Contagion, Tipping and Navigation in Networks

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Contagion, Tipping and Navigation in Networks

• Networked Life
• CIS 112
• Spring 2009
• Prof. Michael Kearns

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What is a Network?

• A collection of individual or atomic entities
• Referred to as nodes or vertices (the dots or
  points)
• Collection of links or edges between vertices
  (the lines)
• Links can represent any pairwise relationship
• Links can be directed or undirected
• Network entire collection of nodes and links
• might sometimes be annotated by other info
  (weights, etc.)
• For us, a network is an abstract object (list of
  pairs) and is separate from its visual layout
• that is, we will be interested in properties that
  are layout-invariant
• Extremely general, but not everything
• e.g. menage a trois
• may lose information by pairwise representation
• We will be interested in properties of networks
• often structural properties
• often statistical properties of families of
  networks

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Some Terminology

• Network size total number of vertices (denoted
  N)
• Maximum possible number of edges N(N-1)/2
  N2/2 ( N)
• Distance between vertices u and v
• number of edges on the shortest path from u to v
• can consider directed or undirected cases
• infinite if there is no path from u to v
• Diameter of a network
• worst-case diameter largest distance between a
  pair
• average-case diameter average distance
• If the distance between all pairs is finite, we
  say the network is connected else it has
  multiple components
• Degree of vertex v number of edges connected to v

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Illustrating the Concepts

• Example scientific collaboration
• vertices math and computer science researchers
• links between coauthors on a published paper
• Erdos numbers distance to Paul Erdos
• Erdos was definitely a hub or connector had 507
  coauthors
• MKs Erdos number is 3, via Kearns ? Mansour ?
  Alon ? Erdos
• how do we navigate in such networks?
• Example real-world acquaintanceship networks
• vertices people in the world
• links have met in person and know last names
• hard to measure
• lets examine the results of our own last-names
  exercise

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average 26.6 min 2 max 114
of individuals
Gaoxiang Hu
Jason Chou
of last names known
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average 30.7 min 0 max 113
of individuals
of last names known
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Structure, Dynamics, and Formation
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Network Structure (Statics)

• Emphasize purely structural properties
• size, diameter, connectivity, degree
  distribution, etc.
• may examine statistics across many networks
• will also use the term topology to refer to
  structure
• Structure can reveal
• community
• important vertices, centrality, etc.
• robustness and vulnerabilities
• can also impose constraints on dynamics
• Less emphasis on what actually occurs on network
• web pages are linked but people surf the web
• buyers and sellers exchange goods and cash
• friends are connected but have specific
  interactions

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Network Dynamics

• Emphasis on what happens on networks
• Examples
• mapping spread of disease in a social network
• mapping spread of a fad
• computation in the brain
• spread of wealth in an economic network
• Statics and dynamics often closely linked
• rate of disease spread (dynamic) depends
  critically on network connectivity (static)
• distribution of wealth depends on network
  topology
• Gladwell emphasizes dynamics
• but often dynamics of transmission
• what about dynamics involving deliberation,
  rationality, etc.?

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Network Formation

• Why does a particular structure emerge?
• Plausible processes for network formation?
• Generally interested in processes that are
• decentralized
• distributed
• limited to local communication and interaction
• organic and growing
• consistent with (some) measurement
• The Internet versus traditional telephony

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Structure and Dynamics Case Study A Contagion
Model of Economic Exchange

• Imagine an undirected, connected network of
  individuals
• no model of network formation
• Start each individual off with some amount of
  currency
• At each time step
• each vertex divides their current cash equally
  among their neighbors
• (or chooses a random neighbor to give it all to)
• each vertex thus also receives some cash from its
  neighbors
• repeat
• A transmission model of economic exchange --- no
  rationality
• Q How does network structure influence outcome?
• A As time goes to infinity
• vertex i will have fraction deg(i)/D of the
  wealth D sum of deg(i)
• degree distribution entirely determines outcome!
• connectors are the wealthiest
• not obvious consider two degree 2 vertices
• How does this outcome change when we consider
  more realistic dynamics?
• e.g. we each have goods available for trade/sale,
  preferred goods, etc.
• What other processes have similar dynamics?
• looking ahead models for web surfing behavior

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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.
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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
• 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 1970s white flight

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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?

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Rates of Growth and Decay
linear
linear
nonlinear, tipping
nonlinear, gradual decay
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Gladwells 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

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Epidemos

• 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 all of 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 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

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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 vs connected
  component?
• this is how much of the forest is going to burn

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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 vs neighbors
• Fraction killed more complex
• depends on both size and structure of vs
  connected component
• Important theme
• mixing regular, local structure with random,
  long-distance connections

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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
• probabilistic nature of connectivity patterns
• probabilistic nature of disease spread
• 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
• p q ? expected fraction burned 9/10

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Small Worlds and the Law of the Few

• Gladwells 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
• 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, wifes
  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

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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

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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
• individual
• large friend degree rarely (
• 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

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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
• TM, Granovetter, Gladwell multiple spaces
  distances
• geographic, professional, social, recreational,
  political,
• we can reason about general principles without
  precise measurement

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The Magic Number 150

• Social channel capacity
• correlation between neocortex size and group size
• Dunbars 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

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A Mathematical Digression

• If theres 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 (D/2)D and 2DD
• So, for example, if N 300M (U.S. population)
• to be certain NW exists, solve N
• if D 4.5
• if D 52
• so these literatures are consistent (whew!)
• More generally multiple structural properties
  may be competing

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• Next up Network Science.