The Science of Social Networks ECECS 728 March 29, 2006

Outline Small Worlds Random Graphs Alpha and Beta Power Laws Searchable Networks Six Degrees of Separation

Outline Small Worlds Random Graphs Alpha and Beta Power Laws Searchable Networks Six Degrees of Separation

Society as a Graph People are represented as nodes.

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

People are represented as nodes. Relationships are represented as edges. (Relationships may be acquaintanceship, friendship, co-authorship, etc.) Allows analysis using tools of mathematical graph theory Society as a Graph

The Kevin Bacon Game Invented by Albright College students in 1994: –Craig Fass, Brian Turtle, Mike Ginelly Goal: Connect any actor to Kevin Bacon, by linking actors who have acted in the same movie. Oracle of Bacon website uses Internet Movie Database (IMDB.com) to find shortest link between any two actors: Boxed version of the Kevin Bacon Game

The Kevin Bacon Game Kevin Bacon An Example Tim Robbins Om Puri Amitabh Bachchan Yuva (2004) Mystic River (2003) Code 46 (2003) Rani Mukherjee Black (2005)

The Kevin Bacon Game Total # of actors in database: ~550,000 Average path length to Kevin: 2.79 Actor closest to “center”: Rod Steiger (2.53) Rank of Kevin, in closeness to center: 876 th Most actors are within three links of each other! Center of Hollywood?

Not Quite the Kevin Bacon Game Kevin Bacon Aidan Quinn Kevin Spacey Kentaro Toyama Bringing Down the House (2004) Cavedweller (2004) Looking for Richard (1996) Ben Mezrich Roommates in college (1991)

Erdős Number Number of links required to connect scholars to Erdős, via co- authorship of papers Erdős wrote papers with 507 co-authors. Jerry Grossman’s (Oakland Univ.) website allows mathematicians to compute their Erdos numbers: Connecting path lengths, among mathematicians only: –average is 4.65 –maximum is 13 Paul Erdős ( )

Outline Small Worlds Random Graphs Alpha and Beta Power Laws Searchable Networks Six Degrees of Separation

Random Graphs G(N,p) N nodes p is probability of edge being in graph. Average degree, k ≈ p(N-1) Poisson degree distribution p k What interesting things can be said for different valuesof p? What is true as N  ∞ Erdős and Renyi (1959) p = 0.0 ; k = 0 N = 12 p = 0.09 ; k = 1 p = 1.0 ; k ≈ ½N 2

Random Graphs Erdős and Renyi (1959) Let’s look at… Size of the largest connected cluster Diameter (maximum path length between nodes) of the largest cluster Average path length between nodes (if a path exists)

Random Graphs Erdős and Renyi (1959) p = 0.0 ; k = 0p = 0.09 ; k = 1p = 1.0 ; k ≈ ½N 2 p = ; k = 0.5 Size of largest component Diameter of largest component Average path length between nodes

Random Graphs If k < 1: –small, isolated clusters –small diameters –short path lengths At k = 1: –a giant component appears –diameter peaks –path lengths are high For k > 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) k phase transition

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 (k >> 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 Erdős and Renyi (1959) David Mumford Peter Belhumeur Kentaro Toya ma Fan Chung

Outline Small Worlds Random Graphs Alpha and Beta Power Laws Searchable Networks Six Degrees of Separation

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) * Same Anatol Rapoport, known for TIT FOR TAT! The Personal Map by MSR Redmond’s Social Computing Group

The Alpha Model Watts (1999) “Preferential Attachment”  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.)

The Beta Model Watts and Strogatz (1998) “Link Rewiring”  = 0  =  = 1 People know others at random. Not clustered, but “small world” People know their neighbors, and a few distant people. Clustered and “small world” People know their neighbors. Clustered, but not a “small world”

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 

Outline Small Worlds Random Graphs Alpha and Beta Power Laws Searchable Networks Six Degrees of Separation

Searchable Networks Just because a short path exists, doesn’t mean you can easily find it. You don’t know all of the people whom your friends know. Under what conditions is a network searchable? Kleinberg (2000)

Searchable Networks a)Variation of Watts’s  model: –Lattice is d-dimensional (d=2). –One random link per node. –Parameter r controls probability of random link – greater for closer nodes. –node u is connected to node v with probability proportional to d(u,v)^-r Kleinberg (2000)

Fundamental consequences of model When long­range contacts are formed independently of the geometry of the grid, short chains will exist but the nodes, operating at a local level, will not be able to find them. When long­range contacts are formed by a process that is related to the geometry of the grid in a specific way, however, then short chains will still form and nodes operating with local knowledge will be able to construct them.

Theorem 1: Effective routing is impossible in uniformly random graphs. When r = 0, the expected delivery time of any decentralized algorithm is at least O(n^2/3), and hence exponential in the expected minimum path length. Theorem 2: Greedy routing is effective in certain random graphs. When r = 2, there is a decentralized (greedy) algorithm, so that the expected delivery time is at most O( logn^2), hence quadratic in expected path length.

Proof Sketch for Lower Bound The impossibility result is based on the fact that the uniform distribution prevents a decentralized algorithm from using any “clues'' provided by the geometry of the grid. Consider the set U of all nodes within lattice distance n^2/3 of destination t. With high probability, the source s will lie outside of U, and if the message is never passed from a node to a long-range contact in U, the number of steps needed to reach t will be at least proportional to n^2/3. But the probability that any message holder has a long-range contact in U is roughly n^(4/3)/n^2 = n^-2/3, so the expected number of steps before a long-range contact in U is found is at least proportional to n^2/3 as well.

Proof Sketch for Upper Bound Th. 2 Greedy algorithm always moves us closer. Consider phases that move the message half the distance to destination. (Recall Zeno’s paradox). Probability of connecting to a node at distance d is ~ 1/(d^2 lgn) and there are ~ d^2 nodes at distance d from destination. Thus ~lg n steps will end the phase. So with lg n phases we are done lg^2 n time

Searchable Networks Watts, Dodds, Newman (2002) show that for d = 2 or 3, real networks are quite searchable. Killworth and Bernard (1978) found that people tended to search their networks by d = 2: geography and profession. Kleinberg (2000) The Watts-Dodds-Newman model closely fitting a real-world experiment

Outline Small Worlds Random Graphs Alpha and Beta Power Laws Searchable Networks Six Degrees of Separation

Power Laws Albert and Barabasi (1999) Degree distribution of a random graph, N = 10,000 p = k = 15. (Curve is a Poisson curve, for comparison.) What’s the degree (number of edges) distribution over a graph, for real-world graphs? Random-graph model results in Poisson distribution. But, many real-world networks exhibit a power-law distribution.

Power Laws Albert and Barabasi (1999) Typical shape of a power-law distribution. What’s the degree (number of edges) distribution over a graph, for real-world graphs? Random-graph model results in Poisson distribution. But, many real-world networks exhibit a power-law distribution.

Power Laws Albert and Barabasi (1999) Power-law distributions are straight lines in log-log space. How should random graphs be generated to create a power-law distribution of node degrees? Hint: Pareto’s* Law: Wealth distribution follows a power law. Power laws in real networks: (a) WWW hyperlinks (b) co-starring in movies (c) co-authorship of physicists (d) co-authorship of neuroscientists * Same Velfredo Pareto, who defined Pareto optimality in game theory.

Power laws, examples cont’ed length of file transfers [Bestavros+] web hit counts [Huberman] magnitude of earthquakes (Guttenberg- Richter law) sizes of lakes/islands (Korcak’s law) Income distribution (Pareto’s law)

Power Laws “The rich get richer!” Power-law distribution of node distribution arises if -- Preferential Attachment (again) –Number of nodes grow; –Edges are added in proportion to the number of edges a node already has. Additional variable fitness coefficient allows for some nodes to grow faster than others. Albert and Barabasi (1999) “Map of the Internet” poster

Outline Small Worlds Random Graphs Alpha and Beta Power Laws Searchable Networks Six Degrees of Separation

The experiment: Random people from Nebraska were to send a letter (via intermediaries) to a stock broker in Boston. Could only send to someone with whom they were on a first-name basis. Among the letters that found the target, the average number of links was six. Milgram (1967) Stanley Milgram ( )

Internet Map [lumeta.com] Food Web [Martinez ’91] Protein Interactions [genomebiology.com] Friendship Network [Moody ’01] Graphs are everywhere!

Applications of Network Theory World Wide Web and hyperlink structure The Internet and router connectivity Collaborations among… –Movie actors –Scientists and mathematicians Sexual interaction Cellular networks in biology Food webs in ecology Phone call patterns Word co-occurrence in text Neural network connectivity of flatworms Conformational states in protein folding

Summary of network ‘laws’ Random and Social Networks have short paths (log n diameter) Social networks have cluster properties New results suggest shrinking diameter (‘<6 degrees’) and densification Power laws for degree distributions

Simulation and Network Generation Wish list for a generator: Power-law-tail in- and out-degrees shrinking/constant or log diameter Densification Power Law communities-within-communities Q: how to achieve all of them?

Credits Albert, Reka and A.-L. Barabasi. “Statistical mechanics of complex networks.” Reviews of Modern Physics, 74(1): (2002) Barabasi, Albert-Laszlo. Linked. Plume Publishing. (2003) Kleinberg, Jon M. “Navigation in a small world.” Science, 406:845. (2000) Watts, Duncan. Six Degrees: The Science of a Connected Age. W. W. Norton & Co. (2003) Slides by Kentaro Toyama - Microsoft Research India