Random Graphs
Liang Li
April 9, 2014
Outline
Objectives
Internet Topology
Melting points [2]
History
Definition
With high probability (whp)
Near clique
Scale free
Models
Erdős-Rényi Model
Random Graphs
Edgar Gilbert Model
Results with classical random graphs
Giant component
Probability methods
Ramsey Number Bound
Hamiltonian paths
Watts and Strogatz Small world Model
"Kavin Bacon game" and "Erdős number"
Small World Model
generating
clustering coefficient
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Random Graphs
Barabási and Albert Preferential attachment Model
generating
properties
Applications
open problems
Some open problems
Homework problems
The average Clustering coefficient
Prove or Disprove
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Random Graphs
Objectives
Internet Topology
When you send or receive data over the internet you computer doesn’t really give
how the data travels. The media (wire, optic fibre, ox cart) and route (via hong
kong or Champaign-Urbana) are irrelevant so long as we don’t mind waiting.
Of course, we do mind so in general routers try to route packets over the fastest
link and shortest distance. A program called traceroute finds out where data is
flowing by sending out suicidal packets of information that self-destruct after they
have seen a set number of computers. Of course some computers don’t care if
the packet dies, some respond with nonsense, some respond too quickly or too
slowly.
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Random Graphs
Figure 1: The internet topology in 2001 taken from https://www.
fractalus.com/steve/stuff/ipmap/
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Random Graphs
https://www.fractalus.com/steve/stuff/ipmap/layout2.gif
https://www.fractalus.com/steve/stuff/ipmap/net-anim.gif
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Random Graphs
Melting points [2]
Think of a solid as a three-dimensional grid of molecules, with neighboring
molecules joined by bonds.
1. Adding energy excites molecules and breaks bonds.
2. Bonds break at random as the temperature (energy level) raises.
3. Break off bonds make the molecules form others, like a liquid or gas.
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Random Graphs
Figure 2: Melting points
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Random Graphs
History
Small world model.
Ramsey number
Survey articles
Watt.
Erdős
Albert
Random graphs BA model
Hamilton path
Erdős
Barabási.
Watt
Szele
1943
1947
1959-1961
1998 1999
2002
2003
Newman
Remco
2006
2014
The theory of random graphs was founded by Erdős and Rényi (1959, 1960,
1961a,b) after Erdős (1947, 1959, 1961) had discovered that probabilistic methods [6, 7] were often useful in tackling extremal problems in graph theory [3].
The small world model [4] of Watts and strogatz(1998) and the preferential
attachment model [5] of Barabási and Albert (1999) [1] have led to an explosion
of research [8].
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Random Graphs
Definition
With high probability (whp)
We say that a graph has a certain property Q, if limn→∞ Pr(Graph has Q) = 1.
Near clique
An undirected graph is a near clique if adding an additional edge would make it a
clique.
Scale free
The degree distribution is almost independent of the size of the graph, and the
proportion of vertices with degree k is close to proportional to P (k) ∼ k −τ ,
typically 2 < τ < 3 for real network [11]. Or Nk ∼ cn k −τ [12].
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Random Graphs
Models
Erdős-Rényi Model
G(n, M ) consists of all graphs with vertex set V = {1, 2, ..., n} having M edges,
in which the graphs have the same probability.
N
Thus with the notations N = n2 , 0 ≤ M ≤ N , G(n, M ) has M
elements and
N −1
every element occurs with probability M
.
M
The random variable G denotes a graph generated in this way.
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Random Graphs
Edgar Gilbert Model
G(n, P (edge) = p) consists of all graphs with vertex set V = {1, 2, ..., n} in
which the edges are chosen independently and with probability p.
In other worlds, if G0 is a graph with vertex set V and it has m edges, then
P ({G0 }) = P (G = G0 ) = pm (1 − p)N −m .
The random variable Gp denotes a graph generated in this way.
For M ' pN , the these two models are almost interchangeable [8].
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Random Graphs
Results with classical random graphs
Giant component
Erdős and Rényi discovered that there was a sharp threshold for the appearance
of many properties [1]. Let c > 0 be a constant and set p = c/n.
• if c < 1 , most of the connected components of the graph are small, which
the largest having only O(log n) vertices, where the O symbol means that
there is a constant C < ∞ so that the Probability (the largest component
is ≤ C log n) tends to 1 as n → ∞.
• if c > 1 there is a constant θ(c) > 0, so that the largest component has
∼ θ(c)n vertices and the second largest component is O(log n). Here
Xn ∼ bn means that Xn /bn converges to 1 in probability as n → ∞.
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Random Graphs
Probability methods
Ramsey Number Bound
The Ramsey number [13] R(m, n) gives the solution to the party problem, which
asks the minimum number of guests R(m, n) that must be invited so that at
least m will know each other or at least n will not know each other.
In the language of graph theory, the Ramsey number is the minimum number of
vertices v = R(m, n) such that all undirected simple graphs of order v contains a
clique of order m or an independent set of order n.
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Random Graphs
S
P
Using the observation that P ( i Ai ) ≤ i P (Ai ).
Theorem (Erdős (1947))
1−(m)
n
2
If m
2
< 1, then R(m, m) > n.
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Random Graphs
Proof. [2]
Define a probability model on graphs with vertex set n by letting each edge
appear independently with probability 0.5. If the probability of the event Q=" no
m-clique or independent m-set" is positive, then the desired graph exists.
m
Each possible p-clique occurs with probability 2−( 2 ) , since obtaining the complete
graph requires obtaining all its edges, and they occur independently. Hence the
−(m)
n
probability of having at least one m-clique is bounded by m
2 2 . The same
bound holds for independent m-sets. Hence the probability of "not Q" is bounded
1−(m)
n
2 , and the given inequality guarantees that P (Q) > 0.
by m
2
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Random Graphs
Hamiltonian paths
A random variable is a function assigning a real number to each element of a
probability space. We use X = k to denote the event consisting of all elements
where variable X has the value k.
P
The expection E(X) of a random variable X is the weighted average k kP (X = k).
The pigeonhole property of the expectation is the statement that there exists
an element of the probability space for which the value of X is as large as (or as
small as) E(X).
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Random Graphs
Theorem (Szele (1943))
Some n vertex tournament has at least n!/2n−1 Hamiltonian
paths.
Proof. [2] Generate tournament on n randomly by choosing i → j or j → i with
equal probability for each pair {i, j}. Let X be the number of Hamiltonian parts;
X is the sums of n! indicator variables for the possible Hamiltonian paths. Each
Hamiltonian path occurs with probability 1/2n−1 , so E(X) = n!/2n−1 . In some
tournament, X is at least as large as the expectation.
This simple bound using expectation gives almost the right answer for the
maximum number of Hamiltonian paths in an n-vertex tournament; Alon[14]
proved that it is at least n!/(2 + o(1))n .
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Random Graphs
Watts and Strogatz Small world Model
"Kavin Bacon game" and "Erdős number"
0
1
1
1673
2
130,851
3
4
349,031
84,615
5
6,718
6
7
788
107
8
11
Table 1: Bacon number
Kevin Bacon number is 2.94; Erdős number is 4.7 with 337,454 authors and
496,489 edges. Facebook released two papers in Nov.2011 that 721 million users
with 69 billion friendship links, average distance is 4.74.
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Random Graphs
Small World Model
The Gp graphs have small diameters, but have very few triangles. (while in social
networks if A and B are friends and A and C are friends, it is fairly that B and C
are also friends.)
To construct a network with small diameter and a positive density of K3 , Watts
and Strogatz started a ring lattice with n vertices and k edges per vertex, where
the construction interpolates between regularity (p = 0) and disorder (p = 1).
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Random Graphs
generating
Figure 3: Generating small world graphs [15]
• Disallow self-edges.
• Disallow multiple edges.
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Random Graphs
clustering coefficient
Denote L(p) be the average distance between two randomly chosen vertices and
define clustering coefficient C(p) to be the fraction of connections that exist
between the k2 pairs of neighbors of a site.
edges between neighbors of v|
Local clustering coefficient of node V: | |actual
possible edges between neighbors of v|
The clustering coefficient for the whole graph is the average of the local values.
Figure 4: C(v) =
4
6
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Random Graphs
A graph is considered small world, if:
• its average clustering coefficient is significantly higher than the one of a
random graph constructed on the same vertex set, and
• it has approximately the same mean shortest path length as its corresponding random graph.
The regular graph has L(0) ∼ n/2k and C(0) ≈ 3/4 if k is large, which the
disorder one has L(1) ∼ (log n)(log k) and C(1) ∼ k/n. Here L(p) decreases
quickly near 0, which C(p) changes slowly so there is a broad interval of p over
which L(p) is almost as small as L(1), yet C(p) is far from 0 [1].
• Small-world networks tend to contain cliques, and near-cliques.
• Most paris of nodes will be connected by at least one short path.
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Random Graphs
Barabási and Albert Preferential attachment
Model
BA model an algorithm for generating
random scale-free networks using a preferential attachment mechanism.
It incorporates two important general
concepts:
Growth means the number of nodes in
the network increases over time.
Preferential attachment means that
the more connected a node is, the
more likely it is to receive new links.
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Random Graphs
generating
The network begins with an initial connected network of m0 nodes.
New nodes are added to the network one at a time with the probability that is
proportional to the number of links that the existing nodes already have:
pi = Pki
j
kj
where ki is the degree of node i and the sum is made over all pre-existing nodes
j.
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Random Graphs
properties
• BA model is scale free. Its power law of the form p(k) ∼ k −3
• The average path length increases approximately with the size of the
network l ∼ lnlnlnNN
• The clustering coefficient with network size C ∼ N −0.75
For example, on the web, very well known sites such as Google or Wikipedia,
rather than to pages that hardly anyone knows will be more likely to be linked. If
someone selects a new page to link to by randomly choosing an existing link, the
probability of selecting a particular page would be proportional to its degree.
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Random Graphs
Applications
World Wide Web...
Internet...
Movie actor collaboration network...
Cellular networks....
Ecological networks...
Phone call network ...
Citation network...
Networks in linguistics...
Power and neural networks...
.....
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Random Graphs
open problems
Random Structures: a model of real world networks, such as Internet, social
network or biological networks it leaves a lot to be desired.
Figure 5: The internet topology in 2001 taken from https://www.
fractalus.com/steve/stuff/ipmap/
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Random Graphs
Some open problems
Is that true that who Gm has δ(Gm )/2 Hamilton cycles?[19]
It is known to be true as long as δ(Gm )/2 = o(average degree).
What is the expected time to for a random walk to get within distance d for every
vertex?
More problems[20]:
Ramsey theory...
Graph coloring problems..
...
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Random Graphs
Homework problems
The average Clustering coefficient
Figure 6:
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Random Graphs
Prove or Disprove
When p is constant, then almost every Gp is has diameter 2 (and Gp is connected).
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Random Graphs
References
[1] Rick Durrett, Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics), Cambridge
University Press, New York, NY, 2006
[2] D. B. West. Introduction to Graph Theory (2nd Edition). Edited by Prentice Hall. Prentice Hall, 2001.
[3] http://en.wikipedia.org/wiki/Random_graph
[4] D. J.Watts, S. H. Strogatz(1998). "Collective dynamics of ’small-world’ networks". Nature 393 (6684): 440Ð442.
[5] R. Albert, A.-L. Barabási (2002). "Statistical mechanics of complex networks". Reviews of Modern Physics 74: 47Ð97.
[6] Erdös, P.; Rényi, A. (1959). "On Random Graphs. I". Publicationes Mathematicae 6: 290Ð297
[7] Erdös, P.; Rényi, A. (1960). "The Evolution of Random Graphs". Magyar Tud. Akad. Mat. Kutató Int. Közl. 5: 17Ð61.
[8] Bollobas, B. and Riordan, O.M.(2003) "Mathematical results on scale-free random graphs" in "Handbook of Graphs and
Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed.
[9] http://en.wikipedia.org/wiki/Small-world_network#Properties_of_small-world_networks
[10] http://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model
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Random Graphs
[11] http://en.wikipedia.org/wiki/Scale-free_network
[12] R. V. D. Hofstad. (2014) "Random Graphs and Complex NetworksÓ. Department of Mathematics and Computer Sciene
Eindhoven University of Technology.
[13] http://mathworld.wolfram.com/RamseyNumber.html
[14] Alon, Noga. (1990)"The maximum number of Hamiltonian paths in tournaments." Combinatorica VOL.10. NO 4. 319324.
[15] http://cs.brynmawr.edu/Courses/cs380/spring2013/section02/slides/06_SmallWorldNetworks.pdf
[16] http://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model#Clustering_coefficient
[17] Albert, RŐka, and Albert-LĞszlŮ BarabĞsi. (2002) "Statistical mechanics of complex networks." Reviews of modern
physics VOL. 74.NO. 1:47-93.
[18] Watts, Duncan J.; Strogatz, Steven H. (June 1998). "Collective dynamics of ’small-world’ networks". Nature 393 (6684):
440Ð442.
[19] http://www.math.cmu.edu/~af1p/Talks/RandomGraphs/rgtalk.pdf
[20] Chung, F. R. K. "Open problems of Paul Erdos in graph theory." Journal of Graph Theory 25.1 (1997): 3-36.
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Questions?