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# 529 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 19 views.

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Date Created: 02/06/15

Properties of real networks degree distribution 03 025 02 A 31 B4 015 01 005 Nodes with small degrees are most frequent The fraction of highly connected nodes decreases but is not zero Look closer use a logarithmic plot semilog Plotting 06 linear power laws and exponentials 20 4O 6O 80 100 The in and outdegree distribution of the VWWV are powerlaws u Pmk 10 a i nodes webpages 1072 B a edges hyperlinks 1oquot 7 O 7 7 7 6 A PWtUC g k 245 10 7 7 7 ll k 21 10 B 3 7 7 10quot 7 7 7 7 1042 i i i 10 2 100 102 10 106 10 2 100 102 10 106 k k R Albert H Jeong AL Barab si Nature 401 130 1999 A Broder eta Comput Netw 33 309 1999 Powerlaw degree distributions were diverse networks n nodes routers 1 edges cables 10quot 10 2 P0010 Pk k39 104 10 5 10 5quot Internet router level Actor collaboration O i i i i i i 39 Go a u b O 7 00 7 00 E 7 00 7 7 RE Q E u Qb 93m 10quot 101 1o2 103 101 102 103 k oundin nodes actors edges cast jointly 7 Pk31k393 R Govindan H Tangmunarunkit IEEE lnfocom 2000 A L Barab si R Albert Science 286 509 1999 Networks of science collaborations also have powerlaw degree distributions u Coauthor HEP Coauthor neurosci 10 l l l l l l 10quot 9o 7 7A 7 10 2 7 o 7 7 7 Plc1o4 7 00 7 7 Pkk39 1 4 0 i Pk slc3921 5 7 7 7 A 10 w 1076 U l 2 l l l 1 2 1 10510 10 10 k M E J Newman Phys Rev E 64 016131 2001 A L Barabasi et al condmat0104162 2001 Metabolic networks have a powerlaw degree distribution 100 m In e 1071 Out 6 Archaeoglobus f 102 39 E coli 3 10 3 Q 104 22 39 P1k k 22 39 39 PM z k bipartite nodes metabolites reactions C39 elegans c 2 directed edges out reactant substrate I in product of reaction I n 102 103 39 im m 39 Hunquot 10 102 103 10 10D 10 k k H Jeong et al Nature 407651 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 Pk K ipm or PkgtK1 m3kK Ex Determine the degree distribution and cumulative degree distribution of the graph on the right If the noncumulative degree distribution is a power law with exponent ocgt1 the cumulative degree distribution will be a power law with exponent oc1 Does not apply for oc1 Probability that a node has a degree bigger than X i PX gt x z cx o 1 Probability that node has degree X Px z cx a V logx Power grid has exponential degree distribution E 1 i t in 7 100 087 D 7 nodes generators O A 7 El 7 0 quot5w 0396 Du power stations E 047 n D 7 E 027 E 39339 7 edges power lines 1047 0 i i i i i i i Hi i o 5 10 15 20 25 k P kgtKocexp05K 25 30 R Albert Albert G L Nakarado Phys Rev E 69 025103R 2004 llogltkgt Path length and order in real networks 15 I I I I 1 0 39food webs neural network 39 X power grid Acollaboration networks 102 VVNW 10 metabolic networks 39 Internet A I A x A x 10394 A 74quot 5 x 10396 0 39 0 f I 10 102 104 106 108 101 10 N I log N logltkgt quot39 A A V x A 5 A A 39food webs neural network metabolic networks Xpower grid Acollaboration networks I waw I llllllll llllllll 100 10 10 106 108 N C oc ltkgt Apparent scaling with the network size and average degree as though these different networks were members of the same family Distribution of be weenness centrality b It 2 39 f Wandwide Web Internet AS level P5g gquot Coauthorship Protein interaction Metabolic netw 1338 22 m45 P5gg co 4 in5 mm 4 9 0 w 0 EN 0 AS 3 1076 1039quot 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 PlgtLz 2500Lo397 Q How does the noncumulative distribution look like in the region where the cumulative distribution is almost horizontal the aower grid O llllllll lllllll lllllllll lllllllll lllllM rr G O O 1039 0 101 102 103 104 105 106 107 L R Albert Albert G L Nakarado Phys Rev E 69 025103R 2004 Network Nodes Edges Nreal Ner 3 SD Z score Nreal Nmnd 1 SD Z score Nfeal Nrmd 3 513 Z score Gene regulation X Feed X Bil39an transcription W forward Y loop W Z W Z E Coll 424 519 40 7 t 3 10 203 47 112 13 S cel evisiae 685 1 052 70 11 t 4 14 1812 300 i 40 41 Neumns X Feed X Y Bilan X Bi W forward amp parallel Y loo Y Z w P z w 1 14 Z W C elegansf 252 509 115 90 4 0 37 127 55 i 13 53 227 35 t 10 20 Food webs X Three X Bi W chain u N parallel Y Y Z v N 14 Z W Little Rock 12 984 3219 3120 t 50 21 7295 2220 i 210 25 Electronic circuits X Feed X Y Bll an z X N Bl forward logic chips W forward Y Z parallel Y loop 31 K V z w W I 515850 10 383 14 240 424 2 t 2 285 1040 1 1 1200 480 2 139 1 335 Electronic circuits X TlireeA X Y Bi an X9 Y Four digital fractional multipliers f node node feedback feedback Y 6 2 loop 2 w z w loop 5208 122 189 10 11 9 4 1 11 38 5 11 5 5420 252 399 20 1 1 18 10 1 1 10 11 1 1 11 5838 512 819 4D 11 1 38 22 11 1 20 23 1 E Wolld Wide Web X Feedback X Fully X Uplinked Q with two f N connected 7 mutual mutual Y E E Z triad Y E E Z dyad dyads Z ndcdu 325729 114626 11c5 203 t 122 800 6806 5amp414c2 15000 1266 1c4 252 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 Erik 1211ng ijQtyNYEquotZkirNl 05 Positive correlation assortative Negative correlation disassortative 113mm rk type size n moneti p39ity r error Ur Def pl139SIEE mautlm ship Lll39ul llE39Ec taEd 5393 39EICI EI 03 3 frz a biology mautlm ralti 2 mini i recnail 521 3935 1112 0013 al 1 L39L39Latltematim mwtlmrship Lind irected 39353 339 I130 I lIIE I mial Film actor collatuaratious undirected M9913 02 a compan directom undirected 39T 573 L276 ElJ d student relaticL Isltipe und imamd 573 003 H113 e L email address bunks dimmed THESE 111392 00M iquot 39 power grid ut39idirected i911 IIDE 13013 2 I Mamet Lind irec bed In g 1 E I IlII E 1 cm kg39ml Warm Wide wet dimLed 5m 005 41mm i software depenilermiea directed 3 Iii392 01 11539 13031 j JED39LEI u Interactions Lind Irected 393 I if H156 I 0113 k metabolic netw curlc uncl ireched i I13 10 I DU39 1 biglogical neural networ l directed 3U 03336 I 016 in marine Food wet direched 131 El E I 037 u i Emeltw ziter incd web ill meted 3933 0326 I IS 1 Social networks tend to be assortative technological and biological 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 is a decreasing function usually a powerlaw The betweenness centrality distribution is a power law as well Both indicate heterogeneity and the existence of hubs The distances scale logarithmically with the network size N logN l N logltkgt The clustering coefficient does not seem to depend on the network size C oc k Frequent subgraphs not universal but common to several networks

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