Contagion, Tipping and Navigation in Networks

Networked Life CIS 112 Spring 2008 Prof. Michael Kearns

Gladwell, page 7: “The Tipping Point

epidemics.

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 Ideas and products and messages and behaviors spread just like viruses do…” …on

networks.

Gladwell Tipping Examples

• Hush Puppies: – 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 linear size of police force nonlinear, gradual decay size of police force nonlinear, tipping 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

“Epidemos”

• • Forest fire simulation – connectivity • probability of forest – tip when forest ~ 0.6

: – grid of forest and vacant cells – fire always spreads to adjacent four cells • “perfect” stickiness or infectiousness parameter: – fire will spread to connected component of source – clean mathematical formalization (e.g. fraction burned) Viral spread simulation – stickiness parameter: – connectivity parameter: : – population on a grid network, each with four neighbors • probability of passing disease • probability of rewiring local connections to random long-distance – 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 • Choose a random remaining vertex v – this is an infection; spreads

random probabilistically

destination – this represents “long-distance” connections (chance meetings) 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

– will eventually formalize this model – what about other/more realistic structure?

or

random

• Tipping was inherently a – model

likely

– might let grid size N 

statistical

– probabilistic nature of disease spread properties of a large – is there a threshold value q:

set

phenomenon – probabilistic nature of connectivity patterns of possible outcomes – can model either inherent randomness or variability • Formalizing tipping in the forest fire demo: infinity, look at fixed values of p • p > q  • p < q  expected fraction burned < 1/10 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 ( – inordinate importance for

global

connectivity (   heavy tails) small diameter) • Travers & Milgram 1969: classic early social network study – destination: a Boston stockbroker; lived in Sharon, MA – sources: Nebraska stockowners; 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 stockowners

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

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

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

A Mathematical Digression

• If there’s a “Magic Number 150” (degree bound)… • …and we want networks with small diameter… • … then there 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 (worst-case) 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 ( – if D D /2)^D and 2 – to be certain NW exists, solve N < ( < 150 (e.g. see Gladwell): D > 4.5

D – if D < 6 (e.g. see Travers & Milgram): /2)^D D > 52 D – so these literatures are consistent… (whew!) ^D • So, for example, if N ~ 300M (U.S. population): • More generally: multiple structural properties may be

competing

• Next up: Network Science.