1. Identity and search in social networks Presented by Pooja Deodhar Duncan J. Watts, Peter Sheridan Dodds and M. E. J. Newman

2. Presentation Outline Introduction Contentions – Social Networks Algorithm explanation Our model and Milgram’s findings Further Extensions Applications 2

3. Introduction Social Networks are “Searchable” Our model offers explanation of searchability in terms of recognizable personal identities Personal identities - sets of characteristics in different social dimensions Class of searchable networks and method for searching them applicable to many real world problems 3

4. Introduction Small World Network ◦ Network in which most nodes are not neighbors of each other but most nodes can be reached from every other node by a number of hops 4

5. Introduction Milgram’s Experiment ◦ Short paths exist between individuals in large social network ◦ Ordinary people can find these short paths ◦ People rarely have more than local knowledge about the network 5 Source

6. Introduction Searchability ◦ Property of being able to find a target quickly Shown to exist in networks ◦ With certain fraction of hubs (highly connected nodes which once reached can distribute messages to all parts of the network) ◦ Built upon underlying geometric lattice 6

7. Introduction Limited hubs in social networks Social Networks are more like a peer-to- peer network Need for a hierarchical model Some measure of distance between individuals Can be based on targets identity, friends identity, friend’s popularity 7

8. Contentions – Social Networks Individual identities – sets of characteristics attributed to them by virtue of association, participation in social groups Groups – Collection of individuals with well-defined set of social characteristics 8

9. Contentions – Social Networks Breaking down of world into set of layers Top layer – whole population Lower layers – specific division into groups 9

10. Contentions – Social Networks Similarity x ij – between individuals i, j x ij – Height of the lowest common ancestor level between i and j Individuals in same group are at distance of one from each other 10

11. Contentions – Social Networks Combined social distance y ij = min h x ij In the above figure H = 2 In 1 st heirarchy, y ij = 1 and y jk = 1 in 2 nd But y ik = 4 > y ij + y jk = 2 11

12. Contentions – Social Networks Probability of acquaintance between i and j decreases with decreasing similarity of groups to which they belong Link distance x for individual i has probability p(x) = ce - α x Measure of homophily – tendency of like to associate with like 12

13. Contentions – Social Networks Individuals hierarchically partition the social world in more than one way. ◦ h = 1, …, H hierarchies Node’s identity is the vector ◦ is position of node i in hierarchy h. Social distance 13

14. Contentions – Social Networks At each step the holder i of the message passes it to one of its friends who is closest to the target t in terms of social distance Individuals know the identity vectors of: ◦ themselves ◦ their friends, ◦ the target Two kinds of partial information – social distance and network paths 14

15. Algorithm Explanation Principal objective – determine conditions for average path length L of a message chain is small Define q as probability of an arbitrary message chain reaching a target. Searchable network - Any network for which q ≥ r for a desired r. 15

16. Searchability Searchable networks occupy a broad region of parameter space which are sociologically plausible Searchability is generic property of social networks 16

17. Algorithm Explanation In terms of chain length L, q = (1 - p) L ≥ r L = length of message chain P = message failure probability From above, L can be obtained by the approximate inequality, L <= ln r / ln (1 - p) 17

18. Our model and Milgram’s findings All searchable networks have α > 0, H > 1 Individuals are essentially homophilous but judge similarity along more than one social dimension Best performance is achieved for H = 2 or 3 Thus, use of 2 or 3 dimensions used by individuals in small world experiments when forwarding a message 18

19. Searchable Networks 19 Solid boundary – N=102,400 Dot-dash – N=204800 Dash – N=409,600 p = 0.25, b = 2, g = 100, r = 0.25 at least

20. Our model and Milgram’s findings Increasing number of independent dimensions from H = 1 yields dramatic reduction in delivery time for α > 0 This improvement lost as H is increased further Thus, network ties become less correlated as H increases For large H, network becomes a random graph, search algorithm becomes random walk 20

21. Searchable Networks 21 Probability of message completion when for α = 0 (squares) and for α = 2 (circles) for N = 102,400 Horizontal line – pos of the threshold Open symbols indicate network is searchable – q <= r

22. Our model and Milgram’s data n(L) – no. of completed chains of length L taken from original small world expt. (shown by bar graphs) Taken for example of our model for N = 10^8 individuals and for 42 completed chains shown by filled circles 22

23. Our model and Milgram’s findings Comparison of distribution of chain lengths in our model with that of Travers and Milgram Avg. chain length for Milgrams expt = 6.5 Avg. chain length for our model = 6.7 23

24. Summary Simple greedy algorithm. Represents properties present in real social networks: ◦ Considers local clustering. ◦ Reflects the notion of locality. High-level structure + random links. 24

25. Further Extensions Should we consider other parameters such as friend’s popularity information in addition to homophily? ◦ Allow variation in node degrees? Assume correlation between hierarchies? Are all hierarchies equally important? 25

26. Applications Broad class of decentralized problems ◦ Peer to peer networking Any data structure in which data elements can be judged along more than one dimension Designing of databases ◦ Eg. Music files – same genre/same year 26