1. Modeling, sampling, generating Networks with MRV
Umass Team

2. Agenda sampling and estimation of network degree distributions – Don
effect of MRV on network characteristics – Bo
generative models for MRV - Shan

3. Characterizing Joint Degree Distribution of Directed Networks
Fabricio Murai, Don Towsley
Umass-Amherst

4. Outline RW-based estimation sensitivity of accuracy to
hidden incoming edges
visible incoming edges
sensitivity of accuracy to
heavy tails
reciprocity
correlation

5. Directed networks: hidden edges
outgoing edges visible, incoming edges hidden
random walk (RW) based estimation (INFOCOM’12)
during walk, construct undirected graph consistent with walk
follow outgoing edge on first visit
walk new undirected graph on revisits
use known solution for undirected RW to characterize network
can estimate outgoing degree distribution
impossibility result for joint in/outdegree distribution
lack of statistical information wo sampling most of graph

6. RW-based: Visible Edges
transform digraph to undirected graph
collect samples using RW
𝑠 1 , 𝑠 2 ,…, 𝑠 𝑛 , 𝑠 𝑘 =( 𝑖 𝑘 , 𝑜 𝑘 )
estimate
𝜑 𝑖,𝑗 = 1 𝑛 𝑘 ℎ 𝑖𝑗 ( 𝑠 𝑘 ) 𝜋 ( 𝑠 𝑘 ) ℎ 𝑖𝑗 ( 𝑠 𝑘 ) , 𝑖,𝑗=0,1,…
ℎ 𝑖𝑗 𝑠 𝑘 = 1, 𝑖 𝑘 =𝑖, 𝑜 𝑘 =𝑗 0, otherwise
𝜋 𝑠 𝑘 =𝐶×deg( 𝑠 𝑘 )
deg( 𝑠 𝑘 ) is degree of new undirected graph, 𝐶 chosen to make 𝜋 a distribution

7. RW-based degree distribution estimation
performance on real datasets
vs. uniform vertex sampling
vs. DURW for marginal outdegree distribution
effect of heavy tails?
effect of reciprocity, correlation?

8. Real Datasets in/out degrees appear to be heavy-tailed Don,
The numbers here are not perfectly accurate, since I’m taking the Giant CC.
I’ll try to get the right numbers.

9. Behavior of RW on real datasets
YouTube
NMRSE
sampling budget 10% of graph size (B = 0.1)
NRMSE
indegree
outdegree
log(PMF)
indegree
outdegree
log(NRMSE)
Simulation results.
NRMSE: behaves as expected. Two forces: High probability mass causes upsampling – small error for low total degree; high total degree causes upsampling – errors decrease with total indegree. Middle left area: small pmf, small total degree.

10. RW vs. Vertex Sampling YouTube, log(NRMSE) with B=0.1 Vertex Sampling
indegree
outdegree
Simulation results. B=0.1
Assuming jump cost=10; hence Vertex sampling only has 1/10 samples.
Vertex sampling performs as well as RW for very low total degree; worse everywhere elese

11. Web-Google B=0.1 log(PMF) log(NRMSE) outdegree outdegree indegree
Here the in-degree spans one order of magnitude more. (web pages can’t have too many out-links)

12. Wiki-Talk B=0.1 log(NRMSE) log(PMF) outdegree outdegree indegree
Here the out-degree spans one order of magnitude more.

13. Out-degree distribution estimation: DURW vs. RW
correlation: 0.95
reciprocity: 0.79
RW based on all edges provides (slightly) lower errors
Misusing indegree information?
NRMSE
YouTube: high reciprocity, high correlation
High correlation -> both methods exhibit the same trends
High reciprocity -> DURW “sees” graph as RW
outdegree

14. Other datasets Correlation: 0.13 Correlation: 0.47 Reciprocity: 0.31
outdegree
NRMSE
Correlation: 0.47
Reciprocity: 0.14
outdegree
NRMSE
WikiTalk: medium correlation -> trends are different in the middle
low reciprocity -> NRMSE is one scale of magnitude smaller for RW
Web-Google: low correlation -> nodes with large out-degree have small in-degree and end-up downsampled with RW, thus, larger error
Medium reciprocity -> NRMSEs have about the same order of magnitude

15. Sensitivity of RW method to graph structure
reciprocity
correlation
Methodology:
approximate RW sampling by independent edge sampling
focus on joint distributions with pareto(3) marginals

16. Approximating RW by edge sampling
indegree
outdegree
RW sampling
𝑓 𝑖,𝑗,𝑟 =𝑃(𝑖𝑛=𝑖, 𝑜𝑢𝑡=𝑗, 𝑟𝑒𝑐𝑖𝑝=𝑟)
select random edge
select random endpoint
prob. sample node with indeg i, outdeg j, recip deg r:
produces unbiased estimate
experiments suggest above expression sufficiently accurate
indegree
outdegree
Edge sampling

17. Controlling reciprocity
Goal: given degree distribution 𝜃 𝑖𝑗 , generate
𝜃 𝑖𝑗𝑟 =𝑃(𝑖𝑛=𝑖, 𝑜𝑢𝑡=𝑗, 𝑟𝑒𝑐𝑖𝑝=𝑟)
with arbitrary amount of reciprocity
Model
given in-degree i, out-degree j, number of reciprocated edges is
𝑟~𝐵𝑖𝑛𝑜𝑚 min 𝑖,𝑗 ,𝑝
𝑝 – tuning parameter, min reciprocity (𝑝=0), max reciprocity (𝑝=1)
distr. may not be graphical

18. Controlling correlation
Goal: given marginal distribution 𝑔 𝑖 , 𝑖=0,1,… generate 𝜃 𝑖𝑗 s.t. 𝜃 𝑖∗ = 𝜃 ∗𝑗 = 𝑔 𝑖 with arbitrary nonnegative Neyman-Pearson correlation
Model
mixture of
perfect correlation 𝑓 1 𝑖,𝑗 = 𝑔 𝑖, 𝑖=𝑗 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
no correlation 𝑓 2 𝑖,𝑗 = 𝑔 𝑖 ×𝑔 𝑗
𝜃 𝑖𝑗 =α 𝑓 1 𝑖,𝑗 +(1−α) 𝑓 2 𝑖,𝑗
𝛼 – tuning parameter

19. Pareto: effect of reciprocity
Correlation, 𝛼=0.99
Pareto: effect of reciprocity
indegree
outdegree
log(NRMSE(recip(0)/NRMSE(recip(1)))
Q: effect of reciprocity on RW estimation efficiency
𝜃 𝑖𝑗 𝑝 - in/outdegree distribution under 𝑝-reciprocity model
focus on
𝑁𝑅𝑀𝑆𝐸( 𝜃 𝑖𝑗 0 ) 𝑁𝑅𝑀𝑆𝐸( 𝜃 𝑖𝑗 1 )
reciprocity hurts estimation when degrees correlated
diagonal estimation helped when no correlation
recip(0) better
No correlation, 𝛼=0
indegree
outdegree
log(NRMSE(recip(0)/NRMSE(recip(1)))
When reciprocity is maximum, the proportion of samples along the diagonal does not change. Hence, log-ratio is zero.
recip(0) better
recip(1)
better

20. Pareto: effect of correlation
indegree
outdegree
log(NRMSE(corr(0)/NRMSE(corr(.99)))
Q: how does correlation affect efficiency of RW-based estimation
𝜃 𝑖𝑗 𝛼 - in/outdegree distribution under 𝛼-correlation model
focus on
𝑁𝑅𝑀𝑆𝐸( 𝜃 𝑖𝑗 0 ) 𝑁𝑅𝑀𝑆𝐸( 𝜃 𝑖𝑗 )
increased correlation -> increased accuracy on diagonal
indegree = outdegree
log(NRMSE(corr(0)/NRMSE(corr(.99)))
When reciprocity is maximum, the proportion of samples along the diagonal does not change. Hence, log-ratio is zero.

21. Capturing MRV characteristic in all of above
Roadmap
rewiring algorithms to generate arbitrary reciprocity, correlation in real networks
more flexible correlation model (input covariance matrix, possibly different marginals)
evaluation on networks
other network structural properties
Capturing MRV characteristic in all of above