Network Social Theory
Networked Life CSE 112 Spring 2006 Prof. Michael Kearns
Natural Networks and Universality
Consider the many kinds of networks we have examined: These networks tend to share certain informal properties:
large scale; continual growth distributed, organic growth: vertices decide who to link to interaction (largely) restricted to links mixture of local and long-distance connections abstract notions of distance: geographical, content, social, social, technological, business, economic, content,
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
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 do 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?
A Canonical Natural Network has
Few connected components: Small diameter:
often only 1 or a small number independent of network size often a constant independent of network size (like 6) or perhaps growing only logarithmically with network size typically exclude infinite distances considerably more so than for a random network in tension with small diameter a small but reliable number of high-degree vertices quantifies Gladwells connectors often of power law form
A high degree of clustering:
A heavy-tailed degree distribution:
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 give few components, small diameter and high clustering does not give heavy-tailed degree distributions gives few components, small diameter and heavy-tailed distribution does not give high clustering few components, small diameter, high clustering, heavy-tailed models group-actor formation
Watts-Strogatz and related models: Preferential attachment: Hierarchical networks: Affiliation networks:
Nothing magic about any of the measures or models
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
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
Statistics and Probability Theory: The Absolute, Bare-Minimum Essentials
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 Pr[X = 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
Some Basic Notions and Laws
Independence:
let X and Y be random variables independence: for any x and y, Pr[X = x & Y = y] = Pr[X=x]Pr[Y=y] 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:
Variance of X:
only makes sense for numeric random variables average value of X according to its distribution formally, E[X] = (Pr[X = x] *x), sum is over all x in S often denoted by always true: E[X + Y] = E[X] + E[Y] for independent random variables: E[XY] = E[X]E[Y]
Var(X) = E[(X )^2]; often denoted by ^2 standard deviation is sqrt(Var(X)) =
Convergence to Expectations
Let X1, X2,, Xn be:
independent random variables with the same distribution Pr[X=x] expectation = E[X] and variance ^2 independent and identically distributed (i.i.d.) essentially n repeated trials of the same experiment natural to examine r.v. Z = (1/n) Xi, where sum is over i=1,,n example: number of heads in a sequence of coin flips example: degree of a vertex in the random graph model E[Z] = E[X]; what can we say about the distribution of Z?
Central Limit Theorem:
heres a demo
as n becomes large, Z becomes normally distributed
with expectation and variance ^2/n
The Normal Distribution
The normal or Gaussian density:
applies to continuous, real-valued random variables characterized by mean (average) and standard deviation density at x is defined as
(1/( sqrt(2))) exp(-(x-)^2/2^2) special case = 0, = 1: a exp(-x^2/b) for some constants a,b > 0
peaks at x = , 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
The Binomial Distribution
The binomial distribution:
coin with Pr[heads] = p, flip n times probability of getting exactly k heads:
choose(n,k) p^k (1-p)^(n-k)
for large n and p fixed:
approximated well by a normal with = pn, = sqrt(np(1-p)) / 0 as n grows leads to strong large deviation bounds
The Poisson Distribution
The Poisson distribution:
like binomial, applies to variables taken on integer values > 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 probability of exactly x events:
exp(-) ^x/x! mean and variance are both here are some examples
binomial distribution with n large, p = /n ( fixed)
converges to Poisson with mean
Heavy-tailed Distributions
Pareto or power law distributions:
for variables assuming integer values > 0 probability of value x ~ 1/x^ typically 0 < < 2; smaller 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?
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: Normal: Poisson:
log(Pr[X = x]) = log(1/x^) = - log(x) linear, slope log(Pr[X = x]) = log(a exp(-x^2/b)) = log(a) x^2/b non-linear, concave near mean log(Pr[X = x]) = log(exp(-) ^x/x!) also non-linear
Lets revisit the recent paper on dollar bill migration
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, = 1)
General theme:
rank events by their frequency of occurrence resulting distribution often is a power law! North America city sizes personal income file sizes genus sizes (number of species) lets look at log-log plots of these
Other examples:
People seem to dither over exact form of these distributions (e.g. value of ), but not heavy tails
Models of Network Generation and Their Properties
The Erdos-Renyi (ER) Model (Random Graphs)
A model in which all edges NW size N > 1 and probability p: distribution G(N,p) The usual regime of interest is when p ~ 1/N, N is large
each edge (u,v) chosen to appear with probability p N(N-1)/2 trials of a biased coin flip are equally probable appear independently
Degree distribution of a typical G drawn from G(N,p):
e.g. p = 1/2N, p = 1/N, p = 2/N, p=10/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 draw G according to G(N,p); look at a random vertex u in G what is Pr[deg(u) = k] for any fixed k? Poisson distribution with mean = p(N-1) ~ pN Sharply concentrated; not heavy-tailed
Especially easy to generate NWs from G(N,p)
A Closely Related Model
For any fixed m <= 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/N^2 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)
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
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?
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):
Graph process model:
select exactly m of the N(N-1)/2 edges to appear all sets of m edges equally likely starting with no edges, just keep adding one edge at a time always choose next edge randomly from among all missing edges 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
Threshold or tipping for (say) connectivity:
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
size of police force
prob. NW connected
crime of rate
number edges
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:
Difficult to study emergence of non-monotone properties as the number of edges is increased
what would it mean?
G is connected G has diameter <= d (not exactly d) G has a clique of size >= k (not exactly k) G has chromatic number >= c (not exactly c) G has a matching of size >= m d, k, c, m may depend on NW size N (How?)
Formalizing Tipping: Thresholds for Monotone Properties
Consider Erdos-Renyi G(N,m) model Let P be some monotone property of graphs Let m(N) be some function of NW size N
P(G) = 1 G has the property P(G) = 0 G does not have the property select m edges at random to include in G
Say that m(N) is a threshold function for P if:
formalize idea that property P appears suddenly at m(N) edges 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
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
Ever More Precise
Connected component of size > N/2:
threshold function is m(N) = N (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
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
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!
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 <= c(u) <= 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
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
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 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
But in many settings, the opposite is true:
Watts Caveman world: Watts Solaria world
overall density of edges is low but two vertices with a common neighbor are likely connected overall density of edges low; no special bias towards local edges like Erdos-Renyi
Making it (Somewhat) Precise: the -model
The -model has the following parameters or knobs: N: size of the network to be generated
k: the friendship threshold parameter p: the default probability two vertices are connected : adjustable parameter for bias towards local connections
For any vertices u and v:
Key quantity: the propensity R(u,v) of u to connect to v
define m(u,v) to be the number of common neighbors (so far)
if m(u,v) >= 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)^ (1-p) here are some plots for different (see Watts page 77)
Generate NW incrementally
Note: = infinity is like Erdos-Renyi (but not exactly)
using R(u,v) as the edge probability; details omitted
Small Worlds and Occams Razor
For small , 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 -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 -model
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 -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, is better than p
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
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
Case 3: C. Elegans Nervous System
Vertices: neurons in the C. elegans worm Edges: axons/synapses between neurons Here is the network and data
Two More Examples
M. Newman on scientific collaboration networks
coauthorship networks in several distinct communities d...