Exploring Networks: From Structure to Dynamics

1. Networks • Networks are everywhere • Internet • Neurons is brains • Social networks • Transportation • Networks have been studied long time • Euler (1736): Bridges of Königsberg  theory of graphs, which is now a major (and difficult! – or almost obvious) branch in mathematics

2. So what is new? • Watts, Barabási, etc during the last 10 or so years • Previously networks have been viewed as objects of pure structure whose properties are fixed in time • Real networks represent populations of individual components doing something • Networks themselves also evolve and change in time

3. The small world hypothesis • The biggest social network: entire population of the world • How far am I from any one individual? • ”Six degrees of separation” • ..if I know 100 people, each of whom know 100 people ... in 6 steps = 1006 • ..but often friends of my friends are my friends • so? • Similar problem for travellers: • How many legs of journey do I need to have to travel to anywhere in the world?

4. Random graphs (Erdös, Rényi 1959) • Nodes connected by links in a purely random fashion • How large is the largest connected component? (as a fraction of all nodes) • Depends on the number of 1-1 connections we have made, varies from 0 to 1

5. Connectivity of a random graph 1 Disconnected phase Fraction of all nodes in largest component Conected phase 0 1 Average number of links per node

6. Social networks • One line of thought in sociology: • links an individual has and the groups to which she belongs to, are the signature of her social identity • network structure is closely related to social structure • Another line of thought: • individuals’ social role depends on his relations and to what information these links give him access to and how he can exert influence along these links

7. The strength of weak ties • Granovetter (1973): effective social coordination does not arise from densely interlocking strong ties, but derives from the occasional weak ties • this is because valuable information comes from these relations (it is valuable if/because it is not available to other individuals in your immediate network)

8. Structure and dynamics • The case of centrality • centers are in networks • by design (central control, dictatorship) • by non-design (unnoticed critical resources, informal groups) • or they emerge as a consequence of certain events • ”he was at the right place at a right time” • clapping in unison

9. Away from randomness we find emergence • Rapoport (1950’s): random-biased networks • an attempt to study formally nonrandom networks • in relation to research on spreading of diseases among human populations • ”if A knows B and B knows C, then it is likely that eventually A will know C”

10. Evolution of a random-biased net B A C D

11. Two extreme types of social networks • Caveman’s world • people live in isolated communities • probability meeting a random person is high if you have mutual friends and very low if you don’t • Solaria • people live isolated from each other but with supreme communication capabilities • your social history is irrelevant to your future

12. =0 =1 = Caveman world Likelihood that A meets B Solaria world Number of mutual friends shared by A and B

13. Alpha network • Watts (1999) • simulations with alpha networks • Clustering coefficient • probability that people you know, know also each other • Path length • Number of people in a chain between you and any other person

14. Clustering coefficient C Small- world net- works Fragmented networks Alpha network Path length L critical 

15. Beta network • Watts and Strogatz (1998) • In ring lattices • Original state: each node is connected to a fixed number of nearest neighbors • ”Rewiring”: if node A is linked to node B, the link is changed into a link from A to a randomly chosen node B’ • Each link is rewired with probability beta (), =0: nothing happens, =1: the network becomes completely random

16. Beta network 1 C Small- world networks 0 L 0 1 

17. Centrality again • All nodes in alpha and beta networks are equal in the sense that the number of connections each nodes has is not very far from the average • Watts and Strogatz had used normal distribution • Real world is not like that • Sizes of cities, Wealth of individuals in USA, Hubs in transportation systems • Barabási and Albert (1999) • Scale-free networks, whose connectivity is defined by a power-law distribution

18. The Pareto Distribution • The Pareto distribution gives the probability that a person's income is greater than or equal to x and is expressed as

19. log-log plot Pareto distribution, m=10000, k=1 Pareto distribution is said to be scale-free because it lacks a characteristic length scale

20. Social structure in networks • Two types of nodes: individuals and affiliations • an affiliation is a thing in common, a group to which a number of individuals belong to • Bipartite networks • individuals can only connect to affiliations and vice versa • Watts et al: random affiliation networks will always be small-world networks

21. Getting practical: search in networks • A node may be linked to another node via a short path but what does it matter if you cannot find the path? • In alpha and beta networks there is no notion of distance, therefore directed searches cannot utilize the shortcuts • Kleinbergs (2000) gamma network • probability of a new random link between two nodes (in 2-dimensional lattice) given their distance is controlled by a parameter ()

22. short paths cannot be found no short paths Typical length of directed search 2  =0 increasing  log p(r) log r

23. When gamma is at its critical value two, the resulting network has the peculiar property that nodes possess the same number of ties at all length scales

24. More hierarchy • Kleinberg’s model has only one distance measure, geographical • In human society the social distance is multidimensional • if A is close to B and C is close to B but in different dimension then A and C can be very far from each other • ”violation of the triangle inequality” • but multidimensionality may enable messages to be transmitted in networks very efficiently

25. Watts et al (2002) search in social networks • = homophily, the tendency of like to associate with like H=number of dimensions along which individuals measure similarity 6  Searchable networks 0 1 10 Kleinberg condition H

26. Cascades in networks • Epidemic diseases • Infectiousness • Kermack and Kendrick (1927) SIR model: nodes are either susceptible, infected, or removed • Flory and Stockmayer (1940’s) Percolation theory: each node is susceptible or not (with occupation probability) each link is open or closed (with probability which is equivalent to infectiousness) • percolating cluster: a single cluster of susceptible sites connected by open bonds that permeates the entire population

27. Failures in scale-free networks • Albert and Barabási (2000) • Scale-free networks are far more resistant to random failures than ordinary random networks • because of hubs • But hubs can be vulnerable or targets of deliberate attacks • which may make scale-free networks less resistant • Cascades of failures

28. 1 Probability of infection 0 Number of infected neighbors Threshold models of decisions 1 Probability of choosing option A 0 Critical Threshold Fraction of neighbors choosing A over B Standard disease spreading model Social decision making