A random graph model
for massive graphs
William Aiello
Fan Chung Graham
Lincoln Lu
SDSC, skitter (July 1998)
What are the properties of the
WWW Graph?
• Is the World Wide Web connected?
• If not, how large is the largest component,
the second largest component, etc.?
• Can these questions be answered exactly?
• Probably not! The WWW is changing
constantly. Even a “snapshot” of the Web
is too large to handle.
An important observation
WWW graph has a power law degree
distribution
Discovered by several groups independently
• Broder, Kleinberg, Kumar, Raghavan, Rajagopalan
aaand Tomkins, 1999.
• Barabási, Albert and Jeung, 1999.
Power Law Graphs
Power law decay of the degree distribution:
The number of vertices of degree d is
proportional to 1/db where b is some constant > 0.
Let y(d) be the number of nodes of degree d
y ~ 1/db
log y = a – blog d
Power Law Graphs Robust and
Ubiquitous
•
•
•
•
•
Internet Router Graph
Power Grid Graph
Phone Call Graph
Scientific Citation Graph
Co-Stars Graph (e.g. the six degrees of
Kevin Bacon)
• The power in the power law stays constant
even as the graphs grow and change.
What does a massive graph look like?
sparse
clustered
small diameter
prohibitively large
dynamically changing
incomplete information
Hard to describe !
Harder to analyze !!
Don’t worry about exact answers—
Use Models Instead
• Data sets too large and dynamic for exact
analysis occur in many other areas: the
physical, biological, and social sciences and
engineering.
• Progress in understanding often made by
iterative interplay between modeling and
experimental data, where both often have a
random or statistical nature.
Modeling Power Law Graphs
• Develop model of Power Law Graphs
• Analyze properties of model, e.g.,
connected component structure
• Compare results to experimental data
• Our model will be of variant of an
important model in graph theory
called Random Graphs
Random Graphs
• G(n,e)
– n nodes
– all graphs with e edges have uniform
probability
H(3,1) prob 1/3
H(3,2) prob 1/3
Random Graphs
• G(n,p)
– n nodes
– each edge is included with probability p
– expected degree = p(n-1)
(1-p)3
p(1-p)2
p3
p2(1-p)
..
/
Paul Erdos and A. Renyi,
On the evolution of random graphs
Magyar Tud. Akad. Mat. Kut. Int. Kozl. 5 (1960) 17-61.
The evolution of random graphs G(n,p)
0
p
disjoint union of trees
c/n, 0<c<1
cycles of any size
1/n
the double jumps
c’/n, c>1
one giant component, i.e., size (n),
other components are o(n)-sized trees
log n/n
G(n,p) is connected
w log n/n, w
connected and almost regular,
expected degree ~ w log n
Random Graphs and Degree
Distributions
• H(n,s)
– n nodes
– s = (y(1), y(2), … , y(n-1)), where y(i) is the
number of nodes with degree i.
– all graphs with degree distribution s have
uniform probability
Random Graphs and Degree
Distributions
H(4,s), s = (1,2,1). All have prob. 1/12
Random Power Law Graphs
A power law degree distribution can be
described by two parameters: a, b
y = ea/xb
log y = a – b log x
where y is the number of nodes of degree x
A new random graph model: P(a,b).
P(a,b) assigns uniform probability to all graphs
with degree distribution y = ea/xb
A few facts about P(a,b):
•The maximum degree is ea/b.
• The number of vertices n is
n =  ea/xb ~ z(b) ea , 1  x  ea/b,
where z(b) = S1/xb the Reimann Zeta function.
• The number of edges E is
E = 1/2  ea/xb-1 ~ z(b-1) ea/2
• The density E/n = z(b-1)/z(b) is controlled by b.
Facts on P(a,b):
0
1
connected
not connected—unique giant component of size (n)
smaller components are of size O(1).
2
b
smaller components are of size O(log n/log log n). For
any x, 2<≤x<O(log n/log log n), there is a component
of size x.
The second largest components are of size O(log n).
For any x, 2<x<O(log n), there is a component of
size x.
3.478... a root of ς(b-2)=2 ς(b-1)
no giant component
How do Power Law Graphs Arise?
• The previous model takes the power law
degree distribution as a given.
• It does not explain how such graphs arise.
• Results which hold in the model with high
probability (e.g., our connected component
results) will apply to the vast majority of
power law graphs regardless of the
particulars of the evolution process.
Yet Another Random Graph Model
G(n) is a random graph evolution:
• Let Kn be the set of all possible edges
• Let Et be the edges chosen in steps 1
through t.
• At time step t+1 choose uniformly one
of the edges in Kn – Et
• Add this edge to Et to get Et+1.
• Study what structures appear with
high probability as a function of t.
Need a new idea
 G(n) fixes the set of nodes and then
adds edges.
 Can show that to get a power law,
need to add both nodes and edges.
 G(n) chooses uniformly among all
eligible edges
 Can show that selecting edges
uniformly will not yield a power law.
A Graph Evolution Process
• At each time step t, toss a biased coin having
heads with probability p.
• “tails” -> add a new vertex with a self-loop.
• “heads” -> add a new edge between the
existing set of nodes:
– Select a vertex u with probability proportional to the
the degree of u, i.e., Pr[ u chosen ] = deg(u)/2|E|.
– Independently select vertex v with probability
proportional to deg v.
– Add the edge {u,v}.
A Graph Evolution Process
p
1-p
v
u
Gt
• The number of nodes grows with time
• Edges are not added uniformly
• Nodes which are added early have an “advantage”
over nodes added late
• Gives a power law degree distribution
y ~ 1/d1+1/p
Comparisons
From real data
From simulation
using Model B
Evolution Process for Directed Graphs
• Select a vertex u with probability proportional to
the the out degree of u, i.e., Pr[ u chosen ] = outdeg(u)/|E|.
• Select a vertex v with probability proportional to
the the in degree of v.
• Flip two coins; heads with prob p1 and p2.
–
–
–
–
Heads, heads -> add an edge from u to v.
Heads, tails -> add an edge from u to a new node.
Tails, heads -> add an edge from a new node to v.
Tails, Tails -> add a directed self-loop to a new node.
• # nodes w/outdegree d ~ 1/d1+1/p1
• # nodes w/indegree of d ~ 1/d1+1/p2
Massive Graphs
Random graphs
Similarities: Adding one (random) edge at a time.
Differences: Random graphs <-- almost regular.
Massive graphs <-- uneven degrees.
Correlations.
The advantages of power law models
• Approximating real data graphs.
• Possible to analyze rigorously—discover
implicit structure of massive graphs
• Models for generating network topologies
Methods:
• Erdös and Réyni’s seminal papers.
• Martingales.
• Concentration bounds.
• Molloy+Reed’s results on random graphs with
. given degree squences.
Future directions
The evolution of power graphs concerning
---- diameters of connected components
luuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuLu’s thesis
-- - frequency of occurrences of certain subgraphs
- power law of eigenvalues
- scaling behavior of power law graphs
- “signatures” in graphs to distinguish models
A JAVA generation/simulation of power graphs
Can be found at http://math.ucsd.edu/~llu