jason quinley roland m uhlenbernd seminar f ur

SLANGSession 4

Jason QuinleyRoland Muhlenbernd

Seminar fur SprachwissenschaftUniversity of Tubingen

Overview

Network propertiesDegree Density and DistributionClustering and Connections

Network formationRandom network formationStrategic network formation

Network properties: Degree Density and Distribution

Degree Density: How can we measure the proportion ofconnections a network could have?

Density =Avg .Degree

n − 1=

TotalDegree

n(n − 1)

Degree Distribution: How likely is a node to have a degree d in arandom network of n nodes?

Pr(d |n, p) =

(n − 1d

)pd(1− p)n−1−d

Network properties: Two Distribution Types

Observe the difference between the random (binomial/ Poisson)network and the scale-free network. What do you see?

Network properties: Poisson Distribution

For a given node, how manylinks do we expect it to have?EX = (n − 1)p Why?

Poisson Distribution

λde−λ

d!

For λ = (n − 1)p, we have

(n − 1)pde−(n−1)p

d!

Useful for:

I Approximating binomialdistribution

I Large n or small p

I Counting improbableevents

Network properties: Scale-Free Distribution 1

Scale free networks follow apower law distribution. Thatmeans, for a given node,

P(d) = cd−γ (1)

Natural phenomena like

I Social and SexualNetworks

I Collaborative Networks(Authors, Actors)

I Word Length Distribution

I Wealth, Earthquakes,Insurgent Attacks

Network properties: Scale-Free Distribution 2

Why do we say scale free?

P(d) = cd−γ (2)

Consider P(2)P(1) vs. P(20)

P(10)

I What do you notice?

I Why are scale-freenetworks more stable tomutation?

I What does this say abouttheir formation?

Network properties: Clustering and Cliques 1

Individual Clustering Suppose that i and j are linked. What is theprobability i and k are linked?Clustering Coefficient(Overall)

Cl(g) =∑

i :unique i ,j ,k

gijgikgjkgijgik

Intuition: #(Triangles)/#(Open Jaws)

Individual Clustering

Cli (g) =∑

unique i ,j ,k

gijgikgjkgijgik

Average Clustering

ClAVG (g) =∑i

Cli (g)

n

Network properties: Clustering 2

Compute ClAVG (g) and Cl(g) for this network.

What happens as the network grows large?

Network properties: Other Connection Properties

1

I Homophily: Birds of a Feather

I Weak Ties: How frequent are interactions?

I Diameter: Longest Path1Weinen et al. Optimal partition and effective dynamics of complex networks

(2007)

Network formation: Random networks

I For a network with n nodes there are n(n− 1)/2 possible linksI For a random network with n nodes and probability p, the

expected number of links is p × n(n − 1)/2I Example: n = 50, p = 0.01I Result: n(n − 1)/2 = 1225, p × n(n − 1)/2 = 12.25


I Probability p = ln(n)/n is a threshold for which isolated nodesshould disappear, i.e. the network becomes connected

I Example: If n = 50, set p > 0.078


Given: n nodes, p = ln(n)/n

I Total possible links: n(n − 1)/2

I Expected links: ln(n)(n − 1)/2

Number of links for a complete network and a ran-dom network which is expected to be connected

Pr(d)

50%

0 1 2 3 4 5 6 7 8 9 d

Degree distribution for a random networkwith n = 10 and p = ln(n)/n = .23

Network formation: Small World networks

The ”Six degrees of separation” (F. Karinthy, 1929)


The Small world experiment is a collection of several experimentsexamining the average path length for social networks of people inthe United States.

Basic procedure: Postcard questioning (S. Milgram, 1967)

I Starting point S : Omaha, Nebraska and Wichita, Kansas

I End point E : Boston, Massachusetts

I Ask a random s ∈ S : Do you know (random) e ∈ E?

I If not, do you know x who could know e ∈ E?

I Ask x : Do you know e ∈ E?

I etcetera

Result: Average path length of around 5.5


I The ”Six degrees of Kevin Bacon”Example: Elvis Presley played togetherwith Edward Asner in ”Change of habit”(1969). Edward Asner played with KevinBacon in ”JFK” (1991). Elvis Presley hasBacon-Number 2.Result: As of December 2010, the highestfinite Bacon number reported by theOracle of Bacon is 9.

I ”Erdos number”...describes the ”collaborative distance”between a person and mathematicianPaul Erdos, as measured by authorship ofmathematical papers.Result: Erdos had 511 direct collaborators(1). In 2007 there were 8,162 people withErdos number 2.


Properties of a small world networkI Large networkI Small diameterI Small average path length


Starting network: A double connected ringI High degree of clustering: .5I High diameter: bn/4cI No variance in the degree distribution


I Idea: Rewiring the network: Much smaller diameter but stillexhibits substantial clustering.

I Example: Rewiring 6 links changes the diameter from 6 to 5,but has minimal impact on the clustering.

Network formation: Strategic networks

Basics

I There are many settings in which not only chance but choiceplays a central role in determining relationships.

I Two central challenges in modelling strategic networks:

1. Explicitly model the costs and benefits arising from severalnetworks

2. Make a prediction of how individual incentives translate intonetwork outcomes

I Comparison between networks formed by individual incentivesand networks maximizing overall society welfare


Basic concepts:

I Utility function: ui : G (N)→ RExample: Distance-based utility of the Symmetric connectionmodel

I Pairwise stability: A network g is pairwise stable relative to(u1 . . . un) iff

1. ∀ij ∈ g : ui (g) > ui (g − ij) and uj(g) > uj(g − ij)2. ∀ij 6∈ g : if ui (g + ij) > ui (g) then uj(g + ij) < uj(g)

I Efficiency: A network g is efficient relative to (u1 . . . un) iff

∀g ′ ∈ G :∑i

ui (g) ≥∑i

ui (g′)


Symmetric connection model

ui (g) =∑j 6=i

δ`ij (g) − di (g)× c

The unique efficient network structure in the distance-based utilitymodel is:

1. The complete network if c < δ1 − δ2

2. The empty network if δ1 + (n − 2)/2× δ2 < c

3. A star network if δ1 − δ2 < c < δ1 + (n − 2)/2× δ2


Extended version: Islands-Connections model

ui (g) =∑

j 6=i :`ij≤Dδ`ij (g) −

∑j :ij∈g

cij

if i and j on the same island cij = c , else cij = C (C > c > 0)

Results:

1. Low costs to nearby players lead to high clustering

2. High value of linking to other islands (accessing many otherplayers) leads to low average path length

3. The high costs of linking to other islands leads to few linksacross islands

The resulting network has small world properties


Externalities:

1. There are nonnegative externalities under u = (u1 . . . un) if

ui (g + jk) ≥ ui (g)

2. There are nonpositive externalities under u = (u1 . . . un) if

ui (g + jk) ≤ ui (g)


The Coauthor Model

I Individuals benefit from interacting with others (for instancein collaborating on a research project)

I The benefit is measured in time, individuals put into a project

I The spent time is anti-proportional to the number of projectsa author has, i.o.w. the degree (e.g. ti = 1/di (g))

I There is also a benefit in form of synergy, proportional to theto the product to the amount of time, both authors devote tothe project

I

ui (g) =∑j :ij∈g

(1

di (g)+

1

dj(g)+

1

di (g)× dj(g)

)for di (g) > 0, and ui (g) = 1 if di (g) = 0


The Coauthor Model

ui (g) =∑j :ij∈g

(1

di (g)+

1

dj(g)+

1

di (g)× dj(g)

)for di (g) > 0, and ui (g) = 1 if di (g) = 0

1

2 3

Round Choice u1(g) u2(g) u3(g)0 Init 1 1 11 1:(1− 2) 3 3 12 2:(2− 3) 2 4 23 3:(3− 1) 2.5 2.5 2.54 1:NOOP 2.5 2.5 2.5

Costs are implicit in the diluted synergy when ef-forts are spread among more coauthors.

Wrapup: What did we discuss?

Network properties

I Degree density

I Degree distribution

I Clustering coefficient

I Homophily, weak ties, diameter

Important network types

I Scale-free network

I Small world network

Strategic networks

I Utility functions

I Resulting networks (stability, efficiency)

jason quinley roland m uhlenbernd seminar f ur

Documents