SLANGSession 3

Jason QuinleyRoland Muhlenbernd

Seminar fur SprachwissenschaftUniversity of Tubingen

Overview

PrefaceSociolinguistic SurveyHomework Review

Network formationRandom Network FormationStrategic Network Formation

Network propertiesNetwork RepresentationConnectionsCentrality

Exercises

Subjects: Language Variation and Change

1. Language VariationI Variation across social network typesI Variation as a Result of LearningI Dialects

2. Language ChangeI Strategic language changeI Information / Opinion spreadI Language contactI Language evolution

Subjects: Sociolinguistic Topics

1. Prestige and PolitenessI Politeness and trust gamesI Reputation and powerI Prestige and language evolutionI Turn-taking

2. Registers and DialectsI Dialect formationI Registers as strategiesI Language death

Homework

1. Network from 2.13

I APL: 1,2,6,7 = 1+1+2+3+4+46 = 15/6

I APL: 3,5 = 1+1+1+2+3+36 = 11/6

I APL: 4 = 1+1+2+2+2+26 = 10/6

I APL(N): = 4(15)+2(11)+107(6) = 46/21

I N1(g) = {2, 3, 4}, N2(g) = {1, 3, 4}, N3(g) = {1, 2, 4, 5},N4(g) = {1, 2, 3, 5, 6, 7}, N5(g) = {3, 4, 6, 7},N6(g) = {4, 5, 7}, N7(g) = {4, 5, 6}

2. Language vs. Dialect

I Why is the ”army and a navy” distinction given?I How do we distinguish them?

3. Networks

I Why study them? Lattice?I Superorganism

Network formation: Classical network structures

Star network

9

12

3

45

6

7

8

Ring network

12

3

45

6

7

8

Tree network

1

2 3

4 5 6 7

Complete network

1 2

3 4

Network formation: Random networks

I To create a random network we need:

I A set of nodes N = {1, . . . , n}I p: Probability for a link between any nodes i , j ;

0 < p < 1

I Exercise: Create & analyse a random network

1. For N = {1, 2, 3, 4, 5} find all possible links2. Set each possible link with probability p = .53. Calculate the average degree of the network4. Let’s collect the degrees of all your nodes

1

2 3

4 5

Network formation: Random networks

I The degree distribution of a random network describes theprobability that any given node will have a degree d .

I The probability that any given node i has degree d is

Pr(d |n, p) =

(n − 1d

)× pd × (1− p)n−1−d

I With n = 5 and p = .5: Pr(d) =

(4d

)× .5d × .54−d

I

Pr(d)

50%

0 1 2 3 4 d

Strategic network formation: Florentine Marriages

Source: Jackson, M. O. (2008), Social and Economic Networks, page 4

Network formation: Symmetric connection model

I Like in the Florentine marriage example links are chosen bythe agents in the network

I Players benefit from direct and indirect connections, butdeterioration over distances

I Benefit factor δd with 0 < δ < 1, d = distance between nodes

I Cost value c for maintaining a direct relationship

I For a given network g the utility for agent i is:

ui =∑j 6=i

δ`ij − di × c

Network formation: Symmetric connection model

Example: N = {1, 2, 3}, δ = .5, c = .3

1

2 3

Round Choice u1(g) u2(g) u3(g)0 Init 0 0 01 1:(1− 2) .2 .2 02 2:(2− 3) .45 .4 .453 3: Noop .45 .4 .45

Pairwise stability:

1. No agent can benefit by deleting a link he is involved in

2. No two agents can both benefit by adding a link betweenthemselves

Network Representation: Graphs and Matrices

I Graph G = (N, g)

I N = {1, 2, 3, 4}

I g =

0 1 0 10 0 0 11 1 0 00 0 1 0

1 2

3

4

I Graph G = (N, g)

I N = {1, 2, 3, 4}

I g =

0 1 1 01 0 1 01 1 0 10 0 1 0

Network Representation: Edges and Graphs

I Graph G = (N, g)

I N = {1, 2, 3, 4}I < 1, 2 >< 1, 4 >

I < 2, 4 >

I < 3, 1 >< 3, 2 >

I < 4, 3 >

1 2

3

4

I Graph G = (N, g)

I N = {1, 2, 3, 4}I {1,2} {1,3} {2,3} {3,4}

Network Representation: Edges and Matrices

I Graph G = (N, g)

I N = {1, 2, 3, 4}I < 1, 2 >< 1, 4 >

I < 2, 4 >

I < 3, 1 >< 3, 2 >

I < 4, 3 >

Matrix

I g =

0 1 0 10 0 0 11 1 0 00 0 1 0

I Graph G = (N, g)

I N = {1, 2, 3, 4}I {1,2} {1,3} {2,3} {3,4}

Matrix

I g =

0 1 1 01 0 1 01 1 0 10 0 1 0

Connections: Directed Graphs

Why directed graphs?

I Information Spread

I Multi-agentCommunication

Important Notions

I In- vs. Out-Degree

I Cycles, Paths, Walks

I What about the Matricesand Edge descriptions?

Connections: Paths and Cycles

I A walk is a sequence of links connecting asequence of nodes

I A path is a walk in which a node appearsat most once in the sequence

I A cycle is a walk in which each nodeappears at most once in the sequence,except the starting node, which alsoappears as the ending node

1 2

3 4

Connections: Euler Problem

How do we solve the Bridges of Konigs-berg problem?

I Goal: Walk the entire network withoutusing a bridge twice.

I Think about Degree.I The walk is directed. Think in terms

of In and Out.I What do you notice about the

degrees of each node?

I Theorem: An Eulerian path ispossible only if and only if there are atmost two nodes of odd degree.

I What happens with two nodes?Zero?

I Why can’t we have a single node ofodd degree?

Connections: Subgraphs and Cliques

I A subgraph/ subnetwork is asubset of the original graph’sconnections.

I A clique is a maximallyconnected subgraph. Howmany edges will a clique have?

I A component is a connectedsubgraph that is notconnected to other subgraphsof the network.

Centrality Measures: Going beyond Degree

How can we measure how central a node is?

I Degree

I APL

I Betweenness

I Neighborhood/ Prestige

Centrality Measures: Betweenness

Source: Jackson, M. O. (2008), Social and Economic Net-works, page 4

I P(ij): Number of shortestpaths between node i and j

I Pk(ij): Number of thosepaths including node k

I Betweenness of k: B(k) =∑ij :i 6=j,k 6∈{i,j}

Pk(ij)/P(ij)

(n − 1)(n − 2)/2

I Some resultsI B(Medici) = .522I B(Guadagni) = .255I B(Strozzi) = .103

Wrapup: What did we discuss today?

How can we synthesize networks?

I Randomly

I Strategically

How can we analyze networks?

I Node/ Local properties

I Global Properties

I Statistically

I Analytically

How can we represent networks?

I Graphs

I Links

I Matrices