interacting networks Beate Schmittmann Department of Physics, - PowerPoint PPT Presentation

A first attempt at characterizing interacting networks Beate Schmittmann Department of Physics, Virginia Tech with Wenjia Liu, Shivakumar Jolad and Royce Zia Large Fluctuations in Nonequilibrium Systems MPI f ü r Physik komplexer Systeme Dresden July 3 – 15, 2011 Funded by ICTAS, Virginia Tech, and the Division of Materials Research, NSF

Shivakumar Jolad Wenjia Liu Royce Zia

Outline: • Networks in science • “Adaptive” and “interacting” networks • Preferred degree networks:  Single community  Two communities • Findings, conclusions, and outlook.

Examples of networks • Physical – critical infrastructures: transportation, power, communications , water/sewer, … Guimerá and Amaral, EPJB 38, 381 (2004)

Examples of networks • Biological – neural networks, food webs, reaction networks, … E. Coli: Metabolites are linked if they White matter tracts in the brain. participate in same reaction. Red: left-right, blue: superior-inferior, green: anterior-posterior. Courtesy of D. Bassett (2010) Marta Sales-Pardo et al, PNAS 104, 15224 (2007)

Examples of networks • Social – author networks, online communities, insurgent groups… Sunni insurgent groups in Iraq. Michael Gabbay (2008) Econometrica (left) Astrophysical Journal (right) Roger Guimera, Northwestern

Nationalist vs Jihadist Factional maps: • Joint communications • Joint operations

Questions? • Static networks and graph theory  Types of networks, structure, and connectivity • Statistical mechanics and dynamics on networks  Order/disorder transitions, diffusion processes, epidemic spreading, opinion dynamics, … • Statistical mechanics and dynamics of networks  Growth, shrinkage, and rewiring ; stability with respect to different perturbations (local vs global, random vs intentional)

More recent questions • Adaptive (co-evolving) networks:  Opinion dynamics: make new connections, break old ties  Epidemics: relationships depend on prevalence of disease • Interacting networks  Interacting infrastructure networks., e.g., internet – power grid  Interacting social networks, e.g., school – Facebook

Simple model • Two communities: Preferred degree networks!  Introverts and extroverts, or few vs many friends  “Natives” and “immigrants”, or “we” vs “them” • Each group creates or removes connections, seeking to maintain a preferred degree • Interactions: connections between members of different communities Link • Network:  Nodes individuals  Links relationships between individuals

Single community • Dynamics:  Select random node, find its degree , k  Create a link, with rate w + ( k ); 1.0 destroy a link, with rate w  ( k ) w + (k)  For simplicity: w  ( k ) = 1  w + ( k ) 0.5 • Note: Receiving node is passive. 0.0 200 220 240 260 280 300 κ k

Single community • Dynamics: Quantities of interest:  Select random node, find its degree , k  Create a link, with rate w + ( k );  Degree distribution  ( k ) 1.0 destroy a link, with rate w  ( k ) average number of nodes with degree k w + (k)  For simplicity: w  ( k ) = 1  w + ( k )  Clustering, connectivity, topology, … 0.5 0.0 200 220 240 260 280 300 κ k

Degree distribution ρ ( k ) Double Gaussian → exponential exponential tails 1 Rigid Inflexible  (k) 1.0 0.1 w + (k) 0.5 0.01 0.0 200 220 240 260 280 300 k 1E-3 Tolerant 230 240 250 260 270 Easy going k N = 1000, κ = 250

Analytic approach • Approximate master equation:     1 1             ( k ) w ( k ) ( k ) w ( k 1 ) ( k 1 ) ...       t     2 2     1 1         ... w ( k ) ( k ) w ( k 1 ) ( k 1 )           2 2

1.0 w + (k) Steady state 0.5 0.0 200 220 240 260 280 300 k 4  (k) /  (k- 1 ) Simulation results Analytical approach 3 2    ( k ) 1 / 2 w ( k 1 ) 1      ( k 1 ) 1 / 2 w ( k )  235 240 245 250 255 260 265 k

Two communities • Many different ways of coupling two networks:  Different rates w + , w  for each community; different  – extrovert vs introvert  Different preferences for creating links inside/outside one’s own community – us vs them  ... • Two versions so far:  After deciding to create/remove link, select ( S ) internal vs external partner Large fluctuations in the number of cross links  After deciding to create/remove link, respect specified ratio ( R ) of crosslinks Fluctuations in the number of cross links suppressed

Two communities More quantities of interest:  Degree distributions for all links, internal links, and crosslinks:  ( k ) ,  ( i ) ( k ) , and  ( c ) ( k )  Dynamics and fluctuations of cross links  Clustering, connectivity, topology, …

Version 1 • Dynamics – new parameter S :  Select a node at random, counts its degree, k  Decide, with rate w + , whether to create or destroy a link  With rate S , select a partner from the other community (with 1  S , from own community )  Can have S 1 ≠ S 2 ,  1 ≠  2

Version 1 – total degree distribution 1.0 w + (k) 0.5 0.0 κ 200 220 240 260 280 300 k

Version 1 – total degree distribution 1 1.0  (k) w + (k) 0.1 also works well here 0.5 0.01 0.0 κ 200 220 240 260 280 300 k N 1 = N 2 = 1000 1E-3 Κ 1 κ 1 = κ 2 = 250 235 240 245 250 255 260 265 k S 1 = 0.8 , S 2 = 0.2   1       1 S S w ( k 1 )   k 1 2 Mean-field for 2          1 1 k steady state:    1 S S w k  1 2 2

For comparison – different   k  =  1 for network 1  k  =  for both networks  k  =  2 for network 2 Total links N 1 = N 2 =1000  1 = 150,  2 = 250  1 =  2 = 250 S 1 = S 2 = 0.5 S 1 =0.8, S 2 =0.2 1 1  (k)  (k) 0.1 0.1 0.01 0.01 1E-3 1E-3 140 160 180 200 220 240 260 235 240 245 250 255 260 265 k k

Other degree distributions N 1 = N 2 = 1000  k  =  k ( i )  +  k ( c )  =  κ 1 =150, κ 2 = 250  k (c  1 =  k ( c )  2 S 1 = S 2 = 0.5 Degree distribution of Degree distribution of cross links “internal” links 0.1 0.1  1 (k) i  2 (k) i  1 (k) c  2 (k) c 0.01 0.01 1E-3 1E-3 1E-4 1E-4 60 80 100 120 140 30 60 90 120 150 180 k k

Properties of cross link distribution • Distribution settles very slowly into steady state • Total number of cross links “diffuses” slowly • On short time scales:  Total number of cross links approximately constant, say, M .   M  Then, probability for a node to have k cross links is simply a binomial: 1 1      ρ k M k ( k ) ( ) ( 1 )     k N N   M 1 1      ρ k M k ( k ) ( ) ( 1 )     k N N N 1 = N 2 = 1000  Gives qualitatively correct behavior, but contains no information about S or   1 =  2 = 250 in which N is the total number of nodes in each network. S 1 = S 2 = 0.5

Properties of cross link distribution Total number of links ~ N  = 2500 • On long time scales, small systems: Number of cross links ~ N  /2 = 1250 N 1 = N 2 = 100 0.1 So, average is understandable  1 =  2 = 25 Histogram But top and bottom boundaries? S 1 = S 2 = 0.5 0.01 2000 number of cross links   M 1 1 1600      ρ k M k ( k ) ( ) ( 1 )   1E-3   k N N 1200 1E-4 800 400 800 1200 1600 2000 M N c 400 100 30000 60000 90000 t (MCS)

Power spectrum 2     i t I ( ) N ( t ) e Consistent with random walk of N c in a potential c t Explores flat bottom for shorter times; 24 bounded by walls for larger times 0.5 0.8 N 1 = N 2 = 100 0.2  1 =  2 = 25 20 0.05 lnI(  ) S 1 = S 2 0.99 0.01 1/x^2 1024   16 M 1 1      ρ k M k ( k ) ( ) ( 1 )     k N N 12 8 512  10 4 MCS 0 1 2 3 4 5 6 7 data taken every 100MCS ln  1024 data points in each time series averaged over 50 series

Consistency check? • Assuming the fraction of cross links,  = N c / N , performs a random walk in a potential V (  ), write Fokker-Planck equation for probability P (  , t ):     0.1          P ( , t ) V ' ( ) P ( , t )     t   0.01   M 1 1      ρ k M k ( k ) ( ) ( 1 )     k N N 1E-3 with stationary solution P* (  )  exp[  V (  )/  ]. 1E-4 400 800 1200 1600 2000 • Now, extract V (  ) from histogram M N  and simulate a random walker in this V – will this process reproduce the cross link dynamics?