“The Tipping Point” and the Networked Nature of Society Michael Kearns Computer and Information Science Penn Reading Project 2004.

“The Tipping Point” and the

Networked Nature of Society

Michael KearnsComputer and Information

SciencePenn Reading Project 2004

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.

The Networked Nature of Society

International Trade

[Krempel&Pleumper]

Corporate Partnerships

[Krebs]

Gnutella

Internet Routers

Artist Mark Lombardi

An Emerging Science

An Emerging Science• Examining apparent similarities between many

human and technological systems & organizations

• Importance of network effects in such systems• How things are connected matters greatly• Structure, asymmetry and heterogeneity• Details of interaction matter greatly• The metaphor of viral spread• Qualitative and quantitative; can be very subtle• A revolution of

– measurement– theory– breadth of vision

Who’s Doing All This?

• Computer Scientists– Understand and design complex, distributed

networks– View “competitive” decentralized systems as

economies

• Social Scientists, Psychologists, Economists– Understand human behavior in “simple” settings– Revised views of economic rationality in humans– Theories and measurement of social networks

• Physicists and Mathematicians– Interest and methods in complex systems– Theories of macroscopic behavior (phase transitions)

• All parties are interacting and collaborating

“Real World” Social Networks• Example: Acquaintanceship networks

– vertices: people in the world– links: have met in person and know last names– hard to measure– let’s 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– MK’s Erdos number is 3, via Mansour Alon Erdos– how do we navigate in such networks?

Update: MK’s Friendster NW, 1/19/03

• If you didn’t get my email invite, let me know– send mail to [email protected]

• 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: [thanks Albert Ip!]

– AOL IM Buddyzoo

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)– we’ll return to this in a

moment…

Universality and Generative Models

A “Canonical” Natural Network has…

• Few connected components:– often only 1 or a small number independent of network size

• Small diameter:– often a constant independent of network size (like 6)– or perhaps growing only logarithmically with network size– typically exclude infinite distances

• A high degree of clustering:– considerably more so than for a random network– in tension with small diameter

• A heavy-tailed degree distribution:– a small but reliable number of high-degree vertices– quantifies Gladwell’s connectors– often of power law form

Some Models of Network Generation

• Random graphs (Erdos-Renyi models):– gives few components and small diameter– does not give high clustering and heavy-tailed degree

distributions– is the mathematically most well-studied and understood model

• Watts-Strogatz and related models:– give few components, small diameter and high clustering– does not give heavy-tailed degree distributions

• Preferential attachment:– gives few components, small diameter and heavy-tailed

distribution– does not give high clustering

• Hierarchical networks:– few components, small diameter, high clustering, heavy-tailed

• Affiliation networks:– models group-actor formation

• Nothing “magic” about any of the measures or models

So Which Properties Tip?• Just about all of them!• The following properties all have threshold functions:

– having a “giant component”– being connected– having a perfect matching (N even)– having “small” diameter

• Demo: look at the following progression– giant component connectivity small diameter– in graph process model (add one new edge at a time)– [example 1] [example 2] [example 3] [example 4] [example 5]

• With remarkable consistency (N = 50):– giant component ~ 40 edges, connected ~ 100, small

diameter ~ 180

“Epidemos”[Thanks to Sangkyum Kim]

• 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 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 adding random (long-distance) connections– no long distance connections: tip at stickiness ~ 0.3– at rewiring = 0.5, often tip at stickiness ~ 0.2

Incorporating Strategic and Economic Behavior

Examples from Schelling and Beyond

• Going to the beach or not– too few you’ll go, making it more crowded– too many you won’t go, or will leave if you’re there

• Sending Christmas cards– people send to those they expect will send to them– everybody hates it, but no individual can break the cycle

• Investing in an apartment fire sprinkler– only worth it if enough people do it– insurance companies won’t discount for it

• Choosing where to sit in the Levine Auditorium

Local Preferences and Segregation

• Special case of preferences: housing choices• Imagine individuals who are either “red” or “blue”• They live on in a grid world with 8 neighboring cells• Neighboring cells either have another individual or are empty• Individuals have preferences about demographics of their

neighborhood• Here is a very nice simulator

A Sample Network and Equilibrium

• Solid edges:– exchange at equilibrium

• Dashed edges:– competitive but unused

• Dotted edges:– non-competitive prices

• Note price variation– 0.33 to 2.00

• Degree alone does not determine price!– e.g. B2 vs. B11– e.g. S5 vs. S14

The Internet as Society

The Internet: What is It?

• The Internet is a massive network of connected but decentralized computers

• Began as an experimental research NW of the DoD (ARPAnet) in the 1970s

• All aspects (protocols, services, hardware, software) evolved over many years

• Many individuals and organizations contributed• Designed to be open, flexible, and general from the

start• Completely unlike prior centralized, managed NWs

– e.g. the AT&T telephone switching network

Hubs and Authorities• Suppose we have a large collection of pages on some topic

– possibly the results of a standard web search

• Some of these pages are highly relevant, others not at all• How could we automatically identify the important ones?• What’s a good definition of importance?• Kleinberg’s idea: there are two kinds of important pages:

– authorities: highly relevant pages– hubs: pages that point to lots of relevant pages– (I had these backwards last time…)

• If you buy this definition, it further stands to reason that:– a good hub should point to lots of good authorities– a good authority should be pointed to by many good hubs– this logic is, of course, circular

• We need some math and an algorithm to sort it out

• Networked Life (CSE 112) web site:– www.cis.upenn.edu/~mkearns/teaching/NetworkedLife– these slides:

• www.cis.upenn.edu/~mkearns/teaching/NetworkedLife/prp.ppt

• Feel free to contact me at– [email protected]