Company LOGO 1 Identity and Search in Social Networks D.J.Watts, P.S. Dodds, M.E.J. Newman Maryam Fazel-Zarandi

Company LOGO 2 Outlines  Introduction  The Hierarchical Model  Discussion

Company LOGO 3 Introduction

Company LOGO 4 Milgram’s Experiment  Short chains of acquaintances exist.  People are able to find these chains using only local information. Source

Company LOGO 5 Results in Literature  Connected random networks have short average path lengths:  x ij   log(N)  N = population size,  x ij = distance between nodes i and j.

Company LOGO 6 Results in Literature  Kleinberg (2000) demonstrated that emergence of the second phenomenon requires special topological structure.  For each node i:  local edges d(i,j) ≤ p  long-range directed edges to q random nodes Pr(i  j) ~ d(i,j) -a

Company LOGO 7 Results in Literature  If networks have a certain fraction of hubs can also search well.  Basic idea: get to hubs first  Hubs in social networks are limited.

Company LOGO 8 The Hierarchical Model

Company LOGO 9 Hierarchical Model – Why? How?  Basic idea: impose some high-level structure, and fill in details at random.  Incorporate identity.  Need some measure of distance between individuals.  Some possible knowledge:  Target's identity, friends' identities, friends' popularity, where the message has been.

Company LOGO 10 Hierarchical Network Construction  x ij = the height of the lowest common ancestor level between i and j  z connections for each node with probability: p(x) = ce -αx Hierarchical template for the network Network constructed from template

Company LOGO 11 Hierarchical Network Construction  Individuals hierarchically partition the social world in more than one way.  h = 1, …, H hierarchies  Identity vector  is position of node i in hierarchy h.  Social distance:

Company LOGO 12 Directing Messages  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.

Company LOGO 13 Expected Number of Steps  What is the expected number of steps to forward a message from a random source to a random target?  Define q as probability of an arbitrary message chain reaching a target.  Searchable network: Any network for which q ≥ r for a desired r.

Company LOGO 14 Number of Steps - Results  If message chains fail at each node with probability p, require where L = length of message chain.  Approximation:  L   ln r / ln (1 - p) q =  (1 - p) L  ≥ r

Company LOGO 15 Searchable Network Regions  In H-α space  p = 0.25, r = 0.05  b = 2  g = 100, z = 99  N=  N=  N=409600

Company LOGO 16 Probability of Message Completion  α = 0 (squares) versus α = 2 (circles)  N =  q ≥ r q < r r = 0.05

Company LOGO 17 Milgram's Data  N = 10 8  b = 10  g = 100  z = 300  L model   6.7  L data   6.5  α = 1, H = 2

Company LOGO 18 Discussion

Company LOGO 19 Is this an acceptable model?  Simple greedy algorithm.  Represents properties present in real social networks:  Considers local clustering.  Reflects the notion of locality.  High-level structure + random links.

Company LOGO 20 Can the Model be Extended?  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?

Company LOGO 21 Applications  Can solutions to sociology problems inform other areas of research?  Suggested applications:  Construction of peer-to-peer networks.  Search in databases.

Company LOGO 22 Thank You! Any Questions???