Social contagion models on hypergraphs

PHYSICAL REVIEW RESEARCH 2, 023032 (2020) Social con agion models on hype g aphs Guilhe me Fe az de A uda ,1Gio anni Pe i,1and Yami Mo eno 2,3,1 1ISI Founda ion, Via Chisola 5, 10126 To ino, I aly 2Ins i u e o Biocompu a ion and Physics o Complex Sys ems (BIFI), Uni e si y o Za agoza, 50018 Za agoza, Spain 3Depa men o Theo e ical Physics, Uni e si y o Za agoza, 50018 Za agoza, Spain (Recei ed 4 Oc obe 2019; accep ed 10 Ma ch 2020; published 10 Ap il 2020) Ou unde s anding o he dynamics o complex ne wo ked sys ems has inc eased signi ican ly in he las wo decades. Howe e , mos o ou knowledge is buil upon assuming pai wise ela ions among he sys em’s componen s. This is o en an o e simpli ica ion, o ins ance, in social in e ac ions ha occu equen ly wi hin g oups. To o e come his limi a ion, he e we s udy he dynamics o social con agion on hype g aphs. We de elop an analy ical amewo k and p o ide nume ical esul s o a bi a y hype g aphs, which we also suppo wi h Mon e Ca lo simula ions. Ou analyses show ha he model has a as pa ame e space, wi h i s - and second- o de ansi ions, bis abili y, and hys e esis. Phenomenologically, we also ex end he concep o la en hea o social con ex s, which migh help unde s anding oscilla o y social beha io s. Ou wo k un olds he esea ch line o highe -o de models and he analy ical ea men o hype g aphs, posing new ques ions and pa ing he way o modeling dynamical p ocesses on highe -o de s uc u es. DOI: 10.1103/PhysRe Resea ch.2.023032 Ne wo k science has had a adical impac on ou knowl- edge abou c i ical dynamics in complex sys ems. In pa - icula , new and ele an phenomena a ise when in es iga - ing social and biological con agion p ocesses [1–7]. Fo in- s ance, while homogeneous sp eading models p edic ini e c i ical poin s [2,6,7], he e ogeneous ne wo ks o en p esen anishing ansi ions [1,2,6–8], suppo ing he p edic ions in eal-wo ld ne wo ks [9–12]. Con agion models co e many aspec s, om di e en ypes o iche subs a es unde lying he p ocess i sel . A ele an de elopmen is he ex ension o con agion p ocesses o mul ilaye ne wo ks, leading he way o combina o ial highe -o de models. Indeed, mul ilaye s’ s uc u al [13–17], sp eading [7,14], and di usion p ope - ies [13,18] ha e new and iche phenomenology. Ne e he- less, as ecen ly a gued in Re . [19], eal da a a e e ealing ha pai wise ela ionships— he undamen al in e ac ion uni s o ne wo ks—do no cap u e complex dependencies. Indeed, mode n messaging sys ems (e.g., Wha sApp, Tele- g am, and Facebook Messenge , among o he s) allow use s o communica e in g oups, which c ea es a di ec channel among all membe s. In o he wo ds, mode n in o ma ion sp eading is o en a one- o-many p ocess. Addi ionally, biological and eam collabo a ions a e also inhe en ly g oup s uc u ed, sim- ila ly o some ypes o molecula in e ac ions [20]. Comple- men a y, he e is e idence om social and biological s udies indica ing ha highe -o de s uc u es ha e c ucial dynamical e ec s [21–24]. The e o e, highe -o de in e ac ions a e ubiq- ui ous, and unde s anding hei p ope ies and impac s is o pa amoun impo ance. No ably, he sizes o such g oups can Published by he Ame ican Physical Socie y unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s) and he published a icle’s i le, jou nal ci a ion, and DOI. span o de s o magni ude. Thus, g aph-p ojec ion-based ap- p oaches migh no be su icien o desc ibe sys ems in ol ing in e ac ions o e many di e en scales and o de s. Combina o ial highe -o de models [19] o e a way o desc ibe hese sys ems as hey o e come he limi a ions o lowe -o de ne wo kmodels.Ina i s a emp ,Bodóe al.[21] p oposed an SIS disease sp eading in a hype g aph. Nex , Iacopini e al. [22] p esen ed a model o social con agion de ined on simplicial complexes and p o ided app oxima e solu ions o complexes o o de h ee. Thei model p esen ed new phenomenological pa e ns associa ed wi h he c i ical p ope ies o he dynamics. Howe e , hei p oposed model is s ill e y cons ained, bo h s uc u ally and dynamically. He e we ex end hei model bo h s uc u ally and dynami- cally. S uc u ally, we adop hype g aphs, which gene alize he concep o g aphs, by allowing an edge o ha e an a bi a y numbe o nodes (see Fig. 1 o an illus a ion). Hype g aphs elax he s uc u al es ic ions equi ed by simplicial com- plexes as hey impose i ually no limi a ion on he ype, size, and mu ual inclusion o in e ac ions, hus, ep esen ing mo e ai h ully and na u ally eal sys ems. F om he dynami- cal iewpoin , we inco po a e explici c i ical-mass dynamics (each hype edge is an independen c i ical-mass p ocess), gene alizing he one modeled in Re . [22]. The esul ing model displays a ich complex phenomenology, emaining e y lexible and able o co e a wide ange o sys ems. We unco e he p esence o discon inuous ansi ions and bis a- bili y led by highe -o de in e ac ions and c i ical-mass dy- namics. No ably, hese c i ical p ope ies con as wi h con a- gion models on g aphs-models, which ins ead usually display con inuous ansi ions, e.g., SIS o SIR disease sp eading, ega dless o he s uc u al con igu a ion. The e o e, assuming a g aph p ojec ion o a hype g aph migh lead o w ong esul s. He e we epo analy ical and nume ical analyses o he heo e ical amewo k ha we in oduce as well as esul s o se e al limi ing cases and hype g aph s uc u es. We ound 2643-1564/2020/2(2)/023032(6) 023032-1 Published by he Ame ican Physical Socie y DE ARRUDA, PETRI, AND MORENO PHYSICAL REVIEW RESEARCH 2, 023032 (2020) FIG. 1. G aphical ep esen a ion o a hype g aph. Ma hema i- cally, V={ 1, 2, 3, 4, 5, 6, 7, 8},E={e1,e2,e3,e4},whe e he hype edges a e e1={ 1, 2, 3},e2={ 3, 4, 5, 6},e3= { 6, 7}and e4={ 8}. o he pape by discussing se e al implica ions o ou s udy and, mos no ably, he ole o c i ical mass dynamics in social con agion, p o iding insigh s ha could help explain epo ed di e ences in expe imen al esul s [24–27]. Le us i s in oduce some o mal de ini ions. A hype - g aph is de ined as a se o nodes, V={ i}, whe e N=|V| is he numbe o nodes, and a se o hype edges E={ej}, whe e ejis a subse o Vwi h a bi a y ca dinali y |ej|.I max (|ej|)=2 we eco e a g aph. We ema k ha we do no explo e his scena io since i eco e s s anda d models. On he o he hand, i o each hype edge wi h |ej|>2 i s subse s a e also con ained in E, we eco e a simplicial complex ( o mo e on he hype g aph s uc u e, see he Supplemen al Ma e ial [28], Sec. I). Figu e 1shows an example o a hy- pe g aph. In an a bi a y hype g aph, we associa e wi h each indi idual ia Be noulli andom a iable Yi(complemen a y Xi). I he node iis ac i e Yi=1(Xi=0), o he wise Yi=0 (Xi=1). To each ac i e node, we associa e a deac i a ion mechanism, modeled as a Poisson p ocess wi h pa ame e δi, Nδi i(Yiδi −→ Xi). Fo each hype edge, j, we de ine a andom a iable Tj=k∈ejXk, which is he numbe o ac i e nodes in he hype edge. I Tjis equal o abo e a gi en h eshold, j, we associa e a Poisson p ocess wi h pa ame e λj,Nλj j ( ha is, i Tj⩾j, hen Xk λj −→ Yk,∀k∈ej). In o he wo ds, he dynamics is gi en by a h eshold p ocess ha becomes ac i e only abo e a c i ical mass o ac i a ed nodes. Mo eo e , i |ej|=2, we assume di ec ed Poisson p ocesses, implying ha i is no a h eshold p ocess anymo e. This de ini ion allows o eco e ing adi ional SIS con agion models. While he p oposed model is gene al, allowing o a bi a y he e o- genei y in pa ame e s, we ocus on mo e s aigh o wa d, bu ep esen a i e, cases. We assume ha δi=δand λj= (|ej|), whe e is an a bi a y unc ion o he ca dinali y o he hype edge. I is also con enien o de ine =∗N, whe e he pa ame e ∗is a eal numbe ep esen ing he ac ion o ac i e nodes. The exac equa ion ha desc ibes he a o emen ioned dynamics can be w i en as dE(Yi) d =E⎛ ⎝−δYi+[1−Yi] ej∩{ i}=∅ λj B 1{Yi=0,Tj⩾j}⎞ ⎠, (1) whe e he i s summa ion is o e all hype edges con aining iand he second o e all he possible dynamical mic os a es inside he hype edge ej, deno ed by he se B. Fu he mo e, 1{Tj⩾j,Yi=0}is an indica o unc ion ha is 1 i Yi=0 and he c i ical mass in he hype edge is eached and 0 o he wise. Na u ally, he o de pa ame e is de ined as he expec ed ac ion o ac i e nodes, i.e., ρ=1 NiE(Yi). Al hough Eq. (1) cap u es he exac p ocess, i canno be nume ically sol ed. Thus, assuming ha he andom a iables a e independen and deno ing yi=E(Yi), we ob ain he i s - o de app oxima ion, gi en as dyi d =−δyi+λ(1−yi) ej∩{i}=∅ |ej|  k=j λ∗(|ej|)Pej(K=k), (2) whe e we assume ha he sp eading a e is composed by he p oduc o a ee pa ame e and a unc ion o he ca dinali y, i.e., λj=λ×λ∗(|ej|). In his o mula ion, he expec a ion o he indica o unc ion ollows a Poisson binomial dis ibu ion ( o mo e on his app oxima ion, see Re . [28], Sec. II). Fo mally, E(1{(Tj−Yk)⩾j})≈ |ej|  m=j Pej(K=l),(3) Pej(K=l)= A∈Fl i∈A yi j∈Ac (1 −yj),(4) whe e Flis he se o all subse s o kin ege s om {1,2,...,n=|ej|},Ais one o hose se s, and Acis i s complemen a y. In ui i ely, Aaccoun s o he possibly ac i e nodes and Ac he possibly inac i e ones. Thus, he summa ion o e Flconside s all possible nodal s a e con igu a ions in a gi en hype edge. Equa ion (4) is no nume ically s able i |ej| is la ge [29]. Conside ing he disc e e Fou ie ans o m, we ob ain a nume ic s able solu ion as [29] Pej(K=k)=1 n+1 n  l=0 C−lk n  m=1 [1 +(Cl−1)ym],(5) whe e C=exp ( 2iπ n+1), which hen allows one o compu e he solu ion o a bi a ily la ge hype edges. In e es ingly, al hough he whole a gumen is qui e in ica e, Eq. (5)is simple, allowing he nume ical e alua ion o Eq. (2). Ou main esul is ha a ich and di e se phase space, gene ally popula ed by con inuous and discon inuous an- si ions and hys e e ic beha io s, cha ac e izes con agion on hype g aphs. In pa icula , we ha e analy ically obse ed discon inui y and bis abili y in he o de pa ame e on op o some egula s uc u es. We p o ide ull de ails o he calcula ions in Re . [28] (see Secs. III and IV) o wo lim- i ing cases, namely, a hype g aph composed o a hype edge con aining all nodes in addi ion o (1) a andom egula ne wo k (which we call a hype blob) and (2) a s a ( e e ed o as a hype s a ). Fo he sake o cla i y, le us show he main esul s o he hype blob. In his case, we can exploi he s uc u al symme ies o sol e ρ(λ, λ∗,δ), ob aining wo locally s able solu ions. Speci ically, conside a hype g aph buil up as a 023032-2 SOCIAL CONTAGION MODELS ON HYPERGRAPHS PHYSICAL REVIEW RESEARCH 2, 023032 (2020) homogeneous se o pai wise in e ac ions wi h a e age deg ee kand a single addi ional hype edge con aining all nodes. In his case, he o de pa ame e can be sol ed as ( o he ull de i a ions, see Re . [28], Sec. III) ρLowe =1−δ kλ,i λ δ⩾1 k 0,o he wise (6) ρUppe =−δ+kλ−λ∗λ+4kλ∗λ2+[δ+(−k+λ∗)λ]2 (2kλ),(7) whe e a second-o de phase ansi ion o ρLowe is na u ally ob ained as λ δ⩾1 k[7]. Fu he mo e, he discon inui ies can also be calcula ed as λL c=δ k−∗k,(8) λU c=− δ∗ λ∗∗−λ∗+(∗)2k−∗k.(9) Phenomenologically, a discon inui y implies ha ou sys em possesses a “social la en hea ” ha is eleased o accumula ed a a cons an alue o λ. Mo e speci ically, be o e he discon inui y, “ene gy” has been s o ed in he pa ial ac i a ion o he hype edges. A he discon inui y his “ene gy” is abso bed ( eleased) a once o a cons an alue o λ. In ac , he social la en hea can be exp essed as QlλX c=[ρUppe (λ, δ, λ∗,N)−ρLowe (λ, δ, λ∗,N)]λ=λX c,(10) whe e Ql(λX c) can be Ql(λL c) (ene gy abso bed) o Ql(λU c) (ene gy eleased). The e o e, o his s uc u e, he la en hea is exp essed as QlλX c=δ−λ(λ∗+k)+[δ+λ(λ∗−k)]2+4λ∗kλ2 2kλλ=λX c ,(11) whe e λX ccan be (λL co λU c). In ac , his exp ession is ue o any alue o λ, bu i s physical in e p e a ion is alid only nea he discon inui y, which in u n depends on ρc=∗. We e e he eade o Re . [28] o mo e de ails. Figu e 2shows he gene al phenomenology o he sys- em ob ained om he analy ical solu ion, i.e., he i s -o de app oxima ion, o he equa ions desc ibing he con agion dynamics o he hype blob. As can be seen in Fig. 2(a), 0.0 0.2 0.4 0.6 0.8 1.0 λ 0.0 0.2 0.4 0.6 0.8 1.0 ρ Ql(λL c) Ql(λU c) λc (a) ρLowe ρUppe ρc(Θ∗,λ ∗(|e|)) ρ ρ∗ 0.0λcλ λ 0.0 0.2 0.4 0.6 0.8 1.0 Θ∗ (b) I II III IV V FIG. 2. Resul s o he hype blob. (a) Possible solu ions o a ixed ∗=0.5. In ed and blue a e he uppe and lowe solu ions (b anches), espec i ely. The ansi ion om he lowe o he uppe solu ion (uppe o lowe ) occu s a he in e sec ion o he lowe (uppe ) solu ion wi h a alue o ρcin which he uppe solu ion became s able (uns able). The discon inui y is cha ac e ized by he la en hea , Ql(λL c)o Ql(λU c). A λc=0.2, he lowe solu ion shows a second-o de phase ansi ion. (b) Schema ic o he pa ame e space: Region I: he abso bing s a e o bo h he lowe and uppe solu ion; Region II: only he lowe solu ion is s able ( he global c i ical mass is no eached, ρ<ρ c); Region III: ρUppe is s able and ρLowe =0 (below he c i ical poin ); Region IV: ρUppe >ρ Lowe >0 and bo h a e s able (bis able); Region V: only he uppe solu ion is s able ( he global c i ical mass was eached, ρ⩾ρc). 023032-3 DE ARRUDA, PETRI, AND MORENO PHYSICAL REVIEW RESEARCH 2, 023032 (2020) he e a e wo possible solu ions, ρLowe and ρUppe .Theso- lu ion depends on he ini ial condi ions and he h eshold, ∗, which, oge he wi h he s uc u e, de ines a alue ρc whe e he dynamics exhibi a discon inui y. I ρ( =0) ⩾ρc, he solu ion is gi en by ρ=ρUppe ( o wa d phase diag am). On he o he hand, i ρ( =0) <ρ cand ρ( =0) = 0, hen ρ=ρLowe (backwa d phase diag am). The a ows show hese solu ions as he size o he jump (i.e., he magni ude o he la en hea ). No e ha he lowe solu ion can exhibi a second- o de phase ansi ion. Nex , in Fig. 2(b) we ins ead ep esen he co esponding pa ame e space, which is composed o i e dis inc egions, as explained in he igu e cap ion. We assumed he mos gene al case, whe e he lowe solu ion has a ansi ion om he abso bing s a e o an ac i e s a e, he e a λc. We ema k ha , o some s uc u es, he lowe solu ion migh ha e a anishing c i ical poin (e.g., he hype s a ), i.e., λc→0, hus sligh ly changing his pic u e. Finally, we ha e compa ed bo h analy ic and nume ic es ima es o he la en hea o his hype g aph s uc u e. The esul s show ha he absolu e e o be ween analy ical and nume ical simula ions is o o de 10−2–10−3in hype g aphs wi h N=104(see Re . [28], Sec. VI.E and Table I), indica ing ha he i s -o de app oxima ion is accu a e. Thus, gene ically, he solu ions o a social con agion dynamics on hype g aphs can be ma hema ically exp essed as ρ=ρLowe i ρLowe <ρ c ρUppe i ρLowe ⩾ρc,(12) ρ∗=ρUppe i ρUppe ⩾ρc ρLowe i ρUppe <ρ c,(13) whe e ρis ob ained i ρ( =0) <ρ c, and ρ∗i ρ( = 0) ⩾ρc, whe e ρcis a global c i ical mass, i.e., he alue o ρa which he discon inui y appea s. As be o e, we no e ha he lowe solu ion (b anch) migh also exhibi a second- o de (con inuous) phase ansi ion, deno ed by λcin Fig. 2, om he abso bing s a e (ρ=0) o he ac i e s a e (ρ>0). Fu he mo e, o a gi en hype g aph wi h ixed δand λ∗, he discon inui y poin s a e o mally de ined as λL c=a gλ[ρLowe (λ, δ, λ∗,N)=ρc],(14) λU c=a gλ[ρUppe (λ, δ, λ∗,N)=ρc],(15) hus, also de ining he bis able egion, (λU c,λ L c). Al hough a closed solu ion o he gene al case is no pos- sible, Mon e Ca lo simula ions and nume ical e alua ion o Eq. (2) a e easonable al e na i es o cha ac e ize ou sys em (see Re . [28], Sec. VI). He e we ocus on a hype g aph wi h an exponen ial dis ibu ion o ca dinali ies (i.e., he numbe o nodes inside a hype edge), P(|e|)∼μexp (−μ|e|) wi h he cons ain ha |e|⩾2. Dynamically, we se λ∗=log2(|e|). This choice is a bi a y, bu we choose he e he log2(|e|) unc- ion because i g ows sublinea ly. No e ha , i a hype edge ca dinali y goes o in ini y, he a e age sp eading alue ends o ze o: lim|e|→∞ log2(|e|) |e|=0. The impac o such a unc ion is ye unknown, and we lea e his analysis o u u e wo k. Figu es 3(a) and 3(b) show ha he o de pa ame e and he suscep ibili y ollow he pa e ns expec ed o a i s -o de ansi ion, i.e., bo h a e discon inuous. Mo eo e , he o de FIG. 3. Es ima ion o ρand χusing he QS me hod in a hype - g aph wi h an exponen ial dis ibu ion o hype edge ca dinali ies and N=104. The dynamical pa ame e a e: δ=1.0, λ∗=log2(|ej|), and ∗=0.1,0.2. (a) The suscep ibili y; (b) he o de pa ame e . We conside ed wo ini ial condi ions o he QS me hod, ρQS( =0) = 0.01, da ke colo s, and ρQS( =0) =1.00 ligh e colo s. (c) The dis ibu ion o ac i e node es ima ed using he QS me hod a λ= 0.086 and ∗=0.2 [ he c ossing be ween he wo suscep ibili y cu es, in Fig. 3(a)]. pa ame e is bis able, implying he p esence o a hys e esis loop. This phenomenon is opposed o an SIS on a g aph. The SIS has a second-o de phase ansi ion, cha ac e ized by a con inuous beha io o he o de pa ame e and a di e ging suscep ibili y in he he modynamic limi . Complemen a ily, Fig. 3(c) shows he dis ibu ion o ac i e nodes in he uppe and lowe b anches. In he o me , we ha e a bell-shaped dis ibu ion, simila o he supe c i ical egime o an SIS p ocess [7,30]. In he la e , we ha e a dis ibu ion peaked a one, simila o he subc i ical egime (abso bing s a e) o an SIS p ocess [7,30]. We emphasize ha Fig. 3(c) displays he dis ibu ion o ac i e nodes o he uppe (le panel) and lowe ( igh panel) b anches and ha he comple e dis ibu ion o a gi en λin a egion whe e bo h solu ions exis is bimodal. In ui i ely, one expec s ha lowe ca dinali y hype edges a e esponsible o he lowe b anch as hey a e easie o ac i a e han he highe ca dinali y ones. No e ha , in egula cases, such as he hype blob (see Re . [28], Secs. III and IV), 023032-4 SOCIAL CONTAGION MODELS ON HYPERGRAPHS PHYSICAL REVIEW RESEARCH 2, 023032 (2020) he gian pai wise componen has Nnodes. Howe e , o he exponen ially dis ibu ed ca dinali y scena io, his is no he case. Al hough P(|e|=2) ⩾P(|e|=k), k=3,4...,N (exponen ial dis ibu ion), he la ges connec ed componen , is e y small, six nodes in he simula ed hype g aph. The e o e, in his scena io, he lowe b anch is de e mined by hype edges wi h ca dinali y lowe bu g ea e han wo. The gene ali y o he epo ed phenomenological beha io sugges s ha he obse ed dynamics a e a consequence o g oup-g oup in e - ac ions. Addi ional expe imen s ( epo ed in Re . [28]) o a hype g aph wi h a powe -law dis ibu ion o ca dinali y u he co obo a ed hese esul s. In all sys ems, we ound simila quali a i e beha io o ini e ne wo ks. Ou esul s a e ele an because hey p o ide a heo- e ical ounda ion o , and a phenomenological explana ion o, seemingly di e en expe imen al indings [24–27,31–33]. These wo ks epo ed c i ical mass le els needed o change an es ablished equilib ium o 10% in some expe imen s and 30%–40% in o he s, in appa en con adic ion. The o malism he e de eloped na u ally b ings o h plausible hypo heses o hese obse a ions and show ha bo h anges a e possible. On he one hand, s udies based on a single g oup sugges a h eshold be ween 30%–40%, a si ua ion ha can be modeled as a single hype edge in ou o malism. On he o he hand, a c i ical mass o 10% would co espond o a popula ion ha is composed o g oups o di e se sizes, each one wi h a (la ge ) ac i a ion h eshold. In o he wo ds, i is possible o ha e indi idual g oups exhibi ing a h eshold ∗be ween 30%–40%, and a he same ime, a global c i ical mass, ρc, o he whole popula ion o abou 10% due o g oup in e sec ions. A second eason ha could explain he expe imen al indings is e en mo e s aigh o wa d: admi edly, he ac ha ou model shows bis abili y also enables, o a gi en λ, wo possible solu ions o ρco esponding o he lowe and he uppe b anches. Tha is, he sys em migh be ope a ing in he egion whe e bo h solu ions a e la ge han ze o and s able. In summa y, in his pape , we ha e de eloped a amewo k ha allows ex ending he s udy o social con agion models when g oup in e ac ions a e ele an . This is achie ed by conside ing hype g aphs as he subs a es ha cap u e such many- o-many in e ac ions. Fi s , ou wo k opens he pa h o deal wi h new dynamical p ocesses on op o highe -o de models and speci ically on hype g aphs. Second, we showed ha simple dynamical p ocesses could exhibi e y ich dy- namics, wi h di e en ansi ions, bis abili y, and hys e esis. Hype g aphs a e ubiqui ous, and ou heo y sugges s ha such a s uc u e allows o he phase diag am epo ed he e. Se e al indings suppo he ele ance o his me hodology. We ema k ha , depending on he s uc u e, adi ional g aph- p ojec ed models may lead o w ong esul s. Ul ima ely, he unco e ed phenomenology allows explaining seemingly con- adic o y expe imen al indings in which g oup in e ac ions play a majo ole. We also men ion ha many in e es ing ques ions a ise om ou wo k. Fo ins ance, i one assumes ha ene gy is p opo ional o ρ, ou model migh display phenomena eminiscen o a Ca no cycle o social con ex s, which migh help o unde s and ab up changes and oscilla- o y pa e ns in social beha io s. G.F.A. hanks E. A iges, H. F. de A uda, J. P. 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