Download - Discrete Mathematics in the Modern World
Discrete Mathematics in the
Modern World
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Mathematics - Driven by Needs
BC: calendar - astronomy
architecture - geometry
navigation - trigonometry
Middle Ages: currency conversion - algebra
introduction of arabic numberals
Rennaissance: first printed maths book:
Peurbach’s Theoricae nova planetarum (1472)
16th -19th century: science - calculus
gambling - probability, combinatorics
20th century: economics - game theory
efficiency - linear programming
Computer age: algorithmic theory, numerical
maths, cryptography, finite mathematics,
graph theory
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Graphs
Def: A graph is an object consisting of
(i) points in the plane (the vertices)
(ii) lines joining the points (the edges)
Rem: Often used synonymously: network
Clarification: A graph is not
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Ex: A map with cities and freeways is a graph
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Ex: Consider only cities and freeways
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Ex: London Underground is a graph
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Ex: The structural formula of Butane is a
graph
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Ex: (i) network of metabolic pathways
(ii) study of genes
(iii) computer networks
(iv) telephone networks
(v) social networks (friendship graph)
Ex: Characterisation of interval graphs led to
Nobel Prize for Microbiology for Benzer’s work
on the fine structure of genes.
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Def: Distance between vertices a and b:
dist(a, b) = #steps needed to get from a to b.
Ex: Graph below: d(a, b) = 1 and d(a, c) = 2.
Rem: If a graph models a transportation net-
work, then
dist(a, b) ∼ travel time from a to b
Def: diameter = largest of all distances.
Ex: Above: diam(G) = 2.
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Rem: In a transportation network:
Diameter ∼ maximum travel time.
Rem: In a sociological network:
Diameter ∼ measure of cohesion.
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Rem: The friendship graph F :
Vertices = people, edges = friendships.
Rem: Very big, hard to study F .
Q: Diameter of F?
Experiment: (S. Milgram, 1967)
(i) starter receives folder with name + address
of target,
(ii) hands folder to someone closer to target,
(iii) many folders reached targets in ≤ 6 steps.
Conclusion: diam(G) is about 6,
the SIX DEGREES OF SEPARATION.
Rem: Some objections, but more or less ac-
cepted.
Mathematics says...
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Def: The degree of a vertex is the number of
vertices it is joined to.
Ex: Graph below: deg(a) = 3 and deg(c) = 2.
The overall average degree is 3.2.
Rem: Friendship graph: degree = # friends.
Reasoning: We know:
(i) F has, say, 5.000.000.000 vertices,
(ii) F has average degree about, say, 42,
(iii) 99% of all graphs satisfying (i) and (ii)
have diameter about 6.
so we conclude
probably diam(F ) ≈ 6.
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Erdos, Renyi: Theory of Random Graphs:
Many properties hold for either close to 100%
of all graphs, or for close to 0%, depending on
the average degree.
Theo: Of all graphs with n vertices and av-
erage degree d, where d ≥ logn, almost 100%
have
diam(G) ≈ constant×logn
log d.
Rem: logn is much smaller than n,
logn ≈ # digits of n
Cor: Most likely diam(F ) is very small.
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Power Law Distributions
Lotka’s Law (1926): Let A(k) = # authors
who published k scientific articles. Then
A(k) ≈ constant×1
k2.
Let A(k) be the number of authors who pub-
lished exactly k articles. If, say, 1000 authors
wrote one paper, then approximately
A(1) A(2) A(3) A(4) A(5) . . .
1000 10004
10009
100016
100025 . . .
1000 = 250 = 111 = 64 = 40 . . .
A(k) follows a power law with exponent 2.
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Rem: Typical for power law: many authors
published 1 paper, fewer published 2, even fewer
published 3,...
Rem: Power law =“heavy tail distribution”
(polynomial, not exponential)
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Zipf’s Law (1952): Suppose all English words
are listed in order of frequency: w1 being the
most common word, w2 the second most com-
mon word, etc. If
W (k) = # occurrences of wk per 100 words
of standard text,
then
W (k) follows a power law with exponent 1:
W (k) ≈ const×1
k.
Rem: Similar for all human languages and
some programming languages.
Awerbach (1913) City sizes follow a power
law.
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Def: Let G be a large graph. Let
Deg(k) = #vertices of degree k.
If Deg(k) follows a power law, then we say that
G is a power law graph.
Observation Many graphs are power law.
Year Network # vert. d exp.
Social:1999 phone calls 47 million 3.16 2.1
2002 emails 59912 1.44 1.5
1998 film actors 449.913 3.48 2.3
Information:
1999 www.nd.edu 269.504 5.55 2.1
2005 the web 53 billion 2.1
2002 word co-occurr. 460902 70.1 2.7
1998 citation netw. 783.339 8.57 3.0
Biological:
2000 metabolic netw. 765 9.64 2.2
2001 protein interact. 2115 2.12 2.4
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The Web
Rem: Prime example of a PLG: WWW
Rem: Important pages have large in-degree.
indeg(google) = 4, indeg(P D home) = 1.
Rem: WWW grows by preferential attach-
ment:
A new page is more likely to be linked to pages
that already have many links.
Rem: Graphs that grow by preferential attach-
ment are usually PLG.
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Theo: Of all PLG with n vertices and given
average degree d, almost 100% have
diam(G) ≈ constantd × log logn.
Meaning: PLG have extremely small diame-
ter.
Study: The web has diameter about 19.
Rem: F also grows by preferential attach-
ment. So F is also a power law graph.
Corollary: If F is a PLG, then probably diam(F )
is extremely small.
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Searching the Web
Rem: search engines consist of 3 parts:
crawler: surfs the web and sends data on the
content of web pages to the search engine
indexer: builds an index (list of key words of
each page)
query engine: checks which pages have rele-
vant content, then ranks the pages found.
Difficult part: Ranking
Rem: Old search engines (AltaVista, Lycos)
were text based.
Google uses the structure of the web graph.
Vast improvement!
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Bad idea: Use in-degree for ranking.
Solution: PageRank algorithm
(L. Page, S. Brin, 1998)
Tool: Use random walks along edges:
If we are at the School of Maths page then
Prob(SoM −→ SAMS) =1
outdeg(SoM)=
1
4.
Idea: Rank according to # visits.
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Def: For a web page A define visits(A) as
visits(A) =# times A is visited
total number of steps
of a long random walk.
Idea: Rank pages according to visits.
Determine visits: Discrete Markov chains with
transition matrix P where
Pi,j =
{ 1outdeg(i) if i links to j,
0 otherwise,
but if vertex i has outdeg(i) = 0, then let
ith row = (1
n,1
n,1
n, . . . ,
1
n)
to avoid getting stuck.
Add, with 15% probability, a random jump
from vertex i to any vertex. New transition
matrix
Q = 0.85P + 0.15J,
where J is the ‘all 1’ n× n matrix.
Qt is ≥ 0 and primitive. By Perron-Frobenius
it has a unique eigenvector E > 0. If |E| = 1
then E corresponds to a stationary state:
visit(i) = Ei.
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Ex: A typical random graph with most vertices
having the same degree:
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Ex: A typical power law graph with many ver-
tices of small degree and few vertices of large
degree :
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