The architecture of complexity Dana Farber Cancer Research Institute Harvard University and Center for Complex Networks Research University of Notre Dame Complex systems Business ties in US biotech-industry Nodes: companies investment pharma research labs public biotechnology Links: collaborations financial R&D http://ecclectic.ss.uci.edu/~drwhite/Movie Erdös-Rényi model (1960) # !" ! ' Nodes: # # # Links: $%& ! " collects all URL’s found in a document and follows them recursively - / γ ( %' () * +( ' &, ( ( $%& " - .../ * 0 ! ) Metabolic Network Protein Interactions Online communities Nodes: Links: 6 4 ( 4( 5 $ 4 ( 1 ( + .(. 4 2( , 2 3( %14 4 ) (1 +( & 7 (4 4 ! + , - -8 9 :;< => .? &7 6 2 / 6 < ;:6 4 Scale-free model ! " #$# # + , %& ' & &( ) # * * Π- / = Σ 12-34-!5 , @ A' ( ./0 .- .../ Can Latecomers Make It? Fitness Model =8 % 8 - B B -η ) B Π- / ≅ η β(η) η/C η η β(η) β(η) ηρ -η/ η− = +/ Bose-Einstein Condensation in Evolving Networks Network η Π = Ση η -η / ρ -η / Fit-gets-rich Bose gas − βε -ε / = -ε / − βε − Bose-Einstein condensation G. Bianconi and A.-L. Barabási, Physical Review Letters 2001; cond-mat/0011029 Spatial Distributions % Spatial Distribution of Routers Fractal set Box counting: N( ) ≡ No. of boxes of size that contain routers N( ) ~ =1.5 Preferential Attachment • Compare maps taken at different times (∆ ∆t = 6 months) • Measure ∆ ( ), increase in No. of links for a node with links Preferential Attachment: ∆ ( )~ Modeling the effect of spatial distribution ) 6 " Π- ( / = 61 7 α # σ - + 6 6 ( 9<&2=3 9:. 9;. 8 12-3 12-3 ! INTERNET ( ) =1.5 ∆ - / α α=1 - / σ σ=1 * 0 0 ! '> ) ? Robustness and fragility node failure Attacks Failures Albert, Jeong, Barabasi, Nature 406 378 (2000) * 0 0 ! '> ) ? 0 - 2A @ B? 3 A 0 ++ * C + Northeast Division > Northwest Division 1 > ' 0/ %0D&%5 2.%%53@E 2.%%.3 1 > ' 0/ %0D&%5 2.%%53 E 2.%%.3 + > ! %5%/.&F2.%%53 A 0 ++ * C + 0, C C > +( 2> + 2 - 50 Northeast Division © 2003 Spoke Software, Inc. West Division A 0 ++ * C + 0, C C > +( 2> + 2 0G C ' +( # 33 + Viral marketing ? H B > - http://www.orgnet.com black: opinion leaders red: influenced green: uninfluenced grey: undecided 500 randomly chosen clusters Day 8 1 2 3 4 5 6 7 500 largest clusters Day 8 1 2 3 4 5 6 7 Conclusions WWW Science collaboration Language ! Citation pattern 0D 2 9 2 > + 2 2 > 2 E 0! + B F G> '& , Cell Internet + 2 B ( 2 & 9 B & > + E B E "4 :" ( 4 ; 4 -4 / %D%D%/F6J 3F 3( ! %D%D5F&/6H I 3 %D%0&.0 ( Zoltán N. Oltvai, U of Pittsburgh Med. School Hawoong Jeong, KAIST, Corea Réka Albert, Penn State Ginestra Bianconi, Trieste Erzsébet Ravasz, Los Alamos Stefan Wuchty, Northwestern U Eivind Almaas, Notre Dame Livermore Zoltan Dezso, Notre Dame Gabor Szabo, Notre Dame Alexei Vazquez, Notre Dame Paris Joao Oliveira, Notre Dame (Aveiro) http://www.nd.edu/~networks