The Networked Natureof Society

Networked LifeCSE 112

Spring 2007Prof. Michael Kearns

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• 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 Notions• 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 v: number of edges connected to v

Types of Networks

“Real World” Social Networks

• 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– MK’s Erdos number is 3, via Kearns Mansour Alon Erdos– how do we navigate in such networks?

• Example: acquaintanceship networks– 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

Online Social Networks• Now outdated and discredited example: Friendster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: thefacebook• 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

MK’s Friendster NW, 1/19/04

• 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– 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

• Of course, every good scandal needs a network– vertices: CEOs, spies, stock brokers, other shifty characters– links: co-occurrence in the same article– content network social network

• 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

– 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– 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 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

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.?

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

Structure, Dynamics, Formation: Two Brief Case Studies

Case Study 1: A “Contagion” 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:

– 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

Case Study 2: Grandmother Cells,Associative Memory, and Random

Networks

• A little more on the human brain:– (neo)cortex most “recently” evolved– memory and higher brain function– closest to a crude “random network”

• all connections equally likely

• Localist or grandmother cell theory:– individual cells respond to complex stimuli

• Hebbian learning:– cells learn to fire when highly correlated

neighboring cells fire– entirely decentralized allocation process

• The problem of associative memory:– consider “Pelican Brief” or “Networked Life”– requires that neurons representing

“pelican” and “brief” have a common neighbor

A Back-of-the Envelope Analysis• Let’s try assuming:

– all connections equally likely– independent with probability p

• 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 connectivity– 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 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)