The Networked Nature of Society

1
The Networked Natureof Society

• Networked Life
• CSE 112
• Spring 2004
• Prof. Michael Kearns

2
Course News and Notes 1/20

• Course mailing list CSE112-001-04A_at_lists.upenn.ed
  u
• Updates to course web site
• Recitation sections
• Tuesday (today) 6-7 PM room announced later
  today
• Wednesday 5-6 in Towne 313

3
Course News and Notes 1/15

• Please give last-names exercise to Nick now
• If you missed the first lecture, please get the
  handouts
• Extra credit for identifying papers, web sites,
  demos or other material related to class, if it
  gets used
• Prof. Kearns office hours today 1030-1130
• Tuesday recitation may be moved to 6 PM stand by
• Please start reading The Tipping Point
• required chapters 1,2,4,5

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What is a Network?

• A collection of individual or atomic entities
• Referred to as nodes or vertices
• Collection of links or edges between vertices
• Links represent pairwise relationships
• Links can be directed or undirected
• Network entire collection of nodes and links
• Extremely general, but not everything
• actors appearing in the same film
• lose information by pairwise representation
• We will be interested in properties of networks
• often statistical properties of families of
  networks

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Some Definitions

• Network size total number of vertices (denoted
  N)
• Maximum number of edges N(N-1)/2 N2/2
• 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 u number of edges

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Examples of Networks
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Real World Social Networks

• Example Acquaintanceship networks
• vertices people in the world
• links have met in person and know last names
• hard to measure
• lets do our own Gladwell estimate
• 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 Mansour ? Alon ?
  Erdos
• how do we navigate in such networks?

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Online Social Networks

• A very recent example Friendster
• vertices subscribers to www.friendster.com
• links created via deliberate invitation
• Heres an interesting visualization by one user
• Older example social interaction in LambdaMOO
• LambdaMOO chat environment with emotes or
  verbs
• vertices LambdaMOO users
• links defined by chat and verb exchange
• could also examine friend and foe
  sub-networks

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Update MKs Friendster NW, 1/19/03

• If you didnt get my email invite, let me know
• send mail to mkearns_at_cis.upenn.edu
• Number of friends (direct links) 8
• NW size (lt 4 hops) 29,901
• 134 29,000
• But lets look at the degree distribution
• So a random connectivity pattern is not a good
  fit
• What is???
• Another interesting online social NW thanks
  Albert Ip!
• AOL IM Buddyzoo

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Content Networks

• Example document similarity
• vertices documents on the web
• links defined by document similarity (e.g.
  Google)
• heres a very nice visualization
• not the web graph, but an overlay content network
• Of course, every good scandal needs a network
• vertices CEOs, spies, stock brokers, other
  shifty characters
• links co-occurrence in the same article
• Then there are conceptual networks
• vertices concepts to be discussed in NW Life
• links arbitrarily determined by Prof. Kearns
• Update here are two more examples thanks Hanna
  Wallach!
• a thesaurus defines a network
• so do the interactions in a mailing list

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Business and Economic Networks

• Example eBay bidding
• vertices eBay users
• links represent bidder-seller or buyer-seller
• fraud detection bidding rings
• Example corporate boards
• vertices corporations
• links between companies that share a board
  member
• Example corporate partnerships
• vertices corporations
• links represent formal joint ventures
• Example goods exchange networks
• vertices buyers and sellers of commodities
• links represent permissible transactions

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Physical Networks

• Example the Internet
• vertices Internet routers
• links physical connections
• vertices Autonomous Systems (e.g. ISPs)
• links represent peering agreements
• latter example is both physical and business
  network
• Compare to more traditional data networks
• Example the U.S. power grid
• vertices control stations on the power grid
• links high-voltage transmission lines
• August 2003 blackout classic example of
  interdependence

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Biological Networks

• Example the human brain
• vertices neuronal cells
• links axons connecting cells
• links carry action potentials
• computation threshold behavior
• N 100 billion
• typical degree sqrt(N)
• well return to this in a moment

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Network 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
• 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

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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 measurement
• The Internet versus traditional telephony

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Course News and Notes 1/22

• Prof. Kearns Friendster NW please send mail to
  mkearns_at_cis.upenn.edu if you want to join
• Prof. Kearns office hours today 1030-12
• Two new articles added to required readings
• Homework 1 distributed today, due in class Feb 5
• NO CLASS on Feb 3
• Todays agenda
• recap of brain analysis from NW Nature of
  Society lectures
• further examples
• Contagion, Tipping and NW material contd
• distribution and discussion of HW 1

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Brief Case Study Associative Memory, Grandmother
Cells, and Random Networks

• A little more on the human brain
• (neo)cortex most recently evolved
• memory and higher brain function
• long distance connections
• pyramidal cells, majority of cortex
• white matter as a box of cables
• closet to a crude random network
• Associative memory
• consider the Pelican Brief
• or Networked Life
• Grandmother cells
• localist vs. holistic representation
• single vs. multi-cell recognition

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A Back-of-the Envelope Analysis

• Lets try assuming
• all connections equally likely
• independent with probability p
• grandmother cell representation
• decentralized learning
• correlated learning of conjunctions
• So
• at some point have learned pelican and brief
  in separate cells
• need to have cells connected to both
• but not too many such cells!
• In this model, p 1/sqrt(N) balances
  near-certain upper and lower bounds on common
  neighbors for random pairs
• Broadly consistent with biology
• A little more on the human brain
• (neo)cortex most recently evolved

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Remarks

• Network formation
• random long-distance
• is this how the brain grows?
• Network structure
• common neighbors for arbitrary cell pairs
• implications for degree distribution
• Network dynamics
• distributed, correlation-based learning
• There is much that is broken with this story
• But it shows how a set of plausible assumptions
  can lead to nontrivial constraints

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Recap

• We chose a particular, statistical model of
  network generation
• each edge appears independently and with
  probability p
• why? broadly consistent with long-distance cortex
  connectivity
• a statistical model allows us to study variation
  within certain constraints
• We were interested in the NW having a certain
  global property
• any pair of vertices should have a small number
  of common neighbors
• why? corresponds to controlled growth of learned
  conjunctions, in a model assuming distributed,
  correlated learning and grandmother cells
• We asked whether our NW model and this property
  were consistent
• yes, assuming that p 1/sqrt(N)
• this implies each neuron (vertex) will have about
  pN sqrt(N) neighbors
• and this is roughly what one finds biologically
• Note this statement is not easy to prove

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Another Example

• We choose a particular, statistical model of
  network generation
• each edge appears independently and with
  probability p
• why? might propose it as a first guess as to how
  social networks form
• We are interested in the NW having a certain
  global property
• that the network be connected, yet not have large
  (say, log(N)) cliques
• why? want a connected society without overly
  powerful subgroups
• We ask whether our NW model and this property are
  consistent
• I dont know the answer to this one yet
• Well follow this kind of questioning and
  analysis frequently

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A Non-Statistical Example

• Often interested in 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
• let D be a bound on the diameter of the network
• 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 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 2DD gt N get D gt 23