Small World Networks. Small World Phenomenon. Small World Phenomenon. Interpreting Milgram’s experiment

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 Small World Networks

Adapted from slides by Lada Adamic, UMichigan

Outline

nn  Small world phenomenon

nn  Milgram’s small world experiment

nn  Small world network models:


nn  Watts & Strogatz (clustering & short paths) nn  Kleinberg (geographical) nn  Watts, Dodds & Newman (hierarchical)

nn  Small world networks: why do they arise? nn  efficiency nn  navigation

Small World Phenomenon: 

Milgram’s Experiment

Instructions:

Given a target individual (stockbroker in Boston), pass the message to a person you correspond with who is “closest” to the target.

20% of initiated chains reached target average chain length = 6.5                  ß “Six degrees of separation”

Source: undetermined

Small World Phenomenon:

Milgrams Experiment Repeated

email experiment by  

  Dodds, Muhamad, Watts;  Science 301, (2003)

(reading linked on website)

•18  targets

•13  different countries

•60  ,000+ participants

•24  ,163 message chains 

•384  reached their targets

•Average path length = 4.0  

Source: NASA, U.S. Government; http://visibleearth.nasa.gov/view_rec.php?id=2429

Small World Phenomenon:

Interpreting Milgram’s experiment

nn   Is 6 is a surprising number?

nn  In the 1960s? Today? Why? nn  If social networks were random… ?

nn  Pool and Kochen (1978) - ~500-1500 acquaintances/person nn  ~ 1,000 choices 1st  link

nn  ~ 10002 = 1,000,000 potential 2nd links nn  ~ 10003 = 1,000,000,000 potential 3rd  links

nn  If networks are completely cliquish:

nn  all my friends’ friends are my friends nn  What would happen?

Small world experiment:

Accuracy of distances

nn  Is 6 an accurate number?

nn  What bias is introduced by uncompleted chains? nn  are longer or shorter chains more likely to be completed? nn  if each person in the chain has 0.5 probability of passing the letter on, what is the likelihood of a chain being completed nn  of length 2? nn  of length 5?

Small world experiment accuracy:

Attrition rate is approx. constant

Source: An Experimental Study of Search in Global Social Networks: Peter Sheridan Dodds, Roby Muhamad, and Duncan J. Watts (8 August 2003); Science 301 (5634), 827.

Small world experiment accuracy: Estimating true distance distribution

nn  observed chain lengths

nn  ‘recovered’ histogram of path lengths

inter-country intra-country 

Source: An Experimental Study of Search in Global Social Networks: Peter Sheridan Dodds, Roby Muhamad, and Duncan J. Watts (8 August 2003); Science 301 (5634), 827.

Small world experiment:

Accuracy of distances

nn  Is 6 an accurate number?

nn  Do people find the shortest paths? nn  Killworth, McCarty ,Bernard, & House (2005, optional): nn  less than optimal choice for next link in chain is made ½ of the time

Current Social Networks

nn  Facebook's data team released two papers in Nov. 2011 nn  721 million users with 69 billion friendship links nn  Average distance of 4.74

nn  Twitter studies nn  Sysomos reports the average distance is 4.67  (2010) nn  50% of people are 4 steps apart, nearly everyone is 5 steps or less

nn  Bakhshandeh et al. (2011) report an average distance of 3.435 among 1,500 random Twitter users


Business applications?

“Social Networking” as a Business:

•  Facebook, Google+, Orkut, Friendster  entertainment, keeping and finding friends

•  LinkedIn:

•more traditional networking for jobs  

•  Spoke, VisiblePath

•helping businesses capitalize on existing  client relationships

Applicable to other kinds of networks

Same pattern:

high clustering            Cnetwork >>Crandom graph

low average shortest path lnetwork ≈ ln(N)

" neural network of C. elegans,  " semantic networks of languages,  " actor collaboration graph  " food webs  

Watts/Strogatz model

Reconciling two observations:

•  High clustering: my friends’ friends tend to be my friends

•  Short average paths 

Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:440-442.


Watts-Strogatz model: Generating small world graphs

Select a fraction p of edges

Reposition one of their endpoints

Add a fraction p of additional edges leaving underlying lattice

nn  As in many network generating algorithms nn  Disallow self-edges nn  Disallow multiple edges

Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:440-442.

Watts-Strogatz model:

Generating small world graphs

nn  Each node has K>=4 nearest neighbors (local) nn  tunable: vary the probability p of rewiring any given edge nn  small p: regular lattice nn  large p: classical random graph

Watts/Strogatz model:

What happens in between?

nn  Small shortest path means small clustering? nn  Large shortest path means large clustering?

nn  Through numerical simulation nn  As we increase p from 0 to 1 nn  Fast decrease of mean distance nn  Slow decrease in clustering

Clustering Coefficient

nn  Clustering coefficient for graph:

Each triangle gets

                                                                                                       # triangles x 3                                                   counted 3 times

# connected triples

nn  Also known as the “fraction of transitive triples”

Localized Clustering Coefficient

nn  Clustering for node v:

# actual edges between neighbors of v

# possible edges between neighbors of v

nn  Number of possible edges between k vertices:

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