1. A simple model for studying interacting networks
Wenjia Liu
Shivakumar Jolad, Beate Schmittmann and R.K.P. Zia
Department of Physics, College of Science, Virginia Tech
The 78th Annual Meeting of the Southeastern Section of the American Physical Society
Funded by the Division of Materials Research, NSF, and ICTAS, Virginia Tech

2. Single network model κ Network: Dynamics: w-(k) = 1‒ w+(k)
Nodes – individuals
Links – relationships between individuals
Rate function w+(k)
interpolates between 0 and 1
sets preferred degree κ – the number of friends that an individual prefers to have
Dynamics:
- Select a node at random and find its degree k
- Update its links as follows:
with rate w+(k), create a link
with rate w-(k), destroy a link κ
Choose, for simplicity, the rate for cutting a link:
w-(k) = 1‒ w+(k)
One Monte Carlo step = N attempts

3. Degree distribution ρ (k) average fraction of nodes with k links
Double exponential
Gaussian → exponential tails
N =1000, κ=250
Rigid Inflexible
With the first 1000MCS discarded, we take the average over 20 measurements, separated by 1000MCS. Fermi-Dirac=1/(1+e^beta(k-kappa)), beta=0.1,0.2
Tolerant
Easy going

4. Analytic approach
N =1000, κ=250
An approximate master equation leads to a prediction for the following stationary degree distribution:

5. Two networks model κ Two different groups are coupled together.
different sizes: N1 ≠ N2
different preferred degrees: κ1≠κ2
obviously, can have S1 ≠ S2 κ
Dynamics:
- Select a node at random and find its degree k
- Update its links as follows:
with rate w-(k), destroy a link
with rate w+(k), create a link
Pick a partner from the other network
With probability S
or inside its own network
With probability 1‒S
One Monte Carlo step = N1 + N2 attempts

6. Total degree distribution
N1= N2 = 1000
Κ1κ1 =κ2=250
S1= 0.8, S2 = 0.2
also works well here
Our simple argument for degree distributions in a single network, generalized to include S1 and S2 :

7. Other degree distributions
N1= N2 = 1000
κ1 =150,κ2=250
S1= S2 = 0.5
Degree distribution of cross links
Degree distribution of “internal” links

8. Total number of cross links Nc
N1= N2 = 1000
Κκ1 =κ2=250
S1= S2 = 0.5
N1= N2 = 100
Κκ1 =κ2=25
S1= S2 = 0.5 Nc

9. Nc
Power spectrum
512 104MCS data taken every 100MCS data points in each time series
averaged over 50 series
N1 = N2 = 100
1 = 2 = 25
S1 = S2
lnI()
Consistent with random walk of Nc in a potential Explores flat bottom for shorter times; bounded by walls for larger times
ln

10. Conclusions and work in progress
Single network
- The degree distribution of one single network is well
understood.
Two interacting networks
- With interactions, the total degree distribution of each
individual network can be profoundly affected.
- Other degree distributions (internal and cross)
- Slow dynamics of cross links

11. Conclusions and work in progress
Other types of interactions
- Variable S model
- κin , κcross model
- Labeling nodes in single network 
Thank you!

12. 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): Nc
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?