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The Networked Nature
of Society
Networked Life
CSE 112
Spring 2005
Prof. Michael Kearns
What is a Network?
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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 structural properties
– often statistical properties of families of networks
Some Definitions
• Network size: total number of vertices (denoted N)
• Maximum number of edges: N(N-1)/2 ~ N^2/2
• 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
Examples of Networks
“Real World” Social Networks
• Example: Acquaintanceship networks
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vertices: people in the world
links: have met in person and know last names
hard to measure
let’s examine the results of our own last-names exercise
• Example: scientific collaboration
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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
MK’s Erdos number is 3, via Mansour  Alon  Erdos
how do we navigate in such networks?
Online Social Networks
• A very recent example: Friendster
– vertices: subscribers to www.friendster.com
– links: created via deliberate invitation
– Here’s an interesting visualization by one user
• More recent and interesting: thefacebook
• Older example: social interaction in LambdaMOO
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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
MK’s Friendster NW, 1/19/04
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Number of friends (direct links): 8
NW size (<= 4 hops): 29,901
13^4 ~ 29,000
But let’s look at the degree distribution
So a random connectivity pattern is not a good fit
What is???
Another interesting online social NW:
– AOL IM Buddyzoo
Content Networks
• Example: document similarity
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vertices: documents on the web
links: defined by document similarity (e.g. Google)
here’s 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
– a thesaurus defines a network
– so do the interactions in a mailing list
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
Physical Networks
• Example: the Internet
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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
Agenda: Thursday, Jan 20
• Finish up “Networked Nature…” lectures
• Detail and due date for first network construction project task
• Introduction to Lifester and first assignment
Biological Networks
• Example: the human brain
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vertices: neuronal cells
links: axons connecting cells
links carry action potentials
computation: threshold behavior
N ~ 100 billion
typical degree ~ sqrt(N)
we’ll return to this in a moment…
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:
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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
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
– but often dynamics of transmission
– what about dynamics involving deliberation, rationality, etc.?
Network Formation
• Why does a particular structure emerge?
• Plausible processes for network formation?
• Generally interested in processes that are
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decentralized
distributed
limited to local communication and interaction
“organic” and growing
consistent with measurement
• The Internet versus traditional telephony
Structure, Dynamics, Formation:
Two Case Studies
Case Study 1: A (Brain-Dead) Model of
Economic Exchange
• Imagine an(y) undirected, connected network of individuals
– no model of network formation
• Start each individual off with some amount of currency
• At each time step:
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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
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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…
• A transmission model of economic exchange --- no “rationality”
• Q: How does network structure influence outcome?
• A: As time goes to infinity:
Case Study 2: Grandmother Cells,
Associative Memory, and Random Networks
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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
– closest to a crude “random network”
• all connections equally likely
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Grandmother cells:
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Hebbian learning of correlations:
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Problem of associative memory:
– localist (vs. “holistic”) representation
– single (vs. multi-) cell recognition
– cells learn to fire when highly correlated
neighboring cells fire
– entirely decentralized allocation process
– consider the Pelican Brief
– or Networked Life
A Back-of-the Envelope Analysis
• Let’s try assuming:
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all connections equally likely
independent with probability p
grandmother cell representation
Hebbian learning of conjunctions
• So:
– at some point have learned “pelican” and
“brief” in separate cells
– need to have cells connected to both to
learn conjunction
– but not too many such cells!
• In this model, p ~ 1/sqrt(N) results in
any pair of cells having just a few
common neighbors!
• Broadly consistent with biology
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
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 p*N ~ sqrt(N) neighbors
– and this is roughly what one finds biologically
• Note: this statement is not easy to prove