1 Random graphs with specified degrees

Transcription

1 1 Ranom graphs with spii grs Rall that a vrtx s gr unr th ranom graph mol G(n, p) ollows a Poisson istribution in th spars rgim, whil most ral-worl graphs xhibit havy-tail gr istributions. This irn is on rason w say that G(n, p) is an unralisti mol. In this ltur, w will larn about ranom graph mols that hav a mor lxibl gr strutur. Thr ar two main lavors o ths mols: thos that gnrat ranom graphs with a spii gr squn (a oniguration mol ) or with a spii gr istribution. Ths ranomgraph mols ar typially in in trms o unirt an unwight graphs, but thy ar straightorwar to gnraliz to bipartit ntworks, or irt graphs. Extning thm to multilayr, tmporal, or wight ntworks rquirs aitional assumptions. A oniguration mol an b not as G(n, k) whr k = {k i } is a gr squn on n vrtis, with k i bing th gr o vrtx i. This mol rprsnts a uniorm istribution ovr all graphs with n vrtis, onition on thir having th k gr squn. So long as k i is vn (o you s why?), w ar r to hoos k in any way w lik. It thn only rmains to in how w will sampl iniviual ntworks rom this istribution. 1 Hr ar som spii xampls o spii oniguration mols. I w ix all th grs to b th som onstant k, w hav in a probability istribution ovr all k-rgular graphs. Or, w oul hoos a st o n valus rawn ii rom som gr istribution Pr(k). I that istribution is Poisson with man /(n 1), thn w rovr th G(n, p) mol. Or, w oul raw th grs rom som othr istribution, lik an xponntial, a log-normal or vn a powr law; th lattr as is somtims all a powr-law ranom graph. I th gr istribution that gnrats th gr squn has a mathmatially simpl orm, lik th istributions just mntion, w an otn omput xatly rtain proprtis o th orrsponing nsmbl o graphs, muh lik w i with Erős-Rényi ranom graphs. This provis us with a rih amily o mols to stuy. In prati, howvr, w ar intrst in a partiular ntwork an its gr strutur, an thus a vry ommon stp in ntwork analysis is to hoos k as th mpirially obsrv gr squn o a ral-worl ntwork. For instan, onsir th karat lub again. Th pair o igurs blow show th original ntwork, an a singl xampl o a ntwork rawn rom th nsmbl in by th oniguration mol, paramtriz by n an th mpirial k. Immiatly notiabl is that whil th gr strutur has bn prsrv, th original group strutur has bn ranomiz. 1 How w sampl rom th spii st o graphs turns out to b somwhat subtl, an it an mattr whthr on uss a haky mtho or a provably orrt way o oing it. Ths an rlat qustions ar isuss at lngth in a tour--or artil B. K. Fosik t al., Coniguring Ranom Graph Mols with Fix Dgr Squns. Prprint, arxiv: (2016). 1

2 karat lub oniguration mol 1.1 A null mol or ntwork analysis Th oniguration mol an srv as a null mol or invstigating th strutur o a ral ntwork A. That is, it allows us to quantitativly answr th qustion o How muh o som obsrv pattrn is rivn by th grs alon? Th oniguration mol ins a probability istribution ovr graphs Pr(G k) that has th sam grs as th original ntwork A. Thus, i w an omput a untion on A, w an omput th sam untion on a graph rawn rom this oniguration mol (G). An, baus G is a ranom variabl, w an omput th ntir istribution Pr((G) k). For simpl untions an simpl spiiations o th oniguration mol, w an otn omput ths istributions analytially. For mor ompliat untions or or a oniguration mol spii with an mpirial gr squn, w an omput Pr((G) k) numrially, by rawing many graphs {G 1, G 2,... } rom th mol, omputing on ah, an tabulating th rsults. I th mpirial valu (A) is unusual rlativ to this istribution, w an onlu that it is a proprty o A that is not wll xplain by th grs alon. W will rvisit this ia latr in this ltur. 1.2 Gnrating ntworks using th oniguration mol Th two most ommon mthos or gnrating a ranom graph with partiular gr strutur ar assoiat with th Chung-Lu mol or ranom simpl graphs an th Molloy-R mol or ranom multigraphs. 2

3 1.2.1 Simpl graphs rom lipping oins Th ntral mathmatial proprty o all ranom-graph mols is th probability that two vrtis i an j ar onnt. In th ranom graph mols w onsir hr, this probability pns only on th grs k i an k j o that pair. Thus, rom th prsptiv o i, th probability that on o its gs onnts to j is qual to th ration o th m total gs w hoos that point to j. Baus w hav hosn j s gr, this ration is xatly k j /2m. An, baus w hav also hosn i s gr, this vnt has k i hans to our an th probability that (i, j) xists is ( ) kj p ij = k i 2m = k ik j n l=1 k l. (1) Th Chung-Lu mol taks this probability as a paramtr an simply lips a singl oin or ah o th pairs i, j to gnrat a simpl graph: { 1 with probability pij i>j A ij = A ji = 0 othrwis, whr p ij is givn by Eq. (1). Just as with gnrating Erős-Rényi graphs, ah pair is onsir only on; hn, this pross prous a simpl graph, with no sl-loops an no multi-gs. (In ontrast, th Molloy-R mol prous a ranom multigraph, whih may hav multi-gs an sl-loops.) This mtho an also b us to gnrat irt ntworks by irst spiying th in-gr an out-gr squns, subjt to th rquirmnt that i kin i hoos p i j = k out i k in j /m an rop th rquirmnt that A ij = A ji. = j kout j. W thn As a rsult o this orm, th gr o ah vrtx i unr this mtho o gnration quals th spii valu k i only in xptation (an similarly or th in- an out-grs in th irt vrsion). Th obsrv gr or no i in th Chung-Lu nsmbl is a Poisson istribution with man k i (o you s why?). Hn, viations rom th xpt valu ar gnrally small, whn th graph is spars an th maximum gr is n. Notably, rawing ranom graphs rom th Chung-Lu mol is omputationally xpnsiv, spially or larg n, as w n to lip Θ(n 2 ) oins, on or ah possibl pair o vrtis i, j V. This ost is on rason that th Molloy-R mol is mor ommonly us or larg mpirial stuis (but s Fosik t al. [2016]) Multigraphs rom ranom mathings Th stanar mtho or gnrating a Molloy-R ranom multigraph is to hoos a uniormly ranom mathing on th gr stubs (hal gs) o th spii gr squn. Unlik in th Chung-Lu mol srib abov, whih only gnrats simpl graphs by sign, this stub mathing mtho will typially prou som numbr o sl-loops an multi-gs. In prati, 3

4 ths viations rom a simpl graph rprsnt an asymptotially small ration o all gs, an w an simpliy th ntwork by isaring sl-loops an ollapsing multi-gs, an potntially also isaring isonnt omponnts. 2 Givn a gr squn k = {k 1, k 2,..., k n }, w say that ah vrtx i has a numbr o stubs qual to its gr. Evry mathing on ths stubs, in whih w rpatly hoos an unmath stub on som vrtx i an onnt it with som unmath stub on vrtx j, rprsnts a ntwork. Unr this mtho o gnrating a graph, w will hoos suh a mathing uniormly at ranom rom among all suh mathings. Eah possibl mathing thus ours with qual probability; howvr, ah ntwork with th spii gr squn os not our with qual probability unr this mol, as som mathings prou th sam ntwork. To illustrat this ia, onsir th st o mathings on thr vrtis, ah with gr 2, that rsult in a triangl. Th ollowing igur shows th istint lablings, an hn istint mathings, that orm a triangl. In th oniguration mol, w hoos ah o ths with qual probability. a b b a a b b a a b b a a b b a Howvr, ths ar not th only possibl mathings on ths six g stubs. Th ollowing igur shows thr othr istint mathings, whih prou non-simpl ntworks, i.., ntworks with slloops an/or multi-gs. a b b a a b In prati, th ration o gs involv in ithr sl-loops or multi-gs is vanishingly small in th larg-n limit, an thus w may gnrally ignor thm without muh impat. (Howvr, thr ar appliations in whih ths aturs ar important, an so it is worth rmmbring that thy xist.) 2 Ths prours o hang th graph strutur slightly, an a sar approah is to us Fosik t al. s (2016) mthos to sampl irtly rom th simpl graph nsmbl. 4

5 On a gr squn k has bn hosn,.g., by taking th gr squn in som mpirial ntwork or by rawing a squn rom a gr istribution, to raw a ntwork rom th orrsponing oniguration mol, w simply n an iint mtho by whih to hoos a uniormly ranom mathing on th i k i stubs. Lt v b an array o lngth 2m an lt us writ th inx o ah vrtx i xatly k i tims in th vtor v. Eah o ths ntris will rprsnt a singl g stub attah to vrtx i. Thn, w tak a ranom prmutation o th ntris o v an ra th ontnts o th array in orr, in pairs. For ah pair that w ra, w a th orrsponing g to th ntwork. (On w hav taking a ranom prmutation o th stubs, w oul hoos th pairs in any othr way, but raing thm in orr allows us to writ own th ull ntwork with only a singl pass through th array.) For instan, th igur abov shows an xampl o this stub mathing onstrution o a oniguration mol ranom graph. On th lt is shown both th vrtis with thir stubs, whih shows th gr squn graphially, an th initial ontnts o th array v. On th right is shown th wir up ntwork in by th in-orr squn o pairs givn in th array, whih has bn rpla with a ranom prmutation o v. In this as, th ranom prmutation prous both on sl-loop an on multi-g. Stanar mathmatial libraris otn provi th untionality to slt a uniormly ranom prmutation on th ontnts o v. Howvr, it is straightorwar to o it manually, as wll: to ah ntry v i, assoiat a uniormly ranom variabl r i U(0, 1) (whih most goo psuoranom numbr gnrators will prou). Sorting th r i valus prous a ranom prmutation 3 on th 3 Eah o ths prmutations ours with probability qual to 1/n!, an baus thr ar n! suh prmutations, w ar hoosing uniormly rom among thm. I w hoos th r i valus uniormly at ranom, thn th probability that any partiular lmnt v i has th smallst r i is 1/n. Similarly, th probability that v i has th ith smallst valu is 1/(n i + 1). By inution, th probability o hoosing any partiular orring is n i=1 (n i + 1) 1 = 1/n!. It is possibl to hoos a ranom prmutation in O(m) tim using an in-pla ranomizr. Insta o sorting th uniorm viats, w insta loop rom i = 1 to n within v an swap v i with a uniormly ranomly hosn lmnt v j whr 5

6 assoiat v i valus, whih an b on using QuikSort in tim O(m log m), or you an gnrat a ranom prmutation irtly in O(m) tim. 1.3 Mathmatial proprtis Th pris mathmatial proprtis or th oniguration mol pn on th hoi o gr squn. In th vrsion o th mol whr w raw th gr squn rom som istribution, w may otn alulat proprtis o th oniguration mol nsmbl analytially (otn using powrul thniqus all gnrating untions). 4 Unr th ranom mathing approah or onstruting an instan o th mol, or a partiular stub attah to vrtx i, thr ar k j possibl stubs, out o 2m 1 (xluing th stub on i unr onsiration), attah to j to whih it oul onnt. An, thr ar k i hans that this oul happn. Thus, th probability that i an j ar onnt is p ij = k ik j 2m 1 k ik j 2m, (2) whr th son orm hols in th limit o larg m. Noti that this immiatly implis that th highr th grs ar o i an j, th gratr th probability that thy onnt unr th oniguration mol Expt numbr o multi-gs Eq. (2) givs th probability that on g appars btwn i an j. A losly rlat quantity is th probability that a son g appars btwn i an j, an this quantity allows us to alulat th xpt numbr o multi-gs in th ntir ntwork. Th onstrution is vry similar to that abov, xpt that w must upat our ounts o stubs to aount or th xistn o th irst g btwn i an j. Th probability that a son g appars is (k i 1)(k j 1)/2m, baus w hav us on stub rom ah o i an j to orm th irst g. Thus, th probability o both a irst an a son g apparing is k i k j (k i 1)(k j 1)/(2m) 2. Summing this xprssion ovr all istint pairs givs us i j n. Th proo that this prous a ranom prmutation ollows th proo abov. 4 For a goo introution to this thniqu, s Wil gnratinguntionology, AK Ptrs (2006). 6

7 th xpt numbr o multi-gs in th ntir ntwork: ( ) ( ) ki k j (ki 1)(k j 1) = 1 1 n 2m 2m 2 (2m) 2 k i (k i 1) istint i,j i=1 j=1 ( ) 1 = 2 k 2 n 2 ki 2 k i i j ( = k 2 n = ( k 2 k ) 2 ki 2 1 n i i n k j (k j 1) kj 2 k j k i ) 1 n kj 2 1 n 2 k 2 [ k 2 ] 2 k. (3) j j k j = 1 2 k In this rivation, w us svral intitis: 2m = k n, whih rlats th numbr o g stubs to th man gr an numbr o vrtis, an k m = 1 n i km i, whih is th mth (unntr) momnt o th gr squn. Th rsult, Eq. (3), is a ompat xprssion that pns only on th irst an son momnts o th gr squn, an not on th siz o th ntwork. Thus, th xpt numbr o multi-gs is a onstant 5 implying a vanishingly small O(1/n) ration o all gs in th larg-n limit Expt numbr o sl-loops This argumnt works almost th sam or sl-loops, xpt that th numbr o pairs o possibl onntions is ( k i ) 2 insta o ki k j. Thus, th probability o a sl-loop is p ii = k i (k i 1) /4m, an th xpt numbr o sl-loops is k 2 k 2 k, whih is a onstant pning only on th irst an son momnts o th gr squn. Thus, just as with multi-gs, sl-loops ar a vanishingly small O(1/n) ration o all gs in th larg-n limit whn k 2 is init. 5 So long as th irst an son momnts o th istribution prouing k ar init, whih is not th as i th gr istribution ollows a powr law with α < 3. W ar also ignoring th at that w trat th i = j as, i.., sl-loops, intially to th i j as, but this irn is small, as th nxt stion shows. 7

8 1.3.3 Expt numbr o ommon nighbors Givn a pair o vrtis i an j, with grs k i an k j, how many ommon nighbors n ij o w xpt thm to hav? For som l to b a ommon nighbor o a pair i an j, both th (i, l) g an th (j, l) gs must xist. As with th multi-g alulation abov, th orrt alulation must aount or th rution in th numbr o availabl stubs or th (j, l) g on w onition on th (i, l) g xisting. Thus, th probability that l is a ommon nighbor is th prout o th probability that l is a nighbor o i, whih is givn by Eq. (2), an th probability that l is a nighbor o j, givn that th g (i, l) xists, whih is also givn by Eq. (2) xpt that w must rmnt th stub ount on l. ( ) ( ) ki k l kj (k l 1) n ij = l = ( ki k j 2m 2m ) = p ij k 2 k k l 2m k l (k l 1) k n. (4) Thus, th probability that i an j hav a ommon nighbor is proportional to th probability that thy thmslvs ar onnt (whr th onstant o proportionality again pns on th irst an son momnts o th gr squn) Th xss gr istribution Many quantitis about th oniguration mol, inluing th lustring oiint, an b alulat using somthing all th xss gr istribution, whih givs th gr istribution o a ranomly hosn nighbor o a ranomly hosn vrtx, xluing th g ollow to gt thr. This istribution also shows us somthing slightly ountrintuitiv about oniguration mol ntworks. Lt p k b th ration o vrtis in th ntwork with gr k, an suppos that ollowing th g brings us to a vrtx o gr k. What is th probability that vnt? To hav arriv at a vrtx with gr k, w must hav ollow an g attah to on o th n p k vrtis o gr k in th ntwork. Baus gs ar a ranom mathing onition on th vrtx s grs, th n point o vry g in th ntwork has th sam probability k/2m (in th limit o larg m) o onnting to on o th stubs attah to our vrtx. 8

9 Thus, th gr istribution o a ranomly hosn nighbor is p nighbor has k = k 2m n p k = k p k k. (5) Although th xss gr istribution is losly rlat to Eq. (5), thr ar a w intrsting things this ormula implis that ar worth sribing. From this xprssion, w an alulat th avrag gr o suh a nighbor, as k nighbor = k k p nighbor has k = k 2 / k, whih is stritly gratr than th man gr itsl k (o you s why?). Countrintuitivly, this mans that your nighbors in th ntwork tn to hav a gratr gr than you o. This happns baus high-gr vrtis hav mor gs attah to thm, an ah g provis a han that th ranom stp will hoos thm. Rturning to th xss gr istribution, not that baus w ollow an g to gt to our inal stination, its gr must b at last 1, as thr ar no gs w oul ollow to arriv a vrtx with gr 0. Th xss gr istribution is th probability o th numbr o othr gs attah to our stination, an thus w substitut k + 1 or k in our xprssion or th probability o a gr k. This yils q k = (k + 1)p k+1 k. (6) Expt lustring oiint Th lustring oiint C is th avrag probability that two nighbors o a vrtx ar thmslvs nighbors o ah othr, whih w an alulat now using Eq. (6). Givn that w start at som vrtx v (whih has gr k 2), w hoos a ranom pair o its nighbors i an j, an ask or th probability that thy thmslvs ar onnt. Th gr istribution o i (or j), howvr, is xatly th xss gr istribution, baus w hos a ranom vrtx v an ollow a ranomly hosn g. Th probability that i an j ar thmslvs onnt is k i k j /2m, an th lustring oiint is givn by this probability multipli by th probability that i has xss gr k i an that j has 9

10 xss gr k j, an summ ovr all hois o k i an k j : C = k i =0 k j =0 = 1 2m = = [ q k k k=0 1 2m k 2 1 2m k 2 q ki q kj k i k j 2m ] 2 [ ] 2 k(k + 1)p k+1 k=0 [ ] 2 k(k 1)p k k=0 [ ] 2 1 = 2m k 2 k 2 p k k p k k=0 k=0 [ k 2 k ] 2 = 1 n k 3. (7) whr w hav us th inition o th mth momnt o a istribution to ru th summations. Lik th xprssion w riv or th xpt numbr o multi-gs, th xpt lustring oiint is a vanishing ration O(1/n) in th limit o larg ntworks, so long as th son momnt o th gr istribution is init Expt lustring oiint (altrnativ) It shoul also b possibl to alulat th xpt lustring oiint unr th oniguration mol by starting with th xpt numbr o ommon nighbors n ij or som pair i, j, whih w riv in Eq. (4). In partiular, givn th rsult riv in Eq. (7), w an xprss th lustring oiint in trms o n ij an p ij : C = 1 2m (Can you xplain why this ormula is orrt?) ( nij Th giant omponnt, an ntwork iamtr p ij ) 2. (8) Just as with th Erős-Rényi ranom graph mol, th oniguration mol also xhibits a phas transition or th apparan o a giant omponnt. Th most ompat alulation uss gnrating untions, an is givn in Chaptr 13.8 in Ntworks. Th rsult o ths alulations is a simpl 10

11 ormula or stimating whn a giant omponnt will xist, whih, lik all o our othr rsults, pns only on th irst an son momnts o th gr istribution: k 2 2 k > 0. (9) Unlik our prvious rsults, howvr, this quation works vn whn th son momnt o th istribution is ininit. In that as, th rquirmnt is trivially tru. A orollary o th xistn o th giant omponnt in this mol is th impliation that th iamtr o th ntwork grows logarithmially with n, whn a giant omponnt xists. As with G(n, p), th oniguration mol is loally tr-lik (whih is onsistnt with th vanishingly small lustring oiint riv abov), implying that th numbr o vrtis within a istan l o som vrtx v grows xponntially with l, whr th rat o this growth again pns on th irst two momnts o th gr istribution (whih ar thmslvs rlat to th numbr o irst- an son-nighbors o v). 1.4 Dirt ranom graphs All o ths rsults an b gnraliz to th as o irt graphs, an th intuition w built rom th unirt as gnrally arris ovr to th irt as, as wll. Thr ar, o ours, small irns, as now w must onrn ourslvs with both th in- an out-gr istributions, an th rsults will pn on son momnts o ths. (Th irst momnts o th in- an out-gr istributions must b qual. Do you s why?) Construting irt ranom graphs using th oniguration mol is also analogous, but with on small variation. Now, insta o maintaining a singl array v ontaining th nams o th stubs, w must maintain two arrays, v in an v out, ah o lngth m, whih ontain th in- an out-stubs rsptivly. Th uniormly ranom mathing w hoos is thn btwn ths arrays, with th bginning o an g hosn rom v out an th ning o an g hosn rom v in. 2 A null mol or mpirial ntworks Th most ommon us o th oniguration mol in analyzing ral-worl ntworks is as a null mol, i.., as an xptation against whih w masur viations. Rall rom th last ltur our xampl o th karat lub, an an instan rawn rom th orrsponing oniguration mol. Using th oniguration mol to gnrat many suh instans, w an us ah ntwork as input to our strutural masurs. This prous a istribution o masurs, whih w an thn ompar irtly to th mpirial valus. Eah o th mini-xprimnts blow us 1000 instans o th oniguration mol, an whr multi-gs wr ollaps an sl-loops isar. 11

12 Th gr istribution (shown blow as both p an ) is vry similar, but with a w notabl irns. In partiular, thr th highst-gr vrtis in th mol hav slightly lowr gr valus than obsrv mpirially. This is rlts th at that both multi-gs an sl-loops hav a highr probability o ourring i k i is larg, an thus onvrting th gnrat ntwork into a simpl ntwork tns to rmov gs attah to ths high-gr vrtis. Othrwis, th gnrat gr istribution is vry los to th mpirial on, as w xpt Karat lub oniguration mol Pr(k) Pr(K k) gr, k gr, k Both th istribution o pairwis gosi istans an th ntwork s iamtr ar auratly rprou unr th oniguration mol, iniating that nithr o ths masurs o th ntwork ar partiularly intrsting as pattrns thmslvs. That is, thy ar about what w woul xpt 6 It is possibl to hang th oniguration mol slightly in orr to liminat sl-loops an multi-gs, by lipping a oin or ah pair i, j (whr i j) with bias xatly k ik j/2m. In this mol, th xpt gr w gnrat is istribut as ˆk i Poisson(k i), whih assums th spii valu in xptation. 12

13 or a ranom graph with th sam gr istribution. On ni atur o th oniguration mol s pairwis istan istribution is that it both ollows an xtns th mpirial pattrn out to gosi istans byon what ar obsrv in th ntwork itsl Karat lub oniguration mol Pr() 10 2 Pr(iamtr) gosi istan, iamtr W may also xamin vrtx-lvl masurs, suh as masurs o ntrality. From th gosi istans us in th prvious igurs, w may also stimat th man harmoni ntrality o ah vrtx. Th irst igur blow plots both th mpirial harmoni ntralitis (in orr o vrtx labl, rom 1 to 34) an th man valus unr th oniguration mol. Th various ntrality sors ar now pla in ontxt, showing that thir sors ar largly rivn by th assoiat vrtx gr, as monstrat by th similar ovrall pattrn sn in th oniguration mol ntworks. 7 But, not all o th valus ar xplain by gr alon. Th son igur plots th irn btwn th obsrv an xpt ntrality sors, whr th lin = 0 iniats no irn btwn obsrv an xpt valus. I an obsrv valu is abov this lin, thn it is mor ntral than w woul xpt bas on gr alon, whil i it is blow th lin, it is lss ntral. Whn making suh omparisons, howvr, it is important to rmmbr that th null mol ins a istribution ovr ntworks, an thus th irn is also a istribution. Fortunatly, howvr, omputing th xpt ntrality sors by rawing many instans rom th oniguration mol also prous th istribution o ntrality sors or ah vrtx, whih provis us with a quantitativ notion o how muh varian is in th oniguration mol valu. Th gry sha rgion shows th 25 an 75% quantils on th istribution o ntrality sors or ah vrtx. Whn th = 0 lin is outsi o this rang, w may laim with som onin that th obsrv valu is 7 Rall also that th Parson orrlation oiint or harmoni ntrality an gr was larg r 2 = 0.83, a at that rinors our onlusion hr. 13

14 irnt rom th xpt valu Karat lub oniguration mol harmoni ntrality irn vrtx labl vrtx labl This analysis shows that th main vrtis (1 an 34, th prsint an instrutor) ar somwhat mor ntral than w woul xpt just bas on thir gr alon. In at, most vrtis ar mor ntral than w woul xpt, on is lss ntral than w xpt, an about a thir o th vrtis all in lin with th xptation. 3 Taking stok o our ranom graph mols Th oniguration mol is rtainly an improvmnt ovr th simpl ranom graph mol in that it allows us to spiy its gr strutur. As a null mol, this proprty is otn suiint or us to us th mol to i whthr som othr proprty o a ntwork oul b xplain by its gr strutur alon. Mor gnrally, th oniguration mol shars many proprtis with th simpl ranom graph mol. For instan, in th spars rgim an whn th gr istribution is wll bhav (i.., whn it has a init son momnt) oniguration mol ntworks hav loally tr-lik strutur. This proprty implis its iamtr is O(log n), it has a O(1/n) lustring oiint, an O(1/n) riproity (whr th pris valus o ths proprtis pn on th strutur o th gr istribution, as w saw abov). Th Tabl blow summarizs ths proprtis an ompars thm with th simpl ranom graph mol. As w vlop mor sophistiat ranom-graph mols throughout th smstr, w will xpan this tabl. 14

15 ntwork proprty ral-worl Erős-Rényi oniguration gr istribution havy tail Poisson( k ) spii iamtr small ( log n) O(log n) O(log n) lustring oiint soial: morat non-soial: low O(1/n) O(1/n) riproity high O(1/n) O(1/n) giant omponnt vry ommon k > 1 k 2 2 k > 0 4 At hom 1. Ra Chaptr (pags ) in Ntworks 15