News and Notes: Feb 9

1
News and Notes Feb 9

• Watts talk reminder
• tomorrow at noon, Annenberg School (3620 Walnut),
  Room 110
• extra credit reports
• Turn in revisions of NW Construction Project,
  Task 1
• MK will review quickly
• deadline for Task 2 set shortly start working!
• Description of Tuesday class experiments
• Social Network Theory, continued

2
Collective Human Computation in Networks Beyond
Shortest Paths

• Travers and Milgram, Dodds et al., Kleinberg,
• human networks can efficiently route messages
• using only local topology and info on target
• What about other computations?
• minimum coloring
• maximum matching
• maximum independent set
• Participation on Tuesday is for course credit
• Start at 1205 sharp
• You will be given a score for each experiment
• but as long as you participate, you will receive
  full credit
• 50 cash prize will be split between those with
  the highest total score
• An experimental investigation of the Price of
  Anarchy
• comparison of centralized social optimum and
  decentralized greedy solutions

3
Graph Colorings

• A coloring of an undirected graph is
• an assignment of a color (label) to each vertex
• such that no pair connected by an edge have the
  same color
• chromatic number of graph G fewest colors needed
• Example application
• classes and exam slots
• chromatic number determines length of exam period
• Heres a coloring demo
• Computation of chromatic numbers is hard
• (poor) approximations are possible
• Interesting fact the four-color theorem for
  planar graphs
• Here is a description of our Lifester Coloring
  Experiment

4
Matchings in Graphs

• A matching of an undirected graph is
• a subset of the edges
• such that no vertex is touched more than once
• perfect matching every vertex touched exactly
  once
• perfect matchings may not always exist (e.g. N
  odd)
• maximum matching largest number of edges
• Can be found efficiently here is a perfect
  matching demo
• Example applications
• pairing of compatible partners
• perfect matching nobody left out
• jobs and qualified workers
• perfect matching full employment, and all jobs
  filled
• clients and servers
• perfect matching all clients served, and no
  server idle
• Here is a description of our Lifester Matching
  Experiment

5
Cliques and Independent Sets

• A clique in a graph G is a set of vertices
• informal that are all directly connected to each
  other
• formal whose induced subgraph is complete
• all vertices in direct communication, exchange,
  competition, etc.
• the tightest possible social structure
• an edge is a clique of just 2 vertices
• generally interested in large cliques
• Independent set
• set of vertices whose induced subgraph is empty
  (no edges)
• vertices entirely isolated from each other
  without help of others
• Maximum clique or independent set largest in the
  graph
• Maximal clique or independent set cant grow any
  larger
• Here is a description of our Lifester Independent
  Set Experiment

6
The Results
7
The chromatic number of the Lifester network is
4...
8
and the 43 class members present computed a
legal 5-coloring.
9
The Lifester network has a maximum independent
set of size 16...
10
and the class computed a maximal independent
set of size 13. (mean degree of winners 4 mean
degree of losers 5.3)
11
The Lifester network has a maximum matching of
size 21 and the class found one. (mean degree of
score 2 5 mean degree of others 3.8)
12
Just 40 More Times and You Can Buy a Share of
Google
CHEN,CHARLENE CHENG,ZAISHAO FAULKNER,ELIZABETH FRA
NK,WILLIAM GROFF,MAX JOHNNIDIS,CHRISTOPHER LAWEE,
AARON LEIKER,MATTHEW MUTREJA,MOHIT RYTERBAND,JASON
SILENGO,MICHAEL SWANSON,EDWARD
Post-experiment analysis assignment due in class
Tuesday!
13
Social Network Theory

• Networked Life
• CSE 112
• Spring 2005
• Prof. Michael Kearns

14
Natural Networks and Universality

• Consider the many kinds of networks we have
  examined
• social, technological, business, economic,
  content,
• These networks tend to share certain informal
  properties
• large scale continual growth
• distributed, organic growth vertices decide
  who to link to
• interaction restricted to links
• mixture of local and long-distance connections
• abstract notions of distance geographical,
  content, social,
• Do natural networks share more quantitative
  universals?
• What would these universals be?
• How can we make them precise and measure them?
• How can we explain their universality?
• This is the domain of social network theory
• Sometimes also referred to as link analysis

15
Some Interesting Quantities

• Connected components
• how many, and how large?
• Network diameter
• maximum (worst-case) or average?
• exclude infinite distances? (disconnected
  components)
• the small-world phenomenon
• Clustering
• to what extent to links tend to cluster
  locally?
• what is the balance between local and
  long-distance connections?
• what roles do the two types of links play?
• Degree distribution
• what is the typical degree in the network?
• what is the overall distribution?

16
A Canonical Natural Network has

• Few connected components
• often only 1 or a small number independent of
  network size
• Small diameter
• often a constant independent of network size
  (like 6)
• or perhaps growing only logarithmically with
  network size
• typically exclude infinite distances
• A high degree of clustering
• considerably more so than for a random network
• in tension with small diameter
• A heavy-tailed degree distribution
• a small but reliable number of high-degree
  vertices
• quantifies Gladwells connectors
• often of power law form

17
Some Models of Network Generation

• Random graphs (Erdos-Renyi models)
• gives few components and small diameter
• does not give high clustering and heavy-tailed
  degree distributions
• is the mathematically most well-studied and
  understood model
• Watts-Strogatz and related models
• give few components, small diameter and high
  clustering
• does not give heavy-tailed degree distributions
• Preferential attachment
• gives few components, small diameter and
  heavy-tailed distribution
• does not give high clustering
• Hierarchical networks
• few components, small diameter, high clustering,
  heavy-tailed
• Affiliation networks
• models group-actor formation
• Nothing magic about any of the measures or
  models

18
Approximate Roadmap

• Examine a series of models of network generation
• macroscopic properties they do and do not entail
• pros and cons of each model
• Examine some real life case studies
• Study some dynamics issues (e.g. navigation)
• Move into in-depth study of the web as network

19
Probabilistic Models of Networks

• All of the network generation models we will
  study are probabilistic or statistical in nature
• They can generate networks of any size
• They often have various parameters that can be
  set
• size of network generated
• average degree of a vertex
• fraction of long-distance connections
• The models generate a distribution over networks
• Statements are always statistical in nature
• with high probability, diameter is small
• on average, degree distribution has heavy tail
• Thus, were going to need some basic statistics
  and probability theory

20
Statistics and Probability TheoryThe Absolute,
Bare Minimum Essentials
21
Probability and Random Variables

• A random variable X is simply a variable that
  probabilistically assumes values in some set
• set of possible values sometimes called the
  sample space S of X
• sample space may be small and simple or large and
  complex
• S Heads, Tails, X is outcome of a coin flip
• S 0,1,,U.S. population size, X is number
  voting democratic
• S all networks of size N, X is generated by
  preferential attachment
• Behavior of X determined by its distribution (or
  density)
• for each value x in S, specify PrX x
• these probabilities sum to exactly 1 (mutually
  exclusive outcomes)
• complex sample spaces (such as large networks)
• distribution often defined implicitly by simpler
  components
• might specify the probability that each edge
  appears independently
• this induces a probability distribution over
  networks
• may be difficult to compute induced distribution

22
Some Basic Notions and Laws

• Independence
• let X and Y be random variables
• independence for any x and y, PrX x Y y
  PrXxPrYy
• intuition value of X does not influence value of
  Y, vice-versa
• dependence
• e.g. X, Y coin flips, but Y is always opposite of
  X
• Expected (mean) value of X
• only makes sense for numeric random variables
• average value of X according to its
  distribution
• formally, EX S (PrX x X), sum is over all
  x in S
• often denoted by m
• always true EX Y EX EY
• true only for independent random variables EXY
  EXEY
• Variance of X
• Var(X) E(X m)2 often denoted by s2
• standard deviation is sqrt(Var(X)) s
• Union bound
• for any X, Y, PrXx or Yy lt PrXx PrYy

23
Convergence to Expectations

• Let X1, X2,, Xn be
• independent random variables
• with the same distribution PrXx
• expectation m EX and variance s2
• independent and identically distributed (i.i.d.)
• essentially n repeated trials of the same
  experiment
• natural to examine r.v. Z (1/n) S Xi, where sum
  is over i1,,n
• example number of heads in a sequence of coin
  flips
• example degree of a vertex in the random graph
  model
• EZ EX what can we say about the
  distribution of Z?
• Central Limit Theorem
• as n becomes large, Z becomes normally
  distributed
• with expectation m and variance s2/n
• heres a demo

24
The Normal Distribution

• The normal or Gaussian density
• applies to continuous, real-valued random
  variables
• characterized by mean (average) m and standard
  deviation s
• density at x is defined as
• (1/(s sqrt(2p))) exp(-(x-m)2/2s2)
• special case m 0, s 1 a exp(-x2/b) for some
  constants a,b gt 0
• peaks at x m, then dies off exponentially
  rapidly
• the classic bell-shaped curve
• exam scores, human body temperature,
• here are some examples
• remarks
• can control mean and standard deviation
  independently
• can make as broad as we like, but always have
  finite variance

25
The Binomial Distribution

• The binomial distribution
• coin with Prheads p, flip n times
• probability of getting exactly k heads
• choose(n,k) pk (1-p)(n-k)
• for large n and p fixed
• approximated well by a normal with m pn, s
  sqrt(np(1-p))
• s/m ? 0 as n grows
• leads to strong large deviation bounds

26
The Poisson Distribution

• The Poisson distribution
• like binomial, applies to variables taken on
  integer values gt 0
• often used to model counts of events
• number of phone calls placed in a given time
  period
• number of times a neuron fires in a given time
  period
• single free parameter l
• probability of exactly x events
• exp(-l) lx/x!
• mean and variance are both l
• here are some examples
• binomial distribution with n large, p l/n (l
  fixed)
• converges to Poisson with mean l

27
Heavy-tailed Distributions

• Pareto or power law distributions
• for variables assuming integer values gt 0
• probability of value x 1/xa
• typically 0 lt a lt 2 smaller a gives heavier tail
• here are some examples
• 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?

28
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(PrX x) log(1/xa) -a log(x)
• linear, slope a
• Normal
• log(PrX x) log(a exp(-x2/b)) log(a)
  x2/b
• non-linear, concave near mean
• Poisson
• log(PrX x) log(exp(-l) lx/x!)
• also non-linear

29
Zipfs 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, a 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)
• lets look at log-log plots of these
• People seem to dither over exact form of these
  distributions (e.g. value of a), but not heavy
  tails

30
Models of Network Generationand Their Properties
31
The Erdos-Renyi (ER) Model(Random Graphs)

• A model in which all edges
• are equally probable
• appear independently
• NW size N gt 1 and probability p distribution
  G(N,p)
• each edge (u,v) chosen to appear with probability
  p
• N(N-1)/2 trials of a biased coin flip
• The usual regime of interest is when p 1/N, N
  is large
• e.g. p 1/2N, p 1/N, p 2/N, p10/N, p
  log(N)/N, etc.
• in expectation, each vertex will have a small
  number of neighbors
• will then examine what happens when N ? infinity
• can thus study properties of large networks with
  bounded degree
• Degree distribution of a typical G drawn from
  G(N,p)
• draw G according to G(N,p) look at a random
  vertex u in G
• what is Prdeg(u) k for any fixed k?
• Poisson distribution with mean l p(N-1) pN
• Sharply concentrated not heavy-tailed
• Especially easy to generate NWs from G(N,p)

32
A Closely Related Model

• For any fixed m lt N(N-1)/2, define distribution
  G(N,m)
• choose uniformly at random from all graphs with
  exactly m edges
• G(N,m) is like G(N,p) with p m/(N(N-1)/2)
  2m/N2
• this intuition can be made precise, and is
  correct
• if m cN then p 2c/(N-1) 2c/N
• mathematically trickier than G(N,p)

33
Another Closely Related Model

• Graph process model
• start with N vertices and no edges
• at each time step, add a new edge
• choose new edge randomly from among all missing
  edges
• Allows study of the evolution or emergence of
  properties
• as the number of edges m grows in relation to N
• equivalently, as p is increased
• For all of these models
• high probability ?? almost all large graphs of
  a given density

34
The Evolution of a Random Network

• We have a large number n of vertices
• We start randomly adding edges one at a time
• At what time t will the network
• have at least one large connected component?
• have a single connected component?
• have small diameter?
• have a large clique?
• have a large chromatic number?
• How gradually or suddenly do these properties
  appear?

35
Recap

• Model G(N,p)
• select each of the possible edges independently
  with prob. p
• expected total number of edges is pN(N-1)/2
• expected degree of a vertex is p(N-1)
• degree will obey a Poisson distribution (not
  heavy-tailed)
• Model G(N,m)
• select exactly m of the N(N-1)/2 edges to appear
• all sets of m edges equally likely
• Graph process model
• starting with no edges, just keep adding one edge
  at a time
• always choose next edge randomly from among all
  missing edges
• Threshold or tipping for (say) connectivity
• fewer than m m(N) edges ? graph almost
  certainly not connected
• more than m m(N) edges ? graph almost certainly
  is connected
• made formal by examining limit as N ? infinity

36
Combining and Formalizing Familiar Ideas

• Explaining universal behavior through statistical
  models
• our models will always generate many networks
• almost all of them will share certain properties
  (universals)
• Explaining tipping through incremental growth
• we gradually add edges, or gradually increase
  edge probability p
• many properties will emerge very suddenly during
  this process

prob. NW connected
number of edges
37
Monotone Network Properties

• Often interested in monotone graph properties
• let G have the property
• add edges to G to obtain G
• then G must have the property also
• Examples
• G is connected
• G has diameter lt d (not exactly d)
• G has a clique of size gt k (not exactly k)
• G has chromatic number gt c (not exactly c)
• G has a matching of size gt m
• d, k, c, m may depend on NW size N (How?)
• Difficult to study emergence of non-monotone
  properties as the number of edges is increased
• what would it mean?

38
Formalizing TippingThresholds for Monotone
Properties

• Consider Erdos-Renyi G(N,m) model
• select m edges at random to include in G
• Let P be some monotone property of graphs
• P(G) 1 ? G has the property
• P(G) 0 ? G does not have the property
• Let m(N) be some function of NW size N
• formalize idea that property P appears suddenly
  at m(N) edges
• Say that m(N) is a threshold function for P if
• let m(N) be any function of N
• look at ratio r(N) m(N)/m(N) as N ? infinity
• if r(N) ? 0 probability that P(G) 1 in
  G(N,m(N)) ? 0
• if r(N) ? infinity probability that P(G) 1 in
  G(N,m(N)) ? 1
• A purely structural definition of tipping
• tipping results from incremental increase in
  connectivity

39
So Which Properties Tip?

• Just about all of them!
• The following properties all have threshold
  functions
• having a giant component
• being connected
• having a perfect matching (N even)
• having small diameter
• Demo look at the following progression
• giant component ? connectivity ? small diameter
• in graph process model (add one new edge at a
  time)
• example 1 example 2 example 3 example 4
  example 5
• With remarkable consistency (N 50)
• giant component 40 edges, connected 100,
  small diameter 180

40
Ever More Precise

• Connected component of size gt N/2
• threshold function is m(N) N/2 (or p 1/N)
• note full connectivity impossible
• Fully connected
• threshold function is m(N) (N/2)log(N) (or p
  log(N)/N)
• NW remains extremely sparse only log(N) edges
  per vertex
• Small diameter
• threshold is m(N) N(3/2) for diameter 2 (or p
  2/sqrt(N))
• fraction of possible edges still 2/sqrt(N) ? 0
• generate very small worlds

41
Other Tipping Points?

• Perfect matchings
• consider only even N
• threshold function is m(N) (N/2)log(N) (or p
  log(N)/N)
• same as for connectivity!
• Cliques
• k-clique threshold is m(N) (1/2)N(2 2/(k-1))
  (p 1/N(2/k-1))
• edges appear immediately triangles at N/2 etc.
• Coloring
• k colors required just as k-cliques appear

42
Erdos-Renyi Summary

• A model in which all connections are equally
  likely
• each of the N(N-1)/2 edges chosen randomly
  independently
• As we add edges, a precise sequence of events
  unfolds
• graph acquires a giant component
• graph becomes connected
• graph acquires small diameter
• etc.
• Many properties appear very suddenly (tipping,
  thresholds)
• All statements are mathematically precise
• But is this how natural networks form?
• If not, which aspects are unrealistic?
• maybe all edges are not equally likely!

43
The Clustering Coefficient of a Network

• Let nbr(u) denote the set of neighbors of u in a
  graph
• all vertices v such that the edge (u,v) is in the
  graph
• The clustering coefficient of u
• let k nbr(u) (i.e., number of neighbors of u)
• choose(k,2) max possible of edges between
  vertices in nbr(u)
• c(u) (actual of edges between vertices in
  nbr(u))/choose(k,2)
• 0 lt c(u) lt 1 measure of cliquishness of us
  neighborhood
• Clustering coefficient of a graph
• average of c(u) over all vertices u

k 4 choose(k,2) 6 c(u) 4/6 0.666
44
Erdos-Renyi Clustering Coefficient

• Generate a network G according to G(N,p)
• Examine a typical vertex u in G
• choose u at random among all vertices in G
• what do we expect c(u) to be?
• Answer exactly p!
• In G(N,m), expect c(u) to be 2m/N(N-1)
• Both cases c(u) entirely determined by overall
  density
• Baseline for comparison with more clustered
  models
• Erdos-Renyi has no bias towards clustered or
  local edges

45
Caveman and Solaria

• Erdos-Renyi
• sharing a common neighbor makes two vertices no
  more likely to be directly connected than two
  very distant vertices
• every edge appears entirely independently of
  existing structure
• But in many settings, the opposite is true
• you tend to meet new friends through your old
  friends
• two web pages pointing to a third might share a
  topic
• two companies selling goods to a third are in
  related industries
• Watts Caveman world
• overall density of edges is low
• but two vertices with a common neighbor are
  likely connected
• Watts Solaria world
• overall density of edges low no special bias
  towards local edges
• like Erdos-Renyi

46
Making it (Somewhat) Precise the a-model

• The a-model has the following parameters or
  knobs
• N size of the network to be generated
• k the average degree of a vertex in the network
  to be generated
• p the default probability two vertices are
  connected
• a adjustable parameter dictating bias towards
  local connections
• For any vertices u and v
• define m(u,v) to be the number of common
  neighbors (so far)
• Key quantity the propensity R(u,v) of u to
  connect to v
• if m(u,v) gt k, R(u,v) 1 (share too many
  friends not to connect)
• if m(u,v) 0, R(u,v) p (no mutual friends ? no
  bias to connect)
• else, R(u,v) p (m(u,v)/k)a (1-p)
• here are some plots for different a (see Watts
  page 77)
• Generate NW incrementally
• using R(u,v) as the edge probability details
  omitted
• Note a infinity is like Erdos-Renyi (but not
  exactly)

47
Small Worlds and Occams Razor

• For small a, should generate large clustering
  coefficients
• we programmed the model to do so
• Watts claims that proving precise statements is
  hard
• But we do not want a new model for every little
  property
• Erdos-Renyi ? small diameter
• a-model ? high clustering coefficient
• etc.
• In the interests of Occams Razor, we would like
  to find
• a single, simple model of network generation
• that simultaneously captures many properties
• Watts small world small diameter and high
  clustering
• here is a figure showing that this can be
  captured in the a-model

48
Meanwhile, Back in the Real World

• Watts examines three real networks as case
  studies
• the Kevin Bacon graph
• the Western states power grid
• the C. elegans nervous system
• For each of these networks, he
• computes its size, diameter, and clustering
  coefficient
• compares diameter and clustering to best
  Erdos-Renyi approx.
• shows that the best a-model approximation is
  better
• important to be fair to each model by finding
  best fit
• Overall moral
• if we care only about diameter and clustering, a
  is better than p

49
Case 1 Kevin Bacon Graph

• Vertices actors and actresses
• Edge between u and v if they appeared in a film
  together
• Here is the data

50
Case 2 Western States Power Grid

• Vertices power stations in Western U.S.
• Edges high-voltage power transmission lines
• Here is the network and data

51
Case 3 C. Elegans Nervous System

• Vertices neurons in the C. elegans worm
• Edges axons/synapses between neurons
• Here is the network and data

52
Two More Examples

• M. Newman on scientific collaboration networks
• coauthorship networks in several distinct
  communities
• differences in degrees (papers per author)
• empirical verification of
• giant components
• small diameter (mean distance)
• high clustering coefficient
• Alberich et al. on the Marvel Universe
• purely fictional social network
• two characters linked if they appeared together
  in an issue
• empirical verification of
• heavy-tailed distribution of degrees (issues and
  characters)
• giant component
• rather small clustering coefficient

53
One More (Structural) Property

• A properly tuned a-model can simultaneously
  explain
• small diameter
• high clustering coefficient
• But what about heavy-tailed degree distributions?
• a-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

54
Preferential Attachment

• Start with (say) two vertices connected by an
  edge
• For i 3 to N
• for each 1 lt j lt i, let d(j) be degree of vertex
  j (so far)
• let Z S 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
• Vertices j with high degree are likely to get
  more links!
• Rich get richer
• 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

55
Two Out of Three Isnt 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 well stop here
• soon there will be a fourth property anyway

56
Two Out of Three Isnt 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 well stop here
• soon there will be a fourth property anyway

57
The Midterm

• Midterm date this Thursday, March 3
• Exam handed out beginning at 12 sharp
• Pencils down at 120 sharp
• Closed-book exam only exams and pencils
• no books, papers, notes, devices, etc.
• Exam covers everything to date
• all assigned readings in books and papers
• all lectures, including todays
• all assignments and experiments
• Todays agenda
• short lecture on search and navigation
• quick midterm review
• NW Construction Project Task 2 due at midnight

58
Search and Navigation
59
Finding Short Paths

• Milgrams experiment, Columbia Small Worlds,
  a-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
• This is not a structural question, but an
  algorithmic one
• statics vs. dynamics
• Navigability may impose additional restrictions
  on model!
• Briefly investigate two alternatives
• variation on the a-model
• a social identity model

60
Kleinbergs Model

• Similar in spirit to the a-model
• Start with an n by n grid of vertices (so N
  n2)
• 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/dr
• so full model given by choice of p, q and r
• small r heavy bias towards more local
  long-distance connections
• large r approach uniformly random
• Kleinbergs 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

61
Kleinbergs Result

• Intuition
• if r is too small (strong local bias), then
  long-distance connections never help much
  short paths may not even exist
• if r is too large (no local bias), we may quickly
  get close to the target but then well 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 Nc
  for 0 lt c lt 1
• a critical value phenomenon
• Note locality of information crucial to this
  argument
• centralized algorithm may compute short paths at
  large r
• can recognize when backwards steps are
  beneficial

62
Navigation via Identity

• Watts et al.
• we dont navigate social networks by purely
  geographic information
• we dont 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 Kleinbergs model

63
Next Up The Web as Network