Loops and Self-Reference in the Construction of Dictionaries

Dictionaries link a given word to a set of alternative words (the definition) which in turn point to further descendants. Iterating through definitions in this way, one typically finds that definitions loop back upon themselves. We demonstrate that such definitional loops are created in order to introduce new concepts into a language. In contrast to the expectations for a random lexical network, in graphs of the dictionary, meaningful loops are quite short, although they are often linked to form larger, strongly connected components. These components are found to represent distinct semantic ideas. This observation can be quantified by a singular value decomposition, which uncovers a set of conceptual relationships arising in the global structure of the dictionary. Finally, we use etymological data to show that elements of loops tend to be added to the English lexicon simultaneously and incorporate our results into a simple model for language evolution that falls within the “rich-get-richer” class of network growth.

Popular Summary

Humans can communicate complex thoughts and ideas owing to a shared vocabulary and rules for stringing words together in meaningful sentences. These vocabulary words are often interrelated and evolve over time in response to the changing needs of communication. Such relations are embedded in the structure of dictionaries in which language is fixed in time and words are used to define other words. By applying the tools of statistical physics and graph theory, we have discovered new features of word connections and, in particular, that some aspects of vocabularies, such as apparently circular definitions (or “definitional loops”), are not defects but essential to the growth of language.

In our paper we report a quantitative analysis achieved by viewing a dictionary as a network of words: Every word has links pointing to each of the words in its definition. In turn, each of those words points to the words in its definition, and so on. Iterating through the whole dictionary, we find that a complicated directed graph appears. The directed trajectories converge to a small set of fundamental words, strongly connected with a high density of loops, from which all others can in principle be defined.

The graph of the dictionary must be cyclic to some extent, given the requirement that every word have a definition. However, we show here that definitional loops arise not as simple artifacts of the dictionary’s construction but rather as the vehicle by which new concepts are introduced to language. When a totally new concept is created, the dictionary can accommodate it only by revising the topology of the graph.

Our analysis further reveals that short loops describe parts of homogeneous subjects, which then overlap to form larger components that represent in turn whole semantic ideas. Finally, we show that elements of loops tend to be added to the English lexicon simultaneously, and we incorporate our results into a simple model for language evolution.

Figure 1

Graph structure examples. Let ${\mathbb{W}}_{t+1}=\{w_a\}$, ${\mathbb{W}}_t=\{w_b\}$, and ${\mathbb{W}}_{t-1}=\{w_c,w_d\}$, and consider the definability of $w_a$. In (a), we can substitute for $w_b$ its definition $w_b\to D_0(w_b)$, and use it in the definition of $w_a$ so that $D(w_a)=\{w_c,w_d\}$. Since there is no directed path from $w_c$ or $w_d$ to $w_b$, we see that $w_a$ was actually definable before time $t$; no new concept was created by $w_b$. In (b), any definition of $w_a$ must include $w_b$, $w_c$, or $w_d$. Since $w_b$ is a descendant of all of these words, $w_a$ is not definable before $t$, implying that a new concept was formed. In fact, despite having originally been introduced at $t-1$, $w_c$ and $w_d$ are now undefinable before $t$, their meanings having changed with the addition of $w_b$.

Figure 2

Definitional iteration of words in the dictionary. Using a random sample of 100 words, the number of unique nodes that could be reached within the given directed distance of each node is recorded. Nearly all starting points lead to a strongly connected component of 6296 words labeled as the core.

Figure 3

Distribution of definitional loops in the dictionary. The data represent counts of links in the core indexed by the shortest loop in which they appear. For the randomization, links are randomly redrawn between nodes while keeping the in-degree and out-degree distributions of the graph constant.

Figure 4

Distribution of definitional loops for several dictionaries. Both the Wiktionary and WordNet 3.0 graphs are constructed by considering only the first sense of a word in the event of polysemy. However, in WordNet, the ordering of senses is determined empirically according to usage frequencies in written texts, while in Wiktionary, the ordering of senses is determined somewhat arbitrarily, with the definition page as a whole representing a general consensus of users.

Figure 5

An example of a large, strongly connected component in the decomposition. Arrows are drawn from a node to words in its definition. Red links appear first in two-loops, green in three-loops, blue in four-loops, and orange in five-loops.

Figure 6

Dates of origin of words in loops. For each loop, the dates of origin of its element words (in the desired sense) were looked up in the Etymology Dictionary. Compound words and proper nouns were ignored, as well as polysemous words. The median pairwise distance of elements (a) and the mean date of origin (b) were calculated for each of the 310 distinct loops in our analysis.

Figure 7

Extraction of model parameters. Probability distributions were measured for (a) in-degree, (b) out-degree, and (c) strongly connected component sizes for the empirical dictionary graph. The dashed lines in (a) and (c) have slopes 2.1 and 2.9, respectively, while the Poisson fit in (b) has parameter 3.5. In the model, these statistics correspond to a new word attractiveness $a=0.4$ and a probability of new SCC formation $p=0.4$.