The Science of Networks 6.1

Overview

Social Goal. Explain why information and disease spread so quickly in social networks.

Mathematical Approach. Model social networks as random graphs and argue that they are likely to have low diameter.

The Science of Networks 6.2

Definition: The clustering coefficient of a node v is the fraction of pairs of v’s friends that are connected to each other by edges.

Clustering Coefficient = 1/2

The higher the clustering coefficient of a node, the more strongly triadic closure is acting on it

The Science of Networks 6.3

Erdos-Renyi Graph models

Randomly choose How well does the E-R model characterize

real networks? high school friendship networks road networks peer-to-peer filesharing networks product co-purchase networks (people who

bought X also bought Y) other?

The Science of Networks 6.4

Random graph diameter

Technique. Grow trees to bound path lengths.

1. Show when trees are small enough (< √n), # of leaves doubles.

2. Grow small trees (< √n) around a pair of nodes.

3. Use birthday paradox to argue trees prob. intersect.

Conclude diameter is 2 log √n = log n.

The Science of Networks 6.5

Small world phenomenon

Milgram’s experiment (1960s).

Ask someone to pass a letter to another person via friends knowing only the name, address, and occupation of the target.

The Science of Networks 6.6

Small world phenomenon

Bob, a farmer in Nebraska

David, mayor of Bob’s town

Bernard, David’s cousin who went to college with

Maya, who grew up in Boston

With Lashawn

The Science of Networks 6.7

Last time: short paths exist.(argument: flood the network)

The Science of Networks 6.8

This time: and people can find short paths!(without flooding the network)

The Science of Networks 6.9

The Milgram experiment

Experiment:

We choose a random “source” and “target”.

Your goal: pass ball from “source” to “target” by throwing it to people you know on a first-name basis.

The Science of Networks 6.10

How did you find short paths?

The Science of Networks 6.11

What information did you use to choose the next hop?

The Science of Networks 6.12

Social network models

Random rolodex model

Random wiring model

The Science of Networks 6.13

Random wiring model

People have a predictable structure of local links (e.g., neighbors, colleagues)

And a few random long-range links (e.g., someone you meet on a trip)

The Science of Networks 6.14

Random wiring model

Local links have grid-like structure (you know the person on your left/right and front/back)

n

n

The Science of Networks 6.15

Random wiring model

Local links represent homophily, the idea that we know people similar to us

n

n

The Science of Networks 6.16

Random wiring model

Long-range links are random – each node chooses one long-range link

n

n

The Science of Networks 6.17

Random wiring model

Long-range links represent weak ties, the links to acquantainces that would otherwise be far away

n

n

The Science of Networks 6.18

Random wiring model

What do you expect the diameter to be?Do you expect people to find short paths?

If so, how short?

The Science of Networks 6.19

We’ll end up repeatedly

over-shooting target.

What if long-range links are

uniformly random?

Short paths?

Problem: navigational clues lost in long-range links.

The Science of Networks 6.20

We’ll end up taking forever

to get anywhere.

What if long-range links are very localized?

Short paths?

Problem: increases path-length.

The Science of Networks 6.21

Decentralized search

Idea: Suppose long-range links are just slightly more likely to be to close nodes.

Result: Then decentralized search finds short paths.

The Science of Networks 6.22

Tradeoff

Discovered path length

uniform

path length

Actual path length

highly local

The Science of Networks 6.23

Optimal tradeoff

Suppose links are proportional to (1/distance)2,

i.e., inverse square.

Inverse square?? WTF?

The Science of Networks 6.24

Inverse square intuition

“Scales of resolution”:

LSRC D235450 Research

Dr.Durham, NC

USA

Room numbe

r

BuildingStreet

City StateCount

ry

The Science of Networks 6.25

Scales of resolution

Street: location within

2 miles

The Science of Networks 6.26

Scales of resolution

City: location within 4 miles

The Science of Networks 6.27

Scales of resolution

County: location within

8 miles

The Science of Networks 6.28

Scales of resolution

Each new scale doubles distance from the center.

The Science of Networks 6.29

Scales of resolution

Long-range links equally likely to connect to each different scale of resolution!

(allows people to make progress towards destination no matter how

far away they are)

The Science of Networks 6.30

How many people do you know?

The Science of Networks 6.31

How many people do you know?

The Science of Networks 6.32

How many people do you know?

The Science of Networks 6.33

How many people do you know?

The Science of Networks 6.34

How many people do you know?

The Science of Networks 6.35

How many people do you know?

The Science of Networks 6.36

How many people do you know?

The Science of Networks 6.37

How well did this work?

For me:Neighborhood (Trinity Heights) Region (RTP)State (NC)SoutheastEastern USUnited StatesWorld

The Science of Networks 6.38

Next topic

decentralized search

The Science of Networks 6.39

How to route

Problem. How can I get this message

from me to the far-away target?

Solution. Pass message to a friend.

closer

(sub)

The Science of Networks 6.40

Scales of resolution

Each new scale doubles distance from the center.

The Science of Networks 6.41

Long-range links

Suppose each person has a long-range friend in each scale of resolution.

The Science of Networks 6.42

How to route

Algorithm. Pass the message to your farthest friend that is to the left of the target.

The Science of Networks 6.43

Trace of route

The Science of Networks 6.44

Analysis

old dist.

1 2 4 2j 2j+

1

new dist.

The Science of Networks 6.45

Distance is cut in half every step!

The Science of Networks 6.46

Analysis

1. Original distance is ?2. Distance is cut in half every step (at

least).3. Number of steps is ?

at most n.

at most log n.

The Science of Networks 6.47

And in real life …

The Science of Networks 6.48

Finding the Short Paths Milgram’s experiment, Columbia Small Worlds,

E-R, a-model… all emphasize existence of short paths between pairs

How do individuals find short paths? in an incremental, next-step fashion using purely local information about the NW and

location of target note: shortest path might require taking steps “away”

from the target! This is not (only) a structural question, but an

algorithmic one statics vs. dynamics

Navigability may impose additional restrictions on formation model!

Briefly investigate two alternatives: a local/long-distance mixture model [Kleinberg] a “social identity” model [Watts, Dodd, Newman]

The Science of Networks 6.49

Kleinberg’s Model Start with an n by n grid of vertices (so N =

n^2) add some long-distance connections to each vertex:

• k additional connections• probability of connection to grid distance d: ~ (1/d)^r

– c.f. dollar bill migration paper so full model given by choice of k and r large r: heavy bias towards “more local” long-distance

connections small r: approach uniformly random

Kleinberg’s question: what value of r permits effective navigation? # hops << N, e.g. log(N)

Assume parties know only: grid address of target addresses of their own immediate neighbors

Algorithm: pass message to nbr closest to target in grid

The Science of Networks 6.50

Kleinberg’s Result Intuition:

if r is too large (strong local bias), then “long-distance” connections never help much; short paths may not even exist (remember, grid has large diameter, ~ sqrt(N))

if r is too small (no local bias), we may quickly get close to the target; but then we’ll have to use grid links to finish

• think of a transport system with only long-haul jets or donkey carts

effective search requires a delicate mixture of link distances

The result (informally): r = 2 is the only value that permits rapid navigation

(~log(N) steps) any other value of r will result in time ~ N^c for 0 < c

<= 1• N^c >> log(N) for large N

a critical value phenomenon or “knife’s edge”; very sensitive

Note: locality of information crucial to this argument centralized, “birds-eye” algorithm can still compute

short paths at r < 2! can recognize when “backwards” steps are beneficial

The Science of Networks 6.51

Navigation via Identity Watts et al.:

we don’t navigate social networks by purely “geographic” information

we don’t use any single criterion; recall Dodds et al. on Columbia SW

different criteria used at different points in the chain Represent individuals by a vector of attributes

profession, religion, hobbies, education, background, etc…

attribute values have distances between them (tree-structured)

distance between individuals: minimum distance in any attribute

only need one thing in common to be close! Algorithm:

given attribute vector of target forward message to neighbor closest to target

Permits fast navigation under broad conditions not as sensitive as Kleinberg’s model

all jobs

scientistsathletes

chemistryCS

soccertrack