SOST71032 Social Network Analysis
Conditional UniformGlobalisation and Corporate Governance Assignment
Graph Distributions
Dr Termeh Shafie
Department of Social Statistics
School of Social Sciences
The University of Manchester
Day 4
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hypothesis testing
null hypothesis
H0: observed network is created from specified model that does X
alternative hypothesis
H1: observed network is not created from specified model that does X
if H0 is true we expect to see networks like those we simulate from model so:
decision rule
if simulated networks look like the observed in only a(100%) of the cases
we would reject H0 on the a(100%) significance level
otherwise, we cannot reject the H0
observed network
distribution of measure under null model
summary measure
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creating a null model
create a null model (H0) to which we can test our observed network
H0 corresponds to a world of hypothetical networks
distribution of chosen statistic under H0 is generated by simulations
we can check where the observed count falls in this distribution
does the observed value dier significantly from the expected?
if yes =) conclude there is a social phenomenon at play =) reject H0
observed network
distribution of measure under null model
summary measure
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conditional uniform graph distributions
creating the null distribution:
tests based on shuling edges randomly while fixing
I the number of edges/density: UjL or UjE(L)
I the degree distribution: Ujd
I number of null, asymmetric and mutual dyads: UjMAN
I
. . .or some other summary measure
statistical inference
relies on the assumption that there is some randomness in the data
we need to model this randomness
example.
a uniform distribution conditional on observed network’s number of edges:
I graphs with specified number of edges are equally probable to appear
I graphs without specified number of edges have a probability of zero to appear
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examples: conditional uniform graph distributions
the Coleman data
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coleman example: UjE(L)
uniform graph distribution given expected density
I calculate the density of the Coleman fall network ≈ 0.046
I generate one random graph with the same density on average
as the observed fall network
the observed and random network may not have the exact same number of edges
but stochastically, it has the same density
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coleman example: UjL
uniform graph distribution given number of edges
I calculate the number of edges in the Coleman fall network ≈ 243
I generate one random graph with the exact same number of edges
as the observed one
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coleman example: UjL
compare the out-degree distribution for observed to random network
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coleman example: UjL
in-degree out-degree assortativity for observed and random network
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coleman example: UjL
compare dyad census for observed to random network
even though the random network has the same density as the observed,
we have a completely dierent count of reciprocated ties
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coleman example: UjMAN
uniform graph distribution given dyad census (MAN)
I generate a random graph with the exact same number of mutual,
asymmetric and null dyads as the observed network
I observed: mutual = 62, asymmetric = 119, null = 2447
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coleman example: UjMAN
compare triad census for observed to random network
interpretation:
had allocation of ties in the network been completely random given ’dyadic
processes’, it would be unlikely to observe any complete triangles
but so far we have only looked at one random graph…
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the world of hypothetical networks
coleman example: UjL
one random network with the exact same density as the observed network
had a very dierent count of reciprocated ties
I was this just a coincidence?
I are most random networks dierent in this way?
to answer this we need to generate more random networks given density
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coleman example: UjL
to see just how unusual mutual ties are in the alternative world
I generate 1000 random given observed number of ties
I plot distribution of mutual ties for each of the 1000 random networks
do any of the 1000 random networks have
as large a number of mutual dyads as the observed?
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coleman example: UjMAN
compare transitivity for observed to 1000 random networks
H0: observed transitivity eect is created from UjMAN
H1: observed transitivity eect is not created from UjMAN
distribution of number of transitive triads under H0 is generated by
simulating 1000 random networks with
the same number of dierent dyads as the observed one
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a cautionary tale
subgraphs are necessarily nested in each other
and if we do not control for one when we count the other,
our results may be confounded
more on this in the coming lectures…
the R codes for replicating the coleman examples
in this lecture will be made available on blackboard
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