Transcript The Networked Nature of Society Networked Life CIS 112 Spring 2008 Prof. Michael Kearns What is a Network? • • • • • • A collection of individual or atomic entities Referred to.

The Networked Nature
of Society
Networked Life
CIS 112
Spring 2008
Prof. Michael Kearns
What is a Network?
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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
Some Handy Definitions
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Network size: total number of vertices (denoted N)
Maximum possible number of edges: N(N-1)/2 ~ N^2/2 (>> N)
Distance between vertices u and v:
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Diameter of a network:
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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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– 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
– worst-case diameter: largest distance between a pair
– average-case diameter: average distance
Types of Networks
“Real World” Social Networks
• 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 Kearns  Mansour  Alon  Erdos
how do we navigate in such 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: acquaintanceship networks
# of last names known
Nechemya Kagedan
# of individuals
Geoffrey Kiderman
average = 30.7
min = 0
max = 113
Online Social Networks
• Now outdated and discredited 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: Facebook
• 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
Prof. Kearns’ Facebook Friends Wheel
Ariana Ireland’s Friends Wheel
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
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Example: document similarity
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Of course, every good scandal needs a network
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Then there are conceptual networks
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vertices: documents on the web
links: defined by document similarity (e.g. Google’s related search)
here’s a very nice visualization
not the web graph, but an overlay content network
– vertices: CEOs, spies, stock brokers, other shifty characters
– links: co-occurrence in the same article
– content network  social network
– 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
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)
Structure, Dynamics, Formation
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:
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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
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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.?
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 (some) measurement
• The Internet versus traditional telephony
Structure, Dynamics, Formation:
Two Brief Case Studies
Case Study:
A “Contagion” Model of Economic Exchange
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Imagine an(y) undirected, connected network of individuals
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Start each individual off with some amount of currency
At each time step:
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A transmission model of economic exchange --- no “rationality”
Q: How does network structure influence outcome?
A: As time goes to infinity:
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How does this outcome change when we consider more “realistic” dynamics?
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What other processes have similar dynamics?
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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…
– e.g. we each have goods available for trade/sale, preferred goods, etc.
– looking ahead: models for web surfing behavior
Case Study:
Associative Memory and Random Networks
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Human brain:
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Localist or grandmother cell theory:
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~ 100 billion neurons
higher cognitive function in neocortex
“random” long-distance connectivity
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individual cells respond to complex stimuli
e.g. one cell fires when you hear “networked”,
another when you hear “life”
assume entirely decentralized allocation
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consider binding new cell to the phrase
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no “interference” between meanings
requires that neurons representing “networked”
and “life” have a common neighbor
The problem of associative memory:
“Networked Life”
A Back-of-the Envelope Analysis
• Let’s try assuming:
– all connections equally likely
– independent with probability p
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– at some point have learned “networked”
and “life” 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!
• Average degree ~ p*N = sqrt(N)
• Broadly consistent with biology
Remarks
• Network formation:
– random connectivity
– but 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 plausible assumptions yield 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 structural property
– any pair of vertices should have a small number of common neighbors
– corresponds to controlled growth of learned conjunctions, in a model
assuming distributed, correlated learning
• 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)