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Supporting Information: On the Universal Structure of Human Lexical Semantics Hyejin Youn,1, 2, 3 Logan Sutton,4 Eric Smith,3, 5 Cristopher Moore,3 Jon F. Wilkins,3, 6 Ian Maddieson,7, 8 William Croft,7 and Tanmoy Bhattacharya3, 9 1 Institute for New Economic Thinking at the Oxford Martin School, Oxford, OX2 6ED, UK 2 3 4 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA American Studies Research Institute, Indiana University, Bloomington, IN 47405, USA 5 Earth-Life Sciences Institute, Tokyo Institute of Technology, 2-12-1-IE-1 Ookayama, Meguro-ku, Tokyo, 152-8550, Japan 6 7 Department of Linguistics, University of New Mexico, Albuquerque, NM 87131, USA 8 9 Ronin Institute, Montclair, NJ 07043 Department of Linguistics, University of California, Berkeley, CA 94720, USA MS B285, Grp T-2, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 2 CONTENTS I. Methodology for Data Collection and Analysis 3 A. Criteria for selection of meanings 3 B. Criteria for selection of languages 4 C. Semantic analysis of word senses 6 D. Bidirectional translation, and linguists’ judgments on aggregation of meanings 9 E. Trimming, collapsing, projecting II. Notation and Methods of Network Representation A. Network representations 15 16 16 1. Multi-layer network representation 17 2. Directed-hyper-graph representation 19 3. Projection to directed simple graphs and aggregation over target languages 19 B. Model for semantic space represented as a topology 20 1. Interpretation of the model into network representation 21 2. Beyond the available sample data 22 C. The aggregated network of meanings 23 D. Synonymous polysemy: correlations within and among languages 23 E. Node degree and link presence/absence data 26 F. Node degree and Swadesh meanings 26 III. Universal Structure: Conditional dependence 27 A. Comparing semantic networks between language groups 28 1. Mantel test 28 2. Hierarchical clustering test 29 B. Statistical significance 32 C. Single-language graph size is a significant summary statistic 32 D. Conclusion 33 IV. Model for Degree of Polysemy 33 A. Aggregation of language samples 33 B. Independent sampling from the aggregate graph 34 1. Statistical tests 34 3 2. Product model with intrinsic property of concepts 36 3. Product model with saturation 37 C. Single instances as to aggregate representation 41 1. Power tests and uneven distribution of single-language p-values 42 2. Excess fluctuations in degree of polysemy 43 3. Correlated link assignments 45 References I. 48 METHODOLOGY FOR DATA COLLECTION AND ANALYSIS The following selection criteria for languages and words, and recording criteria from dictionaries, were used to provide a uniform treatment across language groups, and to compensate where possible for systematic variations in documenting conventions. These choices are based on the expert judgment of authors WC, LS, and IM in typology and comparative historical linguistics. A. Criteria for selection of meanings Our translations use only lexical concepts as opposed to grammatical inflections or function words. For the purpose of universality and stability of meanings across cultures, we chose entries from the Swadesh 200-word list of basic vocabulary. Among these, we have selected categories that are likely to have single-word representation for meanings, and for which the referents are material entities or natural settings rather than social or conceptual abstractions. We have selected 22 words in domains concerning natural and geographic features, so that the web of polysemy will produce a connected graph whose structure we can analyze, rather than having an excess of disconnected singletons. We have omitted body parts—which by the same criteria would provide a similarly appropriate connected domain—because these have been considered previously [1–4]. The final set of 22 words are as follows: • Celestial Phenomena and Related Time Units: STAR, SUN, MOON, YEAR, DAY/DAYTIME, NIGHT • Landscape Features: SKY, CLOUD(S), SEA/OCEAN, LAKE, RIVER, MOUNTAIN 4 • Natural Substances: STONE/ROCK, EARTH/SOIL, SAND, ASH(ES), SALT, SMOKE, DUST, FIRE, WATER, WIND B. Criteria for selection of languages The statistical analysis of typological features of languages inevitably requires assumptions about which observations are independent samples from an underlying generative process. Since languages of the world have varying degrees of relatedness, language features are subject to Galton’s problem of non-independence of samples, which can only be overcome with a full historical reconstruction of relations. However, long-range historical relations are not known or not accepted for most language families of the world [5]. It has become accepted practice to restrict to single representatives of each genus in statistical typological analyses [6, 7].1 In order to minimize redundant samples within our data, we selected only one language from each genus-level family [8]. The sample consists of 81 languages chosen from 535 genera in order to maximize geographical diversity, taking into consideration population size, presence or absence of a written language, environment and climate, and availability of a good quality bilingual dictionary. The list of languages in our sample, sorted by geographical region and phylogenetic affiliation is given in Table I, and the geographical distribution is shown in Fig. 1. The contributions of languages to our dataset, including numbers of words and of polysemies, are shown as a function of language ranked by each language’s number of speakers in Fig. 2. FIG. 1. Geographical distribution of selected languages. The color map represents the number of languages included in our study for each country. White indicates that no language is selected and the dark orange implies that 18 languages are selected. For example, the United States has 18 languages included in our study because of the diversity of Native American languages. 1 As long as the proliferation of members within a language family is not correlated with their typological characteristics, this restriction provides no protection against systematic bias, and in general it must be weighed against the contribution of more languages to resolution or statistical power. 5 Region Africa Family Khoisan Genus Northern Central Southern Niger-Kordofanian NW Mande Southern W. Atlantic Defoid Igboid Cross River Bantoid Nilo-Saharan Saharan Kuliak Nilotic Bango-Bagirmi-Kresh Afro-Asiatic Berber West Chadic E Cushitic Semitic Eurasia Basque Basque Indo-European Armenian Indic Albanian Italic Slavic Uralic Finnic Altaic Turkic Mongolian Japanese Japanese Chukotkan Kamchatkan Caucasian NW Caucasian Nax Katvelian Kartvelian Dravidian Dravidian Proper Sino-Tibetan Chinese Karen Kuki-Chin-Naga Burmese-Lolo Naxi Oceania Hmong-Mien Hmong-Mien Austroasiatic Munda Palaung-Khmuic Aslian Daic Kam-Tai Austronesian Oceanic Papuan Middle Sepik E NG Highlands Angan C and SE New Guinea West Bougainville East Bougainville Australian Gunwinyguan Maran Pama-Nyungan Americas Eskimo-Aleut Aleut Na-Dene Haida Athapaskan Algic Algonquian Salishan Interior Salish Wakashan Wakashan Siouan Siouan Caddoan Caddoan Iroqoian Iroquoian Coastal Penutian Tsimshianic Klamath Wintuan Miwok Gulf Muskogean Mayan Mayan Hokan Yanan Yuman Uto-Aztecan Numic Hopi Otomanguean Zapotecan Paezan Warao Chimúan Quechuan Quechua Araucanian Araucanian Tupı́-Guaranı́ Tupı́-Guaranı́ Macro-Arawakan Harákmbut Maipuran Macro-Carib Carib Peba-Yaguan Language Ju|’hoan Khoekhoegowab !Xóõ Bambara Kisi Yorùbá Igbo Efik Swahili Kanuri Ik Nandi Kaba Démé Tumazabt Hausa Rendille Iraqi Arabic Basque Armenian Hindi Albanian Spanish Russian Finnish Turkish Khalkha Mongolian Japanese Itelmen (Kamchadal) Kabardian Chechen Georgian Badaga Mandarin Karen (Bwe) Mikir Hani Naxi Hmong Njua Sora Minor Mlabri Semai (Sengoi) Thai Trukese Kwoma Yagaria Baruya Kolari Rotokas Buin Nunggubuyu Mara E and C Arrernte Aleut Haida Koyukon Western Abenaki Thompson Salish Nootka (Nuuchahnulth) Lakhota Pawnee Onondaga Coast Tsimshian Klamath Wintu Northern Sierra Miwok Creek Itzá Maya Yana Cocopa Tümpisa Shoshone Hopi Quiavini Zapotec Warao Mochica/Chimu Huallaga Quechua Mapudungun (Mapuche) Guaranı́ Amarakaeri Piro Carib Yagua TABLE I. The languages included in our study. Notes: Oceania includes Southeast Asia; the Papuan languages do not form a single phylogenetic group in the view of most historical linguists; other families in the table vary in their degree of acceptance by historical linguists. The classification at the genus level, which is of greater importance to our analysis, is more generally agreed upon. 6 FIG. 2. Vocabulary measures of languages in the dataset ranked in descending order of the size of the speaker populations. Population sizes are taken from Ethnologue. Each language is characterized by the number of meanings in our polysemy dataset, of unique meanings, of non-unique meanings defined by exclusion of all single occurrences, and of polysemous words (those having multiple meanings), plotted in blue, green, red, and cyan, respectively. We find a nontrivial correlation between population of speakers and data size of languages. C. Semantic analysis of word senses All of the bilingual dictionaries translated object language words into English, or in some cases, Spanish, French, German or Russian (bilingual dictionaries in the other major languages were used in order to gain maximal phylogenetic and geographic distribution). That is, we use English and the other major languages as the semantic metalanguage for the word senses of the object language words used in the analysis. English (or any natural language) is an imperfect semantic metalanguage, because English itself has many polysemous words and divides the space of concepts in a partly idiosyncratic way. This is already apparent in Swadesh’s own list: he treated STONE/ROCK and EARTH/SOIL as synonyms, and had to specify that DAY referred to DAYTIME as opposed to NIGHT, rather than a 24-hour period. However, the selection of a concrete semantic domain including many discrete objects such as SUN and MOON allowed us to avoid the much greater problems of semantic comparison in individuating properties and actions or social and psychological concepts. We followed lexicographic practice in individuating word senses across the languages. Lexicographers are aware of polysemies such as DAYTIME vs. 24 HOUR PERIOD and usually indicate these semantic distinctions in their dictionary entries. There were a number of cases in which 7 different lexicographers appeared to use near-synonyms when the dictionaries were compared in our cross-linguistic analysis. We believe that these choices of near-synonyms in English translations may not reflect genuine subtle semantic differences but may simply represent different choices among near-synonyms made by different lexicographers. These near-synonyms were treated as a single sense in the polysemy analysis; they are listed in Table II. 8 anger ASH(ES) bodily gases celebrity country darkness darkness dawn debris EARTH/SOIL evening feces fireplace flood flow ground haze heat heaven liquid lodestar mark mist mold MOUNTAIN mountainous region NIGHT noon passion pile pond slope spring steam storm stream sunlight swamp time world fury, rage cinders fart, flatulence, etc. famous person, luminary countryside, region, area, territory, etc. [bounded area] dark (n.) dark daybreak, sunrise rubbish, trash, garbage dirt, loam, humus [= substance] twilight, dusk, nightfall dung, excrement, excreta hearth deluge flowing water land [= non-water surface] smog warmth heavens, Heaven, firmament, space [= place, surface up above] fluid Pole star dot, spot, print, design, letter, etc. steam, vapor, spray mildew, downy mildew mount, peak mountain range nighttime midday ardor, fervor, enthusiasm, strong desire, intensity heap, mound pool [= small body of still water] hillside water source vapor gale, tempest brook, creek [small flowing water in channel] daylight, sunshine marsh time of day (e.g. ‘what time is it?’) earth/place TABLE II. Senses treated as synonyms in our study. 9 D. Bidirectional translation, and linguists’ judgments on aggregation of meanings For each of the initial 22 Swadesh entries, we have recorded all translations from the metalanguage into the target languages, and then the back-translations of each of these into the metalanguage. Back-translation results in the additional meanings beyond the original 22 Swadesh meanings. A word in a target language is considered polysemous if its back-translation includes multiple words representing multiple senses as described in subsection I C. In cases where the backtranslation produces the same sense through more than one word in the target language, we call it synonymous polysemy, and we take into account the degeneracy of each such polysemy in our analysis as weighted links. The set of translations/back-translations of all 22 Swadesh meanings for each target language constitutes our characterization of that language. The pool of translations over the 81 target languages is the complete data set. The dictionaries used in our study are listed below. 1. Dickens, Patrick. 1994. English-Ju|’hoan, Ju|’hoan-English dictionary. Köln: Rüdiger Köppe Verlag. 2. Haacke, Wilfrid H. G. and Eliphas Eiseb. 2002. A Khoekhoegowab dictionary, with an English-Khoekhoegowab index. Windhoek: Gamsberg Macmillan. 3. Traill, Anthony. 1994. A !Xóõ dictionary. Köln: Rüdiger Köppe Verlag. 4. Bird, Charles and Mamadou Kanté. Bambara-English English-Bambara Student Lexicon. Bloomington: Indiana University Linguistics Club. 5. Childs, G. Tucker. 2000. A dictionary of the Kisi language, with an English-Kisi index. Köln: Rüdiger Köppe Verlag. 6. Wakeman, C. W. (ed.). 1937. A dictionary of the Yoruba language. Ibadan: Oxford University Press. 7. Abraham, R. C. 1958. Dictionary of modern Yoruba. London: University of London Press. 8. Welmers, Beatrice F. & William E. Welmers. 1968. Igbo: a learners dictionary. Los Angeles: University of California, Los Angeles and the United States Peace Corps. 9. Goldie, Hugh. 1964. Dictionary of the Efik Language. Ridgewood, N.J. 10. Awde, Nicholas. 2000. Swahili Practical Dictionary. New York: Hippocrene Books. 11. Johnson, Frederick. 1969. Swahili-English Dictionary. New York: Saphrograph. 12. Kirkeby, Willy A. 2000. English-Swahili Dictionary. Dar es Salaam: Kakepela Publishing Company (T) LTD. 10 13. Cyffer, Norbert. 1994. English-Kanuri Dictionary. Köln: Rüdiger Köppe Verlag. 14. Cyffer, Norbert and John Hutchison (eds.). 1990. A Dictionary of the Kanuri Language. Dordrecht: Foris Publications. 15. Heine, Bernd. 1999. Ik dictionary. Köln: Rüdiger Köppe Verlag. 16. Creider, Jane Tapsubei and Chet A. Creider. 2001. A Dictionary of the Nandi Language. Köln: Rüdiger Köppe Verlag. 17. Palayer, Pierre, with Massa Solekaye. 2006. Dictionnaire démé (Tchad), précédé de notes grammaticales. Louvain: Peeters. 18. Delheure, J. 1984. Dictionnaire mozabite-français. Paris: SELAF. [Tumzabt] 19. Abraham, R. C. 1962. Dictionary of the Hausa language (2nd ed.). London: University of London Press. 20. Awde, Nicholas. 1996. Hausa-English English-Hausa Dictionary. New York: Hippocrene Books. 21. Skinner, Neil. 1965. Kamus na turanci da hausa: English-Hausa Dictionary. Zaria, Nigeria: The Northern Nigerian Publishing Company. 22. Pillinger, Steve and Letiwa Galboran. 1999. A Rendille Dictionary. Köln: Rüdiger Köppe Verlag. 23. Clarity, Beverly E., Karl Stowasser, and Ronald G. Wolfe (eds.) and D. R. Woodhead and Wayne Beene (eds.). 2003. A dictionary of Iraqi Arabic: English-Arabic, Arabic-English. Washington, DC: Georgetown University Press. 24. Aulestia, Gorka. 1989. Basque-English Dictionary. Reno: University of Nevada Press. 25. Aulestia, Gorka and Linda White. 1990. English-Basque Dictionary. Reno:University of Nevada Press. 26. Aulestia, Gorka and Linda White. 1992. Basque-English English-Basque Dictionary. Reno: University of Nevada Press. 27. Koushakdjian, Mardiros and Dicran Khantrouni. 1976. English-Armenian Modern Dictionary. Beirut: G. Doniguian & Sons. 28. McGregor, R.S. (ed.). 1993. The Oxford Hindi-English Dictionary. Oxford: Oxford University Press. 29. Pathak, R.C. (ed.). 1966. Bhargavas Standard Illustrated Dictionary of the English Language (Anglo-Hindi edition). Chowk, Varanasi, Banaras: Shree Ganga Pustakalaya. 30. Prasad, Dwarka. 2008. S. Chands Hindi-English-Hindi Dictionary. New Delhi: S. Chand & Company. 11 31. Institut Nauk Narodnoj Respubliki Albanii. 1954. Russko-Albanskij Slovar’. Moscow: Gosudarstvennoe Izdatel’stvo Inostrannyx i Natsional’nyx Slovarej. 32. Newmark, Leonard (ed.). 1998. Albanian-English Dictionary. Oxford/New York: Oxford University Press. 33. Orel, Vladimir. 1998. Albanian Etymological Dictionary. Leiden/Boston/Köln: Brill. 34. MacHale, Carlos F. et al. 1991. VOX New College Spanish and English Dictionary. 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A Concise English-Mongolian Dictionary. Indiana University Publications Volume 89, Uralic and Altaic Series. Bloomington: Indiana University. 45. Masuda, Koh (Ed.). 1974. Kenkyushas New Japanese-English Dictionary. Tokyo: Kenkyusha Limited. 46. Worth, Dean S. 1969. Dictionary of Western Kamchadal. (University of California Publications in Linguistics 59.) Berkeley and Los Angeles: University of California Press. 47. Jaimoukha, Amjad M. 1997. Kabardian-English Dictionary, Being a Literary Lexicon of East Circassian (First Edition). Amman: Sanjalay Press. 48. Klimov, G.A. and M.Š. Xalilov. 2003. Slovar Kavkazskix Jazykov. Moscow: Izdatelskaja Firma. 49. Lopatinskij, L. 1890. Russko-Kabardinskij Slovar i Kratkoju Grammatikoju. Tiflis: Tipografija Kantseljarii Glavnonačalstvujuščago graždanskoju častju na Kavkaz. 50. Aliroev, I. Ju. 2005. Čečensko-Russkij Slovar. Moskva: Akademia. 12 51. Aliroev, I. Ju. 2005. Russko-Čečenskij Slovar. Moskva: Akademia. 52. Amirejibi, Rusudan, Reuven Enoch, and Donald Rayfield. 2006. A Comprehensive GeorgianEnglish Dictionary. London: Garnett Press. 53. Gvarjalaze, Tamar. 1974. English-Georgian and Georgian-English Dictionary. Tbilisi: Ganatleba Publishing House. 54. Hockings, Paul and Christiane Pilot-Raichoor. 1992. A Badaga-English dictionary. Berlin: Mouton de Gruyter. 55. Institute of Far Eastern Languages, Yale University. 1966. Dictionary of Spoken Chinese. New Haven: Yale University Press. 56. Henderson Eugénie J. A. 1997. Bwe Karen Dictionary, with texts and English-Karen word list, vol. II: dictionary and word list. London: University of London School of Oriental and African Studies. 57. Walker, G. D. 1925/1995. A dictionary of the Mikir language. New Delhi: Mittal Publications (reprint). 58. Lewis, Paul and Bai Bibo. 1996. Hani-English, English-Hani dictionary. London: Kegan Paul International. 59. Pinson, Thomas M. 1998. Naqxi-Habaq-Yiyu Geezheeq Ceeqhuil: Naxi-Chinese-English Glossary with English and Chinese index. Dallas: Summer Institute of Linguistics. 60. Heimbach, Ernest E. 1979. White Hmong-English Dictionary. Ithaca: Cornell Southeast Asia Program, Linguistic Series IV. 61. Ramamurti, Rao Sahib G.V. 1933. English-Sora Dictionary. Madras: Government Press. 62. Ramamurti, Rao Sahib G.V. 1986. Sora-English Dictionary. Delhi: Mittal Publications. 63. Rischel, Jørgen. 1995. Minor Mlabri: a hunter-gatherer language of Northern Indochina. Copenhagen: Museum Tusculanum Press. 64. Means, Nathalie and Paul B. Means. 1986. Sengoi-English, English-Sengoi dictionary. Toronto: The Joint Centre on Modern East Asia, University of Toronto and York University. [Semai] 65. Becker, Benjawan Poomsan. 2002. Thai-English, English-Thai Dictionary. Bangkok/Berkeley: Paiboon Publishing. 66. Goodenough, Ward and Hiroshi Sugita. 1980. Trukese-English dictionary. (Memoirs of the American Philosophical Society, 141.) Philadelphia: American Philosophical Society. 67. Goodenough, Ward and Hiroshi Sugita. 1990. Trukese-English dictionary, Supplementary volume: English-Trukese and index of Trukese word roots. (Memoirs of the American Philo- 13 sophical Society, 141S.) Philadelphia: American Philosophical Society. 68. Bowden, Ross. 1997. A dictionary of Kwoma, a Papuan language of the north-east New Guinea. (Pacific Linguistics, C-134.) Canberra: The Australian National University. 69. Renck, G. L. 1977. Yagaria dictionary. (Pacific Linguistics, Series C, No. 37.) Canberra: Research School of Pacific Studies, Australian National University. 70. Lloyd, J. A. 1992. A Baruya-Tok Pisin-English dictionary. (Pacific Linguistics, C-82.) Canberra: The Australian National University. 71. Dutton, Tom. 2003. A dictionary of Koiari, Papua New Guinea, with grammar notes. (Pacific Linguistics, 534.) Canberra: Australia National University. 72. Firchow, Irwin, Jacqueline Firchow, and David Akoitai. 1973. Vocabulary of Rotokas-PidginEnglish. Ukarumpa, Papua New Guinea: Summer Institute of Linguistics. 73. Laycock, Donald C. 2003. A dictionary of Buin, a language of Bougainville. (Pacific Linguistics, 537.) Canberra: The Australian National University. 74. Heath, Jeffrey. 1982. Nunggubuyu Dictionary. Canberra: Australian Institute of Aboriginal Studies. 75. Heath, Jeffrey. 1981. Basic Materials in Mara: Grammar, Texts, Dictionary. (Pacific Linguistics, C60.) Canberra: Research School of Pacific Studies, Australian National University. 76. Henderson, John and Veronica Dobson. 1994. Eastern and Central Arrernte to English Dictionary. Alice Springs: Institute for Aboriginal Development. 77. Bergsland, Knut. 1994. Aleut dictionary: unangam tunudgusii. Fairbanks: Alaska Native Language Center, University of Alaska. 78. Enrico, John. 2005. Haida dictionary: Skidegate, Masset and Alaskan dialects, 2 vols. Fairbanks and Juneau, Alaska: Alaska Native Language Center and Sealaska Heritage Institute. 79. Jetté, Jules and Eliza Jones. 2000. Koyukon Athabaskan dictionary. Fairbanks: Alaska Native Language Center. 80. Day, Gordon M. 1994. Western Abenaki Dictionary. Hull, Quebec: Canadian Museum of Civilization. 81. Thompson, Laurence C. and M. Terry Thompson (compilers). 1996. Thompson River Salish Dictionary. (University of Montana Occasional Papers in Linguistics 12.). Missoula, Montana: University of Montana Linguistics Laboratory. 82. Stonham, John. 2005. A Concise Dictionary of the Nuuchahnulth Language of Vancouver Island. Native American Studies 17. Lewiston/Queenston/Lampeter: The Edwin Mellen Press. 14 83. Lakota Language Consortium. 2008. New Lakota Dictionary. Bloomington: Lakhota Language Consortium. 84. Parks, Douglas R. and Lula Nora Pratt. 2008. A dictionary of Skiri Pawnee. Lincoln: University of Nebraska Press. 85. Woodbury, Hanni. 2003. Onondaga-English / English-Onondaga Dictionary. Toronto: University of Toronto Press. 86. Dunn, John Asher. Smalgyax: A Reference Dictionary and Grammar for the Coast Tsimshian Language. Seattle: University of Washington Press. [Coast Tsimshian] 87. Barker, M. A. R. 1963. Klamath Dictionary. University of California Publications in Linguistics 31. Berkeley: University of California Press. 88. Pitkin, Harvey. Wintu Dictionary. (University of California Publications in Linguistics, 95). Berkeley and Los Angeles: University of California Press. 89. Callaghan, Catherine A. 1987. Northern Sierra Miwok Dictionary. University of California Publications in Linguistics 110. Berkeley/Los Angeles/London: University of California Press. 90. Martin, Jack B. and Margaret McKane Mauldin. 2000. A dictionary of Creek/Muskogee. Omaha: University of Nebraska Press. 91. Hofling, Charles Andrew and Félix Fernando Tesucùn. 1997. Itzaj Maya-Spanish-English Dictionary/Diccionario Maya Itaj-Español-Ingles. 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Trimming, collapsing, projecting Our choice of starting categories is meant to minimize culturally or geographically specific associations, but inevitably these enter through polysemy that results from metaphor or metonymy. To attempt to identify polysemies that express some degree of cognitive universality rather than pure cultural “accident”, we include in this report only polysemies that occurred in two or more languages in the sample. The original data comprises 2263 words, translated from a starting list of 22 Swadesh meanings, and 826 meanings as distinguished by English translations. After removal of the polysemies occurring in only a single language, the dataset was reduced to 2257 words and 16 236 meanings. Figure 3 shows that this results in little difference in the statistics of weighted and unweighted degrees. Finally, as detailed below, the most fine-grained representation of the data preserves all English translations to all words in each target language. To produce aggregate summary statistics, we have projected this fine-grained, heterogeneous, directed graph onto the shared English-language nodes, with appropriately redefined links, to produce coarser-level directed and undirected graphs. Specifically, we define a weighted graph whose nodes are English words, where each link has an integer-valued weight equal to the number of translation-back-translation paths between them. We show this procedure in more detail in the next section. heaven month air stream hill ground DUST EARTH/SOIL STONE/ROCK DUST WATER SKY EARTH/SOIL WIND FIRE ASH(ES) MOUNTAIN ... world rain EARTH/SOIL SKY FIRE DAY/DAYTIME WIND WATER STON/ROCK DUST SUN MOUNTAIN ... YEAR divinity WATER stream liquid FIG. 3. Rank plot of meanings in descending order of their degree and strengths. This figure is an expanded version of Fig. 4 from the main text, in which singly-attested polysemies are retained. II. NOTATION AND METHODS OF NETWORK REPRESENTATION A. Network representations Networks provide a general and flexible class of topological representations for relations in data [9]. Here we define the network representations that we construct from translations and 17 back-translation to identify polysemies. 1. Multi-layer network representation We represent translation and back-translation with three levels of graphs, as shown in Fig. 4. Panel (a) shows the treatment of two target languages: Coast Tsimshian and Lakhota, by a multilayer graph. To specify the procedure, the nodes are separated into three types shown in the three layers, corresponding to the input and output English words, and their target-language translations. Two types of links represent translation from English to target languages, and back-translations to English, indicated as arrows bridging the layers.2 Two initial Swadesh entries, labeled S ∈ {MOON, SUN}, are shown in the first row. Words wSL in language L ∈ {Coast Tsimshian, Lakhota} obtained by translation of entry S are shown in the secCoast Tsimshian = {gooypah, gyemk, . . .} and w Lakhota = {haηhépi wı́, haηwı́, wı́, . . .}. ond row, i.e., wMOON MOON Directed links tSw take values tSw 1 if S is translated into w = , 0 otherwise (1) The bottom row shows targets mS obtained by back-translation of all words wSL (fixing S or L as appropriate) into English. Here mMOON = {MOON, month, heat, SUN}. By construction, S is always present in the set of values taken by mS . Back-translation links twm take values twm 1 if w is translated into m = 0 otherwise (2) The sets [tSw ] and [twm ] can therefore be considered adjacency matrices that link the Swadesh list to each target-language dictionary and the target-language dictionary to the full English lexicon.3 We denote the multi-layer network representing a single target language L by G L composed of nodes {S}, {w} and {m} and matrices of links [tSw ] and [twm ] connecting them. Continuing with the example of G Coast Tsimshian , we see that tgooypah,month = 0, while tgooypah,MOON = 1. One such network is constructed for each language, leading to 81 polysemy networks G L for this study. 2 3 In this graph, we regard English inputs and outputs as having different type to emphasize the asymmetric roles of the Swadesh entries from secondary English words introduced by back-translation. The graph could equivalently be regarded as a bipartite graph with only English and non-English nodes, and directed links representing translation. Link direction would then implicitly distinguish Swadesh from non-Swadesh English entries. More formally, indices S, w, and m are random variables taking values, respectively, in the sets of 22 Swadesh entries, target-language entries in all 81 languages, and the full English lexicon. Subscripts and superscripts are then used to restrict the values of these random variables, so that wSL takes values only among the words in language L that translate Swadesh entry S, and mS takes values only among the English words that are polysemes of S in some target language. We indicate random variables in math italic, and the values they take in Roman. 18 (a) Sample level: S MOON translation tSw SUN Words level: wL haŋwí gyemgmáatk wC wí wL gooypah haŋhépi_wí áŋpawí gyemk gimgmdziws backtranslation twm Meaning level: m MOON month m heat SUN Red: Coast_Tsimshian Blue: Lakhota (b) 1 3 MOON (d) (c) Coast_Tsimshian : MOON 1 SUN 1 1 month MOON 2 1 1 heat SUN MOON (e) MOON 6 MOON 2 month 1 3 Bipartite: {S} and {m} 2 Lakhota heat MOON SUN 1 heat 2 2 heat 1 Unipartite: {m} 4 1 2 1 month SUN 1 2 1 SUN 2 month 1 2 SUN 2 SUN (f) kLS1(MOON) kLS3(SUN) Coast Tsimshian Lakhota 6 5 5 4 ... ... ... kLS3(YEAR) ... ... ... kLS4(STAR) ... ... ... ... ... ... ... FIG. 4. Schematic figure of the construction of network representations. Panel (a) illustrates the multilayer polysemy network from inputs MOON and SUN for two American languages: Coast Tsimshian and Lakhota. Panels (b) and (c) show the directed bipartite graphs for the two languages individually, which lose information about the multiple-polysemes “gyemk” and “wı́’ found respectively in Coast Tsimshian and Lakhota. Panel (d) shows the bipartite directed graph formed from the union of links in graphs (b) and (c). Link weights indicate the total number of translation/back-translation paths that connect each pair of English-language words. Panel (e) shows the unipartite directed graph formed by identifying and merging Swadesh entries in two different layers. Link weights here are the number of polysemies across languages in which at least one polysemous word connects the two concepts. Directed links go from the Swadesh-list seed words (MOON and P SUN here) to English words found in the back-translation step. Panel (f) is a table of link numbers nL = and twm are binary (0 or 1) to express, respectively, a link S w,m tSw twm where tSwP from S to w, and from w to m in this paper. w tSw twm gives the number of paths between S and m in network representations. 19 L L The forward translation matrix TL > ≡ [tSw ] has size 22×Y , where Y is the number of distinct translation in language L of all Swadesh entries, and the reverse translation matrix TL < ≡ [twm ] has size Y L × Z L , where Z L is the number of distinct back-translation to English through all targets in language L. For example, Y Coast Tsimshian = 27 and Z Coast Tsimshian = 33. 2. Directed-hyper-graph representation It is common that multipartite simple graphs have an equivalent expression in terms of directed hyper-graphs [10]. A hyper-graph, like a simple graph, is defined from a set of nodes and a collection of hyper-edges. Unlike edges in a simple graph, each of which has exactly two nodes as boundary (dyadic), a hyper-edge can have an arbitrary set of nodes as its boundary. Directed hyper-edges have boundaries defined by pairs of sets of nodes, called inputs and outputs, to the hyper-edge. In a hyper-graph representation, we may regard all English entries as nodes, and words wSL as hyper-edges. The input to each hyper-edge is a single Swadesh entry S, and the outputs are the set of all back-translation mw . It is perhaps more convenient to regard the simple graph in its bipartite, directed form, as the starting point for conversion to the equivalent hyper-graph. A separate hyper-graph may be formed for each language, or the words from multiple languages may be pooled as hyper-edges in an aggregate hyper-graph. 3. Projection to directed simple graphs and aggregation over target languages The hyper-graph representation is a complete reflection of the input data. However, hypergraphs are more cumbersome to analyze than simple networks, and the heterogeneous character of hyper-edges can be an obstacle to simple forms of aggregation. Therefore, most of our analysis is performed on a projection of the tri-partite graph onto a simple network with only one kind of node (English words). The node set may continue to be regarded as segregated between inputs and outputs to (now bidirectional) translation, leaving a bipartite network with two node types, or alternatively we may pass directly to a simple directed graph in which all English entries are of identical type, and the directionality of bidirectional translations carries all information about the asymmetry between Swadesh and non-Swadesh entries with respect to our translation procedure. Directed bipartite graph representations for Coast Tsimshian and Lakhota separately are shown in Fig. 4 (b) and (c), respectively, and the aggregate bipartite network for the two target languages is shown in Fig. 4 (d). 20 Projection of a tripartite graph to a simpler form implicitly entails a statistical model of aggregation. The projection we will use creates links with integer weights that are the sums of link variables in the tripartite graph. The associated aggregation model is complicated to define: link summation treats any single polysemy as a sample from an underlying process assumed to be uniform across words and languages; however, correlations arise due to multi-way polysemy, when a Swadesh word translates to multiple words in a target language, and more than one of these words translates back to the same English word. This creates multiple output-nodes on the boundaries of hyper-edges, rendering these link weights non-independent, so that graph statistics are not automatically recovered by Poisson sampling defined only from the aggregate weights given to links. We count the link polysemy between any Swadesh node S and any English output of bidirectional translation m as a sum (e.g., within a single language L)4 tL Sm = X tSwL twL m S S L = TL > T< Sm L wS B. . (3) Model for semantic space represented as a topology As a mnemonic for the asymmetry between English entries as “meanings” and target-language entries as “words”, we may think of these graphs as overlying a topological space of meanings, and of words as “catching meanings in a set”, analogous to catching fish in the ocean using a variety of nets. Any original Swadesh meaning is a “fish” at a fixed position in the ocean, and each targetn o language word wSL is one net that catches this fish. The back-translations m | twL m = 1 are all S L other fish caught in the same net. If all distinct words wS are interpreted as random samples of nets (a proposition which we must yet justify by showing the absence of other significant sources of correlation), then the relative distance of fish (intrinsic separation of concepts in semantic space) determines their joint-capture statistics within nets (the participation of different concept pairs in polysemies). The “ocean” in our underlying geometry is not 2- or 3-dimensional, but has a dimension corresponding to the number of significant principal components in the summary statistics from our data. If we use a spectral embedding to define the underlying topology from a geometry based on diffusion in Euclidean space, the dimension D of this embedding will equal to the number of 4 Note that for unidirectional links tSw or twm , we need not identify the language explicitly in the notation because that identification is carried implicitly by the word w. For links in projected graphs it is convenient to label with superscript L, because both arguments in all such links are English-language entries. 21 English-language entries recovered in the total sample, and a projection such as multi-dimensional scaling may be used to select a smaller number of dimensions [11, 12]. In this embedding, diffusion is isotropic and all “nets” are spherical. More generally, we could envision a lower-dimensional “ocean” of meanings, and consider nets as ellipsoids characterized by eccentricity and principal directions as well as central locations. This picture of the origin of polysemy from an inherent semantic topology is illustrated in Fig. 5, and explained in further detail in the next section. moonlight luna gooypah satellite gyemk wí MOON heat month SUN Green: Spanish Blue: Lakhota Red: Coast_Tsimshian áŋpawí FIG. 5. Hypothetical word-meaning and meaning-meaning relationships using a subset of the data from Fig. 4. In this relation, translation and back-translation across different languages reveal polysemies through which we measure a distance between one concept and another concept. 1. Interpretation of the model into network representation For an example, consider the projection of the small set of data shown in Fig. 4 (b) and (c). Words in L = Coast Tsimshian are colored red. For these, we find S = MOON is connected to Coast Tshimshian is, hence, m = MOON via the three wSL values gyemgmáatk, gooypah, and gyemk. tMOON MOON Tshimshian = 1 (via gyemk). From the words in L = Lakhota, colored blue in 3, whereas tCoast SUN heat Lakhota Fig. 4(c), we see that again tLakhota MOON MOON = 3, while tSUN heat = 0 because there is no polysemy between these entries in Lakhota. Diffusion models of projection, which might be of interest of diachronic meaning shift in historical linguistics, is mediated by polysemous intermediates, suggest alternative choices of pro jection as well. Instead of counting numbers of polysemes wSL between some S and a given m, a link might be labeled with the share of polysemy of S that goes to that m. This weightCoast Tshimshian = 3/6, because only gyemgmáatk, gooypah, and gyemk are in coming gives tMOON MOON Tshimshian = 1/6, because only gyemk is associated out of six polysemes mon out of 6 and tCoast MOON heat 22 between MOON and heat. In the interpretation of networks as Markov chains of diffusion processes, this weight gives the (normalized) probability of transition to m when starting from S, as P P P t̂L = t t / t t L L L L L L 0 0 0 0 0 Sw w m w m Sm w Sw w m . S S S S S S We may return to the analogy of catching fish in a high-dimensional sea, which is the underlying (geometric or topological) semantic space, referring to Fig. 5. Due to the high dimensionality of this sea, whether any particular fish m is caught depends on both the position and the angle with which the net are cast. When the distance between S and m is very small, the angle may matter little. A cast at a slightly different angle, if it caught S, would again catch m as well. If, instead, m is far from the center of a net cast to catch S, only for a narrow range of angles will both S and m be caught. An integer-weighted network measures the number of successes in catching the fish m as a proxy for relative distance from S. The fractionally-weighted network allows us to consider the probability of success of catching any fish other than S. If we cast a net many times but only one succeeds in producing a polysemy, we should think that other meanings m are all remote from S. Under a fractional weighting, the target language and the English Swadesh categorization may have different rates of sampling, which appear in the translation dictionary. Our analysis uses primarily the integer-weighted network. 2. Beyond the available sample data In the representation of the fine-grained graph as a directed, bipartite graph, English words S and m, and target-language words w, are formally equivalent. The asymmetry in their roles comes only from the asymmetry in our sampling protocol over instances of translation. An ideal, fully symmetric dataset might contain translations between all pairs of languages (L, L0 ). In such a dataset, polysemy with respect to any language L could be obtained by an equivalent projection of all languages other than L. A test for the symmetry of the initial words in such a dataset can come from projecting out all intermediate languages other than L and L0 , and comparing the projected links from L to L0 through other intermediate languages, against the direct translation dictionary. A possible area for future work from our current dataset (since curated all-to-all translation will not be available in the foreseeable future), is to attempt to infer the best approximate translation maps, Coast Tsimshian TLakhota e.g. between Coast Tsimshian and Lakhota, through an intermediate sum T< > analogous to Eq. (3), as a measure of the overlap of graphs G Coast Tsimshian and G Lakhota . 23 EARTH/SOIL SALT SEA/OCEAN DUST ASHES WATER RIVER SMOKE SAND FIRE LAKE CLOUD(S) DAY/DAYTIME STAR SKY MOUNTAIN SUN YEAR MOON STONE/ROCK FIG. 6. Connectance graph of Swadesh meanings excluding non-Swadesh English words. C. The aggregated network of meanings The polysemy networks of 81 languages, constructed in the previous subsection, are aggregated into one network structure as shown in Fig. 2 in the main text. Two types of nodes are distinguished by the case of the label on each node. All-capital labels indicate Swadesh words while all-lowercase indicate non-Swadesh words. The width of each link is the number of polysemes joining the two meanings at its endpoints, including in this count the sum of all synonyms within each target language that reproduce the same polysemy. For example, the thick link between SKY and heaven implies the existence of the largest number of distinct polysemes between these two compared to those between any two other entries in the graph. D. Synonymous polysemy: correlations within and among languages Synonymous polysemy provides the first, in a series of tests that we will perform, to determine whether the aggregate graph generated by addition of polysemy-links is a good summary statistic for the process of word-meaning pairing in actual languages that leads to polysemy. The null model for sampling from the aggregate graph is that each word from a some Swadesh entry S has a fixed probability to be polysemous with a given meaning entry m, independent of the presence or absence of any other polysemes of S with m in the same language. Violations of the null model include excess synonymous polysemy (suggesting, in our picture of an underlying semantic space, that the “proximity” of meanings is in part dynamically created by formation of polysemies, increasing their likelihood of duplication), or deficit synonymous polysemy (suggesting that languages economize 24 1.5 1.4 1.3 1.2 1.1 1.0 0 10 20 30 40 50 60 FIG. 7. The number of synonymous polysemies within a language is correlated with the number of languages containing a given polysemy. The horizontal axis indicates the number of languages (out of 81) in which a Swadesh entry S is polysemous with a given meaning m for meanings found to be polysemous in at least two languages. The vertical axis indicates the average number of synonymous polysemies per language in which the polysemous meaning is represented. Circle areas are proportional to the number of meanings m over which the average was taken. The red line represents the least-squares regression over all (non-averaged) data and has slope and intercept of 0.0029 and 1.05, respectively. on the semantic scope of words by avoiding duplications). The data presented in Fig. 7 shows that if a given polysemy is represented in more languages, it is also more likely to be captured by more than one word within a given language. This is consistent with a model in which proximity relationships among meanings are preexisting. Models in which the probability of a synonymous polysemy was either independent of the number of polysemous languages (corresponding to a slope of zero in Fig. 7) or quadratic in the number of languages were rejected by both AIC and BIC tests. We partitioned the synonymous polysemy data and performed a series of Mann-Whitney U tests. We partitioned all polysemies according to the following scheme: a polysemy was a member of the set cs,p if the language contained the number of polysemies, p, for the given Swadesh word, s of which shared this polysemous meaning. For each category, we constructed a list Ds,p of the numbers of languages in which each polysemous meaning in the set cs,p is found. We then tested all pairs of Ds1 ,p and Ds2 ,p for whether they could have been drawn from the same distribution. 25 D0,1 D1,1 0.167 2.53 × 10−433 D0,2 D1,2 0.333 1.08 × 10−170 D0,2 D2,2 0.0526 1.39 × 10−56 D1,2 D2,2 0.158 2.10 × 10−14 D0,3 D1,3 0.500 7.18 × 10−44 D0,3 D2,3 0.167 1.11 × 10−28 D0,3 D3,3 0.0222 4.81 × 10−9 D1,3 D2,3 0.333 6.50 × 10−5 D1,3 D2,3 0.0444 4.53 × 10−5 D2,3 D3,3 0.133 9.28 × 10−4 D0,4 D1,4 1.00 3.15 × 10−17 D0,4 D2,4 0.0714 1.86 × 10−13 D0,4 D3,4 0.0667 5.63 × 10−5 D1,4 D2,4 0.0714 7.66 × 10−4 D1,4 D3,4 0.0667 0.084 D2,4 D3,4 0.993 1.0 D0,5 D1,5 0.143 1.03 × 10−27 D0,5 D2,5 0.182 6.20 × 10−7 D0,5 D3,5 0.0323 5.09 × 10−6 D1,5 D2,5 1.27 0.35 D1,5 D3,5 0.226 0.15 D2,5 D3,5 0.177 0.06 D0,6 D1,6 1.00 0.15 D0,6 D2,6 0.0247 1.44 × 10−5 D1,6 D2,6 0.0247 0.06 D0,7 D1,7 1.00 0.20 In most comparisons, the null hypothesis that the two lists were drawn from the same distribution was strongly rejected, always in the direction where the list with the larger number of synonymous polysemies (larger values of s) contained larger numbers, meaning that those polysemies were found in more different languages. For a few of the comparisons, the null hypothesis was not rejected, corresponding to those cases where one or both lists included a small number of entries (< 10). 26 The table II D shows all comparisons for lists of greater than length one. The first two columns indicate which two lists are being compared. The third column gives the ratio of the median values of the two lists, with values less than one indicating that the median of the list in the column one is lower than the median of the list in the column two. We return to demonstrate a slight excess of probability to include individual entries in polysemies, in Sec. IV. E. Node degree and link presence/absence data The goodness of this graph as a summary statistic, and the extent to which the heterogeneity of its node degree and the link topology reflect universals in the sense advocated by Greenberg [13], may be defined as the extent to which individual language differences are explained as fluctuations in random samples. We begin systematic tests of the goodness of the aggregate graph with the degrees of its nodes, a coarse-grained statistic that is most likely to admit the null model of random sampling, but which also has the best statistical resolution among the observables in our data. These tests may later be systematically refined by considering the presence/absence statistics of the set of polysemy links, their covariances, or higher-order moments of the network topology. At each level of refinement we introduce a more restrictive test, but at the same time we lose statistical power because the number of possible states grows faster than the data in our sample. We let nL m denote the degree of meaning m—defined as the sum of weights of links to m— in language L. Here m may stand for either Swadesh or non-Swadesh entries ({S} ⊂ {m}). P L nm ≡ L nm is then the degree of meaning m in the aggregated graph of Fig. 2 (main text), P P shown in a rank-size plot in Fig. 3. N ≡ m nm = m,L nL m denotes the sum of all link weights in the aggregated graph. F. Node degree and Swadesh meanings The Swadesh list was introduced to provide a priority for the use of lexical items, which favored universality, stability, and some notion of “core” or “basic” vocabulary. Experience within historical linguistics suggests qualitatively that it satisfies these criteria well, but the role of the Swadesh list within semantics has not been studied with quantitative metrics. We may check the degree to which the items in our basic list are consistent with a notion of core vocabulary by studying their position in the rank-size distribution of Fig. 4 in the main text. 27 Our sampling methodology naturally places the starting search words (capitalized black characters) high in the distribution, such as EARTH/SOIL, WATER and DAY/DAYTIME with more than 100 polysemous words, because they are all connected to other polysemes produced by polysemy sampling. Words that are not in the original Swadesh list, but which are uncovered as polysemes (red), are less fully sampled. They show high degree only if they are connected to multiple Swadesh entries.5 These derived polysemes fall mostly in the power-law tail of the rank-size distribution in Fig. 3. The few entries of high degree serve as candidates for inclusion in an expanded Swadesh list, on the grounds that they are frequently recovered in basic vocabulary. Any severe violation of the segregation of Swadesh from non-Swadesh entries (hence, the appearance of many derived polysemes high in the rank-size distribution) would have indicated that the Swadesh entries were embedded in a larger graph with high clustering coefficient, and would have suggested that the low-ranking Swadesh words were not statistically favored as starting points to sample a semantic network.6 III. UNIVERSAL STRUCTURE: CONDITIONAL DEPENDENCE We performed an extensive range of tests to determine whether language differences in the distribution of 1) the number of polysemies, 2) the number of meanings (unweighted node degree), and 3) the average proximity of meanings (weighted node degree, or “strength”) are correlated with language relatedness, or with geographic or cultural characteristics of the speaker populations, including the presence or absence of a writing system. The interpretation is analogous to that of population-level gene frequencies in biology. Language differences that covary with relatedness disfavor the Greenbergian view of typological universals of human language, and support a Whorfian view that most language differences are historically contingent and recur due to vertical transmission within language families. Differences that covary with cultural or geographical parameters suggest that language structure responds to extra-linguistic conditions instead of following universal endogenous constraints. We find no significant regression of the patterns in our degree distribution on any cladistic, cultural, or geographical parameters. At the same time, we found single-language degree distributions consistent with a model of random sampling (defined below), suggesting that the degree distribution of polysemies is an instance of a Greenbergian universal. Ruling out dummy variables of clade and culture has a second important implication for studies 5 6 Note that two non-Swadesh entries cannot be linked to each other, even if they appear in a multi-way polysemy, because our protocol for projecting hypergraphs to simple graphs only generates links between the (Swadesh) inputs and the outputs of bidirectional translation. With greater resources, a bootstrapping method to extend the Swadesh list by following second- and higher-order polysemes could provide a quantitative measure of the network position of the Swadesh entries among all related words. 28 of this kind. We chose to collect data by hand from printed dictionaries, foregoing the sample size and speed of the many online language resources now available, to ensure that our sample represents the fullest variation known among languages. Online dictionaries and digital corpora are dominated by a few languages from developed countries, with long-established writing systems and large speaker populations, but most of these fall within a small number of European or Asian language families. Our demonstration that relatedness does not produce a strong signal in the parameters we have measured opens the possibility of more extensive sampling from digital sources. We note two caveats regarding such a program, however. First, languages for this study were selected to maximize phylogenetic distance, with no two languages being drawn from the same genus. It is possible that patterns of polysemy could be shared among more closely related groups of languages. Second, the strength of any phylogenetic signal might be expected to vary across semantic domains, so any future analysis will need to be accompanied by explicit universality tests like those performed here. A. Comparing semantic networks between language groups We performed several tests to see if the structure of the polysemy network depends, in a statistically significant way, on typological features, including the presence or absence of a literary tradition, geography, topography, and climate. The geographical information is obtained from the LAPSyD database [14]. We choose the climate categories as the major types A (Humid), B (Arid), and C–E (Cold) from the Köppen-Geiger climate classification [15], where C–E have been merged since each of those had few or no languages in our sample. We list the typological features that are tested, and the numbers of languages for each feature shown in parentheses in the table III 1. Mantel test Given a set S of languages, we define a weighted graph between English words as shown in Fig. 2 in the main text. Each matrix entry Aij is the total number of foreign words, summed over all languages in S, that can be translated or back-translated to or from both ith and j. From this network, we find the commute distance between the vertices. The commute distance is the expected number of steps a random walker needs to take to go from one vertex to another [16]. It is proportional to the more commonly used resistance distance by a proportionality factor of the sum of all resistances (inverse link weights) in the network. 29 Variable Subset Size Americas 29 Eurasia 20 Geography Africa 17 Oceania 15 Humid 38 Climate Cold 30 Arid 13 Inland 45 Topography Coastal 36 Some or long literary tradition 28 Literary tradition No literary tradition 53 TABLE III. Various groups of languages based on nonlinguistic variables. For each variable we measured the difference between the subsets’ semantic networks, defined as the tree distance between the dendrograms of Swadesh words generated by spectral clustering. For the subgroups of languages, the networks are often disconnected. So, we regularize them by adding links between all vertices with a small weight of 0.1/[n ∗ (n − 1)], where n is the number of vertices in the graph, when calculating the resistance distance. We do not include this regularization in calculating the proportionality constant between the resistance and commute distances. Finally, we ignore all resulting distances that are larger than n when making comparisons. The actual comparison of the distance matrices from two graphs is done by calculating the Pearson correlation coefficient, r, between the two. This is then compared to the null expectation of no correlation by generating the distribution of correlation coefficients on randomizing the concepts in one distance matrix, holding the other fixed. The Mantel test p-value, p1 , is the proportion of this distribution that is higher than the observed correlation coefficient. To test whether the observed correlation is typical of random language groups, we randomly sample without replacement from available languages to form groups of the same size, and calculate the correlation coefficient between the corresponding distances. The proportion of this distribution that lies lower than the observed correlation coefficient provided p2 . 2. Hierarchical clustering test The commute measures used in the Mantel test, however, only examine the sets that are connected in the networks from the language groups. To understand the longer distance structure, we instead look at the hierarchical classification obtained from the networks. We cluster the vertices of the graphs, i.e., the English words, using a hierarchical spectral clustering algorithm. Specifically, 30 we assign each word i a point in Rn based on the ith components of the eigenvectors of the n × n weighted adjacency matrix, where each eigenvector is weighted by the square of its eigenvalue. We then cluster these points with a greedy agglomerative algorithm, which at each step merges the pair of clusters with the smallest squared Euclidean distance between their centers of mass. This produces a binary tree or dendrogram, where the leaves are English words, and internal nodes correspond to groups and subgroups of words. We obtained these for all 826 English words, but for simplicity we we show results here for the 22 Swadesh words. Doing this where S is the set of all 81 languages produces the dendrogram shown in Fig. 8. We applied the same approach where S is a subgroup of the 81 languages, based on nonlinguistic variables such as geography, topography, climate, and the presence or absence of a literary tradition. These groups are shown, along with the number of languages in each, in Table III. For each nonlinguistic variable, we measured the difference between the semantic network for each pair of language groups, defined as the distance between their dendrograms. We used two standard tree metrics taken from the phylogenetic literature. The triplet distance Dtriplet [17, 18] is the fraction of the n3 distinct triplets of words that are assigned a different topology in the two trees: that is, such that the trees disagree as to which pair of these three words is more closely related to each other than to the third. The Robinson-Foulds distance DRF [19] is the number of “cuts” on which the two trees disagree, where a cut is a separation of the leaves into two sets resulting from removing an edge of the tree. We then performed two types of bootstrap experiments, comparing these distances to those one would expect under the null hypotheses. First we considered the hypothesis that there is no underlying notion of relatedness between senses—for instance, that every pair of words is equally likely to be siblings in the dendrogram. If this were true, then the dendrograms of each pair of groups would be no closer than if we permuted the senses on their leaves randomly (while keeping the structure of the dendrograms the same). Comparing the actual distance between each pair of groups to the resulting distribution gives the p-values, labeled p1 , shown in Figure 3 in the main text. These p-values are small enough to decisively reject the null hypothesis; indeed, for most pairs of groups the Robinson-Foulds distance is smaller than that observed in any of the 1000 bootstrap trials, making the p-value effectively zero. This gives overwhelming evidence that the semantic network has universal aspects, applying across language groups: for instance, in every group we tried, SEA/OCEAN and SALT are more related than either is to SUN. In the second type of bootstrap experiment, the null hypothesis is that the nonlinguistic variables have no effect on the semantic network, and that the differences between language groups simply 31 SAND EARTHêSOIL ASHHESL DUST FIRE STONEêROCK MOUNTAIN SALT WATER RIVER LAKE SEAêOCEAN MOON DAYêDAYTIME SUN SMOKE CLOUDHSL NIGHT SKY WIND YEAR STAR FIG. 8. The dendrogram of Swadesh words generated from spectral clustering on the polysemy network taken over all 81 languages. The three largest groups are highlighted; roughly speaking, they comprise earth-related, water-related, and sky-related concepts. result from random sampling: for instance, that the distance between the dendrograms for the Americas and Eurasia is what one would expect from any disjoint subsets S1 , S2 of the 81 languages of sizes |S1 | = 29 and |S2 | = 20 respectively. To test this, we generate random pairs of disjoint subsets with the same sizes as the groups in question, and measure the resulting distribution of distances. The resulting p-values are labeled p2 in Table 1. These p-values are not small enough to reject the null hypothesis. Thus, at least given the current data set, it does not appear that these nonlinguistic variables have a statistically significant effect on the semantic network—further supporting our thesis that it is, at least in part, universal. For illustration, in Fig. 3 (main text) we compare the triplet distance between the dendrograms for Americas and Oceania with the distributions from the two bootstrap experiments. These two groups are closer than less than 2% of the trees where senses have been permuted randomly, but 38% of the random pairs of subsets of size 29 and 15 are farther away. Using a p-value of 0.05 as the usual threshold, we can reject the hypothesis that these two groups have no semantic structure in common; moreover, we cannot reject the hypothesis that the differences between them are due to random sampling rather than geographic differences. 32 B. Statistical significance The p-values reported in Fig. 3 have to be corrected for multiple tests. Eleven independent comparisons are performed for each of the metrics, so a low p-value is occasionally expected simply by chance. In fact, under the null hypothesis, a column will contain a single p = 0.01 by chance about 10% of the time. To correct for this, one can employ a Bonferroni correction [20] leading to a significance threshold of 0.005 for each of the 11 tests, corresponding to a test size of p = 0.05. Most of the comparisons in the p1 columns for r and DRF are comfortably are below this threshold, implying that the networks obtained from different language groups are indeed significantly more similar than comparable random networks. A Bonferroni correction, however, is known to be aggressive: it controls the false positive error rate but leads to many false negatives [21], and is not appropriate for establishing the lack of significance for the p2 columns. The composite hypothesis that none of the comparisons are statistically significant leads to the predictions that the corresponding p-values are uniformly distributed between 0 and 1. One can, therefore, test the obtained p-values against this expected uniform distribution. We performed a Kolmogorov-Smirnov test for uniformity for each column of the table. This composite p-value is about 0.11 and 0.27 for the p2 columns corresponding to Dtriplet and DRF , showing that these columns are consistent with chance fluctuations. The p-value corresponding to the p2 column for r is about 0.03, evidence that at least one pair of networks are more dissimilar than expected for a random grouping of languages. This is consistent with the indication from the Bonferroni threshold as well—the comparison of Americas and Eurasia has a significant p-value, as possibly also the comparison between Humid and Arid. Removing either of these comparisons raises the composite p-value to 0.10, showing that such a distribution containing one low p-value (but not two) would be expected to occur by chance about 10% of the time. C. Single-language graph size is a significant summary statistic The only important language-dependent variable not attached to words in the aggregate graph of Fig. 2 (main text), which is a strongly significant summary statistic for samples, is the total link P weight in the language, nL ≡ S nL S . In the next section we will quantify the role of this variable in producing single-language graphs as samples from the aggregate graph, conditioned on the total weight of the language. Whereas we do associate node degree and link weight in the aggregate graph with inherent and 33 universal aspects of human language, we cannot justify a similar interpretation for the total weight of links within each language. The reason is that total weight — which may reflect a systematic variation among languages in the propensity to create polysemous words — may also be affected by reporting biases that differ among dictionaries. Different dictionary writers may be more or less inclusive in the meaning range they report for words. Additional factors, such as the influence of poetic traditions in languages with long written histories, may preserve archaic usages alongside current vernacular, leading to systematic differences in the data available to the field linguist. D. Conclusion By ruling out correlation and dependence on the exogenous variables we have tested, our data are broadly consistent with a Greenbergian picture in which whatever conceptual relations underlie polysemy are a class of typological universals. They are quantitatively captured in the node degrees and link weights of a graph produced by simple aggregation over languages. The polysemes in individual languages appear to be conditionally independent given the graph and a collection of language-specific propensities toward meaning aggregation, which may reflect true differences in language types but may also reflect systematic reporting differences. IV. MODEL FOR DEGREE OF POLYSEMY A. Aggregation of language samples We now consider more formally the reasons sample aggregates may not simply be presumed as summary statistics, because they entail implicit generating processes that must be tested. By demonstrating an explicit algorithm that assigns probabilities to samples of Swadesh node degrees, presenting significance measures consistent with the aggregate graph and the sampling algorithm, and showing that the languages in our dataset are typical by these measures, we justify the use and interpretation of the aggregate graph (Fig. 2 in the main text). We begin by introducing an error measure appropriate to independent sampling from a general mean degree distribution. We then introduce calibrated forms for this distribution that reproduce the correct sample means as functions of both Swadesh-entry and language-weight properties. The notion of consistency with random sampling is generally scale-dependent. In particular, the existence of synonymous polysemy may cause individual languages to violate criteria of randomness, but if the particular duplicated polysemes are not correlated across languages, even small groups 34 of languages may rapidly converge toward consistency with a random sample. Indeed the section II D shows the independence of synonymous polysemy. Therefore, we do not present only a single acceptance/rejection criterion for our dataset, but rather show the smallest groupings for which sampling is consistent with randomness, and then demonstrate a model that reproduces the excess but uncorrelated synonymous polysemy within individual languages. B. Independent sampling from the aggregate graph Figure 2 (main text) treats all words in all languages as independent members of an unbiased sample. To test the appropriateness of the aggregate as a summary statistic, we ask: do random samples, with link numbers equal to those in observed languages, and with link probabilities proportional to the weights in the aggregate graph, yield ensembles of graphs within which the actual languages in our data are typical? 1. Statistical tests The appropriate summary statistic to test for typicality, in ensembles produced by random sampling (of links or link-ends) is the Kullback-Leibler (KL) divergence of the sample counts from the probabilities with which the samples were drawn [22, 23]. This is because the KL divergence is the leading exponential approximation (by Stirling’s formula) to the log of the multinomial distribution produced by Poisson sampling. The appropriateness of a random-sampling model may be tested independently of how the aggregate link numbers are used to generate an underlying probability model. In this section, we will first evaluate a variety of underlying probability models under Poisson sampling, and then we will return to tests for deviations from independent Poisson samples. We first introduce notation: For a single language, the relative degree (link frequency), which is used as the normalization L L of a probability, is denoted as pdata S|L ≡ nS /n , and for the joint configuration of all words in all languages, the link frequency of a single entry relative to the total N will be denoted pdata SL ≡ L L data nL nL /N ≡ pdata S /N = nS /n S|L pL . Corresponding to any of these, we may generate samples of links to define the null model for L a random process, which we denote n̂L S , n̂ , etc. We will generally use samples with exactly the same number of total links N as the data. The corresponding sample frequencies will be denoted L sample L L L by psample ≡ n̂L ≡ n̂L n̂ /N ≡ psample psample , respectively. S /n̂ and pSL S /N = n̂S /n̂ L S|L S|L 35 Finally, the calibrated model, which we define from properties of aggregated graphs, will be the prior probability from which samples are drawn to produce p-values for the data. We denote the model probabilities (which are used in sampling as “true” probabilities rather than sample model model . frequencies) by pmodel S|L , pSL , and pL For nL links sampled independently from the distribution psample for language L, the multinomial S|L L probability of a particular set nS may be written, using Stirling’s formula to leading exponential order, as p nL S |n L ∼e sample model −nL D pS|L pS|L (4) where the Kullback-Leibler (KL) divergence [22, 23] sample X p S|L model ≡ psample log model . D psample pS|L S|L S|L pS|L S (5) For later reference, note that the leading quadratic approximation to Eq. (5) is 2 L pmodel 1 X n̂L − n S S|L model ≈ nL D psample , pS|L S|L model L 2 n pS|L S (6) so that the variance of fluctuations in each word is proportional to its expected frequency. As a null model for the joint configuration over all languages in our set, if N links are drawn , the multinomial probability of a particular set nL independently from the distribution psample S SL is given by p nL S |N ∼e sample model −N D pSL pSL (7) where7 ! psample SL model p SL S,L X model sample sample model = D psample p + p D p p . L S|L L L S|L X model D psample p ≡ psample log SL SL SL L (8) 7 As long as we calibrate pmodel to agree with the per-language link frequencies nL /N in the data, the data will L always be counted as more typical than almost-all random samples, and its probability will come entirely from the KL divergences in the individual languages. 36 Multinomial samples of assignments n̂L S to each of the 22 × 81 (Swadesh, Language) pairs, with N links total drawn from distribution pL S null , will produce KL divergences uniformly distributed in the coordinate dΦ ≡ e−DKL dDKL , corresponding to the uniform increment of cumulative probability in the model distribution. We may therefore use the cumulative probability to the right of model p D pdata (one-sided p-value), in the distribution of samples n̂L SL SL S , as a test of consistency of our data with the model of random sampling. In the next two subsections we will generate and test candidates for pmodel which are different functions of the mean link numbers on Swadesh concepts and the total links numbers in languages. 2. Product model with intrinsic property of concepts In general we wish to consider the consistency of joint configurations with random sampling, as a function of an aggregation scale. To do this, we will rank-order languages by increasing nL , form non-overlapping bins of 1, 3, or 9 languages, and test the resulting binned degree distributions against different mean-degree and sampling models. We denote by nL the average total link number in a bin, and by nL S the average link number per Swadesh entry in the bin. The simplest model, which assumes no interaction between concept and language properties, makes the model probability pmodel a product of its marginals. It is estimated from data without regard to binning, SL as ≡ pproduct SL nL nS × . N N (9) The 22 × 81 independent mean values are thereby specified in terms of 22 + 81 sample estimators. The KL divergence of the joint configuration of links in the actual data from this model, under whichever binning is used, becomes model D pdata p =D SL SL L L ! nS nS n N N N (10) As we show in Fig. 11 below, even for 9-language bins which we expect to average over a large amount of language-specific fluctuation, the product model is ruled out at the 1% level. We now show that a richer model, describing interaction between word and language properties, accepts not only the 9-language aggregate, but also the 3-language aggregate with a small adjustment of the language size to which words respond (to produce consistency with word-size and language-size marginals). Only fluctuation statistics at the level of the joint configuration of Swadeshes 37 Languages product , nsample in accordance with Fig. 4S (f). The colors denote FIG. 9. Plots for the data nL S , N pSL SL corresponding numbers of the scale. The original data in the first panel with the sample in the last panel seems to agree reasonably well. 81 individual languages remains strongly excluded by the null model of random sampling. 3. Product model with saturation An inspection of the deviations of our data from the product model shows that the initial propensity of a word to participate in polysemies, as inferred in languages where that word has few links, in general overestimates the number of links (degree). Put it differently, languages seem to place limits on the weight of single polysemies, favoring distribution over distinct polysemies, but the number of potential distinct polysemies is an independent parameter from the likelihood that the available polysemies will be formed. Interpreted in terms of our supposed semantic space, the proximity of target words to a Swadesh entry may determine the likelihood that they will be polysemous with it, but the total number of proximal targets may vary independently of their absolute proximity. These limits on the number of neighbors of each concept are captured by additional 22 parameters. To accommodate such characteristic, we revise the model Eq. (9) to the following function: AS nL . BS + hnL i 38 25 40 20 nLS=MOON nLS=WATER 30 20 10 15 10 5 0 0 0 20 40 60 0 20 <nL>bin 40 60 <nL>bin FIG. 10. Plots of the saturating function (11) with the parameters given inTable IV, compared to nL S (ordinate) in 9-language bins (to increase sample size), versus bin-averages nL (abscissa). Red line is drawn through data values, blue is the product model, and green is the saturation model. WATER requires no significant deviation from the product model (BWATER /N 20), while MOON shows the lowest saturation value among the Swadesh entries, at BMOON ≈ 3.4. where degree numbers nL S for each Swadesh S is proportional to AS and language size, but is bounded by BS , the number of proximal concepts. The corresponding model probability for each language then becomes psat SL = p̃S pdata (AS /BS )(nL /N ) L ≡ . L 1 + n /BS 1 + pdata L N/BS (11) As all BS /N → ∞ we recover the product model, with pdata ≡ nL /N and p̃S → nS /N . L A first-level approximation to fit parameters AS and BS is given by minimizing the weighted mean-square error E≡ X L !2 AS n L 1 X L nS − . hnL i BS + hnL i (12) S The function (12) assigns equal penalty to squared error within each language bin ∼ nL , proportional to the variance expected from Poisson sampling. The fit values obtained for AS and BS do not depend sensitively on the size of bins except for the Swadesh entry MOON in the case where all 81 single-language bins are used. MOON has so few polysemies, but the MOON/month polysemy is so likely to be found, that the language Itelman, with only one link, has this polysemy. This point leads to instabilities in fitting BMOON in single-language bins. For bins of size 3–9 the instability is removed. Representative fit parameters across this range are shown in Table IV. Examples of the saturation model for two words, plotted against the 9-language binned degree data in Fig. 10, 39 Meaning category Saturation: BS Propensity p̃S STAR 1234.2 0.025 SUN 25.0 0.126 YEAR 1234.2 0.021 SKY 1234.2 0.080 SEA/OCEAN 1234.2 0.026 STONE/ROCK 1234.2 0.041 MOUNTAIN 1085.9 0.049 DAY/DAYTIME 195.7 0.087 SAND 1234.2 0.026 ASH(ES) 13.8 0.068 SALT 1234.2 0.007 FIRE 1234.2 0.065 SMOKE 1234.2 0.031 NIGHT 89.3 0.034 DUST 246.8 0.065 RIVER 336.8 0.048 WATER 1234.2 0.073 LAKE 1234.2 0.047 MOON 1.2 0.997 EARTH/SOIL 1234.2 0.116 CLOUD(S) 53.4 0.033 WIND 1234.2 0.051 TABLE IV. A table of fitted values of parameters BS and p̃S for the saturation model of Eq. (11) . The saturation value BS is an asymptotic number of meanings associated with the entry S, and the propensity p̃S is a rate at which the number of polysemies increases with nL at low nL S. show the range of behaviors spanned by Swadesh entries. The least-squares fits to AS and BS do not directly yield a probability model consistent with the marginals for language size that, in our data, are fixed parameters rather than sample variables P to be explained. They closely approximate the marginal N L psat SL ≈ nS (deviations < 1 link P sat for every S) but lead to mild violations N S pSL 6= nL . We corrected for this by altering the saturation model to suppose that, rather than word properties’ interacting with the exact value nL , they interact with a (word-independent but language-dependent) multiplier 1 + ϕL nL , so that the model for nL S in each language becomes becomes AS 1 + ϕL nL , BS + (1 + ϕL ) nL in terms of the least-squares coefficients AS and BS of Table IV. The values of ϕL are solved with 40 500 450 400 Histogram counts 350 300 250 200 150 100 50 0 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 DKL(samples || saturation model) FIG. 11. Kullback-Leibler divergence of link frequencies in our data, grouped into non-overlapping 9language bins ordered by rank, from the product distribution (9) and the saturation model (11). Parameters AS and BS have been adjusted (as explained in the text) to match the word- and language-marginals. From 10,000 random samples n̂L S , (green) histogram for the product model; (blue) histogram for the saturation model; (red dots) data. The product model rejects the 9-language joint binned configuration at the at 1% level (dark shading), while the saturation model is typical of the same configuration at ∼ 59% (light shading). Newton’s method to produce N P S L psat SL → n , and we checked that they preserve N sat L pSL P ≈ nS within small fractions of a link. The resulting adjustment parameters are plotted versus nL for individual languages in Fig. 12. Although they were computed individually for each L, they form a smooth function of nL , possibly suggesting a refinement of the product model, but also perhaps reflecting systematic interaction of small-language degree distributions with the error function (12). 0.2 0.1 0 ϕL -0.1 -0.2 -0.3 -0.4 -0.5 0 10 20 30 40 50 60 70 nL FIG. 12. Plot of the correction factor ϕL versus nL for individual languages in the probability model used in text, with parameters BS and p̃S shown in Table IV. Although ϕL values were individually solved with Newton’s method to ensure that the probability model matched the whole-language link values, the resulting correction factors are a smooth function of nL . 41 200 180 160 Histogram counts 140 120 100 80 60 40 20 0 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 DKL(samples || saturation model) FIG. 13. The same model parameters as in Fig. 11 is now marginally plausible for the joint configuration of 27 three-language bins in the data, at the 7% level (light shading). For reference, this fine-grained joint configuration rejects the null model of independent sampling from the product model at p-value ≈ 10−3 (dark shading in the extreme tail). 4000 samples were used to generate this test distribution. The blue histogram is for the saturation model, the green histogram for the product model, and the red dots are generated data. With the resulting joint distribution psat SL , tests of the joint degree counts in our dataset for consistency with multinomial sampling in 9 nine-language bins are shown in Fig. 11, and results of tests using 27 three-language bins are shown in Fig. 13. Binning nine languages clearly averages over enough language-specific variation to make the data strongly typical of a random sample (P ∼ 59%), while the product model (which also preserves marginals) is excluded at the 1% level. The marginal acceptance of the data even for the joint configuration of three-language bins (P ∼ 7%) suggests that language size nL is an excellent explanatory variable and that residual language variations cancel to good approximation even in small aggregations. C. Single instances as to aggregate representation The preceding subsection showed intermediate scales of aggregation of our language data are sufficiently random that they can be used to refine probability models for mean degree as a function of parameters in the globally-aggregated graph. The saturation model, with data-consistent marginals and multinomial sampling, is weakly plausible by bins of as few as three languages. Down to this scale, we have therefore not been able to show a requirement for deviations from the independent sampling of links entailed by the use of the aggregate graph as a summary statistic. However, we were unable to find a further refinement of the mean distribution that would 42 reproduce the properties of single language samples. In this section we characterize the nature of their deviation from independent samples of the saturation model, show that it may be reproduced by models of non-independent (clumpy) link sampling, and suggest that these reflect excess synonymous polysemy. 1. Power tests and uneven distribution of single-language p-values To evaluate the contribution of individual languages versus language aggregates to the acceptance or rejection of random-sampling models, we computed p-values for individual languages or language bins, using the KL-divergence (5). A plot of the single-language p-values for both the null (product) model and the saturation model is shown in Fig. 14. Histograms for both single languages (from the values in Fig. 14) and aggregate samples formed by binning consecutive groups of three languages are shown in Fig. 15. For samples from a random model, p-values would be uniformly distributed in the unit interval, and histogram counts would have a multinomial distribution with single-bin fluctuations depending on the total sample size and bin width. Therefore, Fig. 15 provides a power test of our summary statistics. The variance of the multinomial may be estimated from the large-p-value body where the distribution is roughly uniform, and the excess of counts in the small-p-value tail, more than one standard deviation above the mean, provides an estimate of the number of languages that can be confidently said to violate the random-sampling model. From the upper panel of Fig. 15, with a total sample of 81 languages, we can estimate a number of ∼ 0.05 × 81 ≈ 4–5 excess languages at the lowest p-values of 0.05 and 0.1, with an additional 2–3 languages rejected by the product model in the range p-value ∼ 0.2. Comparable plots in Fig. 15 (lower panel) for the 27 three-language aggregate distributions are marginally consistent with random sampling for the saturation model, as expected from Fig. 13 above. We will show in the next section that a more systematic trend in language fluctuations with size provides evidence that the cause for these rejections is excess variance due to repeated attachment of links to a subset of nodes. 43 0 −0.5 −1 log10(P) −1.5 −2 −2.5 −3 −3.5 −4 0 10 20 30 40 50 60 70 80 90 Language (rank) FIG. 14. log10 (p−value) by KL divergence, relative to 4000 random samples per language, plotted versus language rank in order of increasing nL . Product model (green) shows equal or lower p-values for almost all languages than the saturation model (blue). Three languages – Basque, Haida, and Yorùbá – had value p = 0 consistently across samples in both models, and are removed from subsequent regression estimates. A trend toward decreasing p is seen with increase in nL . 2. Excess fluctuations in degree of polysemy If we define the size-weighted relative variance of a language analogously to the error term in Eq. (12), as σ2 L ≡ 2 1 X L L model n − n p , S S|L nL (13) S Fig. 16 shows that − log10 (p−value) has high rank correlation with σ 2 L and a roughly linear regression over most of the range.8 Two languages (Itelmen and Hindi), which appear as large outliers relative to the product model, are within the main dispersion in the saturation model, showing that their discrepancy is corrected in the mean link number. We may therefore understand a large fraction of the improbability of languages as resulting from excess fluctuations of their degree numbers relative to the expectation from Poisson sampling. Fig. 17 then shows the relative variance from the saturation model, plotted versus total average link number for both individual languages and three-language bins. The binned languages show no significant regression of relative variance away from the value unity for Poisson sampling, whereas single languages show a systematic trend toward larger variance in larger languages, a pattern that 8 Recall from Eq. (6) that the leading quadratic term in the KL-divergence differs from σ 2 L in that it presumes Poisson fluctuation with variance nL pmodel at the level of each word, rather than uniform variance ∝ nL across S|L all words in a language. The relative variance is thus a less specific error measure. 44 0.35 Unambiguous excess low-P samples Fraction of languages with probability P 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.8 0.9 1 P (bin center) 0.35 Fraction of languages with probability P 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 P (bin center) FIG. 15. (Upper panel) Normalized histogram of p-values from the 81 languages plotted in Fig. 14. The saturation model (blue) produces a fraction ∼ 0.05 × 81 ≈ 4–5 languages in the lowest p-values {0.05, 0.1} above the roughly-uniform background for the rest of the interval (shaded area with dashed boundary). A further excess of 2–3 languages with p-values in the range [0, 0.2] for the product model (green) reflects the part of the mismatch corrected through mean values in the saturation model. (Lower panel) Corresponding histogram of p-values for 27 three-language aggregate degree distributions. Saturation model (blue) is now marginally consistent with a uniform distribution, while the product model (green) still shows slight excess of low-p bins. Coarse histogram bins have been used in both panels to compensate for small sample numbers in the lower panel, while producing comparable histograms. we will show is consistent with “clumpy” sampling of a subset of nodes. The disappearance of this clumping in binned distributions shows that the clumps are uncorrelated among languages at similar nL . 45 3 2.5 Hindi - log10(P) 2 1.5 1 0.5 Itelmen 0 0 0.5 1 1.5 2 2.5 3 (σ2)L 3.5 3 - log10(P) 2.5 2 1.5 1 0.5 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (σ2)L L FIG. 16. (Upper panel:) − log10 (P ) plotted versus relative variance σ 2 from Eq. (13) for the 78 languages with non-zero p-values from Fig. 14. (blue) saturation model; (green) product model. Two languages (circled) which appear as outliers with anomalously small relative variance in the product model—Itelman and Hindi—disappear into the central tendency with the saturation model. (Lower panel:) an equivalent plot for 26 three-language bins. Notably, the apparent separation of individual large-nL languages into two L groups has vanished under binning, and a unimodal and smooth dependence of − log10 (P ) on σ 2 is seen. 3. Correlated link assignments We may retain the mean degree distributions, while introducing a systematic trend of relative variance with nL , by modifying our sampling model away from strict Poisson sampling to introduce “clumps” of links. To remain within the use of minimal models, we modify the sampling procedure by a single parameter which is independent of word S, language-size nL , or particular language L. We introduce the sampling model as a function of two parameters, and show that one function of these is constrained by the regression of excess variance. (The other may take any interior value, 46 3 (σ2)L = 0.011938 nL + 0.87167 (σ2)L = 0.0023919 nL + 0.94178 2.5 (σ2)L 2 1.5 1 0.5 0 0 10 20 30 40 50 60 70 nL FIG. 17. Relative variance from the saturation model versus total link number nL for 78 languages excluding Basque, Haida, and Yorùbá. Least-squares regression are shown for three-language bins (green) and individual languages (blue), with regression coefficients inset. Three-language bins are consistent with Poisson sampling at all nL , whereas single languages show systematic increase of relative variance with increasing nL . so we have an equivalence class of models.) In each language, select a number B of Swadesh entries randomly. Let the Swadesh indices be denoted {Sβ }β∈1,...B . We will take some fraction of the total links in that language, and assign them only to the Swadeshes whose indices are in this privileged set. Introduce a parameter q that will determine that fraction. We require correlated link assignments be consistent with the mean determined by our model fit, since binning of data has shown no systematic effect on mean parameters. Therefore, for the random choice {Sβ }β∈1,...B , introduce the normalized density on the privileged links pmodel S|L πS|L ≡ PB model β=1 pSβ |L (14) if S ∈ {Sβ }β∈1,...B and πS|L = 0 otherwise. Denote the aggregated weight of the links in the privileged set by W ≡ B X pSβ |L . (15) β=1 Then introduce a modified probability distribution based on the randomly selected links, in the 47 form p̃S|L ≡ (1 − qW ) pS|L + qW πS|L . (16) Multinomial sampling of nL links from the distribution p̃S|L will produce a size-dependent variance of the kind we see in the data. The expected degrees given any particular set {Sβ } will not agree with the means in the aggregate graph, but the ensemble mean over random samples of languages will equal pS|L , and binned groups of languages will converge toward it according to the central-limit theorem. The proof that the relative variance increases linearly in nL comes from the expansion of the expectation of Eq. (13) for random samples, denoted D σ̂ 2 L E * ≡ 2 1 X L L model n̂ − n p S S|L nL + S * i2 1 Xh L L L model = n̂ − n p̃ + n p̃ − p S|L S|L S S|L nL S * + X 1 2 L = n̂L + S − n p̃S|L nL S * + 2 X model L p̃S|L − pS|L . n + (17) S The first expectation over n̂L S is constant (of order unity) for Poisson samples, and the second expectation (over the sets {Sβ } that generate p̃S|L ) does not depend on nL except in the prefactor. Cross-terms vanish because link samples are not correlated with samples of {Sβ }. Both terms in the third line of Eq. (17) scale under binning as (bin-size)0 . The first term is invariant due to Poisson sampling, while in the second term, the central-limit theorem reduction of the variance in samples over p̃S|L cancels growth in the prefactor nL due to aggregation. Because the linear term in Eq. (17) does not systematically change under binning, we interpret the vanishing of the regression for three-language bins in Fig. 17 as a consequence of fitting of the mean value to binned data as sample estimators.9 Therefore, we require to choose parameters B and q so that regression coefficients in the data are typical in the model of clumpy sampling, while regressions including zero have non-vanishing weight in models of three-bin aggregations. Fig. 18 compares the range of regression coefficients obtained for random samples of languages 9 We have verified this by generating random samples from the model (17), fitting a saturation model to binned sample configurations using the same algorithms as we applied to our data, and then performing regressions equivalent to those in Fig. 17. In about 1/3 of cases the fitted model showed regression coefficients consistent with zero for three-language bins. The typical behavior when such models were fit to random sample data was that the three-bin regression coefficient decreased from the single-language regression by ∼ 1/3. 48 with the values nL in our data, from either the original saturation model psat S|L , or the clumpy model p̃S|L randomly re-sampled for each language in the joint configuration. Parameters used were (B = 7, q = 0.975).10 With these parameters, ∼ 1/3 of links were assigned in excess to ∼ 1/3 of words, with the remaining 2/3 of links assigned according to the mean distribution. 7 x 10 5 6 5 4 3 2 1 0 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 FIG. 18. Histograms of regression coefficients for language link samples n̂L either generated by Poisson S sampling from the saturation model pmodel fitted to the data (blue), or drawn from clumped probabilities S|L p̃S|L defined in Eq. (16), with the set of privileged words {Sβ } independently drawn for each language (green). Solid lines refer to joint configurations of 78 individual languages with the nL values in Fig. 17. Dashed lines are 26 non-overlapping three-language bins. The important features of the graph are: 1) Binning does not change the mean regression coefficient, verifying that Eq. (17) scales homogeneously as (bin-size)0 . However, the variance for binned data increases due to reduced number of sample points; 2) the observed regression slope 0.012 seen in the data is far out of the support of multinomial sampling from psat S|L , whereas with these parameters, it becomes typical under p̃S|L while still leaving significant probability for the three-language binned regression around zero (even without ex-post fitting). [1] Brown, C. H., General principles of human anatomical partonomy and speculations on the growth of partonomic nomenclature. Am. Ethnol. 3, 400-424 (1976). [2] Brown, C. H., A theory of lexical change (with examples from folk biology, human anatomical partonomy and other domains). Anthropol. Linguist. 21, 257-276 (1979). 10 Solutions consistent with the regression in the data may be found for B ranging from 3–17. B = 7 was chosen as an intermediate value, consistent with the typical numbers of nodes appearing in our samples by inspection. 49 [3] Brown, C. H. & Witkowski, S. 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