these languages.We then discuss how other network architectures might be applied to these (and related) languages, and elaborate on how architectures other than single-layer networks may be required to advance psycholinguistics and cognitive science, and expand on cognitive network science more broadly.

The Definition of "Connection" and American Sign Language
Connections in networks are typically used to identify the presence (or absence) of a relationship between two entities (i.e., nodes), such as two people being friends or not in a social network [49].These dichotomous, unweighted connections simplify computations and interpretations, but may also obscure relationships that are in fact variable.For example, the concept of how "similar" two entities are to each other might be gradient rather than dichotomous in nature.Using some sort of threshold to turn the gradient measure into a dichotomous measure runs the risk of altering the structure of the network depending on which the threshold is selected [49].In such cases, using connections that are weighted to reflect that variability may be more informative (despite the increase in the complexity of the network that accompanies the use of weighted connections).After providing background on ASL, we consider differences observed in the structure of a network of ASL signs and discuss how certain characteristics of ASL may force us to reconsider the definition of a connection in a single-layer lexical network.
Although spoken English and American Sign Language (ASL) are both used in the USA, ASL is a natural language that is distinct from spoken English.ASL is not simply a manual version of spoken English, but has its own syntax and grammar.Similarly, although English is also spoken and mutually intelligible in the UK, Australia, Canada, and elsewhere in the world, British Sign Language (BSL) and Australian Sign Language (Auslan) are distinct from and not mutually intelligible with ASL.In Canada, ASL is typically used by Anglophone communities and Langue des Signes Québécoise (LSQ) is typically used by Francophone communities.Although ASL and LSQ are both used in Canada, the two languages are distinct from each other and distinct from spoken English and spoken French, respectively.
Instead of using concatenations of spoken sounds (i.e., phonemes) to represent words, ASL uses "signs", or specific body and facial movements, to represent a word or concept.Signs are typically formed with the hands, with hand shape, palm orientation, hand movement, finger orientation, and hand location being some key features that distinguish words from each other.In spoken languages, it is generally believed that there is an arbitrary mapping between form and meaning.However, in ASL (and other sign languages), there are many examples of iconicity, or the sign resembling the referent in some way [50,51].
One could argue that each sign/word is a distinct entity and, therefore, provides a clear definition of what constitutes a node in a network of ASL signs (cf., [49,52]).However, defining the connections between signs/nodes in ASL using a definition of "phonological similarity" similar to the definition previously used in the network of spoken English [3] and other spoken languages [32] leads to a number of problems.We discovered some of these problems when we tried to create a single-layer "phonological" network of the signs found in the publicly available ASL-LEX 2.0 database [53].

Methodology
The 15 features used to phonologically transcribe the 2723 signs in ASL-LEX 2.0 [53] were used to construct two unweighted networks, each with a different threshold to convert the continuous measure of "number of phonological features shared by two signs" (as indicated in the supplementary file NeigborPairs.csv in [53]) into a dichotomous measure of phonological similarity.In the first network, we placed a connection between two signs if they shared any 10 or more of the 15 coded features.Given that most of the previous work on phonological neighborhood density examined monosyllabic CVC words (cf., [38,54]), this value (10/15 = 67% overlap) was comparable to having two out of three phonemes in a spoken word in common, and therefore seemed like a reasonable value with which to start.After examining the network structure of the resulting network, we then constructed a second network in which we placed a connection between two signs if they shared any 12 or more of the 15 coded features (12/15 = 80% overlap) to examine how a different threshold might influence the structure of the network.Gephi (version 0.10.1 [55]) was used to analyze the resulting networks.

Results
In Table 1, we report the summary values for several network measures for the two ASL networks with different numbers of features used as the threshold to create unweighted networks and, for comparison, the (unweighted) network of spoken English [3] constructed using the Levenshtein-like metric.Note: 10+ features = the network with connections placed between signs that shared 10 or more features.12+ features = the network with connections placed between signs that shared 12 or more features.English = the network of spoken English words examined in [3].GC = giant component.See the text for explanation of the network measures.
Network size refers to the number of nodes in each network.# of Connections refers to the number of connections between the nodes in each network.Mean degree refers to the mean number of connections per node.Network diameter refers to the shortest distance between the two most distant nodes in the network.The shortest path length refers to the smallest number of connections that must be traversed between two nodes in the network.Average (Shortest) Path Length refers to the mean value of all possible shortest path lengths.
Components are groups of nodes that are connected to each other in some way (also referred to as lexical islands).A node with no connections is referred to as an isolate (or lexical hermit).Connected Components refers to the number of components in the network.The component with the most nodes in it is referred to as the Giant Component.

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Whereas components contain nodes that are connected to each other but are disconnected from other components, Communities refer to subsets of nodes in the giant component that tend to be more connected with other nodes in the same community than to nodes in other communities [56].The Louvain modularity optimization method [57] was used to compute Modularity, Q, which assesses the well-formedness of the communities.Higher values of Q indicate a higher density of connections within communities compared to the density of connections between communities [56].
The clustering coefficient refers (in a lexical network) to the proportion of phonological neighbors of a target word that are also neighbors of each other (see [13] for a mathematically formal definition).The Average Clustering Coefficient refers to the mean clustering coefficient value of all of the nodes with more than 1 neighbor.
In the ASL network with connections placed between signs with 10 or more features in common, there was one component, meaning that 100% of the nodes were found in the giant component, and there were no smaller components or isolates (i.e., lexical islands or lexical hermits).In the ASL network with connections placed between signs with 12 or more features in common, there were 53 components.The giant component contained 2664 of the 2723 signs (97% of the nodes).The remaining nodes were found in five components/islands containing two nodes connected to each other, one component/island containing three nodes connected to each other (~1% of the nodes in smaller components), and 46 isolates/hermits (2%).For comparison, the English network [3] had 6508 of the 19,340 words in the giant component (34%), 2567 nodes in variously sized components (13%), and 10,265 nodes that were isolates/hermits (53%).

Discussion
The ASL-LEX 2.0 database contains 2723 signs as well as psycholinguistic information for each sign, and a phonological transcription for each word using 15 features to code for the location and shape of the hand [53].The phonological transcriptions include these 15 features: (1) sign type (one-handed, two-handed, symmetrical or alternating, two-handed asymmetrical with the same hand configuration, or two-handed asymmetrical with different hand configurations), (2) major location, (3) minor location, (4) selected fingers, (5) flexion, (6) movement, (7) second minor location, (8) a change in abduction, (9) a change in flexion, (10) ulnar rotation, (11) contact between the hand and major location, (12) thumb position, (13) thumb contact with the selected fingers, (14) the handshape of the dominant hand, and (15) the handshape of the non-dominant hand.These 15 phonological features uniquely identify 70% of the signs, with the remaining signs being homophonous or differing on non-coded features.The lack of distinction among signs brought about by the limited number of coded phonological features and potential differences between signs in non-coded features made it difficult to clearly define the connections in a network of ASL.
Although phonological similarity between words in spoken languages has been defined in a number of ways (e.g., [38,58]), the most common way to define a phonological neighbor (and the method employed in [3,32]) is derived from Levenshtein distance (a metric used in computer science to measure the similarity of two strings of characters).In this approach, a single phoneme in one word is added, deleted, or substituted to form another word [59,60], resulting in a connection being placed between those two words to represent the "phonological similarity" between those two words (see Figure 1).Using this Levenshtein-like definition of phonological similarity, [3] was able to have nodes represent English words and connections be simply present or absent between nodes-referred to as unweighted connections in [49].
However, if the same Levenshtein-like metric is applied to the signs in ASL-LEX 2.0, there would be very few signs with a neighbor due to the lack of "true" minimal pairs (i.e., words that differ by a single phoneme, like man and tan or cat and cap) in many sign languages [61].Therefore, we may need a different definition of "phonological similarity" to determine how we place connections between nodes in a network of ASL signs, such as defining "phonological similarity" with weighted connections.That is, perhaps connections could be placed between every possible pair of signs in the ASL lexicon (to create a fully connected network), with the weight on the connection reflecting the number of phonological features that overlap in the two signs.Thus, two signs that shared 1 of the 15 features would have a weight of (1/15 = 0.07) on the connection between them, two signs that shared 10 of the 15 features would have a weight of (10/15 = 0.67) on the connection between them, two signs that shared 12 of the 15 features would have a weight of (12/15 = 0.8) on the connection between them, etc.
One implication of using weighted connections in a network of ASL signs is that the fully connected network structure that emerges contrasts with the macro-level network structure that was observed in several spoken languages [32].It was found (among other things) in [32] that the phonological networks of five spoken languages contained a giant component that was smaller (containing 34-66% of the nodes) than is typically observed in networks of social or technological systems (where upwards of 90% of the nodes in the network are found in the giant component).Importantly, it was found that words outside of the giant component were produced, perceived, or recalled better than words that were in the giant component [25,26,62].This effect has also been observed in a computer simulation of language processing in a network of English words [63].If a similar behavioral effect exists in ASL, then representing the network as a weighted fully connected network without a (smaller) giant component and lexical islands/hermits might result in a failure to capture certain effects that have been observed in other (spoken) languages.
Using weighted connections between the ASL signs/nodes may have other implications.As noted by [64], several studies have demonstrated that phonologically related signs may prime or inhibit sign production or comprehension, but those studies have often provided conflicting evidence across studies.It was suggested in [64] that the different results across studies may be an artifact of how phonological similarity was defined across the studies (see also [65]).Using weighted connections in a network of ASL signs adds yet another definition of phonological similarity to the literature and does little to clarify the empirical differences that were observed by [64] regarding the influence of phonological similarity on sign processing.
Similarly, a weighted fully connected network for ASL signs may capture distinctions that typical language users do not make.Consider the findings of [66] who examined whether the path length (i.e., the number of connections that must be traversed to move from one node to another) could be used as a measure of the semantic distance between two words in a network of Hebrew words with connections representing semantic relatedness between words.
In one of their experiments, ref. [66] asked participants to judge the semantic relatedness between two words that were separated by varying path lengths.They found that reaction times increased and relatedness judgements decreased as the path length increased (suggesting that the two words were becoming less related as more connections between them were traversed), but only up to a path length of 4. With more than four connections separating the words, the participants responded rapidly (i.e., a significant decrease in reaction time) and dominantly judged the word pairs as unrelated.This response pattern suggests that semantic relatedness was continuous until a certain threshold was reached (i.e., four connections separated the words).Beyond that threshold, semantic relatedness became categorical.(For a similar decrease in influence with the increase in the number of connections between people in a social network, see [67].) If a similar threshold exists for phonological similarity (see [68])-that is, word pairs vary continuously in their similarity up to a certain distance, but beyond a certain threshold similarity, it becomes categorical-then the weights on the connections beyond that threshold may represent information that language users do not actually use.Representing information in the model that language users do not actually use may reduce the value of using a network with weighted connections and distort our understanding of the language processing system.Given the difficulty in defining a connection between signs/nodes to form a singlelayer network of ASL similar to the single-layer phonological network of English [3], researchers interested in using the powerful quantitative tools of cognitive network science to examine ASL might consider alternative network architectures.For example, it was examined in [69] how several different network architectures might account for the wellstudied effects on language processing of phonotactic probability or the frequency with which phonological segments and sequences of phonological segments appear in words (for a review, see [70]).One type of architecture they examined was a bipartite network, a type of multilayer network with two different types of nodes with connections between nodes in one layer to nodes in the other layer, but no connections within the same layer [71].For a different type of multilayer network that might also be useful for representing ASL, see [72].
In the bipartite network used in [69], one layer of nodes represented words, and the other layer of nodes represented phonological segments.Connections were placed from words to the phonological segments they contained (but not between word nodes or between phonological segment nodes), as in Figure 2. Computer simulations on the bipartite network replicated the several well-studied effects of phonotactic probability on the processing of real words and non-words [73,74].In the case of ASL, a bipartite network with nodes in one layer representing words and nodes in the other layer representing the 15 phonological features in the transcriptions from [53] could provide researchers with important insights related to ASL processing and language processing in general.
vary continuously in their similarity up to a certain distance, but beyond a certain threshold similarity, it becomes categorical-then the weights on the connections beyond that threshold may represent information that language users do not actually use.Representing information in the model that language users do not actually use may reduce the value of using a network with weighted connections and distort our understanding of the language processing system.
Given the difficulty in defining a connection between signs/nodes to form a singlelayer network of ASL similar to the single-layer phonological network of English [3], researchers interested in using the powerful quantitative tools of cognitive network science to examine ASL might consider alternative network architectures.For example, it was examined in [69] how several different network architectures might account for the wellstudied effects on language processing of phonotactic probability or the frequency with which phonological segments and sequences of phonological segments appear in words (for a review, see [70]).One type of architecture they examined was a bipartite network, a type of multilayer network with two different types of nodes with connections between nodes in one layer to nodes in the other layer, but no connections within the same layer [71].For a different type of multilayer network that might also be useful for representing ASL, see [72].
In the bipartite network used in [69], one layer of nodes represented words, and the other layer of nodes represented phonological segments.Connections were placed from words to the phonological segments they contained (but not between word nodes or between phonological segment nodes), as in Figure 2. Computer simulations on the bipartite network replicated the several well-studied effects of phonotactic probability on the processing of real words and non-words [73,74].In the case of ASL, a bipartite network with nodes in one layer representing words and nodes in the other layer representing the 15 phonological features in the transcriptions from [53] could provide researchers with important insights related to ASL processing and language processing in general.By considering a language other than English (specifically ASL), we see that the definition of phonological similarity that has been previously used to place connections between words/nodes in cognitive networks of several other spoken languages (e.g., [32]) may not apply across all languages.Although our exploration of ASL suggests that researchers may not be able to use the same methodology and network architecture used previously (which may impede direct comparisons with previous work), there are modeling choices that are computationally and theoretically defendable, as well as alternative network architectures that may still provide insights to researchers interested in studying ASL with the cognitive network science approach.

The Definition of "Node" and Kaqchikel
Nodes in networks are typically used to represent distinct entities in a system [49].Continuing with the example of a friendship network, nodes represent individual people who may or may not be friends with each other (with connections placed between people who are friends).Now consider another example involving people that may interact with each other, namely, the employees of several companies.Collapsing all of the employees of a company into a single node to represent the company obscures the fact that the CEO of one company may interact with the CEO of another company, but the same CEO is unlikely to interact with either the employees working in the warehouse of their own company or the employees working in the warehouse of another company.Collapsing all of the employees into nodes representing the company they work for also obscures the fact that the employees working in the warehouse of one company may interact with the employees working in the warehouse of another company.In addition to obscuring the existence (or lack thereof) of certain interactions between entities, changing the definition of the node from individual people to companies is likely to alter the size and density of the resulting networks, which may also alter other network measures [49].After providing background on Kaqchikel, we discuss how certain characteristics of Kaqchikel may force us to reconsider the definition of a node in a single-layer lexical network.
Kaqchikel is an Indigenous spoken language in the K'ichean branch of the Mayan language family.There are 411,089 speakers of Kaqchikel (4 y.o. or older) found in the central highlands of Guatemala, with most speakers residing in regions between Guatemala City and Lake Atitlán [75].Because of the small number of speakers, Kaqchikel is considered an endangered language.Kaqchikel has a variable word order, although the most common word orders are Verb-Object-Subject (VOS) and Subject-Verb-Object (SVO) [75][76][77].Kaqchikel is also a pro-drop language, meaning that an explicit pronoun can be dropped or omitted, because it can be inferred grammatically or contextually.In contrast, English is not a pro-drop language, meaning that an explicit pronoun must be used (except in limited contexts, such as imperative sentences, e.g., Come here!).
Kaqchikel is also moderately synthetic, meaning that single words can contain multiple affixes.In contrast, English is typically categorized as being more analytic, meaning that it relies more on word order than inflectional morphology to convey meaning.To illustrate this characteristic of Kaqchikel (using examples from [77]), consider the positional root tz'uy [to sit].Positional roots are a special class of consonant-vowel-consonant (CVC) roots that cannot appear underived and can be root intransitive verbs, root transitive verbs, root nouns, and root adjectives [77].To form an intransitive verb form of the positional root tz'uy, the suffix -e' is added: Xitz'uye' (x-i-tz'uy-e') ["I sat down."].To form a transitive verb form of the positional root tz'uy, the suffix -VRb'a' is added (where VR is a copy of the vowel found in the root): Xintz'uyub'a' (X-in-tz'uy-ub'a') ["I sat her/him down."].
Given the synthetic nature of Kaqchikel, what constitutes a word in Kaqchikel differs from what constitutes a word in the more analytic language of English.Indeed, there is no agreed upon definition of "word" that applies cross-linguistically [78].The definition of word in the more analytic language of English lends itself easily to identifying distinct entities to represent as nodes (with connections placed between nodes that are related in some way).The definition of word in the more synthetic language of Kaqchikel challenges our ability to define distinct entities and, therefore, to define a node in a way that does not obscure other information or hide interactions among other units that may be linguistically relevant.
One possibility is to represent Xitz'uye' ["I sat down."] and Xintz'uyub'a' ["I sat her/him down."] as separate nodes.Given all of the possible combinations of prefixes, suffixes, etc. used to derive various forms of the ~300 positional roots in Kaqchikel, the number of nodes that would be required just to represent this unique class of roots would quickly multiply.Alternatively, one could form a single-layer network of positional roots, as we do below, and simply assume that some sort of automatic rule-based morphosyntactic process might exist to derive various forms of the positional and other roots as needed, akin to the way that regular verbs may be conjugated in English [79].

Methodology
We obtained 303 CVC positional roots from [77] and transcribed them to computerreadable characters to represent each phoneme.Using the Levenshtein-like definition of phonological similarity used in [3], we created a network with roots being connected if changing one phoneme resulted in another root.Gephi (version 0.10.1;[55]) was used to visualize and analyze the resulting networks.A subnetwork showing tz'uy (represented as Zuy in the computer readable transcription), the neighbors of tz'uy (cuy and ZuB), and the neighbors of the neighbors of tz'uy (luB, ruB, ZEB, cuK, cly, and cup) is shown in Figure 3.Such subnetworks are referred to as 2-hop ego networks.
suffixes, etc. used to derive various forms of the ~300 positional roots in Kaqchikel, the number of nodes that would be required just to represent this unique class of roots would quickly multiply.Alternatively, one could form a single-layer network of positional roots, as we do below, and simply assume that some sort of automatic rule-based morphosyntactic process might exist to derive various forms of the positional and other roots as needed, akin to the way that regular verbs may be conjugated in English [79].

Methodology
We obtained 303 CVC positional roots from [77] and transcribed them to computerreadable characters to represent each phoneme.Using the Levenshtein-like definition of phonological similarity used in [3], we created a network with roots being connected if changing one phoneme resulted in another root.Gephi (version 0.10.1;[55]) was used to visualize and analyze the resulting networks.A subnetwork showing 'uy (represented as Zuy in the computer readable transcription), the neighbors of 'uy (cuy and ZuB), and the neighbors of the neighbors of 'uy (luB, ruB, ZEB, cuK, cly, and cup) is shown in Figure 3.Such subnetworks are referred to as 2-hop ego networks.

Results
In Table 2, we report the summary values for the same network measures we reported for the ASL networks above.We again include the values from the (unweighted) network of spoken English [3] that was also constructed using the Levenshtein-like metric.Like the ASL networks, the network containing the positional roots in Kaqchikel had a large proportion of nodes in the giant component (95%).The remaining nodes were found in one component/island containing three nodes connected to each other, two components/islands containing two nodes connected to each other (2% of the nodes in smaller components), and eight isolates/hermits (3%).Recall that the English network [3] had 6508 of the 19,340 words in the giant component (34%), 2567 nodes in variously sized components (13%), and 10,265 nodes that were isolates/hermits (53%).

Discussion
At present, there is (to our knowledge) no behavioral data from Kaqchikel on the influence of phonological neighbors in perception or production, or evidence regarding fully derived forms versus roots and affixes being represented in the lexicon.It is also unclear if the location of words in the network (i.e., in vs. out of the giant component) affects processing in other languages as it does in English [25,26,62,63].
Clearly, Kaqchikel and languages that are even more synthetic than Kaqchikel pose a challenge not only to the linguistic definition of "word", but also to the definition of node in cognitive network science.This is especially true in the single-layer architecture used in the English phonological network [3].Perhaps, as with ASL, some alternative network architecture could be employed to model Kaqchikel.If one considers a bipartite network like that proposed for ASL, one could have CVC roots on one level and various affixes on another level.In Kaqchikel, prefixes are inflectional, whereas suffixes can be either derivational or inflectional [77].
However, words in languages like Kaqchikel are often formed by more morphemes than might easily be captured by pairs of nodes interacting with each other.Rather, three or more morphemes/nodes may interact as a group (or a set) to convey meaning (with the "word" emerging from that interaction).In cases where node interactions are more complex than interacting dyads, a hypergraph might be a better architecture to consider [80].In a hypergraph of Kaqchikel, nodes might represent various roots and affixes, and hyperedges would group together nodes of various linguistic classes as well as various linguistic components that form legal "words" in Kaqchikel from their higher-order interaction.Figure 4 shows a simplified hypergraph to illustrate this idea.

Conclusion: Moving Forward with Cognitive Network Science
Our understanding of typical language processing and various speech, language, and hearing disorders has been advanced by the application of cognitive network science methods using relatively simple single-layer networks (e.g., [81,82]).The present analyses

Conclusion: Moving Forward with Cognitive Network
Our understanding of typical language processing and various speech, language, and hearing disorders has been advanced by the application of cognitive network science methods using relatively simple single-layer networks (e.g., [81,82]).The present analyses of ASL and Kaqchikel suggests that there may be a limit to what can be learned by using single-layer networks and highlights the need to apply the cognitive network approach to languages other than English to determine where some of those limits lie.Although ASL is only one example of the numerous sign languages used in the world, and Kaqchikel is only one example from the Mayan language family, we believe other sign languages and other languages from the Mayan family (and non-analytic languages from other language families) will also challenge the (same or different) assumptions of the cognitive network approach examined in this paper.Perhaps different metrics of similarity [58] can resolve some of the problems observed in the present analyses.Alternatively, additional processes, such as an automatic rule-based morphosyntactic process [79], could be added to language networks to overcome other challenges observed in the present study.
We believe that recent work using multilayer networks (e.g., [83,84]), feature-rich networks (e.g., [85]), and other network architectures such as bipartite networks (e.g., [69]) offers researchers a possible direction forward.However, much of the work with these alternative architectures has again been in the English language, further highlighting the need to apply the cognitive network approach to languages other than English to better understand how alternative network architectures might be able to accommodate the linguistic characteristics of other languages.Such work will also increase our understanding of language-related processes and cognitive processing in general [30].Until more research is conducted with alternative network architectures and languages other than English, it will be difficult to determine if this direction is indeed moving the field forward.
An important tenet of the (cognitive) network science approach is that the structure of a network influences processing in that system [13].It was found in [86] in Hebrew speakers that less creative individuals had semantic networks with a rigid structure and few connections between nodes, whereas the semantic networks of more creative individuals had broader connections (connecting seemingly distant concepts) and was more spread out.Looking at novice speakers of a second language, semantic networks with a rigid structure and few connections between nodes were found in [87].The novice L2 speakers also produced less creative/more utilitarian responses in a word association task.In contrast, proficient language users (i.e., L1 speakers) had semantic networks with broader, more spread-out connections and produced more creative responses in a word association task.It was suggested in [87] that the difference in the structure of the semantic networks of L2 vs. L1 speakers might provide an alternative account of the "foreign language effect", where speakers of a foreign language make decisions about moral dilemmas that are more utilitarian in nature than native speakers of a language [88].By looking at other languages, the cognitive network approach may reveal other insights about cognitive processing that conventional approaches are not able to show us.
The cognitive network science approach offers advantages to modeling language processing that other approaches do not offer.Among those advantages is that the cognitive network science approach allows for the integration of various language processes.That is, models of different language processes do not need to be fragmented or independent from each other as is often seen in other modeling approaches, such as separate connectionist models of related language processes, such as word recognition [89] and word production [90].For example, typical and delayed language development can be modeled by a growing network, as in the network of semantic relationships among words examined by [91] or the phonological network examined by [92]; see also [93].Typical changes and acquired disorders that occur with increased age can also be captured by dynamic networks (e.g., [94]) or by damaging the network structure in some way (e.g., [62]).And the interaction of phonological and semantic information, for example, can be modeled in a multilayered network [95].
Future work should also consider how phonological and information interacts with other cognitive processes, such as directed attention or creative processes (as might occur when playing a language-related game), social influences (as might occur when switching from a professional style of speaking to a casual style of speaking based on the interlocutors), and other types of memory (such as motor patterns or episodic memories related to one's personal experiences).Additional research is also required to determine if the diffusion of activation that is commonly used to model cognitive processing in language networks [18] is sufficient to capture other types of cognitive processing or if other mechanisms might be required, such as random walks, directed walks, or a combination of directed and random walks, as discussed in [5].It will also be important for future work to determine how diffusion processes and different types of walks affect processing in the alternative network architectures suggested here.
The equal parts of theory and methodology in the network science approach means that: ". ..[N]etworks offer both a theoretical framework for understanding the world and a methodology for using this framework to collect data, test hypotheses, and draw conclusions" ( [6]; pg.5).We believe that cognitive networks with various architectures hold much promise for increasing our understanding of the variety of languages in the world, while still providing a common theoretical framework [96].Coupling work in cognitive networks with the investigation of languages other than English also holds much promise for increasing our understanding of human cognition in general [30].

Figure 1 .
Figure 1.As in[3], nodes represent words, and connections are placed between words that are phonological similarity to each other as defined by a simple computational metric (add, delete, or substitute a phoneme in a word to form the other word).

Figure 2 .
Figure 2.An example of a bipartite network with nodes in one layer representing words (top layer)and nodes in another layer representing phonological segments (bottom layer).Connections are placed from word nodes to the phonological segments contained in that word, but no connections are placed between word nodes or between phonological segment nodes.

Figure 3 .
Figure 3.A 2-hop ego network for the positional root 'uy [represented as Zuy in the computer readable transcription].

Figure 3 .
Figure 3.A 2-hop ego network for the positional root tz'uy [represented as Zuy in the computer readable transcription].

Figure 4 .
Figure 4.A simplified illustration of a hypergraph.Nodes (black dots) represent various morphemes.Hyperedges encircle morphemes that belong to the same grammatical class or that might interact to form a word that conveys meaning.The top set of nodes represent the class of positional roots.The bo om set of nodes represent various affixes.The middle set of nodes represents the derived form of a root (i.e., the emergent "word").Note that pair-wise connections among nodes would not adequately capture the relationships among nodes represented in the figure.

Figure 4 .
Figure 4.A simplified illustration of a hypergraph.Nodes (black dots) represent various morphemes.Hyperedges encircle morphemes that belong to the same grammatical class or that might interact to form a word that conveys meaning.The top set of nodes represent the class of positional roots.The bottom set of nodes represent various affixes.The middle set of nodes represents the derived form of a root (i.e., the emergent "word").Note that pair-wise connections among nodes would not adequately capture the relationships among nodes represented in the figure.

Table 1 .
Summary of network measures for two ASL networks and the spoken English network.

Table 2 .
Network measures for positional roots in Kaqchikel and the spoken English network.