Supplementary Material for. Robust Reconstruction of Complex Networks from Sparse Data

Transcription

1 Supplmntary Matrial for Roust Rconstruction of Complx Ntworks from Spars Contnts 1 Prformanc assssmnt 2 2 Dtail rsults in aition to Tal I an II in th main txt 2 3 of mpirical ntworks 3 4 Infrring intrinsic iniviual ynamics in ultimatum gams 4 5 Supplmntary Rfrncs 5 6 Supplmntary Figurs 6 1

2 1 Prformanc assssmnt To quantify th prformanc of our rconstruction mtho, w introuc two stanar masurmnt inics, th ara unr th rcivr oprating charactristic curv () an th ara unr th prcision-rcall curv () [S1]. Tru positiv rat (TPR), fals positiv rat (FPR), Prcision an Rcall that ar us to calculat an ar fin as follows: TPR(l) = TP(l) P, (S1) whr l is th cutoff in th g list, TP(l) is th numr of tru positivs in th top l prictions in th g list, an P is th numr of positivs in th gol stanar. FPR(l) = FP(l) Q, (S2) whr FP(l) is th numr of fals positiv in th top l prictions in th g list, an Q is th numr of ngativs in th gol stanar. TP(l) Prcision(l) = TP(l)+FP(l) = TP(l), (S3) l Rcall(l) = TP(l) P, whrrcall(l), which is call snsitivity, is quivalnt totpr(l). (S4) 2 Dtail rsults in aition to Tal I an II in th main txt Tal I in th main txt shows th minimum ata amount that nsur at last 95% an simultanously for iffrnt cass. In Supplmntary Matrials, w provi mor tails of an as a function of ata amount for all th cass prsnt in Tal I in th main txt. Supplmntary Fig. S2- S4 show for ntwork siz N = 100 an avrag no gr k = 6, an as a function of ata amount for iffrnt varianc σ 2 of Gaussian whit nois N(0,σ 2 ) m in th tim sris for otaining vctor Y in th rconstruction form Y = ΦX. For thr typs ynamical procsss, volutionary ultimatum gams, transportation an communications, full rconstruction of ntwork structur can achiv from sufficint ata in thr typs mol ntworks. In th asnc of nois or for small nois varianc, say σ = 5 (Supplmntary Fig. S2 an Fig. S3), full rconstruction can assur y small amounts of ata rlativ to th ntwork siz N. For larg nois varianc, say, σ = 0.3 (Supplmntary Fig. S4), full rconstruction can still achiv as on rlativly larg amounts of ata for iffrnt ntworks, manifsting th strong roustnss of our mtho against nois in tim sris. W hav also tst th roustnss of our rconstruction against th xistnc of xtrnally inaccssil nos. W assum that a fraction of ranomly slct nos cannot masur an thir tim sris ar missing. Supplmntary Fig. S5 an Fig. S6 show an as a function of ata 2

3 mount for n m = 5% an n m = 30% fraction of inaccssil nos, rspctivly. W s that vn for n m = 30%, w can still accuratly rconstruct connctions among availal nos from tim sris of thr ynamical procsss, inicating strong roustnss of our mtho against th missing of partial nos. Furthrmor, as on th rconstruct ntwork, w can intify who has connctions with unosrval nos in volutionary ultimatum gams. This can accomplish y comparing actual Y with rconstruct Y from tim sris. To concrt, w can us th tim sris of playrs stratgis an rconstruct ntwork to calculat payoffs of a playr in iffrnt rouns. Th calculat Y will iffr from th actual Y if th rconstruct no conncts to som of th inaccssil nos, which provis a clu for infrring irct nighors of missing nos. Supplmntary Fig. S7 an Fig. S8 show an as a function of ata amount for avrag no gr k = 12 an k = 18, rspctivly, with N = 100. Th rsults monstrat that for larg valus of k, our mtho can guarant complt intification of all links for iffrnt cominations of ntwork structurs an ynamical procsss. Supplmntary Fig. S9 an Fig. S10 show for k = 6, an as a function of ata amount for th ntwork siz N = 500 an 1000, rspctivly. Supplmntary Fig. S11 shows th rconstruction prformanc of our mtho appli to svral mpirical ntworks. W s that for largr ntwork sizs, our mtho rquirs rlativly lss ata to ascrtain all links. This is u to th fact that th nighoring vctor coms sparsr in largr ntworks. ThL 1 norm in th lasso nsurs th rquirmnt of lss ata for rconstructing a sparsr vctor, allowing us to achiv full rconstruction y using lss ata for largr ntworks. 3 of mpirical ntworks In th main txt, w hav us svral mpirical ntworks to tst th prformanc of our rconstruction mtho. Hr w provi mor tails of ths mpirical ntworks, as isplay in Supplmntary Tal S1. Supplmntary Tal S1: Summary of th mpirical ntworks us in Tal II in th main txt. Hr N nots ntwork sizs, L nots th numr of links an k rprsnts avrag gr of a ntwork. Ntwork ata can ownloa from rlvant rfrncs. Ntworks N L k Class Dscription Karat [S2] Social ntwork Ntwork of frinship in a karat clu Dolphins [S3] Social ntwork Frqunt associations twn 62 olphins Ntscinc [S4] Social ntwork Coauthorship ntwork of scintists working on ntwork IEEE 39 BUS [S5] Powr ntwork 10-machin Nw-Englan Powr Systm ntwork IEEE 118 BUS [S6] Powr ntwork A portion of th Amrican Elctric Powr Systm ntwork IEEE 300 BUS [S7] Powr ntwork Ntwork vlop y th IEEE Tst Systms Task Forc Footall [S8] Social ntwork Ntwork of Amrican collg footall gam Jazz [S9] Social ntwork Ntwork of jazz musicians [S10] Social ntwork Ntwork of mail intrchangs 3

4 4 Infrring intrinsic iniviual ynamics in ultimatum gams Insofar as th ntwork structur of volutionary ultimatum gams ar ascrtain, stratgy upating ruls of playrs accompani with mutation nois can infrr y a phas iagram. To concrt, w introuc th phas iagram y fining two varials, as follows: an p i,max p i (t+1) p max (t) p i,i p i (t+1) p i (t), (S5) (S6) whr rprsnts asolut valu an p max is th proposal associat with maximum payoff in th nighorhoo of noi. Hr p i,max rprsnts th asolut iffrnc twnp i (t+1) anp max (t). If p i,max is sufficintly small,.g., smallr than stratgy upating (SU) nois δ (mutation rat), w can confintly infr that playrilarn th stratgy corrsponing to th maximum payoff ini s nighors at tim t + 1. On th othr han, p i,i that masurs th asolut iffrnc twn p i (t + 1) an p i (t) can us to ascrtain if playr i upat his/hr stratgy at tim t + 1. Th fact that p i,i is smallr than SU nois δ inicats that i in t upat his/hr stratgy at tim t + 1. Thus th twoparamtr phas iagram provis ncssary information for rvaling how playrs upat thir stratgis in th volutionary gams. Sinc w hav alray ha th intractions ntwork amount playrs, y incorporating th tim sris of payoffs an stratgis, w can intify p max (t) of ach playr in ach roun, allowing us to raw th phas iagram. As shown in Supplmntary Fig. S12, th p i,max - p i,i phas iagram is ivi into four rgions y two ounaris of SU nois. In rgion (I) ( p i,max < δ an p i,i > δ), ata points rflct that iniviuals upat thir stratgis y imitating thos of thir nighors with th highst payoffs. In rgion (II) ( p i,i < δ an p i,max > δ), ata points inicat that playrs o not vary thir stratgis in th nxt roun. In rgion (III) ( p i,max < δ an p i,i < δ), on cannot ascrtain th actions of playrs u to th fact that th iffrnc twnp i (t+1) anp max (t) is lss than th noisδ. Howvr, if ata points appar in th iagonal of rgion (III), th playrs corrsponing to th ata points must hol th st stratgis in thir nighorhoos in th last roun. In rgion (IV) ( p i,max > δ an p i,i > δ), ata points inicat playrs upat thir stratgis y larning from thir nighors without highst payoffs. Dspit th xistnc of SU nois, th phas iagram provis sufficint information to uncovr how playrs upat thir stratgis, rgarlss of nois. As shown in Supplmntary Fig. S12, all ata points ar prsnt in rgion (I), (II) an (III), manifsting that thy ithr larn from th st stratgy or kp thir own stratgis in th last roun. Th phas iagram can implmnt for ach singl playr, allowing us to intify a mixtur of playrs with iffrnt larning inclinations. Although th groun truth of SU nois is unknown, th xplicit ounary inuc y SU nois in Supplmntary Fig. S12 yils th SU nois irctly, allowing us to infr intrinsic noal ynamics associat with mutation nois in stratgy upating. If all playrs larn from st stratgis in thir nighorhoo, thr will xist an arupt transition at th ounaris from rgion (I) to rgion (IV) an from rgion (II) to rgion (IV). Th phas transition can simply prov to xist in th phas iagram. 4

5 5 Supplmntary Rfrncs [S1] D. Marach, J. C. Costllo an R. Kúffnr, t al, Nat. Mthos, 9, 796 (2012). [S2] W. W. Zachary, J. Anthropol. Rs. 33, 452 (1977). [S3] D. Lussau, t al, Bhav. Ecol. Socioiol 54, 396 (2003). [S4] M. E. J. Nwman, Phys. Rv. E (2006). [S5] M. A. Pai, Enrgy function analysis for powr systm staility (Springr, 1989). [S6] P. M. Mahav an R. D. Christi, IEEE Trans. Powr Syst. 8, 1084 (1993). [S7] H. Glatvitsch, F. Alvarao, IEEE Trans. Powr Syst. 13, 1013 (1998). [S8] M. Girvan an M. E. J. Nwman, Proc. Natl. Aca. Sci. USA 99, 7821 (2002). [S9] P. Glisr an L. Danon, Av. Complx Syst. 6, 565 (2003). [S10] R. Guimra, L. Danon, A. Diaz-Guilra, F. Giralt an A. Arnas, Phys. Rv. E, 68, , (2003). 5

6 6 Supplmntary Figurs Supplmntary Figur S1: (a) an () of rconstructing rsistor ntworks from iffrnt amounts of. (c) an () of rconstructing communication ntworks from iffrnt amounts of. Watts-Strogatz small-worl ntworks ar us with ntwork siz N = 100, avrag gr k = 6, an rwiring proaility 0.3. is th numr of ata samplings ivi y ntwork siz N. Th ash lins rprsnt th rsults of compltly ranom gusss. 6

7 c f Supplmntary Figur S2: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications. Ntwork siz N is 100. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Avrag gr k = 6 an Rwiring proaility of ntworks is

8 c f Supplmntary Figur S3: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications in th prsnc of nois. Ntwork siz N is 100. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Th avrag gr k = 6. Rwiring proaility of ntworks is 0.3. Th istriution of nois is N(0,5 2 ). 8

9 c f Supplmntary Figur S4: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications in th prsnc of nois. Ntwork siz N is 100. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Th avrag gr k = 6. Rwiring proaility of ntworks is 0.3. Th istriution of nois is N(0,0.3 2 ). 9

10 c f Supplmntary Figur S5: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications in th prsnc of xtrnally inaccssil nos. Th fraction n m of xtrnally inaccssil nos is 5. Ntwork siz N is 100. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Avrag gr k = 6. Rwiring proaility of ntworks is

11 c f Supplmntary Figur S6: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications in th prsnc of xtrnally inaccssil nos. Th fractionn m of xtrnally inaccssil nos is 0.3. Ntwork sizn is 100. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Avrag gr k = 6. Rwiring proaility of ntworks is

12 c f Supplmntary Figur S7: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications. Ntwork sizn is 100 an avrag gr k = 12. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Rwiring proaility of ntworks is

13 c f Supplmntary Figur S8: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications. Ntwork sizn is 100 an avrag gr k = 18. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Rwiring proaility of ntworks is

14 c f Supplmntary Figur S9: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications. Ntwork sizn is 500 an avrag gr k = 6. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Rwiring proaility of ntworks is

15 c f Supplmntary Figur S10: an of rconstructing ranom (), small-worl () an scal-fr () ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications. Ntwork siz N is 1000 an avrag gr k = 6. Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. Rwiring proaility of ntworks is

16 c f Karat Dolphins Ntscinc IEEE 39 BUS IEEE 118 BUS IEEE 300 BUS Footall Jazz Karat Dolphins Ntscinc IEEE 39 BUS IEEE 118 BUS IEEE 300 BUS Footall Jazz Supplmntary Figur S11: an of rconstructing svral mpirical ntworks as on th tim sris otain from (a)-() volutionary ultimatum gams, (c)-() transportation of lctric currnt an ()-(f) communications. Thr mpirical ntworks: Karat, Dolphins an Ntscinc ntworks ar us in (a)-(). Thr mpirical ntworks, IEEE 39 BUS, IEEE 118 BUS an IEEE 300 BUS ntworks ar us in (c)-(). Thr mpirical ntworks, Footall, Jazz an ntworks ar us in ()-(f). Each ata point is otain y avraging ovr 10 inpnnt ralizations. Error ars not th stanar viations. 16

17 Supplmntary Figur S12: Four phass ar intifi in th phas iagram: (I) imitat th st stratgy in nighors, (II) stratgy unchang, (III) inistinguishal an (IV) imitat othr stratgis. p i,max p i (t + 1) p max (t) an p i,i p i (t + 1) p i (t), whr rprsnts asolut valu an p max is th proposal associat with maximum payoff in th nighorhoo of no i. ntworks ar us with ntwork siz N = 100, avrag gr k = 6, an rwiring proaility 0.3. Th ash lins ar p i,max = δ an p i,i = δ. 17