SLANG Session 4. Jason Quinley Roland Mühlenbernd Seminar für Sprachwissenschaft University of Tübingen

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1 SLANG Session 4 Jason Quinley Roland Mühlenbernd Seminar für Sprachwissenschaft University of Tübingen

2 Overview Network properties Degree Density and Distribution Clustering and Connections Network formation Random network formation Strategic network formation

3 Network properties: Degree Density and Distribution Degree Density: How can we measure the proportion of connections 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 a random network of n nodes? Pr(d n, p) = ( n 1 d ) p d (1 p) n 1 d

4 Network properties: Two Distribution Types Observe the difference between the random (binomial/ Poisson) network and the scale-free network. What do you see?

5 Network properties: Poisson Distribution Poisson Distribution For a given node, how many links do we expect it to have? EX = (n 1)p Why? λ d e λ d! For λ = (n 1)p, we have Useful for: (n 1)p d e (n 1)p d! Approximating binomial distribution Large n or small p Counting improbable events

6 Network properties: Scale-Free Distribution 1 Scale free networks follow a power law distribution. That means, for a given node, P(d) = cd γ (1) Natural phenomena like Social and Sexual Networks Collaborative Networks (Authors, Actors) Word Length Distribution Wealth, Earthquakes, Insurgent Attacks

7 Network properties: Scale-Free Distribution 2 Why do we say scale free? Consider P(2) P(1) P(d) = cd γ (2) vs. P(20) P(10) What do you notice? Why are scale-free networks more stable to mutation? What does this say about their formation?

8 Network properties: Clustering and Cliques 1 Individual Clustering Suppose that i and j are linked. What is the probability i and k are linked? Clustering Coefficient(Overall) Cl(g) = i:unique i,j,k g ij g ik g jk g ij g ik Intuition: #(Triangles)/#(Open Jaws) Individual Clustering Cl i (g) = Average Clustering unique i,j,k g ij g ik g jk g ij g ik Cl AVG (g) = i Cl i (g) n

9 Network properties: Clustering 2 Compute Cl AVG (g) and Cl(g) for this network. What happens as the network grows large?

10 Network properties: Other Connection Properties 1 Homophily: Birds of a Feather Weak Ties: How frequent are interactions? Diameter: Longest Path 1 Weinen et al. Optimal partition and effective dynamics of complex networks (2007)

11 Network formation: Random networks For a network with n nodes there are n(n 1)/2 possible links For a random network with n nodes and probability p, the expected number of links is p n(n 1)/2 Example: n = 50, p = 0.01 Result: n(n 1)/2 = 1225, p n(n 1)/2 = 12.25

12 Network formation: Random networks

13 Network formation: Random networks Probability p = ln(n)/n is a threshold for which isolated nodes should disappear, i.e. the network becomes connected Example: If n = 50, set p > 0.078

14 Network formation: Random networks Given: n nodes, p = ln(n)/n Total possible links: n(n 1)/2 Expected links: ln(n)(n 1)/2 50% Pr(d) Number of links for a complete network and a random network which is expected to be connected d Degree distribution for a random network with n = 10 and p = ln(n)/n =.23

15 Network formation: Small World networks The Six degrees of separation (F. Karinthy, 1929)

16 Network formation: Small World networks The Small world experiment is a collection of several experiments examining the average path length for social networks of people in the United States. Basic procedure: Postcard questioning (S. Milgram, 1967) Starting point S: Omaha, Nebraska and Wichita, Kansas End point E: Boston, Massachusetts Ask a random s S: Do you know (random) e E? If not, do you know x who could know e E? Ask x: Do you know e E? etcetera Result: Average path length of around 5.5

17 Network formation: Small World networks The Six degrees of Kevin Bacon Example: Elvis Presley played together with Edward Asner in Change of habit (1969). Edward Asner played with Kevin Bacon in JFK (1991). Elvis Presley has Bacon-Number 2. Result: As of December 2010, the highest finite Bacon number reported by the Oracle of Bacon is 9. Erdos number...describes the collaborative distance between a person and mathematician Paul Erdos, as measured by authorship of mathematical papers. Result: Erdos had 511 direct collaborators (1). In 2007 there were 8,162 people with Erdos number 2.

18 Network formation: Small World networks Properties of a small world network Large network Small diameter Small average path length

19 Network formation: Small World networks Starting network: A double connected ring High degree of clustering:.5 High diameter: n/4 No variance in the degree distribution

20 Network formation: Small World networks Idea: Rewiring the network: Much smaller diameter but still exhibits substantial clustering. Example: Rewiring 6 links changes the diameter from 6 to 5, but has minimal impact on the clustering.

21 Network formation: Small World networks

22 Network formation: Strategic networks Basics There are many settings in which not only chance but choice plays a central role in determining relationships. Two central challenges in modelling strategic networks: 1. Explicitly model the costs and benefits arising from several networks 2. Make a prediction of how individual incentives translate into network outcomes Comparison between networks formed by individual incentives and networks maximizing overall society welfare

23 Network formation: Strategic networks Basic concepts: Utility function: u i : G(N) R Example: Distance-based utility of the Symmetric connection model Pairwise stability: A network g is pairwise stable relative to (u 1... u n ) iff 1. ij g : u i (g) > u i (g ij) and u j (g) > u j (g ij) 2. ij g : if u i (g + ij) > u i (g) then u j (g + ij) < u j (g) Efficiency: A network g is efficient relative to (u 1... u n ) iff g G : i u i (g) i u i (g )

24 Network formation: Strategic networks Symmetric connection model u i (g) = j i δ l ij (g) d i (g) c The unique efficient network structure in the distance-based utility model is: 1. The complete network if c < δ 1 δ 2 2. The empty network if δ 1 + (n 2)/2 δ 2 < c 3. A star network if δ 1 δ 2 < c < δ 1 + (n 2)/2 δ 2

25 Network formation: Strategic networks Extended version: Islands-Connections model u i (g) = δ l ij (g) j i:l ij D j:ij g if i and j on the same island c ij = c, else c ij = 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 other players) leads to low average path length 3. The high costs of linking to other islands leads to few links across islands The resulting network has small world properties c ij

26 Network formation: Strategic networks Externalities: 1. There are nonnegative externalities under u = (u 1... u n ) if u i (g + jk) u i (g) 2. There are nonpositive externalities under u = (u 1... u n ) if u i (g + jk) u i (g)

27 Network formation: Strategic networks The Coauthor Model Individuals benefit from interacting with others (for instance in collaborating on a research project) The benefit is measured in time, individuals put into a project The spent time is anti-proportional to the number of projects a author has, i.o.w. the degree (e.g. t i = 1/d i (g)) There is also a benefit in form of synergy, proportional to the to the product to the amount of time, both authors devote to the project u i (g) = j:ij g ( 1 d i (g) + 1 ) d j (g) + 1 d i (g) d j (g) for d i (g) > 0, and u i (g) = 1 if d i (g) = 0

28 Network formation: Strategic networks The Coauthor Model u i (g) = j:ij g ( 1 d i (g) + 1 ) d j (g) + 1 d i (g) d j (g) for d i (g) > 0, and u i (g) = 1 if d i (g) = Round Choice u 1 (g) u 2 (g) u 3 (g) 0 Init :(1 2) :(2 3) :(3 1) :NOOP Costs are implicit in the diluted synergy when efforts are spread among more coauthors.

29 Wrapup: What did we discuss? Network properties Degree density Degree distribution Clustering coefficient Homophily, weak ties, diameter Important network types Scale-free network Small world network Strategic networks Utility functions Resulting networks (stability, efficiency)