Long Tails and Navigation Networked Life CSE 112 Spring 2007 Prof. Michael Kearns

One More ( Structural ) Property… A properly tuned  -model can simultaneously explain –small diameter –high clustering coefficient –other models can, too (e.g. cycle+random rewirings) But what about connectors and heavy-tailed degree distributions? –  -model and simple variants will not explain this –intuitively, no “bias” towards large degree evolves –all vertices are created equal Can concoct many bad generative models to explain –generate NW according to Erdos-Renyi, reject if tails not heavy –describe fixed NWs with heavy tails all connected to v1; N/2 connected to v2; etc. not clear we can get a precise power law not modeling variation –why would the world evolve this way? As always, we want a “natural” model

Quantifying Connectors: Heavy-Tailed Distributions

Heavy-tailed Distributions Pareto or power law distributions: –for random variables assuming integer values > 0 –probability of value x ~ 1/x^  –typically 0 <  < 2; smaller  gives heavier tail –here are some examplesexamples –sometimes also referred to as being scale-free For binomial, normal, and Poisson distributions the tail probabilities approach 0 exponentially fast Inverse polynomial decay vs. inverse exponential decay What kind of phenomena does this distribution model? What kind of process would generate it?

Distributions vs. Data All these distributions are idealized models In practice, we do not see distributions, but data Thus, there will be some largest value we observe Also, can be difficult to “eyeball” data and choose model So how do we distinguish between Poisson, power law, etc? Typical procedure: –might restrict our attention to a range of values of interest –accumulate counts of observed data into equal-sized bins –look at counts on a log-log plot –note that power law: –log(Pr[X = x]) = log(1/x^  ) = -  log(x) –linear, slope – linear, slope –  Normal: –log(Pr[X = x]) = log(a exp(-x^2/b)) = log(a) – x^2/b –non-linear, concave near meannon-linear, concave near mean Poisson: –log(Pr[X = x]) = log(exp(- ) ^x/x!) –also non-linear Let’s look at the assigned paper on dollar bill migrationdollar bill migration

Zipf’s Law Look at the frequency of English words: –“the” is the most common, followed by “of”, “to”, etc. –claim: frequency of the n-th most common ~ 1/n (power law,  = 1) General theme: –rank events by their frequency of occurrence –resulting distribution often is a power law! Other examples: –North America city sizes –personal income –file sizes –genus sizes (number of species) –let’s look at log-log plots of theselog-log plots People seem to dither over exact form of these distributions –e.g. value of  –but not over heavy tails

Generating Heavy-Tailed Degrees: (Just) One Model

Preferential Attachment Let’s first look at a little “Rich Get Richer” Matlab demo… Start with (say) two vertices connected by an edge For i = 3 to N: –for each 1 <= j < i, let d(j) be degree of vertex j (so far) –let Z =  d(j) (sum of all degrees so far) –add new vertex i with k edges back to {1,…,i-1}: i is connected back to j with probability d(j)/Z might need to do some “smoothing” to deal with zero-degree vertices Vertices j with high degree are likely to get more links! “Rich get richer” or “Matthew Effect” Natural model for many processes: –hyperlinks on the web –new business and social contacts –transportation networks Generates a power law distribution of degrees –exponent depends on value of k Let’s look at the NetLogo simulationsimulation

Two Out of Three Isn’t Bad… Preferential attachment explains –heavy-tailed degree distributions –small diameter (~log(N), via “hubs”) Will not generate high clustering coefficient –no bias towards local connectivity, but towards hubs Can we simultaneously capture all three properties? –probably, but we’ll stop here –soon there will be a fourth property anyway…

Search and Navigation

Finding the Short Paths Milgram’s experiment, Columbia Small Worlds, E-R,  -model… –all emphasize existence of short paths between pairs How do individuals find short paths? –in an incremental, next-step fashion –using purely local information about the NW and location of target –note: shortest path might require taking steps “away” from the target! This is not a structural question, but an algorithmic one –statics vs. dynamics Navigability may impose additional restrictions on formation model! Briefly investigate two alternatives: –a local/long-distance mixture model –a “social identity” model

Kleinberg’s Model Start with an n by n grid of vertices (so N = n^2) –add local connections: all vertices within grid distance p (e.g. 2) –add distant connections: q additional connections probability of connection at distance d: ~ (1/d)^r –so full model given by choice of p, q and r –large r: heavy bias towards “more local” long-distance connections –small r: approach uniformly random Kleinberg’s question: –what value of r permits effective search? Assume parties know only: –grid address of target –addresses of their own direct links Algorithm: pass message to neighbor closest to target

Kleinberg’s Result Intuition: –if r is too large (strong local bias), then “long-distance” connections never help much; short paths may not even exist –if r is too small (no local bias), we may quickly get close to the target; but then we’ll have to use local links to finish think of a transport system with only long-haul jets or donkey carts –effective search requires a delicate mixture of link distances The result (informally): –r = 2 is the only value that permits rapid navigation (~log(N) steps) –any other value of r will result in time ~ N^c for 0 < c <= 1 –N^c >> log(N) for large N –a critical value phenomenon or “knife’s edge”; very sensitive –contrast with 1/d^(1.59) from dollar bill migration paper Note: locality of information crucial to this argument –centralized algorithm may compute short paths at r < 2 –can recognize when “backwards” steps are beneficial Later: What happens when distance-d edges cost d^r?

Navigation via Identity Watts et al.: –we don’t navigate social networks by purely “geographic” information –we don’t use any single criterion; recall Dodds et al. on Columbia SW –different criteria used a different points in the chain Represent individuals by a vector of attributes –profession, religion, hobbies, education, background, etc… –attribute values have distances between them (tree-structured) –distance between individuals: minimum distance in any attribute –only need one thing in common to be close! Algorithm: –given attribute vector of target –forward message to neighbor closest to target Permits fast navigation under broad conditions –not as sensitive as Kleinberg’s model all jobs scientists athletes chemistry CS baseball tennis