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# 666 Class Note for PHYS 597A with Professor Albert at PSU

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This 16 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

Properties of real networks degree distribution Nodes with small degrees are most frequent The fraction of highly connected nodes decreases but is not zero Look closer use a logarithmic plot loglog39 1 0394 Plotting power laws and exponentials Note these are plots of functions and not degree distributions semilog 06 02 fx0x fx cx OS fx 2 cf 20 40 linear 6O 80 100 n and outdegree distribution 039 the VWWV 100 a nodes webpages 102 7 B a 7 7 edges hyperlinks 4 7 O 7 7 777 10 A Pautk k 245 573 1076 7 7 5E 7 21 1 78 n k 10 7 O 7 7 10quot 7 7 7 10712 72 U 2 4 6 72 1o 10 1o 10 1o 10 Usage the degree distribution scales as a power law R Albert H Jeong AL Barabasi Nature 401 130 1999 A Broder eta Comput Netw 33 309 1999 Degree distributions in networks of science collaborations u Coauthor HEP Coauthor neurosci 1O 1 1 1 1 1 1 71 10 1 D 1 2 10 Pk1o Poem 10 7 7 Pkk39 5 7 7 7 A 10 w 10 6 i 1 1 1 1 2 1 10510 10 10 k M E J Newman Phys Rev E 64 016131 2001 A DaraDaSI er al conumat0104162 2001 Powerlaw degree distributions were diverse networks nodes routers 10 edges cables 10quot 10 2 P0010 Pk k39 1043 10 5 10 5quot Internet router level Actor collaboration o o O o O O o O c 93 Q a Qb 93m 10quot 101 1o2 103 k 101 102 1o3 oundin nodes actors edges cast jointly 7 Pk31k393 R Govindan H Tangmunarunkit IEEE lnfocom 2000 A L Barabasi R Albert Science 286 509 1999 Metabolic networks have a powerlaw degree distribution 10 10quot Archaeoglobusf 102 5 1039s 10 wow 10 s I ln e Out El j E coli Bmk42 10539 mm Illiilii 1mm bipartite 10 10quot 10 C elegans Q 1073 104 105 i nodes metabolites reactions directed edges OUtI reactant substrate in product of reaction 67 101 01 H Jeong et al Nature 407 651 102 k 103 10 10 1039 k 102 103 2000 Cleaning up degree distributions Often it is dif cult to determine the best t to the points that make up a degree distribution Methods of data cleanup 1 logarithmic binning bin the k range use bins of exponentially increasing size 2 Display the cumulative degree distribution K Pk K 213k or kkmin PkgtK1 PkgK Ex Determine the degree distribution and cumulative degree distribution of the graph on the right If the noncumulative degree distribution aligns with a power law with exponent ocgt1 the cumulative degree distribution will align with a power law with exponent oc1 Does not apply for oc1 Probability that node has degree X Px z cx Probability that a node has a degree bigger than X h PX gtx z cor 1 lOgiX V P ower grid has exponential degree distribution 2 if il ti inl e 100 187 D 7 nodes generators 00 ED 06 Dun power stations E 047 u 7 g 02 in El edges power lines 2 l i l i l i l ml i 10 00 5 10 15 20 quot 25 k 104 P k gt K cc exp05K 6 i 10 0 5 10 15 20 25 30 K R Albert Albert G L Nakarado Phys Rev E 69 025103R 2004 Path length and order in real networks Vioodmb akmmalnetwom A AA gtlt wa m Astsmm vl x A Owww metzmhcn kmks A A 0mm Q mumps v w 7naumnamm 5 Immmhcmmms Xwnmvam Amthbuumn namms waw to m m m2 tu Hf m N In N l a g c ac k lugltkgt Apparent scaling with the network size and average degree as though these different networks were members ofthe same family Distribution of be weenness centrality 10 u Coauthorship b 10 Protein interaction WorldWlde Web Metabolic netw A T2 Internet A8 level if 103 P 2 22 L 55 g P5 g 5 g w ms 4 9 039 www 0K EN lt AS 3 10396 mm 10quot ID39 102 103 1047 w in2 103 m4 105 g E K I Goh et al PNAS 99 12583 2002 Betweenness centrality load distribution of PlgtL5 2500Lo397 Q How does the noncumulative distribution look like in the region where the cumulative distribution is almost horizontal the aower grid l lllllll l llllllll l llllllll l lllllll l llllM r G O O 1039 100 101 102 103 104 105 106 107 L R Albert Albert G L Nakarado Phys Rev E 69 025103R 2004 Network Nodes Edges le Ner 1 SD Z score Nt a1 Nmud 3 SD Z score Nrcal N rand SD Z score Gene ngulmiou X Feed X Bil39au lrauscrlpllon W forward Y lnop W 2 w Z E ml 424 519 40 7 1 3 10 203 47 1 12 13 S cer ewsiaequot 685 1052 70 111 4 14 1812 300 1 40 41 Neurons X Fccd X Bilan X Bi W forward M M amp parallel Y 00 Y Z w P z w 51 4 Z W C elegans r 252 509 125 90 0 37 127 55 1 13 53 227 35 1 10 20 Food webs X Three X B1 W clmin M N parullcl Y Y Z y N u Z W Lii c Rock 92 984 3219 3120 1 50 21 7295 2220 1 210 25 Electronicdrcuits X Feed X Y BMW M X N Bi forward logic chips W forward Y Z parallel Y loop N Z V z w w Z 515850 10 383 14 240 424 2 1 2 285 1040 1 1 1 1200 480 2 1 1 335 Eleclrollic circuits X Tbree X Y Bil au X9 Y Four dlgital fractional multipliers node node feedback feadback v e z loop 2 w 2 w 00 5208 122 189 10 111 9 4 111 38 5 111 5 5420 252 399 20 11 1 18 10 1 11 10 11 1 11 11 58381 512 819 40 11 1 38 22 1 1 1 20 23 1 1 1 25 World Wide Web X Feedback X Fully X Unlinked will Iwo f N connected mulual mutual Y E E Z mad Y E E Z dyad dyads Z ndcdn 325729 1141626 11c5 203 1 102 800 6866 5c41452 15000 1296 1c4 1 262 5000 Mixing patterns in networks Mixing in social networks assortative people prefer to associate with others who are like them disassortative people prefer to associate with others who are different Mixing with respect of node degree assortative high degree nodes tend to be connected to high degree nodes disassortative high degree nodes tend to be connected to low degree nodes Focus on edge i denote the excess indegree of its starting point with j and the excess outdegree of its endpoint with k Mixing is quantified by the correlation between j and k over all i Ziljiki ZijikN ij QJifNJQTZklz ki2N r 05 Positive correlation assortative Negative correlation disassortative techno legieal biological Social networks tend to be assortative technological and biological network type size ft insetnativity r error or ref phngle coautno rel39n p undirected 529139 41363 1003 a biology coautl39io relli p undirected l 53931 quot351 0137quot I D 1 3 met ltenmti to co amt l39LCIl Sl39ll p undirected 253 339 01213 I DD E b Ftlrn acte r collabor at ions und i rected MEI 913 02118 I DCHII39Z c company directors undirected T 63973 027 Dill1 cl student reletioneltipe undirected 573 E l 11113quot e entail thide books directed 11353 11092 a l lquot power grid undirected 1 El all Ill i 11113 gquot I i1ternet uncl irected 10 g Ill 1 HE D lIlZ l1 I bflclllrrldf Vuch directed 3139501 39EET 1300132 i software dependatom directed E 1173 ill 16 011311 j protein interactions undirected 392 l 15 43156 l k metabolic network und irected T I53 10 I iJIUT l neural networ l di remed 31 02 10le tn marine food web directed 131 oer3 D 113 n freshwater web directed 3933 433 11113 o networks tend to be disassortative Possible causes of assortativity attraction of similars group affiliation Possible cause of disassortativity service relationships eg directories M E J Newman Phys Rev E 2003 Universality in largescale networks The degree distribution follows a decreasing function usually a powerlaw The betweenness centrality distribution is also decreasing Both indicate heterogeneity and the existence of hubs The distances scale logarithmically with the network size logN I z logltkgt The clustering coefficient does not seem to depend on the network size and it seems to be proportional with the average degree C oc k Fre uent subra hs not universal but common to several networks

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