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```Connectivity and the Small World
Overview
Background:
•de Pool and Kochen:
•Random & Biased networks
•Rapoport’s work on diffusion
Travers and Milgram
•Argument
•Method
•Watts
•Argument
•Findings
•Methods:
•Biased Networks
•Reachability Curves
•Calculating L and C
Connectivity and the Small World
Started by asking the probability than any two people
would know each other.
Extended to the probability that people could be
connected through paths of 2, 3,…,k steps
Linked to diffusion processes: If people can reach
others, then their diseases can reach them as well, and
we can use the structure of the network to model the
disease.
The reachability structure was captured by comparing
curves with a random network, which we will do later
today.
Connectivity and the Small World
Travers and Milgram’s work on the small world is
responsible for the standard belief that “everyone is
connected by a chain of about 6 steps.”
Two questions:
Given what we know about networks, what is the longest
path (defined by handshakes) that separates any two
people?
Is 6 steps a long distance or a short distance?
Longest Possible Path:
OH
Hermit
Store
Owner
Truck
Driver
Two Hermits on the opposite side of the country
Manager
Corporate
Manager
Corporate
President
Congress
Rep.
Congress
Rep.
Corporate
President
Corporate
Manager
Manager
Truck
Driver
Store
Owner
Mt.
Hermit
What if everyone maximized
structural holes?
Associates do not know each other:
Results in an exponential growth
curve. Reach entire planet quickly.
What if people know each
other randomly?
Random graph theory shows us that we could
reach people quite quickly if ties were random.
Random Reachability:
By number of close friends
100%
Degree = 4
Degree = 3
Percent Contacted
80%
Degree = 2
60%
40%
20%
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Milgram’s test: Send a packet from sets of randomly selected
people to a stockbroker in Boston.
Experimental Setup:
Arbitrarily select people from 3 pools:
a) People in Boston
b) Random in Nebraska
c) Stockholders in Nebraska
Milgram’s Findings:
Distance to target person, by sending group.
Most chains found their
way through a small
number of
intermediaries.
What do these two findings
tell us of the global
structure of social relations?
Milgram’s Findings:
Length of completed chains
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
Asks why we see the small world pattern and what implications
it has for the dynamical properties of social systems.
His contribution is to show that globally significant changes can
result from locally insignificant network change.
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
Watts says there are 4 conditions that make the small world
phenomenon interesting:
1) The network is large - O(Billions)
2) The network is sparse - people are connected to a small
fraction of the total network
3) The network is decentralized -- no single (or small #) of
stars
4) The network is highly clustered -- most friendship circles
are overlapping
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
Formally, we can characterize a graph through 2 statistics.
1) The characteristic path length, L
The average length of the shortest paths
connecting any two actors.
(note this only works for connected graphs)
2) The clustering coefficient, C
•Version 1: the average local density. That
is, Cv = ego-network density, and C = Cv/n
•Version 2: transitivity ratio. Number of
closed triads divided by the number of
closed and open triads.
A small world graph is any graph with a
relatively small L and a relatively large C.
The most clustered graph
is Watt’s “Caveman”
graph:
Ccaveman
6
 1 2
k 1
Lcaveman
n

2(k  1)
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
C and L as functions of k
for a Caveman graph of n=1000
1.2
140
Clustering Coefficient
100
0.8
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0.6
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0.4
40
0.2
20
0
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60
Degree (k)
80
100
0
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Characteristic Path Length
120
1
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
Compared to random graphs, C is large and L is long. The
intuition, then, is that clustered graphs tend to have (relatively)
long characteristic path lengths. But the small world
phenomenon rests on just the opposite: high clustering and
short path distances. How is this so?
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
A model for pair formation, as a function of mutual contacts
formations.
1
mi , j  k

 m 
Ri , j   ki , j (1  p)  p k  mi , j  0,

p
mi , j  0

 
Using this equation,  produces networks that range from
completely ordered to completely random. (Mij is the number
of friends in common, p is a baseline probability of a tie, and k
is the average degree of the graph)
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
A model for pair formation, as a function of mutual contacts
formations.
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
C=Large, L is Small =
SW Graphs
Why does this work? Key is fraction of shortcuts in the network
In a highly clustered, ordered
network, a single random
connection will create a shortcut
that lowers L dramatically
Watts demonstrates that
Small world graphs occur
in graphs with a small
number of shortcuts
Empirical Examples
1) Movie network: Actors through Movies
Lo/Lr= 1.22 Co/Cr = 2925
2) Western Power Grid:
Lo/Lr= 1.50 Co/Cr = 16
3) C. elegans
Lo/Lr= 1.17
Co/Cr = 5.6
What are the substantive implications?
Return to the initial interest in connectivity: disease diffusion
1) Diseases move more slowly in highly clustered graphs
(fig. 11) - not a new finding.
2) The dynamics are very non-linear -- with no clear pattern
based on local connectivity. Implication: small local
changes (shortcuts) can have dramatic global outcomes
(disease diffusion)
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
How do we know if an observed graph fits the SW model?
Random expectations:
For basic one-mode networks (such as acquaintance
nets), we can get approximate random values for L and C as:
Lrandom ~ ln(n) / ln(k)
Crandom ~ k / n
As k and n get large.
Note that C essentially approaches zero as N increases, and K
is assumed fixed. This formula uses the density-based measure
of C.
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
How do we know if an observed graph fits the SW model?
One problem with using the simple formulas for most extant
data on large graphs is that, because the data result from people
overlapping in groups/movies/publications, necessary
clustering results from the assignment to groups.
G1
G2 G3 G4 G5
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0 1 0
0
1 0 1
0
1 0 1
1
0 0 0
1
0 1 0
0
1 0 1
0
1 0 1
. . . . LINES CUT . . . . .
William
0
1 0 0
Xavier
0
1 0 1
Yolanda
1
0 1 0
Zanfir
0
1 1 1
12 14 9 14
Amy
Billy
Charlie
Debbie
Elaine
Frank
George
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Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
How do we know if an observed graph fits the SW model?
Newman, M. E. J.; Strogatz, S. J., and Watts, D. J.
“Random Graphs with arbitrary degree distributions and
their applications” Phys. Rev. E. 2001
This paper extends the formulas for expected clustering and path
length using a generating functions approach, making it possible
to calculate E(C,L) for graphs with any degree distribution.
Importantly, this procedure also makes it possible to account for
clustering in a two-mode graph caused by the distribution of
assignment to groups.
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
How do we know if an observed graph fits the SW model?
Newman, M. E. J.; Strogatz, S. J., and Watts, D. J. “Random Graphs with
arbitrary degree distributions and their applications” Phys. Rev. E. 2001
log( N / z1 )
l
log( z 2 / z1 )
Where N is the size of the graph, Z1 is the average number of people 1 step away
(degree) and z2 is the average number of people 2 steps away.
Theoretically, these formulas can be used to calculate many properties of the network
– including largest component size, based on degree distributions.
A word of warning: The math in these papers is not simple, sharpen your calculus
pencil before reading the paper…
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
How do we know if an observed graph fits the SW model?
Since C is just the transitivity ratio, there are a number of good
formulas for calculating the expected value. Using the ratio of
complete to (incomplete + complete) triads, we can use the
expected values from the triad distribution in PAJEK for a
simple graph or we can use the expected value conditional on the
dyad types (if we have directed data) using the formulas in
SPAN and W&F.
The line of work most closely related to the small world is that on
biased and random networks. Recall the reachability curves in a
random graph:
Random Reachability:
By number of close friends
Percent Contacted
100%
Degree = 4
Degree = 3
80%
Degree = 2
60%
40%
20%
0
0
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Remove
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For a random network, we can estimate the trace curves with
the following equation:
Pi 1  (1  X i )(1  e
 api
)
Where Pi is the proportion of the population newly
contacted at step i, Xi is the cumulative number contacted
by step i, and a is the mean number of contacts people
have.
This model describes the reach curves for a random
network. The model is based on a, which (essentially)
tells us how many new people we will reach from the new
people we just contacted. This is based on the assumption
that people’s friends know each other at a simple random
rate.
For a real network, people’s friends are not random, but clustered.
We can modify the random equation by adjusting a, such that some
portion of the contacts are random, the rest not. This adjustment is
a ‘bias’ - I.e. a non-random element in the model -- that gives rise to
the notion of ‘biased networks’. People have studied
(mathematically) biases associated with:
•Race (and categorical homophily more generally)
•Transitivity (Friends of friends are friends)
•Reciprocity (i--> j, j--> i)
There is still a great deal of work to be done in this area empirically,
and it promises to be a good way of studying the structure of very
large networks.
Figure 1. Connectivity Distribution for a large Jr. High School
P r o p o r tio n R e a c h e d
"Pine Brook Jr. High"
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Random graph
Observed
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How useful are C & L for characterizing a network?
These two graphs both have high C
To calculate Average Path Length and Clustering in
UCINET
Clustering Coefficient
1)
2)
3)
To keep w. Watts, make the network symmetric
•
Transform > Symmetrasize > Maximum
•
Note what you saved the graph as
Calculate clustering coefficient
•
Network > Network Properties > Clustering Coefficient
•
The local density version is the “overall clustering coefficient”
•
The transitivity version is the “weighted clustering coefficient”
To calculate Average Path Length and Clustering in
UCINET
Average Length
1)
2)
3)