Stphn P. Borgtti Grph Thory A lthough grph thory is on o th youngr rnhs o mthmtis, it is unmntl to numr o ppli ils, inluing oprtions rsrh, omputr sin, n soil ntwork nlysis. In this hptr w isuss th si onpts o grph thory rom th point o viw o soil ntwork nlysis. Grphs Th unmntl onpt o grph thory is th grph, whih (spit th nm) is st thought o s mthmtil ojt rthr thn igrm, vn though grphs hv vry nturl grphil rprsnttion. Grph inition. A grph usully not G(V,E) or G = (V,E) onsists o st o vrtis V togthr with st o gs E. Th numr o vrtis in grph is usully not n whil th numr o gs is usully not m. Vrtis. Vrtis r lso known s nos, points n (in soil ntworks) s tors, gnts or plyrs. Egs. Egs r lso known s lins n (in soil ntworks) s tis or links. An g = (u,v) is in y th unorr pir o vrtis tht srv s its n points. As n xmpl, th grph pit in Figur 1 hs vrtx st V={,,,,.} n g st E = {(,),(,),(,),(,),(,),(,)}.
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. Figur 1. Ajny. Two vrtis u n v r jnt i thr xists n g (u,v) tht onnts thm. Inin. An g (u,v) is si to inint upon nos u n v. Loops. An g = (u,u) tht links vrtx to itsl is known s sl-loop or rlxiv ti. Visuliztion issus. Whn looking t visuliztions o grphs suh s Figur 1, it is importnt to rliz tht th only inormtion ontin in th igrm is jny; th position o nos in th pln (n thror th lngth o lins) is ritrry unlss othrwis spii. Hn it is usully ngrous to rw onlusions s on th sptil position o th nos. For xmpl, it is tmpting to onlu tht nos in th mil o igrm r mor importnt thn nos on th priphris, ut this will otn i not usully mistk. Multigrphs. Whn us to rprsnt soil ntworks, w typilly us h lin to rprsnt instns o th sm soil rltion, so tht i (,) inits rinship twn th prson lot t no n th prson lot t no, thn (,) woul lso init rinship twn n. In ition, w only llow on g twn ny two vrtis. Thus, h istint soil rltion tht is mpirilly msur on th sm group o popl is rprsnt y sprt grphs, whih r likly to hv irnt struturs (tr ll, who tlks to whom is not th sm s who isliks whom).
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. Howvr, in grph thory, no suh limittions r impos. Egs n long to irnt typs, n w n multipl gs twn th sm pir o vrtis. Ajny mtrix. Evry grph hs ssoit with it n jny mtrix, whih is inry n n mtrix A in whih ij = 1 n ji = 1 i vrtx vi is jnt to vrtx vj, n ij = 0 n ji = 0 othrwis. Th nturl grphil rprsnttion o n jny mtrix is tl, suh s shown in Figur 2. 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 Figur 2. Ajny mtrix or grph in Figur 1. Compltnss. Exmining ithr Figur 1 or Figur 2, w n s tht not vry vrtx is jnt to vry othr. A grph in whih ll vrtis r jnt to ll othrs is si to omplt. Dnsity. Th xtnt to whih grph is omplt is init y its nsity, whih is in s th numr o gs ivi y th numr possil. I sl-loops r xlu, thn th numr possil is n(n-1)/2. I sl-loops r llow, thn th numr possil is n(n+1)/2. Hn th nsity o th grph in Figur 1 is 6/15 = 0.40. Sugrphs. A sugrph o grph G is grph whos points n lins r ontin in G. A omplt sugrph o G is stion o G tht is omplt (i.., hs nsity = 1). Cliqus. A liqu is mximl omplt sugrph. A mximl omplt sugrph is sugrph o G tht is omplt n is mximl in th sns tht no othr no o G oul to th sugrph without losing th ompltnss proprty. In Figur 1, th nos {,,} togthr with th lins onnting thm orm liqu. Cliqus hv n sn s wy to rprsnt wht soil sintists hv ll primry groups.
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. Conntnss. Whil not vry vrtx in th grph in Figur 1 is jnt, on n onstrut squn o jnt vrtis rom ny vrtx to ny othr. Grphs with this proprty r ll onnt. Rhility. Similrly, ny pir o vrtis in whih on vrtx n rh th othr vi squn o jnt vrtis is ll rhl. I w trmin rhility or vry pir o vrtis, w n onstrut rhility mtrix R suh s pit in Figur 3. Th mtrix R n thought o s th rsult o pplying trnsitiv losur to th jny mtrix A. g g g Figur 3. Componnt. A omponnt o grph is in s mximl sugrph in whih pth xists rom vry no to vry othr (i.., thy r mutully rhl). Th siz o omponnt is in s th numr o nos it ontins. A onnt grph hs only on omponnt. Wlk. A squn o jnt vrtis v 0,v 1,,v n is known s wlk. In Figur 3, th squn,,,,,g is wlk. A wlk n lso sn s squn o inint gs, whr two gs r si to inint i thy shr xtly on vrtx. Clos. A wlk is los i v o = v n. Pth. A wlk in whih no vrtx ours mor thn on is known s pth. In Figur 3, th squn,,,,, is pth.
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. Tril. A wlk in whih no g ours mor thn on is known s tril. In Figur 3, th squn,,,,,,g is tril ut not pth. Evry pth is tril, n vry tril is wlk. Cyl. A yl n in s los pth in whih n >= 3. Th squn,, in Figur 3 is yl. Tr. A tr is onnt grph tht ontins no yls. In tr, vry pir o points is onnt y uniqu pth. Tht is, thr is only on wy to gt rom A to B. Lngth. Th lngth o wlk (n thror pth or tril) is in s th numr o gs it ontins. For xmpl, in Figur 3, th pth,,,, hs lngth 4. Gosi. A wlk twn two vrtis whos lngth is s short s ny othr wlk onnting th sm pir o vrtis is ll gosi. O ours, ll gosis r pths. Gosis r not nssrily uniqu. From vrtx to vrtx in Figur 1, thr r two gosis:,,,,, n,,,g,,. Distn. Th grph-thorti istn (usully shortn to just istn ) twn two vrtis is in s th lngth o gosi tht onnts thm. I w omput th istn twn vry pir o vrtis, w n onstrut istn mtrix D suh s pit in Figur 4. Th mximum istn in grph ins th grph s imtr. As shown in Figur 4, th imtr o th grph in Figur 1 is 4. I th grph is not onnt, thn thr xist pirs o vrtis tht r not mutully rhl so tht th istn twn thm is not in n th imtr o suh grph is lso not in. g 0 1 2 3 3 4 3 1 0 1 2 2 3 2 2 1 0 1 1 2 1 3 2 1 0 1 2 2 3 2 1 1 0 1 1 4 3 2 2 1 0 2 g 3 2 1 2 1 2 0 Figur 4. Distn mtrix or grph in Figur 3.
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. Powrs. Th powrs o grph s jny mtrix, A p, giv th numr o wlks o lngth p twn ll pirs o nos. For xmpl, A 2, otin y 2 multiplying th mtrix y itsl, hs ntris ij tht giv th numr o wlks o lngth 2 tht join no v i to no v j. Hn, th gosi istn mtrix p D hs ntris ij = p, whr p is th smllst p suh tht ij > 0. (Howvr, thr xist muh str lgorithms or omputing th istn mtrix.) Entriity. Th ntriity (v) o point v in onnt grph G(V,E) is mx (u,v), or ll u V. In othr wors, point s ntriity is qul to th istn rom itsl to th point rthst wy. Th ntriity o no in Figur 3 is 3. Rius & Dimtr.. Th minimum ntriity o ll points in grph is ll th rius r(g) o th grph, whil th mximum ntriity is th imtr o th grph. In Figur 3, th rius is 2 n th imtr is 4. Cntr. A vrtx tht is lst istnt rom ll othr vrtis (in th sns tht its ntriity quls th rius o th grph) is mmr o th ntr o th grph n is ll ntrl point. Evry tr hs ntr onsisting o ithr on point or two jnt points. Dgr. Th numr o vrtis jnt to givn vrtx is ll th gr o th vrtx n is not (v). It n otin rom th jny mtrix o grph y simply omputing h row sum. For xmpl, th gr o vrtx in Figur 3 is 4. Avrg gr. Th vrg gr,, o ll vrtis pit in Figur 3 is 2.29. Thr is irt rltionship twn th vrg gr,, o ll vrtis in grph n th grph s nsity: nsity = n 1 Isolts & pnnts. A vrtx with gr 0 is known s n isolt (n onstituts omponnt o siz 1), whil vrtx with gr 1 is pnnt. Dgr vrin. Holing vrg gr onstnt, thr is tnny or grphs tht ontin som nos o high gr (n thror high vrin
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. in gr) to hv shortr istns thn grphs with lowr vrin, with th high gr nos srving s shortuts ross th ntwork. Cutpoint. A no whos rmovl rom grph isonnts th grph (or, mor gnrlly, inrss th numr o omponnts in th grph) is ll utpoint or n rtiultion point. Th grph in Figur 3 hs thr utpoints, nmly,, n. A onnt, non-trivil grph is ll non-sprl i it hs no utpoints. Blok. A lok or i-omponnt is mximl nonsprl sugrph. Bloks prtition th gs in grph into mutully xlusiv gs. Thy lso shr no nos xpt utpoints. Thus, utpoints ompos grphs into (nrly) non-ovrlpping stions. In loks o mor thn two points, vry pir o points lis long ommon yl, whih mns tht thr is lwys minimum o two wys to gt rom ny point to ny othr. In Figur 3, w in th ollowing loks: {,}, {,}, {,,,g}, {,}. Cutst. Th notion o utpoint n gnrliz to utst, whih is st o points whos joint rmovl inrss th numr o omponnts in th grph. O prtiulr intrst is minimum wight utst, whih is utst tht is s smll s possil (i.., no othr utst hs wr mmrs). Thr n mor thn on istint minimum wight utst in grph. Vrtx onntivity. Th siz o grph s minimum wight utst ins th vrtx onntivity κ(g) o grph, whih is th minimum numr o nos tht must rmov to inrs th numr o omponnts in th grph (or rnr it trivil). Th point onntivity o isonnt grph is 0. Th point onntivity o grph ontining utpoint is no highr thn 1. Th point onntivity o non-sprl grph is t lst 2. W n nlogously in th vrtx onntivity κ(u,v) o pir o points u,v s th numr o nos tht must rmov to isonnt tht pir. Th onntivity o th grph κ (G) is just th minimum κ (u,v) or ll u,v in V. Inpnnt pths. A mous thorm y Mngr pulish in 1929 rlts th vrtx onntivity o pir o nos to th mximum numr o noinpnnt pths onnting thos nos. A st o pths rom sour no s to trgt no t is no-inpnnt i non o th pths shr ny vrtis si rom s n t. Mngr s thorm stts tht or ny sour s n trgt t, th mximum numr o no-inpnnt pths twn s n t is qul to th vrtx onntivity o tht pir i.., th numr o nos tht
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. must rmov to isonnt thm. Hn, thr might mny irnt pths rom s to t, ut i thy ll shr rtin no (i.., r not inpnnt), thn s n t n sily isonnt y liminting just tht no. Thus, w n think o th point onntivity o grph s n initor o th invulnrility o th grph to thrts o isonntion y rmovl o nos. I κ(g) is high, or i th vrg κ (u,v) is high or ll pirs o nos, thn w know tht it is irly iiult to isonnt th nos in th grph y rmoving intrmiris. Brig. Th vrtx-s notions o utpoint, utst, vrtx onntivity n no-inpnnt pth st hv nlogous ountrprts or gs. A rig is in s n g whos rmovl woul inrs th numr o omponnts in th grph. Eg onntivity. Eg onntivity is not λ(g) n th g onntivity o pir o nos is not λ(u,v). A isonnt grph hs λ(g)=0, whil grph with rig hs λ(g)=1. Point onntivity n lin onntivity r rlt to h othr n to th minimum gr in grph y Whitny s inqulity: κ ( G) λ( G) δ ( G) Dirt Grphs Motivtion. As not t th outst, th gs ontin in grphs r unorr pirs o nos (i.., (u,v) is th sm thing s (v,u)). As suh, grphs r usul or noing irtionlss rltionships suh s th soil rltion siling o or th physil rltion is nr. Howvr, mny rltions tht w woul lik to mol r not irtionlss. For xmpl, is th oss o is usully nti-symmtri in th sns tht i u is th oss o v, it is unlikly tht v is th oss o u. Othr rltions, suh s givs vi to r simply non-symmtri in th sns tht i u givs vi to v, v my or my not giv vi to u. Dinition. To mol non-symmtri rltions w us irt grphs, lso known s igrphs. A igrph D(V,E) onsists o st o nos V n st
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. o orr pirs o nos E ll rs or irt lins. Th r (u,v) points rom u to v. Digrphs r usully rprsnt visully lik grphs, xpt tht rrowhs r pl on lins to init irtion (s Figur 5). Whn oth rs (u,v) n (v,u) r prsnt in igrph, thy my rprsnt y oul-h rrow (s in Figur 5), or two sprt rrows (s shown in Figur 5). Figur 5 Figur 5 Dirt Wlk. In igrph, wlk is squn o nos v o,v 1, v n in whih h pir o nos v i, v i +1 is link y n r (v i,v i +1). In othr wors, it is trvrsl o th grph in whih th low o movmnt ollows th irtion o th rs, lik r moving rom pl to pl vi on-wy strts. A pth in igrph is wlk in whih ll points r istint. Smiwlk. A smiwlk is squn o nos v o,v 1, v n in whih h pir o nos v i, v i +1 is link y ithr th r (v i,v i +1) or th r (v i +1,v i ). In othr wors, in smiwlk, th trvrsl n not rspt th irtion o rs, lik r tht rly gos th wrong wy on on-wy strts. By nlogy, w n lso in smipth, smitril, n smiyl.
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. Unrlying grph. Anothr wy to think o smi-wlks is s wlks on th unrlying grph, whr th unrlying grph is th grph G(V,E) tht is orm rom th igrph D(V,E ) suh tht (u,v) E i n only i (u,v) E or (v,u) E. Thus, th unrlying grph o igrph is silly th grph orm y ignoring irtionlity. Strongly onnt. A igrph is strongly onnt i thr xists pth (not smipth) rom vry point to vry othr. Not tht th pth rom u to v n not involv th sm intrmiris s th pth rom v to u. Uniltrlly onnt. A igrph is uniltrlly onnt i or vry pir o points thr is pth rom on to th othr (ut not nssrily th othr wy roun). Wkly onnt. A igrph is wkly onnt i vry pir o points is mutully rhl vi smipth (i.., i th unrlying grph is onnt). Strong omponnt. A strong omponnt o igrph is mximl strongly onnt sugrph. In othr wors, it is sugrph tht is strongly onnt n whih is s lrg s possil (thr is no no outsi th sugrph tht is strongly onnt to ll th nos in th sugrph). A wk omponnt is mximl wkly onnt sugrph. Outgr. Th numr o rs originting rom no v (i.., outgoing rs) is ll th outgr o v, not o(v). Ingr. Th numr o rs pointing to no v (i.., inoming rs) is ll th ingr o v, not i(v). In grph rprsnting rinship lings mong st o prsons, outgr n sn s initing grgriousnss, whil ingr orrspons to populrity. Th vrg outgr o igrph is nssrily qul to th vrg ingr. Dirt jny. Th jny mtrix A o igrph is n n n mtrix in whih ij = 1 i (v i,v j ) E n ij = 0 othrwis. Unlik th jny mtrix o n unirt grph, th jny mtrix o irt grph is not onstrin to symmtri, so tht th top right hl n not qul th ottom lt hl (i.., ij <> ji ). I igrph is yli, thn it is
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. possil to orr th points o D so tht th jny mtrix uppr tringulr (i.., ll positiv ntris r ov th min igonl). Soil Ntwork Extnsions to Grph Thory In this stion w onsir ontriutions to grph thory rom th stuy o soil ntworks. Thr r two min groups o ontriutions: ohsiv susts n rols/positions. Not tht th initions o ohsiv susts ssum grphs, whil thos o rols/position ssum igrphs. Cohsiv Susts It ws mntion rlir tht th notion o liqu n sn s ormlizing th notion o primry group. A prolm with this, howvr, is tht it is too strit to prtil: rl groups will ontin svrl pirs o popl who on t hv los rltionship. A rlxtion n gnrliztion o th liqu onpt is th n-liqu. An n-liqu S o grph is mximl st o nos 1 in whih or ll u,v S, th grph-thorti istn (u,v) <= n. In othr wors, n n-liqu is st o nos in whih vry no n rh vry othr in n or wr stps, n th st is mximl in th sns tht no othr no in th grph is istn n or lss rom vry othr no in th sugrph. A 1-liqu is th sm s n orinry liqu. Th st {,,,,} in Figur 6 is n xmpl o 2-liqu. Figur 6. 1 Cohsiv susts r tritionlly in in trms o sugrphs rthr thn susts o nos. Howvr, sin most popl think out thm in trms o no sts, n us using sugrphs omplits nottion, w us susts hr.
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. Not tht th pth o lngth n or lss linking mmr o th n-liqu to nothr mmr my pss through n intrmiry who is not in th group. In th 2-liqu in Figur 6, nos n r istn 2 only us o, whih is not mmr o th 2-liqu. In this sns, n-liqus r not s ohsiv s thy might othrwis ppr. Th notion o n n-ln vois tht. An n-ln is n n-liqu in whih th imtr o th sugrph G inu y S is lss thn or qul to n. Th sugrph G o grph G inu y th st o nos S is in s th mximl sugrph o G tht hs point st S. In othr wors, it is th sugrph o G otin y tking ll nos in S n ll tis mong thm. Thror, n n-ln S is n n-liqu in whih ll pirs hv istn lss thn or qul to n vn whn w rstrit ll pths to involv only mmrs o S. In Figur 6, th st {,,,,} is 2- ln, ut {,,,,} is not us n hv istn grtr thn 2 in th inu sugrph. Not tht {,,,} is lso ils th 2-ln ritrion us n-lns r in to n-liqus n {,,,} is not 2-liqu (it ils th mximlity ritrion sin {,,,,}). An n-lu orrts this prolm y liminting th n-liqu ritrion rom th inition. An n-lu is sust S o nos suh tht in th sugrph inu y S, th imtr is n or lss. Evry n-ln is oth n n-lu n n n-liqu. Th st {,,,} is 2-lu. Whrs n-liqus, n-lns n n-lus ll gnrliz th notion o liqu vi rlxing istn, th k-plx gnrlizs th liqu y rlxing nsity. A k- plx is sust S o nos suh tht vry mmr o th st is onnt to n-k othrs, whr n is th siz o S. Although not prt o th oiil inition, it is onvntionl to itionlly impos mximlity onition, so tht propr susts o k-plxs r ignor. Thr r som gurnts on th ohsivnss o k-plxs. For xmpl, k-plxs in whih k < (n+2)/2 hv no istns grtr thn 2 n nnot ontin rigs (mking thm rsistnt to ttk y lting n g). In Figur 6, th st {,,,} ils to 2-plx us h mmr must hv t lst 4-2=2 tis to othr mmrs o th st, yt hs only on ti within th group. In th grph in Figur 7, th st {,,,} is 2-plx.
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. Figur 7. Mor ohsiv thn k-plxs r LS sts. Lt H st o nos in grph G(V,E) n lt K propr sust o H. Lt α(k) not th numr o gs linking mmrs o K to V-K (th st o nos not in K). Thn H is n LS st o G i or vry propr sust K o H, α(k) > α(h). Th si i is tht iniviuls in H hv mor tis with othr mmrs thn thy o to outsirs. Anothr wy to in LS sts tht mks this mor vint is s ollows. Lt α(x,y) not th numr o gs rom mmrs o st X to mmrs o st Y. Thn H is n LS st i α(k,h-k) > α(k,v-h). In Figur 7, th st {,,,} is not n LS st sin α({,,},{}) is not grtr thn α({,,},{}). In ontrst, th st {,,,} in Figur 8 os quliy s n LS st. Figur 8. A ky proprty o LS sts is high g onntivity. Spiilly, vry no in n LS st hs highr g onntivity (λ) with othr mmrs o th LS st thn with ny non-mmr. Tking this s th sol ritrion or ining ohsiv sust, lm st is in s mximl sust o nos S suh tht or ll,, S n V-S, λ(,) > λ(,). To th xtnt tht λ is high, mmrs o th sm lm st r iiult to isonnt rom on
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. nothr us λ ins th numr gs tht must rmov rom th grph in orr to isonnt th nos within th lm st. A k-or is mximl sugrph H in whih δ(h) >= k. Hn, vry mmr o 2-or is onnt to t lst 2 othr mmrs, n no no outsi th 2-or is onnt to 2 or mor mmrs o th or (othrwis it woul not mximl). Evry k-or ontins t lst k+1 vrtis, n vrtis in irnt k-ors nnot jnt. A 1-or is simply omponnt. K-ors n sri s loosly ohsiv rgions whih will ontin mor ohsiv susts. For xmpl, vry k-plx is ontin in k-or. Rols/Positions Givn igrph D(V,E), th in-nighorhoo o no v, not Ni(v) is th st o vrtis tht sn rs to v. Tht is, N i (v) = {u: (u,v) E}. Th outnighorhoo o no v, not N o (v) is th st o vrtis tht riv rs rom v. Tht is, N o (v) = {u: (v,u) E}. A olortion C is n ssignmnts o olors to th vrtis V o igrph. Th olor o vrtx v is not C(v) n th st o istint olors ssign to nos in st S is not C(S) n trm th sptrum o S. In Figur 9, olortion o nos is pit y lling th nos with lttrs suh s r or r, n y or yllow. Nos olor th sm r si to quivlnt. r y w r Figur 9. y A olortion is strong struturl olortion i nos r ssign th sm olor i n only i thy hv intil in n out nighorhoos. Tht is, or ll u,v V, C(u) = C(v) i n only i Ni(u) = Ni(v) n No(u) = No(v). Th
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. olortion in Figur 9 is strong struturl olortion. W n hk this y tking pirs o nos n vriying tht i thy r olor th sm (i.., r strongly struturlly quivlnt) thy hv intil nighorhoos, n i thy r not olor th sm, thy hv irnt nighorhoos. For xmpl, n r olor th sm, n oth o thir nighorhoos onsist o {,,}. Not tht in strong struturl olortions, ny two nos tht r olor th sm r struturlly intil: i w rmov th intiying lls rom th intilly olor nos, thn spin th grph roun in sp or pling it k own on th pg, w woul not l to igur out whih o th sm-olor nos ws whih. Consquntly, ny proprty o th nos tht stms rom thir struturl position (suh s xpt tim until rrivl o somthing lowing through th ntwork) shoul th sm or nos tht r quivlnt. A olortion C is rgulr i C(u) = C(v) implis tht C(Ni(u)) = C(Ni(v)) n C(No(u)) = C(No(v)) or ll u, v V. In othr wors, in rgulr olortions, vry pir o nos tht hs th sm olor must riv rs rom nos omprising th sm st o olors n must sn rs to nos omprising th sm st o olors. Evry struturl olortion is rgulr olortion. r y r r y Figur 10. Rgulr olortion. Th olortion in Figur 10 (whih pits th sm igrph s in Figur 9) is rgulr, ut not strongly struturl. To s this, onsir tht vry r no hs n out-nighorhoo ontining only yllow nos (.g., C(N o ()={y}), n n in-nighorhoo ontining only yllow nos, whil vry yllow no hs n out-nighorhoo ontining only r nos n n in-nighorhoo ontining only r nos. Figur 11 pits nothr rgulr olortion. Not tht no g oul not olor th sm s or i,
1 st Drt, Rvision 2, writtn vry quikly. My ontin rrors. B wr. us it hs n outnighorhoo onsisting o whit no, whil n i hv no outnighorhoo t ll. Consquntly no p oul not olor th sm s n, sin p s out-nighorhoo ontins no o irnt olor thn th n. This lso implis tht g nnot olor th sm s n i us it riv ti rom no o irnt olor. r p r g h i y s g y w Figur 11. Rgulr olortion j I grph rprsnts soil ntwork, w n think o th olors s ining mrgnt lsss or typs o popl suh tht i on mmr o rtin lss (lu) hs outgoing tis to mmrs o xtly two othr lsss (yllow n grn), thn ll othr mmrs o tht (lu) lss hv outgoing tis to mmrs o thos sm two lsss (yllow n grn). Thus, rgulr olortions lssiy mmrs o soil ntwork oring to thir pttrn o rltions o othrs, n two popl r pl in th sm lss i thy intrt in th sm wys with th sm kins o othrs (ut not nssrily with sm iniviuls). Just s th vrious gnrliztions o liqus r ttmpts to ptur mthmtilly th notion o soil group, rgulr olortions r n ttmpt to ptur th notion o soil rol systm, in whih popl plying rtin rol hv hrtristi rltions with othrs who r plying omplmntry rols (suh s otors, nurss n ptints).