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

Funded by ICTAS, Virginia Tech, and the Division of Materials Research, NSF

Large Fluctuations in Nonequilibrium Systems

MPI für Physik komplexer Systeme Dresden July 3 – 15, 2011

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.

• 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

participate in same reaction.

Marta Sales-Pardo et al, PNAS 104, 15224 (2007)

Examples of networks

White matter tracts in the brain.

Red: left-right, blue: superior-inferior, green: anterior-posterior. Courtesy of D. Bassett (2010)

Econometrica (left) Astrophysical Journal (right)

Roger Guimera, Northwestern

Examples of networks

• Social – author networks, online communities, insurgent groups…

Sunni insurgent groups in Iraq.

Michael Gabbay (2008)

Factional maps:

• Joint communications
• Joint operations

Nationalist vs Jihadist

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:

 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

• Network:

 Nodes individuals  Links relationships between individuals

Link

Preferred degree networks!

• Dynamics:

 Select random node, find its degree, k  Create a link, with rate w+(k); destroy a link, with rate w (k)  For simplicity: w (k) = 1  w+(k)

• Note: Receiving node is passive.

Single community

κ

200 220 240 260 280 300 0.0 0.5 1.0

w+(k) k

• Dynamics:

 Select random node, find its degree, k  Create a link, with rate w+(k); destroy a link, with rate w (k)  For simplicity: w (k) = 1  w+(k)

Single community

κ

200 220 240 260 280 300 0.0 0.5 1.0

w+(k) k

Quantities of interest:

 Degree distribution  (k)

average number of nodes with degree k

 Clustering, connectivity, topology, …

Degree distribution ρ(k)

N =1000, κ=250 Double exponential Gaussian → exponential tails Tolerant Easy going Rigid Inflexible

200 220 240 260 280 300 0.0 0.5 1.0

w+(k) k

230 240 250 260 270 1E-3 0.01 0.1 1

(k)

k

• Approximate master equation:

Analytic approach

) 1 ( 2 1 ) 1 ( ) ( 2 1 ) ( ...                  

 

k k w k k w  

... ) 1 ( 2 1 ) 1 ( ) ( 2 1 ) ( ) (                     

 

k k w k k w k

t

  

Steady state

200 220 240 260 280 300 0.0 0.5 1.0

w+(k) k

235 240 245 250 255 260 265 1 2 3 4

Simulation results

Analytical approach

(k)/(k-1)

k

) ( 2 / 1 ) 1 ( 2 / 1 ) 1 ( ) ( k w k w k k

 

      

• 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

• ne’s own community – us vs them

 ...

Two communities

• 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, …

• 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 S1 ≠ S2 , 1 ≠ 2

Version 1

Version 1 – total degree distribution

κ

200 220 240 260 280 300 0.0 0.5 1.0

w+(k) k

Version 1 – total degree distribution

N1= N2 = 1000

S1= 0.8, S2 = 0.2

235 240 245 250 255 260 265 1E-3 0.01 0.1 1

(k)

k

κ

200 220 240 260 280 300 0.0 0.5 1.0

w+(k) k

Mean-field for steady state:

also works well here

         

k w S S k w S S k k

 

        

2 1 2 1

1 2 1 ) 1 ( 1 2 1 1  

Total links N1=N2=1000

1 = 2 = 250 S1 =0.8, S2 =0.2 1 = 150, 2 = 250 S1 = S2 = 0.5

140 160 180 200 220 240 260 1E-3 0.01 0.1 1

(k)

k

235 240 245 250 255 260 265 1E-3 0.01 0.1 1

(k)

k

For comparison – different 

k =  for both networks k = 1 for network 1 k = 2 for network 2

Other degree distributions

N1= N2 = 1000

κ1 =150, κ2 = 250

S1= S2 = 0.5

Degree distribution of cross links Degree distribution of “internal” links

60 80 100 120 140 1E-4 1E-3 0.01 0.1

2(k)c 1(k)c

k

30 60 90 120 150 180 1E-4 1E-3 0.01 0.1

2(k)i 1(k)i

k

k = k(i) + k(c)  =  k(c1 = k(c)2

in which N is the total number of nodes in each network.

k M k

N N k M k



          ) 1 1 ( ) 1 ( ) ( ρ

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 .  Then, probability for a node to have k cross links is simply a binomial:  Gives qualitatively correct behavior, but contains no information about S or 

k M k

N N k M k



          ) 1 1 ( ) 1 ( ) ( ρ

N1 = N2 = 1000 1 = 2 = 250 S1 = S2 = 0.5

k M k

N N k M k



          ) 1 1 ( ) 1 ( ) ( ρ

Properties of cross link distribution

• On long time scales, small systems:

100 30000 60000 90000 400 800 1200 1600 2000 number of cross links

t(MCS)

N1 = N2 = 100 1 = 2 = 25 S1 = S2 = 0.5 400 800 1200 1600 2000 1E-4 1E-3 0.01 0.1

M Total number of links ~ N = 2500 Number of cross links ~ N/2 = 1250 Histogram So, average is understandable But top and bottom boundaries? Nc

k M k

N N k M k



          ) 1 1 ( ) 1 ( ) ( ρ

Power spectrum

1024 N1 = N2 = 100 1 = 2 = 25 S1 = S2

2

) ( ) (





t t i c

e t N I





8 12 16 20 24 1 2 3 4 5 6 7 0.5 0.8 0.2 0.05 0.99 0.01 1/x^2

ln lnI()

512 104MCS data taken every 100MCS 1024 data points in each time series averaged over 50 series

Consistent with random walk of Nc in a potential Explores flat bottom for shorter times; bounded by walls for larger times

k M k

N N k M k



          ) 1 1 ( ) 1 ( ) ( ρ

Consistency check?

• Assuming the fraction of cross links,  = Nc/N, performs a

random walk in a potential V( ), write Fokker-Planck equation for probability P(,t):

) , ( ) ( ' ) , ( t P V t P

t

                  

with stationary solution P*()  exp[ V( )/].

• Now, extract V( ) from histogram

and simulate a random walker in this V – will this process reproduce the cross link dynamics?

400 800 1200 1600 2000 1E-4 1E-3 0.01 0.1

M

N

• Dynamics – new parameter R:

 Select a node at random, counts its degree, k  Decide, with rate w+ , whether to create or destroy a link  Evaluate r = (cross links) / (total links) for this node If r < R add cross link or delete internal link If r > R add internal or delete cross link  Can have R1 ≠ R2

Version 2

Version 2 – Degree distributions

N1 = N2 = 1000 1 = 2 = 250 R1 = R2 = 0.301

0.00001 0.0001 0.001 0.01 0.1 1 100 200 300 COM1 COM2 COM1_2 COM2_1 COM1_1 COM2_2

N1 = N2 = 1000 1 = 2 = 250 R1 = R2 = 0.299

0.00001 0.0001 0.001 0.01 0.1 1 100 200 300 COM1 COM2 COM1_2 COM2_1 COM1_1 COM2_2

Conclusions and work in progress

• Single network:

 Degree distribution well understood  Other network characteristics?  Other rates, other degree distributions?

• Two interacting networks:

 Interactions modify total degree distribution of individual networks  Analytic understanding of other distributions (cross links, internal links)?  Analytic understanding of dynamics?  Other types of couplings? Realistic applications?