1. CS 124/LINGUIST 180 From Languages to Information
Christopher Manning
Lecture 19: (Social) Networks:
Fat Tails, Small Worlds, and Weak Ties
Slides from Dan Jurafsky who mainly got them from Lada Adanic; some also from James Moody, Bing Liu, James Moody

2. Networks We’ve looked a lot at information in language
We’ve seen that you can also get information from the patterns of pages across the Web … PageRank
But there are also patterns of people who interact
Maybe it’s also useful to study and understand that?
Many goals:
Sociology …
Better ad targeting
Slide from Chris Manning

3. Social network analysis
Social network analysis is the study of entities (people in an organization), and their interactions and relationships.
The interactions and relationships can be represented with a network or graph,
each vertex (or node) represents an actor and
each link represents a relationship. May be directed or not.
CS583, Bing Liu, UIC
Nov 5, 2009

4. Centrality
Important or prominent actors are those that are involved with other actors extensively.
A person with extensive contacts or communications with many other people in the organization is considered more important
The contacts can also be called links or ties.
A central actor is one involved in many ties.
Degree centrality: The number of direct connections a node has
What really matters is where those connections lead to and how they connect the otherwise unconnected.
CS583, Bing Liu, UIC

5. Prestige
Prestige is a more refined measure of prominence of an actor than centrality.
Distinguish: ties sent (out-links) and ties received (in-links).
A prestigious actor is one who is the object of extensive ties as a recipient.
To compute the prestige: we use only in-links.
PageRank is based on prestige
CS583, Bing Liu, UIC

6. Social Network Analysis
2. Betweenness Centrality:
A node with high betweenness has great influence over what flows in the network indicating important links and single point of failure.
3. Closeness Centrality:
The measure of nodes which are close to everyone else.
The pattern of direct and indirect ties allows the nodes to get to any other node in the network more quickly than anyone else.
They have the shortest paths to all others.
from Amit Sharma UT Austin

7. Outline
Sketch of some real networks
Fat Tails
Small Worlds
Weak Ties

8. High school dating Peter S. Bearman, James Moody and Katherine Stovel
Chains of affection: The structure of
adolescent romantic and sexual networks
American Journal of Sociology 110
44-91 (2004)
Image drawn by Mark Newman
Slide from Drago Radev

10. More networks

11. Europe/Middle East/Central Asia/Africa – Green
5 million edges
Graph Colors:
Asia Pacific – Red
Europe/Middle East/Central Asia/Africa – Green
North America – Blue
Latin American and Caribbean – Yellow
RFC1918 IP Addresses – Cyan
Unknown - White
Slide from Drago Radev

12. The Harlem Shake: Anatomy of a Viral Meme by Gilad Lotan (SocialFlow)

13. Slide from Chris Manning

14. Degree of nodes Many nodes on the internet have low degree
One or two connections
A few (hubs) have very high degree
The number P(k) of nodes with degree k follows a power law:
Where alpha for the internet is about 2.1

15. What is a heavy tailed-distribution?
Unlike a normal distribution, power-law distributions have no “scale” [I’ll explain that later…]
Right skew
normal distribution (not heavy tailed)
Zipf’s or power-law distribution (heavy tailed)
High ratio of max to min
human heights vs.
city sizes
Slide from Lada Adamic

16. Normal (Gaussian) distribution of human heights
average value close to
most typical
distribution close to
symmetric around
average value
Slide from Lada Adamic

17. Power-law distribution
linear scale
log-log scale
high skew (asymmetry)
straight line on a log-log plot
Slide from Lada Adamic

18. Logarithmic axes 20=1, 21=2, 22=4, 23=8, 24=16, 25=32, 26=64,….
powers of a number will be uniformly spaced 1 2 3 10 20 30
100
200
20=1, 21=2, 22=4, 23=8, 24=16, 25=32, 26=64,….
Slide from Lada Adamic

19. Power laws are seemingly everywhere note: these are cumulative distributions
Moby Dick
scientific papers
AOL users visiting sites ‘97
Slide from Lada Adamic
bestsellers
AT&T customers on 1 day
California

20. Yet more power laws wars (1816-1980) Moon Solar flares
Slide from Lada Adamic
richest individuals 2003
US family names 1990
US cities 2003

21. Power law distribution
Straight line on a log-log plot
Exponentiate both sides to get that p(x), the probability of observing an item of size ‘x’ is given by
normalization constant (probabilities over all x must sum to 1)
power law exponent a
Slide from Lada Adamic

22. What does it mean to be scale free?
A power law looks the same no mater what scale we look at it on (2 to 50 or 200 to 5000)
Only true of a power-law distribution!
p(bx) = g(b) p(x) – shape of the distribution is unchanged except for a multiplicative constant
p(bx) = (bx)-a = b-a x-a
log(p(x))
x →b*x
Slide from Lada Adamic
log(x)

23. Many real world networks are power law
exponent a
(in/out degree)
film actors co-appearance
2.3
telephone call graph
2.1
networks
1.5/2.0
sexual contacts
3.2
WWW
2.3/2.7
internet
2.5
peer-to-peer
metabolic network
2.2
protein interactions
2.4
Slide from Lada Adamic

24. Power laws and “fat tails”

25. “Fat tails” in the news

26. Hey, not everything is a power law
number of sightings of 591 bird species in the North American Bird survey in 2003.
cumulative
distribution
another examples:
size of wildfires (in acres)
Slide from Lada Adamic

27. Not every network is power law distributed
address books
power grid
Roget’s thesaurus
company directors…
Slide from Lada Adamic

28. Zipf’s law is a power-law
George Kingsley Zipf
how frequent is the 3rd or 8th or 100th most common word?
Intuition: small number of very frequent words (“the”, “of”)
lots and lots of rare words (“expressive”, “Jurafsky”)
Zipf's law: the frequency of the r'th most frequent word is inversely proportional to its rank: y ~ r -b , with b close to unity.

29. Pareto’s law and power-laws
The Italian economist Vilfredo Pareto was interested in the distribution of income.
Pareto’s law is expressed in terms of the cumulative distribution (the probability that a person earns X or more).
P[X > x] ~ x-k
Here we recognize k as just a -1, where a is the power-law exponent
Slide from Lada Adamic

30. 80/20 rule
The fraction W of the wealth in the hands of the richest P of the population is given by W = P(a-2)/(a-1)
Example: US wealth: a = 2.1
richest 20% of the population holds 86% of the wealth
Slide from Lada Adamic

31. Generative processes for power-laws
Many different processes can lead to power laws
There is no one unique mechanism that explains it all
Slide from Lada Adamic

32. First considered by [Price 65] as a model for citation networks
One mechanism for generating a power-law distribution: Preferential attachment
First considered by [Price 65] as a model for citation networks
each new paper is generated with m citations (mean)
new papers cite previous papers with probability proportional to their in-degree (citations)
what about papers without any citations?
each paper is considered to have a “default” citation
probability of citing a paper with degree k, proportional to k+1
Power law with exponent α = 2+1/m
Slide from Lada Adamic

33. Small worlds
Slide from Lada Adamic

34. Small world experimentsNE MA
Stanley Milgram’s experiment (1967):
Given a target individual and a particular property, pass the message to a person you correspond with who is “closest” to the target.
Slide from Lada Adamic

35. Milgram’s small world experiment
Target person worked in Boston as a stockbroker.
296 senders from Boston and Omaha.
20% of senders reached target.
average chain length = 6.5.
“Six degrees of separation”
Slide from Lada Adamic

36. Milgram’s Findings: Distance to target person, by sending group.
Slide from James Moody

37. 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.
Slide from James Moody

38. 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
Slide from James Moody

39. 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 diameter)
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.
Slide from James Moody

40. Local clustering coefficient (Watts&Strogatz 1998)
For a vertex i
The fraction pairs of neighbors of the node that are themselves connected
Let ni be the number of neighbors of vertex i
number of connections between i’s neighbors
maximum number of possible connections between i’s neighbors
# directed connections between i’s neighbors
ni * (ni -1)
# undirected connections between i’s neighbors
ni * (ni -1)/2
Ci =
Ci directed =
Ci undirected =
Slide from Lada Adamic

41. Local clustering coefficient (Watts&Strogatz 1998)
Average over all n vertices
ni = 4
max number of connections:
4*3/2 = 6
3 connections present
Ci = 3/6 = 0.5 i
link present
link absent
Slide from Lada Adamic

42. The most clustered graph is Watt’s “Caveman” graph:
Slide from Lada Adamic

43. 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
Slide from Lada Adamic

44. Watts and Strogatz model [WS98]
Start with a ring, where every node is connected to the next z nodes ( a regular lattice)
With probability p, rewire every edge (or, add a shortcut) to a uniformly chosen destination.
order
p = 0
0 < p < 1
p = 1
Slide from Lada Adamic
Small world
randomness

45. Clustering and Path Length
Random Graphs have a low clustering coefficient but a low diameter
Regular Graphs have a high clustering coefficient but also a high diameter
Slide from Lada Adamic

46. Weak links
Mark Granovetter (1960s) studied how people find jobs. He found out that most job referrals were through personal contacts
But more by acquaintances and not close friends.
Accepted by the American Journal of Sociology after 4 years of unsuccessful attempts elsewhere.
One of the most cited papers in sociology.
Slide from Drago Radev

47. Mark Granovetter: The strength of weak ties
Strength of ties
amount of time spent together
emotional intensity
intimacy (mutual confiding)
reciprocal services
Many strong ties are transitive
we meet our friends through other friends
if we spend a lot of time with our strong ties, they will tend to overlap
balance theory – if two of my friends do not like each other – we will all be unhappy. Triad closure is the happy solution = Homophily
Slide from James Moody

48. Strength of weak ties Weak ties can occur between cohesive groups
old college friend
former colleague from work
weak ties will tend to have low transitivity
Slide from James Moody

49. Strength of weak ties Evidence from small world experiments
Small world experiment at Columbia: acquaintanceship ties more effective than family, close friends
Slide from James Moody

50. Strength of weak ties – how to get a job
Granovetter: How often did you see the contact that helped you find the job prior to the job search
16.7% often (at least once a week)
55.6% occasionally (more than once a year but less than twice a week)
27.8% rarely – once a year or less
Weak ties will tend to have different information than we and our close contacts do
Long paths rare
39.1 % info came directly from employer
45.3 % one intermediary
3.1 % > 2 (more frequent with younger, inexperienced job seekers)
Compatible with Watts/Strogatz small world model: short average shortest paths thanks to ‘shortcuts’ that are non- transitive
Slide from James Moody

51. Finding paths Watts model shows how these short paths can exist
small world networks
But how do people find the paths?
People seem to be successful by making greedy local decisions
The existence of findable short paths depends on further elucidating the structure of the network
Slide from Lada Adamic

52. Spatial search nodes are placed on a lattice and
Kleinberg, ‘The Small World Phenomenon, An Algorithmic Perspective’ Proc. 32nd ACM Symposium on Theory of Computing, 2000.
(Nature 2000)
“The geographic movement of the [message] from Nebraska to Massachusetts is striking. There is a progressive closing in on the target
area as each new person is added to the chain”
S.Milgram ‘The small world problem’, Psychology Today 1,61,1967
nodes are placed on a lattice and
connect to nearest neighbors
additional links placed with puv~
Slide from Lada Adamic

53. Increasing r favors near nodes
d -r(u,v) =1 , Uniform Distribution
r=0,
Link to each other node equally likely
r=1, inverse of distance
If a node is twice as far away, 1/2 as likely
r=2, inverse squared
If a node is twice as far away, 1/4 as likely
Slide from Lada Adamic

54. Kleinberg’s SW network is Greedy Routable iff r=2
Greedy routing algorithm
using local information only,
find a short path from s to t v u s
When u is the current node, choose next v: the closest to t (use lattice distance) with (u,v) a local or random edge.
(i.e. the view at this node).
Slide from Lada Adamic

55. Kleinberg’s SW network is Greedy Routable iff r=2
A greedy routing algorithm
using local information only, find a short path from s to t v u s
The number of hops is the the ‘delivery time’
This greedy routing achieves
expected ‘delivery time’ of O(log2n), i.e. the st paths have expected length O(log2n).
(i.e. the view at this node).
Slide from Lada Adamic

56. Kleinberg’s SW network is Greedy Routable iff r=2
A greedy routing algorithm
using local information only, find a short path from s to t v u s
This greedy routing achieves
expected `delivery time’ of O(log2n), i.e. the st paths have expected length O(log2n).
This does not work unless r=2 : for r2, >0 such that the expected delivery time of any decentralized algorithm is (n).
(i.e. the view at this node).
Slide from Lada Adamic

57. Overly localized links on a lattice
When r>2 expected search time ~ N(r-2)/(r-1)
Slide from Lada Adamic

58. no locality
When r=0, links are randomly distributed, ASP ~ log(n), n size of grid
When r=0, any decentralized algorithm is at least a0n2/3
When r<2,
expected time at
least arn(2-r)/3
Good paths exist, but are not greedily findable
Slide from Lada Adamic

59. Links balanced between long and short range
When r=2, expected time of a DA is at most C (log N)2
Slide from Lada Adamic

60. Outline
Quick sketch of some networks
Fat Tails
Small Worlds
Weak Ties