Networks and all that

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Networks and all that Brian Williams
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Prevalence of HIV in adults () North
India 0.2 Brazil 0.7 Western Cape Coloured 1.2
? 0.5 South India 1.5 West Africa 3 East
Africa 5 South Africa 21 Western Cape Black 22
? 3 (Median values in Africa)
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Fisher (1937) Diffusion of genes The Fisher
Equation Spread depends on diffusion and
logistic growth. The rate of spread, v is
Distance between neighbours, d, is 1km R0 ? 10
L ? 10 years HIV would spread at a rate of 260
m/year
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Fisher again
Time
Age
Births
Deaths
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Fisher in space
d dist. between partners L life expectancy
(10 years) Fisher with age
? standard deviation of the age of sexual
partners. Spreads out by c years of age for each
calendar year of time. c gt 1 ?? ?? gt 3 years
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If we wish to create an epidemic of HIV we must
have some degree of migration and some degree of
inter-generational sex. The question is how
much?
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Spatial Epidemiology of HIV in a small world
Doubling time 1 year Life expectancy 10 yrs
Number of partners 4 size of population
16k Left
Right Sex with neighbours 90 of sex
with only
neighbours
10 with people
chosen at random
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Without connections across space and across ages
HIV cannot persist. Since HIV does persist, what
is the minimal level of connectedness needed to
sustain an epidemic? Does this help us to
explain different endemic levels of HIV and to
find ways to control HIV?
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Notes and queries based on The structure and
dynamics of networks Mark Newman, Albert-Lázló
Barabási and Duncan Watts
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1951 and 1960
Network of V vertices linked by E edges Degree of
a vertex d number of edges attached Average
degree ?d ? 2E/V Connect the vertices randomly
with equal probability
Branching processes
Mean-field theory applies
Giant component
Isolated islands
R0 lt 1
R0 gt 1
x
Solomonoff Rapoport (1951)

Erdos Renyi (1960) p. 182
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For uniform random graphs the degree distribution
is Poisson On the web the mean degree
distribution is 11.2
Log10 P(d)
Log10(d)
Degree distribution of the world-wide-web
p. 182
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The network structure of the Internet at the
level of autonomous systemslocal groups of
computers each representing hundreds or thousands
of machines.
Newman, M. Society for Industrial and Applied
Mathematics (2003) 45, 167256
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Log10 P(d)
Log10(d)
Degree distribution of the world-wide-web
Albert, Jeong Barabsasi 1999 p. 182
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Giant component present No
giant component
Sizes of non-giant component constant
ln(V)
India? Sweden?
Africa?
Mean degree Variance of degree ?d ? ?
Var?d ? ?
0 1
2 3.48
Power k
Properties of power-law networks as a function of
the power
Aiello, Chung Lu (2000). p. 237
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Giant component in the Web
Strongly connected or giant component 56M nodes
Out 44M nodes
In 44M nodes
Tendrils, tubes and disconnected components 64M
Broder et al. (2000) p. 318
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Power k 2.3
Log10P(d)
?
1 2 4
8 10 15 20
No. sexual partners in the last month, d
Liljeros et al. (2001) p. 227
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• Footnote
• To generate a random network with any degree
  distribution
• Specify the degree distribution
• 2. Set up V vertices with V?P(1) having one stub,
• V?P(2) having two stubs etc. where P(i) is
  the
• degree distribution.
• 3. Connect stubs randomly with equal probability.

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Why are power-law or scale-free networks
ubiquitous? Add vertices and then connect them
to existing vertices with probability
proportional to the degree of each existing
vertex. But the exponent is then always equal
to 3. Try Then a gt 1 one node has very high
degree, the rest have an exponential degree
distribution. a lt 1 all have an exponential
distribution. More complex growing techniques.
1. Add edges, rewire edges, add vertices,
repeat. 2. Give each vertex an intrinsic
fitness p.
343 3. Gene duplication model Copy an
existing vertex with its edges Delete some edges
(deleterious mutations) Add some edges
(favourable mutations)

p. 347
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What can we say about ALL random graphs whatever
the degree distribution? where l is the
average distance between pairs of vertices, V is
the number of vertices, ?d1? is the average
degree or the average number of nearest
neighbours and ?d2? is the average number of
second-nearest neighbours l always scales as
ln(V) (for an ordered graph as Vd) and to
calculate l (which is the key global property)
you only need ?d1? and ?d2? which are local
properties. Graphs with very different degree
distributions but with similar values of ?d1? and
?d2? will have similar properties. For the
Poisson random graph ?d2? ?d1?2 l
ln(V)/ln(d). We need to know about the partners
of partners!
p. 276
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Combining Fisher and Erdos Small worlds 1.
Everyone is connected to everyone else by a
rather short path (random) 2. Your friends tend
to know each other (lattice)
Mixture of a random graph and an ordered
structured lattice.
Problem Sex with your neighbours only ?
linear spread (Fisher) Sex with partners chosen
at random ? exponential spread (SI
models) Exponentials always win you just have to
wait long enough
Everyone here is connected to the Mexican painter
Diego Riviera by not more than 9 people
All of you know someone who has met Mandela, who
spoke to Clinton on the phone, who knew Nixon,
who shook hands with Mao Zedong, who was a buddy
of Stalin, who killed Trotsky, who had an affair
with Frida Kahlo, who was married to Diego
Riviera.
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Small worlds are ubiquitous
Define a clustering co-efficient C as the ratio
of triangles to triples
For a random graph we can show that C ?d ? /V
which is generally very small.
p. 289
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Average value of vertex-vertex distances and
clustering coefficient scaled to the maximum value
One dimensional small world model with V 1000
k 10
Newman (2003) p. 291
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Migration?
Small worlds of migration 1?
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Network with a predetermined power-law degree
distribution and a probability of being connected
that declines with distance?
Small worlds of migration 2?
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Can we invert the problem? Instead of drawing
all possible vertices and then connecting them
with edges why not draw all possible edges and
then decide which ones to activate?
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Women
Men
Men
Men
A
ATA
Adjacency matrix Aij 1 if Vi and Vj are linked
otherwise zero
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Women Men
Iterated
Men and women each have sex with 1 regular
partner and other partners with probability p
0.02 p 0.05 p 0.20
Men Men
Men
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1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
14 pairs 16 triples 6 triangles C
6/16 0.4 CRG 2/6 0.3
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Plot of ordered eigenvalues
Eigenvalue (density) spectrum
Poisson random graph Ordered 1-D lattice

• Eigenvalues of the adjacency matrix
• Lattice with k 8, V 128 Random graph with p
  8/128, V 128

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Poisson network
Power-law network
Proportion immunized
Proportion immunized
Steady state prevalence for an SIS epidemic model
in a small world with a) a Poisson degree
distribution for large d and b) a power law
degree distribution. People are either immunized
at random or from those with the highest to those
with the lowest degree. Pastor-Satorras R,
Vespignani A. Immunization of complex networks
(2004)
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SIR models can be solved analytically on any
network with degree distribution i.e. a
power-law with an exponential cut-off.
Newman M. Spread of epidemic disease on networks
Phys. Rev. E, 66, 016128, (2002)
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For a heterosexual epidemic we have two connected
layers of men and women with possibly quite
different degree distributions. What are the
consequences, if any, for the network structure
and the spread of disease? Can Carletonville
really be explained by the few percent of miners
who have more than 20 partners a month? Perhaps
it is all to do with small worlds and big worlds
i.e. migration? How important is it to target
high risk individuals for vaccination or
circumcision campaigns? How important is it to
consider the strength of various links? I can
find no references to this in the literature. For
men at least the more partners you have the less
sex you are likely to have with each one so that
if we add edges we need to reduce the strength of
each edge. Can we use the Masi and RU data to
collect empirical data on TB and HIV
networks? If we knew the distribution of
second-nearest-neighbour contacts would this be
enough? Can we use this to explain the
difference in prevalence in M an RU? Can we
investigate the network structure of HIV
sub-types in South Africa? What about strain
variation of HIV in infected individuals? How do
we use this in relation to micro-simulation
models? Can we use the RU data with
fingerprinting to construct contact networks?