1. 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
Findings
Methods:
Biased Networks
Reachability Curves

2. 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.

3. 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?

4. Longest Possible Path:
Two Hermits on the opposite side of the country OH
Hermit
Store
Owner
Truck
Driver
Manager
About steps.
Corporate
Manager
Corporate
President
Congress
Rep.
Congress
Rep.
Corporate
President
Corporate
Manager
Manager
Truck
Driver
Store
Owner
Mt.
Hermit

5. What if everyone maximized structural holes?
Associates do not know each other:
Results in an exponential growth curve. Reach entire planet quickly.

6. What if people know each other randomly?
Random graph theory shows us that we could reach people quite quickly if ties were random.

7. By number of close friends
Random Reachability:
By number of close friends
20%
40%
60%
80%
100%
Percent Contacted 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Remove
Degree = 4
Degree = 3
Degree = 2

8. 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

9. Distance to target person, by sending group.
Milgram’s Findings:
Distance to target person, by sending group.

10. 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?

11. 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.

12. 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

13. 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.

14. The most clustered graph is Watt’s “Caveman” graph:

15. for a Caveman graph of n=1000
Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
C and L as functions of k
for a Caveman graph of n=1000
0.2
0.4
0.6
0.8 1
1.2 20 40 60 80
100
120
Degree (k)
Clustering Coefficient
140
Characteristic Path Length

16. 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?

17. Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
A model for pair formation, as a function of mutual contacts formations.
Using this equation, a produces networks that range from completely ordered to completely random.

18. Duncan Watts: Networks, Dynamics and the Small-World Phenomenon
C=Large, L is Small =
SW Graphs

19. 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

20. 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

21. 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)

22. By number of close friends
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:
20%
40%
60%
80%
100%
Percent Contacted 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Remove
Degree = 4
Degree = 3
Degree = 2
Random Reachability:
By number of close friends

23. For a random network, we can estimate the trace curves with the following equation:
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.

24. For a real network, people’s friends are not random, but clustered
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.

25. Figure 1. Connectivity Distribution for a large Jr. High School
(Add Health data)
Random graph
Observed