Contagion Processes
in Complex Networks
Michelle Girvan
University of Maryland
Protein Interaction Networks
Jeong, H., Mason, S., Barabasi, A.-L. & Oltvai, Z. N., ”Lethality and centrality in protein
networks”, Nature, 411, pp 41, 2001.
Wagner, A, “How large protein interaction networks evolve.” Proc. R. Soc. Lond. B 270, 2003. 457-466
The Internet
Human Sexual Contacts
The structure and function of complex networks, M. E. J. Newman, SIAM Review 45, 167-256 (2003)
Contagion of TB
Courtesy of Valdis Krebs, orgnet.com
Status Communication Network
QuickTime™ and a decompre
Flack JC, Girvan M, de Waal FBM, Krakauer DC. 2006. Policing stabilizes construction
of social niches in primates. Nature 439:426-429.
Traditional vs. Complex Systems
Approaches to Networks
Traditional Questions:
Social Networks:
Who is the most
important person in
the network?
Graph Theory:
Does there exist a cycle
through the network that
uses each edge exactly
once?
Complex Systems Questions:
What fraction of edges have to be removed to
disconnect the graph?
What kinds of structures emerge from simple
growth rules?
Areas of Network Research
Structural Complexity
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The wiring diagram could be an intricate tangle, far from perfectly
regular or perfectly random. Example: community structure
The network could include different classes of nodes
The edges could be heterogeneous with different weights, directions
and signs.
Dynamical Complexity
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Dynamics on the network: processes could be taking place on the
fixed network. Examples: disease spread, synchronization
Dynamics of the network: the network itself could be evolving in time.
Airlines
Degree Distributions
Poisson distribution
Exponential Network
Power-law distribution
Scale-free Network
Degree distributions for various networks
(a) World-Wide Web
(b) Coauthorship
networks: computer
science, high energy
physics, condensed
matter physics,
astrophysics
(c) Power grid of the
western United States
and Canada
(d) Social network of 43
Mormons in Utah
Contagion Processes on
Complex Networks
• How does the structure of the network influence the
onset of cascades? degree distributions,
assortativity, clustering
• How do the local rules for interaction change the
effects of network structure?
Ordinary Percolation as a starting point for
understanding contagion in networks
subcritical
critical
supercritical
Percolation on Complex
Networks
bond percolation
Extending Percolation to Contagion
Models
Start with the SIR (susceptible, infected, recovered) model of
epidemic disease. The fraction of individuals in the states S, I,
and R are governed by the equations:
Grassberger (1983) showed that this can be mapped onto a bond percolation
problem with occupation probability T given by the following expression
Calculating the phase transition for networks with
arbitrary degree distributions that are otherwise
random
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Let pk be the probability density
function for the degrees of nodes in
the network.
Let qk be the probability density
function for the degree of a node at
the end of a randomly chosen edge:
qk=k pk/ <k>.
Let f be the occupation probability for
each edge.
Qu ickTim e™ a nd a
TIFF (Un comp resse d) de comp resso r
are need ed to see th is picture.
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Assume that we start with one
node and we want to find the size
of the component connected to
that node.
Let zn be the number of neighbors
reachable in n steps.
zn+1 = zn x the average excess
degree of nodes reachable in n
steps (the expectation value of (k1) over the distribution qk) x the
occupation probability, f
zn+1=f zn(<k2>-<k>)/<k>
Critical value of f:
f=<k>/(<k2>-<k>)
Size of the giant component
Resilience of skewed networks to random
removals
Qui ckTim e™ an d a
TIFF (Uncompressed ) deco mpre ssor
are need ed to s ee thi s pictu re.
Occupation probability ( 1 - probability of node removal)
Size of the giant component as a function of the occupation probability for
three different degree distributions, decreasing skewness from left to right.
D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Network robustness
and fragility: Percolation on random graphs, Phys. Rev. Lett., 85 (2000), pp. 5468–5471.
Assortative Mixing by Degree
• A network is said to be assortatively mixed by degree if high
degree vertices tend to connect to other high degree vertices
• A network is disassortatively mixed by degree if high degree
vertices tend to connect to low degree vertices.
Assortative
Scale-free network
Disassortative
Scale-free network
Measured assortativity for various
networks
Qu ickTim e™ a nd a
TIFF (Un compresse d) decompressor
are need ed to see th is picture.
M.E.J Newman and M. Girvan, Mixing Patterns and
Community Structure in Networks (2002).
How does assortative mixing impact SIR
dynamics?
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In assortatively mixed networks,
the epidemic transition occurs
sooner as the transmissability is
increased. In other words,
epidemics occur at lower
transmissability than in neutral or
disassortative networks.
In parameter regimes where
epidemics readily occur, the
number of infected individuals is
generally lower for assortatively
mixed networks.
The SIRS (Susceptible, Infected,
Recovered, Susceptible) Model
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Rule: An individual who comes in contact with an infected individual gets
infected with probability T if he or she has not been infected in the last t time
steps.
Initial Question: How well does the pathogen persist in the population as the
average connectivity per node (z) is increased? In the SIR model, the
pathogen does better (in terms of larger cascades) when the average
connectivity is increased
Application: Influenza
Two ways to approach modeling:
– Explicitly model pathogen strains and their mutations
– Mean field solution creates a set of difference equations:
Fraction of individuals infected as a function of time, on
a randomly mixed graphs with different connectivity z.
Increasing z tends to lead to pathogen
extinction
• Extreme minimum in oscillations leads to pathogen
extinction
• Pathogen goes extinct more easily in smaller systems
because the fraction of individuals that ‘should’ be infected
falls below 1/N.
How Does Community Structure Impact SIRS Dynamics?
The addition of community structure tends to increase the pathogens ability
to persist because it dampens the oscillations in the number of infecteds.
For very strong community structure, the pathogen’s ability to persist is
actually lessened because it essentially is forced to operate on a smaller
system size.
Limitations of the standard network
approach in contagion modeling
• Percolation looks at a static picture, real networks are
dynamic – example traceroute sampling of the
internet.
• All edges are treated as equal in strength, when they
are probably quite heterogeneous
From Newman 2003
From Bearman, Moody, and Stovel
Consider three different
mechanisms for interactions
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Ordinary imitation: imitate each neighbor
with a given probability. (Traditional SIR
percolation model).
Herd imitation: individuals look to the bulk
behavior of their neighbors. (Threshold
model see Granovetter Am. J. Soc.1978 &
Watts PNAS 2002)
Reciprocal resource-limited imitation:
imitate neighbors based on a “time-sharing”
algorithm for determining connection
strengths.
Casting the mechanisms in
terms of percolation problems
• Ordinary imitation: weight between all pairs is equal.
Wij=1.
Using the tuning parameter f , the probability of each edge being
occupied is f *Wij=f.
• Herd imitation: the influence on node i by node j is given by
Wij=1/kj
• Reciprocal resource-limited imitation: the influence on node i by
node j is given by:
Wij=min(1/ki , 1/kj)
Elucidating Model Differences
• Ordinary Imitation: all links are taken as equal.
• Herd Imitation: neighborhoods are equal. Even with
undirected edges, influence is not reciprocal
• Reciprocal Imitation: nodes are equal, in the sense that
they have equal time to distribute
Effect of degree
distributions
• Ordinary Imitation:
phase transition
happens at lower f for
more highly skewed
degree distributions
• Herd Imitation: phase
transition happens
independent of skew, if
average degree held
constant.
• Reciprocal Imitation:
lower transition for
poisson distribution .
Targeting strategies
• Ordinary Imitation:
targeting high degree
nodes is highly
advantageous
• Herd Imitation: targeting
high degree nodes is
slightly advantageous
• Reciprocal imitation:
targeting high degree
nodes is
disadvantageous
Interpolating between modes
• In herd imitation, the influence of node i on node
j is given by
• In persuasion, the mutual influence btw i and j is
Interpolating:
Extending the Models
• Popularity enhanced imitation.
• Including broadcast modes that may or
may not be connected to network
properties.
• Individuals can have strategies for
splitting their time, and hence
connection strengths to others.
Conclusions
• Network properties like degree distribution,
assortativity, and clustering can strongly
influence contagion processes
• These effects may change significantly with
alteration of the local rules for transmission.
• Percolation with bond strengths dependent
on node properties is good starting point for
exploring these phenomena.
• However, consideration of dynamics is
sometimes necessary for understanding
certain classes of contagion problems.
Collaborators
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Mark Newman, University of Michigan
Steve Strogatz, Cornell University
Duncan Callaway, Cornell University, UC Davis
Lauren Ancel Myers, University of Texas, Austin
Erica Jen, Santa Fe Institue
Jessica Flack, Santa Fe Institute
David Krakauer, Santa Fe Institute
Ole Peters, Santa Fe Institute, UCLA
Shane McCarthy, University of Maryland
Danny Bates, MIT