Network Science:
“Universal” Structure and
Models of Formation
Networked Life
CIS 112
Spring 2009
Prof. Michael Kearns
“Natural” Networks and Universality
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Consider the many kinds of networks we have examined:
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These networks tend to share certain informal properties:
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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 network science
– social, technological, business, economic, content,…
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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,…
Some Interesting Quantities
• Connected components:
– how many, and how large?
• Network diameter:
– 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?
• Etc. etc. etc.
A “Canonical” Natural Network has…
• Few connected components:
– often only 1 or a small number (compared to network size)
• Small diameter:
– often a constant independent of network size (like 6…)
– or perhaps growing only very slowly with network size
– typically look at average; exclude infinite distances
• A high degree of edge 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 Gladwell’s connectors
– often of power law form
Some Models of Network Formation
• Random graphs (Erdos-Renyi model):
– gives few components and small diameter
– does not give high clustering or 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
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
crime rate
prob. NW connected
– we gradually add edges, or gradually increase edge probability p
– many properties will emerge very suddenly during this process
size of police force
number of edges
Approximate Roadmap
• Examine a series of models of network formation
– macroscopic properties they do and do not entail
– tipping behavior during network formation
– pros and cons of each model
• Examine some “real life” case studies
• Study some dynamics issues (e.g. seach/navigation)
• Move on to an in-depth study of the web as network
Probabilistic Models of Networks
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Network formation models we will study are probabilistic or statistical
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They can generate networks of any size
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They often have various parameters that can be set/chosen:
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The models each generate a distribution over networks
Statements are always statistical in nature:
– later in the course: economic formation models
– we will typically ask what happens when N is very large or N  infinity
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size of network generated
probability of an edge being present or absent
fraction of long-distance vs. local connections
etc. etc. etc.
– with high probability, diameter is small
– on average, degree distribution has heavy tail
Optional Background on
Probability and Statistics
[Next three slides.]
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 Erdos-Renyi
• Behavior of X determined by its distribution (or density)
– for each specific value x in S, specify Pr[X = x]
– these probabilities sum to exactly 1 (mutually exclusive outcomes)
– complex sample spaces (such as large networks):
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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:
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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, and 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
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formally, E[X] = S (Pr[X = x] *x), sum is over all x in S
often denoted by m
always true: E[X + Y] = E[X] + E[Y]
for independent random variables: E[XY] = E[X]E[Y]
• Variance of X:
– Var(X) = E[(X – m)^2]; often denoted by s^2
– standard deviation is sqrt(Var(X)) = s
Convergence to Expectations
• Let X1, X2,…, Xn be:
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independent random variables
with the same distribution Pr[X=x]
expectation m = E[X] and variance s^2
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 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:
– as n becomes large, Z becomes normally distributed
• with expectation m and variance s^2/n
– here’s a demo
The Erdos-Renyi Model
The Erdos-Renyi (E-R) Model
(Random Networks)
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A model in which all edges:
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Two parameters: NW size N > 1 and edge probability p:
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About the simplest (dumbest?) imaginable formation model
The usual regime of interest is when p ~ 1/N, N is large
– are equally probable and appear independently
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each edge (u,v) appears with probability p, is absent with probability 1-p
N(N-1)/2 independent trials of a biased coin flip
results in a probability distribution over networks of size N
especially easy to generate networks from this distribution
– e.g. p = 1/2N, p = 1/N, p = 2/N, p=150/N, p = log(N)/N, etc.
– in expectation, each vertex will have a “small” number of neighbors (~ pN)
• Gladwell’s “Magic Number 150” and cognitive bounds on degree
• mathematical interest: just near the boundary of connectivity
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– will then examine what happens when N  infinity
– can thus study properties of large networks with bounded degree
Degree distribution of a typical E-R network G:
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draw G according to E-R with N, p; look at a random vertex u in G
what is Pr[deg(u) = k] for any fixed k? (or histogram of degrees)
Poisson distribution with mean l = p(N-1) ~ pN
Sharply concentrated; not heavy-tailed
The Poisson Distribution
• The Poisson distribution:
– 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) l^x/x!
• mean and variance are both l
• here are some examples; again compare to heavy tails
– similar to a normal (bell-shaped) distribution, but only takes on
positive, integer values
Another Version of Erdos-Renyi
• In Erdos-Renyi:
– expected number of edges in the network = pN(N-1)/2 = m
– actual number of edges will be ”extremely close” to m
– so suppose we instead of fixing p, we fix the number of edges m
• Incremental Erdos-Renyi model:
– start with N vertices and no edges
– at each time step, add a new edge, up to m edges total
– 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 (in relation to N)
– again, let’s look at Erdos-Renyi giant component demo
• For our purposes, these models are equivalent under pN(N-1)/2 = m
The Evolution of a Random Network
• We have a large number N of vertices
• We start randomly adding edges one at a time (or increasing p)
• At what point will the network:
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have at least one “large” connected component?
have a single connected component?
have “small” diameter?
have a “large” clique?
• How gradually or suddenly do these properties appear?
Monotone Network Properties
• Often interested in monotone network properties:
– suppose G has the property (e.g. G is connected)
– now add edges to G to obtain G’
– then G’ must have the property also (e.g. G’ is connected)
• Examples:
– G is connected
– G has diameter <= d (not exactly d)
– G has a clique of size >= k (not exactly k)
• Interesting/nontrivial monotone properties:
– G has no edges  G does not have the property
– G has all edges (complete)  G has the property
– so we know as p goes from 0 or 1, property emerges
Formalizing Tipping
for Monotone Properties
• Consider the standard Erdos-Renyi model
– each edge appears with probability p, absent with probability 1-p
• Pick a monotone property P of networks (e.g. being connected)
• Say that P has a tipping point at q if:
– when p < q, probability network obeys P is ~ 0
– when p > q probability network obeys P is ~ 1
• Aside to math weenies:
– formalize by asking that probabilities converge to 0 or 1 as N  infinity
• Incremental E-R version:
– replace q by “tipping” number of edges
• A purely structural definition of tipping
– tipping results from incremental increase in connectivity
• No obvious reason any given property should tip
So… Which Properties Tip?
• The following properties all have tipping points:
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having a “giant component”
being connected
having “small” diameter
in fact…
• 1996: All monotone network properties have tipping points!
– So at least in one setting, tipping is the rule, not the exception
• Demo: look at the following progression
– giant component  connectivity  small diameter
– in Incremental Erdos-Renyi model (add one new edge at a time)
– with remarkable consistency (N = 50):
• giant component ~ 40 edges, connected ~ 100, small diameter ~ 180
• Number of possible edges = N(N-1)/2 = 1225
– [example 1] [example 2] [example 3] [example 4] [example 5]
More Precise…
• Connected component of size > N/2:
– tipping point p ~ 1/N
– note: full connectivity virtually impossible
• Fully connected:
– tipping point is p ~ log(N)/N
– NW remains extremely sparse: only ~ log(N) edges per vertex
• Small diameter:
– tipping point is p ~ 2/sqrt(N) for diameter 2
– fraction of possible edges still ~ 2/sqrt(N)  0
– generates very small worlds
• Upshot: right around/beyond p ~ 1/N, lots suddenly happens
Other Tipping Points
• Perfect matchings
– consider only even N
– tipping point is p ~ log(N)/N
– same as for connectivity!
• Cliques
– k-clique tipping point is p ~ 1/N^(2/k-1)
– edges appear immediately; triangles at N/2; etc.
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:
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network acquires a giant component
network becomes connected
network acquires small diameter
etc. etc. etc.
• Properties appear very suddenly (tipping, thresholds)
– … and this is the rule, not the exception!
• All statements are mathematically precise
• All happen shortly around/after edge density p ~ 1/N
– very efficient use of edges!
• 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 network
– all vertices v such that the edge (u,v) is in the graph
• The clustering coefficient of u:
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let k = |nbr(u)| (i.e., number of neighbors of u = degree 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 u’s 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…
u
Erdos-Renyi: Clustering Coefficient
• Generate a network G according to Erdos Renyi with 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 E-R, typical c(u) entirely determined by overall density
• Baseline for comparison with “more clustered” models
– Erdos-Renyi has no bias towards clustered or local edges
• Clustering coefficient meaningless in isolation
• Must compare to the “background rate” of connectivity
“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
Making it More Precise: the a-model
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An incremental formation model
Pick network size N
Throw down a few random “seed” edges
Then for each pair of vertices u and v:
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compute probability of adding edge between u and v
probability will depend on current network structure
the more common neighbors u and v have, more likely to add edge
provide knobs that let us adjust how weak/strong the effect is
Making it More Precise: the a-model
1.0
smaller a
y = probability of
connecting u & v
= p + (1-p)*(x/N)^a
larger a
“default” probability p
x = number of current
common neighbors of u & v network size N
Small Worlds and Occam’s Razor
• For small a, should generate large clustering coefficients
– after all, we “programmed” the model to do so!
• But we do not want a new model for every little property
– Erdos-Renyi  small diameter
– a-model  high clustering coefficient
– etc. etc. etc.
• In the interests of Occam’s Razor, we would like to find
– a single, simple model of network generation…
– … that simultaneously captures many properties
• Watt’s small world: small diameter and high clustering
– here is a figure showing that this can be captured in the a-model
An Alternative Model
• The a-model programmed high clustering into the formation process
– and then we got small diamter “for free” (at certain a)
• A different model:
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– start with all vertices arranged on a ring or cycle
– connect each vertex to all others that are within k steps
– with probability p, rewire each local connection to a random vertex
Initial cyclical structure models “local” or “geographic” connectivity
Long-distance rewiring models “long-distance” connectivity
p=0: high clustering, high diameter
p=1: low clustering, low diameter (E-R)
In between: look at this simulation
Which of these models do you prefer?
– sociology vs. math
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:
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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 E-R
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
– 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