Social network and disease spread
Laurens Bakker, Philippe Giabbanelli
Outline
▪ What is a social network?
▪ Measures
▪ Disease spread
▪ Three case studies
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What is a social network?
How does it form?
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But really, how does it form?
People go places…
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and meet in the process
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But really, how does it form?
People want things…
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and use others
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But really, how does it form?
People have things in common… and express their commonalities
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What is a social network?
How does it form?
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Fluffy theories
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If we want to do science…
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we need something with teeth!
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Network definition
• Actor => Vertex/Node
– Boundary
• Connection => Edge/Link
– Interaction
– Dynamic social networks
• Observability
– Degree (Dombrowski 2007)
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Measures
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Motifs – Clustering – Average distance – Degree distribution
Global
Degree
distribution
Average
distance
Clustering
Motifs
Local
(Giabbanellli 2011)
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Motifs – Clustering – Average distance – Degree distribution
2
1
1
0
3
0
2
Given a graph G…
a motif is a subgraph
that appears at a ‘very’
different frequence in
G than in S.
0
and a set S of random graphs of
the same size and average degree,
(Milo 2004)
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Motifs – Clustering – Average distance – Degree distribution
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Motifs – Clustering – Average distance – Degree distribution
For a given node i , we denote its neighborhood by Ni.
The clustering coefficient Ci of i is the edge density of its neighborhood.
Here, there are two edges
between nodes in Ni.
Ci = 2.2/(5.4) = 0.2
If a graph
has ithigh
clustering
coefficient,
then
there
arei-1)
communities
At most,
would
be a complete
graph
with
Ni.(N
edges.
(i.e., cliques) in this graph.
People tend to form communities so they are common in social networks.
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Motifs – Clustering – Average distance – Degree distribution
The distance is the number of edges to go from one node to another.
The average distance is the average of
the distance between all pairs of nodes.
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Motifs – Clustering – Average distance – Degree distribution
The average
distance l is:
∙ small if
l∝ln(n)
∙ ultrasmall if
l∝ln(ln(n))
(Newman 2003)
(Cohen 2003)
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Motifs – Clustering – Average distance – Degree distribution
History (the Hype)
• Milgram (Milgram 1969)
– Small world
• Watts & Strogatz (Watts 1998)
– “Small Worlds” & “6 Degrees”
• Barabasi & Albert (Barabasi 1999)
– Power Law (scale free)
• Newman (Newman 2003)
– Review
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Motifs – Clustering – Average distance – Degree distribution
Many measured phenomena are centered around a particular value.
(Newman 2005)
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Motifs – Clustering – Average distance – Degree distribution
Many measured phenomena are centered around a particular value.
There also exists numerous phenomena with a heavy-tailed distribution.
lets plot it on a
log-log scale
(Newman 2005)
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Motifs – Clustering – Average distance – Degree distribution
We also
say that
thisnumerous
distribution
follows awith
power-law,
with exponent
α.
There
exists
phenomena
a heavy-tailed
distribution.
The equation of a line is p(x) = -αx + c.
Here we have a line on a log-log scale:
ln p(x) = -α ln x + c
apply exponent e
c -α
p(x) = ecx
(Newman 2005)
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Motifs – Clustering – Average distance – Degree distribution
We say that this distribution follows a power-law, with exponent α.
computer files
people’s incomes
Keep in mind that this is quite common.
moon craters
visits on web pages
(Li 2005)
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Disease spread
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Thresholds – Variations – Immunization
A ‘threshold’ is the extent to which a disease must be infectious before
you can’t stop it from spreading in the population.
Very famous claim: scale-free networks have no thresholds! It will spread!
(Wikipédia: modèles compartimentaux en épidémiologie)
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Thresholds – Variations – Immunization
Very famous claim: scale-free networks have no thresholds! It will spread!
« in a scale-free network there is no epidemic
threshold thus eliminating a sexually
transmissible disease is impossible »
(Kretschmar 2007, opening of Networks in Epidemiology)
That’s actually sort of false…
…it needs additional conditions, that may not exist.
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Thresholds – Variations – Immunization
Depending on the diseases, there are several epidemiological classes:
infected (I), recovered (R), carriers (C)…
It may be interesting to see how the properties of the network influences
the number of individuals in each class over time.
order
randomness
(Kuperman 2001; Crepey 2006)
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Thresholds – Variations – Immunization
There are four broad approaches (Giabbanelli 2011).
Is the disease spreading at the same time?
Yes
We can immunize anybody
We must follow social links
Global competitive Global preventive
= network game
= separator problem
NP-hard
NP-complete
(Kostka 2008)
(Rosenberg 2001)
Local competitive
Local preventive
Agents that fight…
…and explore
(Giabbanelli 2009)
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No
(Stauffer 2006)
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Case Study #1
Measuring what matters
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Example #1: Social networks
Measure: distance
Property: average distance
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Example #2:
Obesity map
Measure: Centrality
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Example #3: Backbone network
We do not care about clustering or whether the network is scale-free.
(Giabbanelli 2010)
Measure betweenness and average distance.
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What can we measure in a network?
Network
Process
Measures
Social
network
Disease spread
Average distance
Factors incluencing
obesity
Obesity level
Centrality
Backbone network
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Deploying equipment
Betweenness centrality
Average distance
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How do we find out what we should measure?
▪ Know the properties of the network you are studying.
→ Network analysis
▪ Generate many of them using appropriate stochastic models.
→ Network generation
▪ Record several measures, and the value of the outcome process.
→ Possibly optimization
▪ Analyze which measures are linked to the outcome.
→ Data mining
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Case Study #2
Health & Social
Networks
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« People are interconnected, and so their health is interconnected. »
« … there has been growing conceptual and empirical attention over the
past decade to the impact of social networks on health. »
(Smith 2008)
Christakis&Fowler have used social networks to show that people are
correlated in weight status, smoking, and… happiness!
http://www.ted.com/talks/lang/eng/nicholas_christakis_the_hidden_influence_of_social_networks.html
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The basic idea
A long imbalance between energy intake&output yields obesity.
What spread between people are behaviours impacting intake&output.
Eating
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Exercising
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How we modelled it
We used social networks.
Each individual has a
level of physical activity
and an energy intake.
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How we modelled it
We also modelled
human metabolism.
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Results from Phase 1
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Results from Phase 1
Presented at ICO
8.6% acceptance
Positive reactions
Journal on its way
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Case Study #3
Homeless in the tri-cities
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Homeless in the Tri-Cities (I)
• Hope for Freedom Society
• Vertex definition
– Boundary: existence of client file
• Edge definition
– Interaction: co-observation
• Time!
– Connection: repeated interaction
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Homeless in the Tri-Cities (II)
• Descriptives:
– 2 years
– ~250 actors
– ~3000 observations
• Statistical Models
– Static: PNET = ERGM = logit p* (Hunter 2006)
– Dynamic: SIENA (Snijders 2006)
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References
Barabasi 1999
AL Barabasi, R Albert, Emergence of
Scaling in Random Networks, Science, 1999
Cohen 2003
R Cohen, S Havlin, Scale-free networks are
ultrasmall, Physical Review Letters, 2003.
Crepey 2006
P Crepey et al, Epidemic variability in complex
networks, Phys. Rev. E, 2006.
Drombrowski 2007
K Dombrowski, R Curtis, SR Friedman,
Injecting drug user network topologies and
infectious disease tranmission: suggestive
findings, Working Paper 2007
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References
Giabbanelli 2009
PJ Giabbanelli, Self-improving immunization
policies for complex networks, MSc
Thesis@SFU, 2009
Giabbanelli 2010
PJ Giabbanelli, Impact of complex network
properties on routing in backbone networks,
CCNet 2010 (IEEE Globecom)
Giabbanelli 2011
PJ Giabbanelli, JG Peter, Complex networks
and epidemics, TSI, 2011, to appear.
Hunter 2006
D Hunter, Exponential Random Graph Models
for Network Data, Talk, 2006,
http://www.stat.psu.edu/~dhunter/talks/ergm.pdf
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References
Kostka 2008
J Kostka et al., Word of Mouth : Rumor
Dissemination in Social Networks, Lecture
Notes in Computer Science, 2008.
Kretzschmar 2007
M Kretzschmar, J Wallinga, Networks in
Epidemiology, Mathematical Population Studies,
2007
Kuperman 2001
M Kuperman, G Abramson, Small World
Effect in an Epidemiological Model, Physical
Review Letters, 2001.
Li 2005
L Li et al., Towards a Theory of Scale-Free
Graphs : Definition, Properties and
Implications, Internet Mathematics, 2005.
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References
Milgram 1969
J Travers, S Milgram, An Experimental Study of
the Small World Problem, Sociometry, 1969
Milo 2004
R Milo, et al., Superfamilies of Evolved and
Designed Networks, Science, 2004.
Newman 2003
MEJ Newman, The structure and function of
complex networks, SIAM Review, 2003.
Newman 2005
MEJ Newman, Power laws, Pareto distributions
and Zipf’s law, Contemporary Physics, 2005.
Rosenberg 2001
AL Rosenberg, Graph Separators, with
Applications, Kluwer Academic, 2001
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References
Smith 2008
KP Smith, NA Christakis, Social networks and
health, Annu Rev Social, 2008
Snijders 2006
TAB Snijders, Statistical Methods for Network
Dynamics, Proceedings of the XLIII Scientific
Meeting of the Italian Statistical Society, 2006
Stauffer 2006
AO Stauffer et al, A dissemination strategy for
immunizing scale-free networks, Phys. Rev. E,
2006.
Watts 1998
DJ Watts, SH Strogatz, Collective dynamics of
'small-world' networks, Nature, 1998
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