Improving Acquaintance Immunization via
Strategic Synchronous Snowball Sampling
Samuel F. Rosenblatt1,2, Jeffrey A. Smith3, G. Robin Gauthier3, & Laurent Hébert-Dufresne1,2
• Acquaintance Immunization: Immunize
random neighbors of random nodes [1]
• Leverages the “friendship paradox” to select
node proportional to their degree, leading to
selection of high degree nodes with no prior
network data [2]
1 Department of Computer Science, University of Vermont, 2 Vermont Complex Systems Center, University of Vermont, 3 Department of Sociology, University of Nebraska-Lincoln
Correspondence:
[email protected]
0.0 0.5 1.0 1.5 2.0
0
25
50
% Outbreak Reduction
vs. Classic Acquaintance
Hamilton-Like Networks - Assortativity = 0.429
Avg Outbreak Size
w/ Classic Acquaintance
(% of Network)
0.0 0.5 1.0 1.5 2.0
0
25
50
% Outbreak Reduction
vs. Classic Acquaintance
Bowdoin-Like Networks - Assortativity = 0.261
Avg Outbreak Size
w/ Classic Acquaintance
(% of Network)
0.0 0.5 1.0 1.5 2.0
0
25
50
% Outbreak Reduction
vs. Classic Acquaintance
Haverford-Like Networks - Assortativity = 0.257
Avg Outbreak Size
w/ Classic Acquaintance
(% of Network)
0.0 0.5 1.0 1.5 2.0
0
25
50
% Outbreak Reduction
vs. Classic Acquaintance
Amherst-Like Networks - Assortativity = 0.253
Avg Outbreak Size
w/ Classic Acquaintance
(% of Network)
0.0 0.5 1.0 1.5 2.0
0
25
50
% Outbreak Reduction
vs. Classic Acquaintance
USFCA-Like Networks - Assortativity = 0.251
Avg Outbreak Size
w/ Classic Acquaintance
(% of Network)
0.0 0.5 1.0 1.5 2.0
0
25
50
% Outbreak Reduction
vs. Classic Acquaintance
Swarthmore-Like Networks - Assortativity = 0.180
Avg Outbreak Size
w/ Classic Acquaintance
(% of Network)
0.0 0.5 1.0 1.5 2.0
0
25
50
% Outbreak Reduction
vs. Classic Acquaintance
Simmons-Like Networks - Assortativity = 0.124
Avg Outbreak Size
w/ Classic Acquaintance
(% of Network)
• In assortative networks, neighbors of neighbors
of random nodes can have even higher degree
than random neighbors because of both the
friendship paradox and assortativity
• Thus, beginning with a small subsample and
adding any (previously unknown) neighbors of
nodes encountered during the acquaintance
immunization process to the sample leads to a
positive degree bias in the sampling frame
• Forming the sample can be done synchronously
with immunization for rapid response and
efficient resource allocation
• This leverages work already necessary in classic
acquaintance immunization yet yields an
immunized portion with a much higher average
degree
• Unlike acquaintance immunization, this
method requires no complete sampling frame
(census) of the population prior to beginning [3]
References:
1. Cohen, Havlin, and Ben-Avraham,
Physical Review Letters (2003).
2. Christakis and Fowler, PLoS ONE
(2010).
Concurrent Sampling Process
1. Start with a small percent of all nodes as
seeds.
2. Interview a random node in your sample
and ask them to list all* their contacts.
3. If any named contacts are unknown to your
sample, add them.
4. Immunize a random one of the nodes given
by the node in step 2.
5. Repeat steps 2-4 till you reach your desired
immunization level.
*Alternative variant tested as well with asking for 1 contact, with
interesting, but more nuanced, results
• Used the CCM model from [4] to create 1000
random networks for each of our seed networks
• Hard constraint on degree distribution seed
network, soft constraint on degree-degree
correlation matrix, which allows us to create
random networks with positive assortativity
• Used networks between 1K and 2K nodes from
FB100 dataset, plus PGP network
Synthetic Networks
The increased degree of nodes selected for
immunization leads to substantially lower outbreak
sizes. In the figures above and to the right, we
measure the percentage difference between classic
acquaintance immunization and our method over a
range of infection rates, β, with fixed to 1.
(B)
(A)
0 500 1000 1500 2000
Position in Immunization Sequence
8
10
12
14
16
18
20
22
Degree
Average Degree of the xth Immunized Node:
0.1 % Starting Sample
Classic Acquaintance
Updating Sample Acquaintance
(B)
100 101 102
Degree, k
10 5
10 4
10 3
10 2
10 1
100
CCDF
Degree Distributions
PGP-like Networks
All nodes:
< k >=4.48
Classic Acquaintance
immunized nodes:
< k >=11.4
(A)
3. Rosenblatt et al., PLoS
Computational Bio. (2020).
4. Newman, Physical review letters.
(2002).
2 4 6 8
0
20
40
60
% Outbreak Reduction
vs. Classic Acquaintance
PGP-Like Networks - Assortativity = 0.238
% Reduction in outbreak size
Avg Outbreak Size
w/ Classic Acquaintance
(% of Network)