1. Small World Model Original slides from Kentaro Toyama, Microsoft Research, India and Dr. Rao

2. People are represented as nodes. Relationships are represented as edges. (Relationships may be acquaintanceship, friendship, co-authorship, etc.)‏ Society as a Graph

3. Properties Size –Number of nodes Density –Number of ties that are present the amount of ties that could be present Out-degree –Sum of connections from a node to others In-degree –Sum of connections to a node

4. Distance Walk –A sequence of nodes and relations that begins and ends with nodes Geodesic distance –The number of relations in the shortest possible walk from one node to another Maximum flow –The amount of different nodes in the neighborhood of a source that lead to pathways to a target

5. Some Measures of Power & Prestige (based on Hanneman, 2001)‏ Degree –Sum of connections from or to an node Transitive weighted degree  Authority, hub, pagerank Closeness centrality –Distance of one node to all others in the network Betweenness centrality –Number that represents how frequently an node is between other nodes’ geodesic paths

6. Cliques and Social Roles (based on Hanneman, 2001)‏ Cliques –Sub-set of nodes More closely tied to each other than to nodes who are not part of the sub-set –(A lot of work on “trawling” for communities in the web-graph)‏ –Often, you first find the clique (or a densely connected subgraph) and then try to interpret what the clique is about Social roles –Defined by regularities in the patterns of relations among nodes

7. Small world model (Watts and Strogatz 1998)‏ Small world effect : Most pairs of vertices are connected by a short path through the network Vertex-vertex distance in the network increases logarithmically with the total number of vertices in the network High clustering or 'transitivity' – High probability that two vertices will be connected directly if they have another neighboring vertex in common

8. Random Graphs N nodes A pair of nodes has probability p of being connected. Average degree, z ≈ pN What interesting things can be said for different values of p or z ? (that are true as N  ∞ )‏ Erdős and Renyi (1959)‏ p = 0.0 ; z = 0 N = 12 p = 0.09 ; z = 1 p = 1.0 ; z ≈ ½N 2

9. Random Graphs Erdős and Renyi (1959)‏ p = 0.0 ; z = 0p = 0.09 ; z = 1p = 1.0 ; z ≈ Np = 0.045 ; z = 0.5 Size of largest component Diameter of largest component Average path length between (connected) nodes 1 5 11 12 0 4 7 1 0.0 2.0 4.2 1.0

10. Random Graphs If z < 1: –small, isolated clusters –small diameters –short path lengths At z = 1: –a giant component appears –diameter peaks –path lengths are high For z > 1: –almost all nodes connected –diameter shrinks –path lengths shorten Erdős and Renyi (1959)‏ Percentage of nodes in largest component Diameter of largest component (not to scale)‏ 1.0 0 z phase transition

11. Random Graphs What does this mean? If connections between people can be modeled as a random graph, then… –Because the average person easily knows more than one person (z >> 1), –We live in a “small world” where within a few links, we are connected to anyone in the world. –Erdős and Renyi showed that average path length between connected nodes is (ln N) / (ln z )‏ Erdős and Renyi (1959)‏

12. Random vs. Real Social networks Random network models introduce an edge between any pair of vertices with a probability p –The problem here is NOT randomness, but rather the distribution used (which, in this case, is uniform)‏ Real networks are not exactly like these –Tend to have a relatively few nodes of high connectivity (the “Hub” nodes)‏ –These networks are called “Scale-free” networks Macro properties scale- invariant

13. Degree Distribution & Power Laws Degree distribution of a random graph, N = 10,000 p = 0.0015 k = 15. (Curve is a Poisson curve, for comparison.)‏ But, many real-world networks exhibit a power-law distribution.  also called “Heavy tailed” distribution Typically 2<r<3. For web graph r ~ 2.1 for in degree distribution 2.7 for out degree distribution Note that poisson decays exponentially while power law decays polynomially Long tail k -r Sharp drop Rare events are not so rare!

14. Measure of clustering – Clustering coefficients Watts and Strogatz (1998)‏ Newman et al.(2002)‏

15. Clustering Clustering measures the fraction of neighbors of a node that are connected themselves Regular Graphs have a high clustering coefficient –but also a high diameter Random Graphs have a low clustering coefficient –but a low diameter Both models do match the properties expected from real networks! Random Graph (z=4)‏ Short path length –L~log k N Almost no clustering –C~z/n Regular Graph (z=4)‏ Long paths –L ~ n/(2z) Highly clustered –C~3/4 Base metwork is circle

16. Properties – Small world Model On a random graph p = z/n for all vertex pairs C rg = z/n Watts and Strogatz(1998) defined a network to have high clustering if C >> C rg Small world network if Mean vertex-vertex distance l is comparable with that on a random graph l/ l rg ~ 1 Clustering coefficient is much greater than that for a random graph C/C rg >> 1

17. 0.0650.228.7134Food web 0.0900.5928.3315Metabolic network 0.0490.2814.0282Neural network 0.000150.4470.1460902Word occurrence 0.00190.5914.47673Company directors 0.000250.20113.4449913Film actor collaborations 0.0000150.153.9253339Mathematics collaborations 0.0000100.08115.51520251Biology collaborations 0.000540.0802.74941Power grid 0.000230.1135.2153127World Wide Web 0.000600.243.86374Internet Random graphMeasuredznNetwork Clustering coefficient C

18. Natural networks Between regular grids and totally random graphs Need for parametrized models: –Regular -> natural -> random Watts –Alpha model ( not intuitive)‏ –Beta rewiring model 

19. Models for natural networks Watts’ models are based on four elements. Social networks consist of many small overlapping groups that are densely internally connected and that overlap by virtue of individuals having multiple affiliations Social networks are not static objects Not all potential relationships are equally likely Occasional random relationships (Eg. Occasionally actors do things that are derived entirely from actors’ preferences and characteristics, and these preferences may lead actors to meet new people who have no connection to their previous friends at all)‏

20. The Alpha Model The people you know aren’t randomly chosen. People tend to get to know those who are two links away (Rapoport *, 1957). The real world exhibits a lot of clustering. Watts (1999)‏ The Personal Map by MSR Redmond’s Social Computing Group

21. Alpha network Alpha (  ) distance parameter –  =0 : if A and B have a friend in common, they know each other (Caveman world)‏ –  =∞ : A & B don’t know each other, no matter how many common friends they have (Solarian world)‏

22. The Alpha Model Watts (1999)‏  model: Add edges to nodes, as in random graphs, but makes links more likely when two nodes have a common friend. For a range of  values: –The world is small (average path length is short), and –Groups tend to form (high clustering coefficient). Probability of linkage as a function of number of mutual friends (  is 0 in upper left, 1 in diagonal, and ∞ in bottom right curves.)‏

23. The Alpha Model Watts (1999)‏  Clustering coefficient / Normalized path length Clustering coefficient (C) and average path length (L)‏ plotted against   model: Add edges to nodes, as in random graphs, but makes links more likely when two nodes have a common friend. For a range of  values: –The world is small (average path length is short), and –Groups tend to form (high clustering coefficient).

24. Random rewiring Random rewiring of regular graph (by Watts and Strogatz) –With probability p (or  ) rewire each link in a regular graph to a randomly selected node –Resulting graph has properties, both of regular and random graphs High clustering and short path length –FreeNet has been shown to result in small world graphs

25. The Beta Model Watts and Strogatz (1998)‏  = 0  = 0.125  = 1 People know others at random. Not clustered but low average path length People know their neighbors, and a few distant people. Clustered and low average path length People know their neighbors. Clustered and high average path length

26. The Beta Model First five random links reduce the average path length of the network by half, regardless of N! Both  and  models reproduce short-path results of random graphs, but also allow for clustering. Small-world phenomena occur at threshold between order and chaos. Watts and Strogatz (1998)‏ Clustering coefficient / Normalized path length Clustering coefficient (C) and average path length (L) plotted against 

27. Scale-free Networks Scale-free networks also exhibit small-world phenomena However, scale-free networks tend to be more brittle –You can drastically reduce the connectivity by deliberately taking out a few nodes This can also be seen as an opportunity.. –Disease prevention by quarantaining super-spreaders As they actually did to poor Typhoid Mary..

28. Attacks vs. Disruptions on Scale-free vs. Random networks Disruption –A random percentage of the nodes are removed How does the diameter change? –Increases monotonically and linearly in random graphs –Remains almost the same in scale-free networks Since a random sample is unlikely to pick the high- degree nodes Attack –A precentage of nodes are removed willfully (e.g. in decreasing order of connectivity)‏ How does the diameter change? –For random networks, essentially no difference from disruption All nodes are approximately same –For scale-free networks, diameter doubles for every 5% node removal! This is an opportunity when you are fighting to contain spread…

29. Exploiting/Navigating Small-Worlds Case 1: Centralized access to network structure –Paths between nodes can be computed by shortest path algorithms E.g. All pairs shortest path –..so, small-world ness is trivial to exploit.. This is what Facebook, Orkut, Friendster etc are trying to do.. Case 2: Local access to network structure –Each node only knows its own neighborhood –Search without children- generation function  –Idea 1: Broadcast method Obviously crazy as it increases traffic everywhere –Idea 2: Directed search But which neighbors to select? Are there conditions under which decentralized search can still be easy? How does a node in a social network find a path to another node?  6 degrees of separation will lead to n 6 search space (n=num neighbors)‏  Easy if we have global graph.. But hard otherwise Computing one’s Erdos number used to take days in the past! There are very few “fully decentralized” search applications. You normally have hybrid methods between Case 1 and Case 2

30. Searchability in Small World Networks Searchability is measured in terms of Expected time to go from a random source to a random destination –We know that in Smallworld networks, the diameter is exponentially smaller than the size of the network. –If the expected time is proportional to some small power of log N, we are doing well Qn: Is this always the case in small world networks? To begin to answer this we need to look generative models that take a notion of absolute (lattice or coordinate-based) neighborhood into account Kleinberg experimented with Lattice networks (where the network is embedded in a lattice—with most connections to the lattice neighbors, but a few shortcuts to distant neighbors)‏ and found that the answer is “Not always” Kleinberg (2000)‏

31. Neighborhood based random networks Lattice is d-dimensional (d=2). One random link per node. Probability that there is a link between two nodes u and v is r(u,v) -  –r(u,v) is the “lattice” distance between u and v (computed as manhattan distance)‏ As against geodesic or network distance computed in terms of number of edges –E.g. North-Rim and South-Rim  determines how steeply the probability of links to far away neighbors reduces

32. Searcheability in lattice networks For d=2, dip in time-to-search at  =2 –For low , random graph; no “geographic” correlation in links –For high , not a small world; no short paths to be found. Searcheability dips at  =2 (inverse square distribution), in simulation Corresponds to using greedy heuristic of sending message to the node with the least lattice distance to goal For d-dimensional lattice, minimum occurs at  =d

33. Searchable Networks Watts, Dodds, Newman (2002) show that for d = 2 or 3, real networks are quite searchable.  the dimensions are things like “geography”, “profession”, “hobbies” Killworth and Bernard (1978) found that people tended to search their networks by d = 2: geography and profession. The Watts-Dodds-Newman model closely fitting a real-world experiment

34. ..but didn’t Milgram’s letter experiment show that navigation is easy? …may be not –A large fraction of his test subjects were stockbrokers So are likely to know how to reach the “goal” stockbroker –A large fraction of his test subjects were in boston As was the “goal” stockbroker –A large fraction of letters never reached Only 20% reached So how about (re)doing Milgram experiment with emails? –People are even more burned out with (e)mails now Success rate for chain completion < 1% !

35. Summary A network is considered to exhibit small world phenomenon, if its diameter is approximately logarithm of its size and exhibits high clustering Most uniform random networks have low diameter Most real world networks are not uniform random –Their in degree distribution exhibits power law behavior –However, most power law random networks also exhibit small world phenomena –But they are brittle against attack The fact that a network exhibits small world phenomenon doesn’t mean that an agent with strictly local knowledge can efficiently navigate it (i.e, find paths that are O(log(n)) length –It is always possible to find the short paths if we have global knowledge This is the case in the FOAF (friend of a friend) networks on the web