The S-Curve Model: How Lexical Diffusion Maps the Gradual Spread of Sound Change Through a Language

The S-curve model reveals that sound changes do not hit a language all at once; instead, they incubate slowly in a handful of high-frequency words before exploding into rapid, systemic adoption and eventually tapering off. Lexical diffusion maps this exact mathematical progression, proving that human speech evolves word by word rather than sound by sound. The traditional assumption among linguists was that when a vowel or consonant shifted, it changed everywhere simultaneously, like a software update applied to an entire operating system. The reality is far messier, resembling a viral trend that starts in a small neighborhood, suddenly sweeps through the city, and leaves a few stubborn holdouts untouched.[3][5]

To understand why this mathematical model upended linguistics, you have to look at the dogma it replaced. In the late nineteenth century, a group of scholars known as the Neogrammarians established a strict rule: sound changes operate without exception. If the Old English long 'a' shifted to a long 'o' (turning 'stan' into 'stone'), it had to happen to every single applicable word in the lexicon at the exact same time. Any exceptions were dismissed as dialect mixing or analogical borrowing. It was an elegant, frictionless theory that made historical linguistics look like a hard science, but it ignored the chaotic reality of how people actually talk.[1][5]

The breaking point for the Neogrammarian view arrived in the 1960s, driven by the sheer computational power of early mainframes. In 1962, Peking University published the Hanyu Fangyin Zihui, a massive dataset containing transcriptions of 2,444 morphemes across 17 modern varieties of Chinese. At the University of California, Berkeley, the linguist William S-Y. Wang and his team digitized this data to apply traditional comparative methods at a scale previously impossible. What they found was a landscape riddled with pervasive irregularities that the old exceptionless rules simply could not explain.[5]

Wang’s team identified stark contradictions in the data. For instance, Middle Chinese words in the third tone class with voiced initials had split into two entirely different forms in the modern Teochew dialect, with no phonetic factor to explain the divergence. They documented 12 specific pairs of words that were perfectly homophonous in Middle Chinese but had evolved completely different modern pronunciations. Under the old rules, this was impossible. Wang realized that these irregularities were not mistakes or dialect anomalies; they were the permanent residue of a sound change that had run out of momentum before it could infect the entire vocabulary.[1][5]

In his landmark 1969 paper, 'Competing Changes as a Cause of Residue,' Wang proposed a radical new framework: lexical diffusion. He argued that a phonetic change is phonetically abrupt but lexically gradual. When a new pronunciation enters a language, it does not instantly overwrite the old one across the board. Instead, it takes root in a small cluster of words—often those most frequently used by a specific subculture or demographic. From there, it spreads word by word, competing with the older forms until it either takes over the lexicon or stalls out, leaving a trail of linguistic fossils behind.[1][3]

This word-by-word spread is not linear; it follows a distinct mathematical trajectory known as the S-curve. The horizontal axis represents time, while the vertical axis measures the number of words in the language that have adopted the new pronunciation. The curve begins with a long, flat latency phase. During this period, the innovation is confined to a tiny fraction of the vocabulary—perhaps 5 to 15 percent—and its growth is agonizingly slow. It is a private, localized quirk, hovering on the edge of extinction as speakers test the new variant in isolated social pockets.[4][5]

If the innovation survives this incubation period and hits a critical mass, the curve suddenly steepens. This is the period of rapid spread, where the new pronunciation cascades through the lexicon. Words fall like dominoes. The social pressure to adopt the new form accelerates, and what was once a marginal variant becomes the dominant standard. This explosive middle phase accounts for the vast majority of the vocabulary shifting in a relatively compressed window of time, creating the steep vertical spine of the mathematical model.[2][4]

Finally, the curve flattens out again at the top, entering the period of maintenance or completion. The change has infected 85 to 90 percent of the applicable words, but the momentum slows drastically. The last few holdouts—often highly salient words, or conversely, extremely rare ones—resist the change. Some may eventually succumb, but others never do. These permanent holdouts become the residue that Wang identified in the Chinese dialects, the irregular verbs and bizarre spellings that frustrate language learners centuries later.[1][4]

While Wang built his theory on historical Chinese data, modern digital communication has allowed linguists to watch the S-curve unfold in real time. A 2013 study presented at the Berkeley Linguistics Society tracked the diffusion patterns of lexical innovations on Twitter. By analyzing massive datasets of social media posts, researchers could map exactly how a new slang term or phonetic spelling incubates within a dense, highly connected social network before breaking out into the broader platform. The digital data perfectly mirrored the S-curve trajectory that Wang had theorized four decades earlier.[2]

The mechanics driving this curve are fundamentally social. As the linguist William Croft noted in 2000, the time course of language change typically follows an S-curve because of 'the desire of hearers to identify with the community' to which a speaker belongs. In the early stages, adopting the new pronunciation is a marker of in-group identity. As it spreads, it becomes a marker of broader cultural relevance. By the time the curve hits its steep upward trajectory, the social cost of not adopting the change outweighs the effort of learning it.[4]

The S-curve is so robust that linguists now debate whether it applies beyond phonetics to syntax and grammar. Historical data on early Modern English morphosyntax shows the same jagged but recognizable S-shaped forms. For example, the spread of specific sentence constructions—like the shift in how negative imperatives were formed between the twelfth and fifteenth centuries—followed a similar path of slow incubation, rapid adoption, and eventual plateau. Whether this represents true lexical diffusion or a different mechanism entirely remains a point of contention, but the mathematical shape of the change is undeniable.[4]

The mathematical elegance of the S-curve lies in how it accounts for human stubbornness. It explains why a modern English speaker still says 'child' and 'children' instead of applying a uniform plural, or why a Teochew speaker maintains two distinct pronunciations for words that were identical a thousand years ago. The model proves that language does not upgrade all at once. It evolves exactly as people do: in fits and starts, adopting the new when it becomes socially necessary, but always keeping a few irregular fossils safely preserved in the plateau.[1][5]