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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