Contagion and Tipping in Networks
Networked Life
CSE 112
Spring 2006
Prof. Michael Kearns
Gladwell, page 7:
“The Tipping Point is the biography of the idea…
that the best way to understand the emergence of
fashion trends, the ebb and flow of crime waves, or
the rise in teen smoking… is to think of them as
epidemics. Ideas and products and messages and
behaviors spread just like viruses do…”
…on networks.
Some Tipping Examples
• Hush Puppies:
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almost dead in 1994; > 10x sales increase by ’96
no advertising or marketing budget
claim: “viral” fashion spread from NY teens to designers
must be certain connectivity and individuals
• NYC Crime:
– 1992: > 2K murders; < 770 five years later
– standard socio-economic explanations:
• police performance, decline of crack, improved economy, aging…
– but these all changed incrementally
– alternative: small forces provoked anti-crime “virus”
• Technology tipping: fax machines, email, cell phones
• “Tipping” origins: 1970’s “white flight”
Key Characteristics of Tipping
(according to Gladwell)
• Contagion:
– “viral” spread of disease, ideas, knowledge, etc.
– spread is determined by network structure
– network structure will influence outcomes
• who gets “infected”, infection rate, number infected
• Amplification of the incremental:
– small changes can have large, dramatic effects
• network topology, infectiousness, individual behavior
• Sudden, not gradual change:
– phase transitions and non-linear phenomena
• How can we formalize some of these ideas?
Rates of Growth and Decay
linear
crime rate
crime rate
linear
size of police force
size of police force
nonlinear, tipping
crime rate
crime rate
nonlinear, gradual decay
size of police force
size of police force
Gladwell’s Three Sources of Tipping
• The Law of the Few (Messengers):
– Connectors, Mavens and Salesman
– Hubs and Authorities
• The Stickiness Factor (Message):
– The “infectiousness” of the “message” itself
– Still largely treated as a crude property of transmission
• The Power of Context:
– global influences affecting messenger behavior
Case Study: Baltimore Syphilis Epidemic
• Mid-90’s: sudden increase in syphilis in Baltimore
• Three plausible explanations:
– increased crack usage  altered sexual behavior
• an incremental change in overall context of behavior
– diminished medical services  longer time to treatment
• an incremental change in the stickiness of the disease
– housing project demolition  migration of infected
• an incremental change in connectivity (law of the few)
• Any or all could be right
“Epidemos”
[Thanks to Sangkyum Kim]
• Forest fire simulation:
– grid of forest and vacant cells
– fire always spreads to adjacent four cells
• “perfect” stickiness or infectiousness
– connectivity parameter:
• probability of forest
– fire will spread to connected component of source
– tip when forest ~ 0.6
– clean mathematical formalization (e.g. fraction burned)
• Viral spread simulation:
– population on a grid network, each with four neighbors
– stickiness parameter:
• probability of passing disease
– connectivity parameter:
• probability of adding random (long-distance) connections
– no long distance connections: tip at stickiness ~ 0.3
– at rewiring = 0.5, often tip at stickiness ~ 0.2
“Mathematizing” the Forest Fire
• Start with a regular 2-dimensional grid network
– this represents a complete forest
• Delete each vertex (and its edges) with probability p (independently)
– this represents random “clear-cutting” or natural fire breaks
• Choose a random remaining vertex v
– this is my campsite
• Q: What is the expected size of v’s connected component?
– this is how much of the forest is going to burn
“Mathematizing” the Epidemic
• Start with a regular 2-dimensional grid network
– this represents a dense population with “local” connections (neighbors)
• Rewire each edge with probability p to a random destination
– this represents “long-distance” connections (chance meetings)
• Choose a random remaining vertex v
– this is an infection; spreads probabilistically to each of v’s neighbors
• Fraction killed more complex:
– depends on both size and structure of v’s connected component
• Important theme:
– mixing regular, local structure with random, long-distance connections
Some Remarks on the Demos
• Connectivity patterns were either local or random
– will eventually formalize this model
– what about other/more realistic structure?
• Tipping was inherently a statistical phenomenon
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probabilistic nature of disease spread
probabilistic nature of connectivity patterns
model likely properties of a large set of possible outcomes
can model either inherent randomness or variability
• Formalizing tipping in the forest fire demo:
– might let grid size N  infinity, look at fixed values of p
– is there a threshold value q:
• p > q  expected fraction burned < 1/10
• p < q  expected fraction burned > 9/10
Small Worlds and the Law of the Few
• Gladwell’s “Law of the Few”:
– a “small” number of “highly” connected vertices ( heavy tails)
– inordinate importance for global connectivity ( small diameter)
• Travers & Milgram 1969: classic early social network study
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destination: a Boston stockbroker; lived in Sharon, MA
sources: Nebraska stockbrokers; Nebraska and Boston “randoms”
forward letter to a first-name acquaintance “closer” to target
target information provided:
• name, address, occupation, firm, college, wife’s name and hometown
• navigational value?
• Basic findings:
– 64 of 296 chains reached the target
– average length of completed chains: 5.2
• interaction of chain length and navigational difficulties
– main approach routes: home (6.1) and work (4.6)
– Boston sources (4.4) faster than Nebraska (5.5)
– no advantage for Nebraska stockbrokers
The Connectors to the Target
• T & M found that many of the completed chains passed through a very
small number of penultimate individuals
– Mr. G, Sharon merchant: 16/64 chains
– Mr. D and Mr. P: 10 and 5 chains
• Connectors are individuals with extremely high degree
– why should connectors exist?
– how common are they?
– how do they get that way? (see Gladwell for anecdotes)
• Connectors can be viewed as the “hubs” of social traffic
• Note: no reason target must be a connector for small worlds
• Two ways of getting small worlds (low diameter):
– truly random connection pattern  dense network
– a small number of well-placed connectors in a sparse network
• Let’s revisit our Gladwell estimate of class connectivity and MK’s NW
A Mathematical Digression
• We’ve been bouncing around the idea that connectors (~high degree
vertices) lead to small diameter
• Obviously, if everyone is connected to everyone else…
• But there’s often a limit to the largest possible degree
– you can’t have an unbounded number of friends, colleagues, etc.
• May be constraints on the mere existence of certain NWs
– let D be the largest degree allowed
• why? e.g. because there is a limit to how many friends you can have
– suppose we are interested in NWs with diameter D (or less)
• why? because many have claimed that D is often small
– let N(D,D) = size of the largest possible NW obeying D and D
• Exact form of N(D,D) is notoriously elusive
– but known that it is between (D/2)^D and 2D^D
• So, for example, if we want N ~ 300M (U.S. population):
– if D = 150 (e.g. see Gladwell) and D = 6 (6 degrees): NW exists
– D = 6, N = 300M, solve 2D^D > N: get D > 23
Small Worlds: A Modern Experiment
• The Columbia Small Worlds Project:
– considerably larger subject pool, uses email
– subject of Dodds et al. assigned paper
• Basic methodology:
– 18 targets from 13 countries
– on-line registration of initial participants, all tracking electronic
– 99K registered, 24K initiated chains, 384 reached targets
• Some findings:
– < 5% of messages through any penultimate individual
– large “friend degree” rarely (< 10%) cited
– Dodds et al:  no evidence of connectors!
• (but could be that connectors are not cited for this reason…)
– interesting analysis of reasons for forwarding
– interesting analysis of navigation method vs. chain length
The Strength of Weak Ties
• Not all links are of equal importance
• Granovetter 1974: study of job searches
– 56% found current job via a personal connection
– of these, 16.7% saw their contact “often”
– the rest saw their contact “occasionally” or “rarely”
• Your “closest” contacts might not be the most useful
– similar backgrounds and experience
– they may not know much more than you do
– connectors derive power from a large fraction of weak ties
• Further evidence in Dodds et al. paper
• T&M, Granovetter, Gladwell: multiple “spaces” & “distances”
– geographic, professional, social, recreational, political,…
– we can reason about general principles without precise measurement
Stickiness
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Generalizes the notion of infectiousness
A common marketing term
How likely is the idea, fashion, etc. to be transmitted?
Can also view as the “failure probability” of a link
Some examples:
– “Winston tastes good…”
– Columbia Record Club Gold Box
– Sesame Street and Blues Clues
• In modern times, stickiness may trump raw exposure
– supplementary readings on “viral” and “word-of-mouth” marketing
• Persuasion (messenger) and stickiness (message) related
• Making technology sticky:
– “The Media Equation”, by Nass and Reeves
– human reaction to technology design and interface
The Power of Context
• Transmission on a network (social or otherwise) can be
modulated by subtle changes in global influences
– heat waves: crime, the power grid, and Amtrak
• Broken Windows and the NYC squeegee crackdown
• Gladwell: “…the premise that an epidemic can be reversed,
can be tipped, by tinkering with the smallest details of the
immediate environment.”
• Human behavior strongly and subtly influenced by context
• The Stanford Prison Experiment
• The Good Samaritans of Princeton
The Magic Number 150
• Social channel capacity
– correlation between neocortex
size and group size
– Dunbar’s equation: neocortex
ratio  group size
• Clear implications for many
kinds of social networks
• Again, a topological
constraint on typical degree
• From primates to military
units to Gore-Tex
• Next up: let’s make some of these ideas more precise
– graph theory: the mathematical language of networks
– social network theory