Transcript here

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
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
About 12-13 steps.
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%
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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?
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.
2) The clustering coefficient, C
Is the average local density. That is, Cv =
ego-network density, and C = Cv/n
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
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Clustering Coefficient
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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.
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)
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%
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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
(Add Health data)
P r o p o r tio n R e a c h e d
"Pine Brook Jr. High"
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Observed
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