Jo Ellis-MonaghanSt. Michaels College, Colchester, VT 05439e-mail: jellis-monaghan@smcvt.eduwebsite: http://academics.smcvt.edu/jellis-monaghan

EARLY MATHEMATICSEARLY MATHEMATICS----ARITHMETICARITHMETIC

The Mathematics of: Counting, Money, Inventory, Taxes, Census, Apportionment, Calendars

Enumeration

Cavemen

Prehistory

www.cartoonstock.com www.cartoonstock.com

www.valdostamuseum.org/hamsmith/eghier.html

CLASSICAL MATHEMATICSCLASSICAL MATHEMATICS----GEOMETRYGEOMETRY

The Mathematics of: Measurement, Surveying, Architecture, Astronomy, Volumes and Areas

Static Measurement

Euclid of Alexandria(325 BC-265 BC)

The Mathematics of:Projectiles, Optimization, Engineering, Machines, Rockets, Planetary Motion, Gravity…

Motion in time and space

MODERN MATHEMATICSMODERN MATHEMATICS----CALCULUSCALCULUS

Sir Isaac Newton

(1643-1727)

MATHEMATICS OF THE FUTUREMATHEMATICS OF THE FUTURE——GRAPH THEORYGRAPH THEORY

The Mathematics of:Computer Chips, the Internet, Electrical Circuits, Cell Phone Coverage, Transportation Networks, Social Interactions, Genetics, Food Webs….

Interconnections and Relations

William Tutte

(1917- 2002)

The edge e = (u,u) is called a loop.

A graph, G, is a bunch of dots, called vertices, together with a (possibly empty) bunch of lines connecting the vertices of G called edges.

DEFINITIONSDEFINITIONS

The edge f = (u,v) is said to join the vertices u and v. The vertices u and v are adjacent to each other, while f is said to beincident with u and v. If e1 and e2 are distinct edges of G with a common vertex, then e1 and e2 are said to be adjacent.

e1

e2

eu

v fA digraph, i.e. a directed or oriented graph.

A FEW COMMON GRAPHSA FEW COMMON GRAPHS

C6 (cycle)

K3,4 (bipartite)

E

BF

A

CD

1 2

34

56A planar (multi) graph

(tree) K6(complete)

A vertex of degree 3

KONIGSBERG BRIDGE PROBLEMKONIGSBERG BRIDGE PROBLEM

Can you take a walk crossing each bridge exactly once and returning to your starting place?

A map of Konigsberg

in Euler’s time….

Leonard Euler (1707-1783)

KONIGSBERGEKONIGSBERGE BRIDGE WALK IS NOT POSSIBLEBRIDGE WALK IS NOT POSSIBLE……..

Theorem: A graph has a Euler cycle if and only if all the vertices have even degree (an even number of edges attached).

No…. All vertices have odd degree!

Yes…. All vertices have even degree!

Solution from Fleury’s algorithm, O(E2) or better.

PROBLEMPROBLEM

Say we want to read this piece of single-stranded DNAWe can’t read it all in one piece, so we break it up into l-length “fragments,”and then piece it back together to reconstruct the DNA sequence.Here, we’ll use fragments of 4 nucleotides.For the sequence above, we would get fragments of:

ATCGACTATAAGGCATCGAA

TCGA

CGAC GACTACTA

CTATTATA

ATAA

TAAGAAGGAGGC

GGCA GCAT

CATC ATCG

TCGA CGAA

ATCG

A T C G A T C A T A A G G C A T C G A A

CONSTRUCTING THE CONSTRUCTING THE DEBRUJINDEBRUJIN GRAPHGRAPH

The set of fragments can be represented by a graph.Each fragment has length 4. The “head” vertex contains the first three of one fragment and the “tail” vertex contains the last three. A directed edge joins the vertices.Example:

GGCA = GGC GCA

TCGA CGACGACTACTA

CTATTATA

ATAA

TAAG

AAGG

AGGC

GGCA

GCAT

CATC ATCG

TCGA

CGAA

ATCG

CONSTRUCTING THE CONSTRUCTING THE DEBRUIJNDEBRUIJN GRAPH CONT.GRAPH CONT.

Do for all fragments – connect with directed edges.Above is a construction of the DeBruijn graph for the subsequence of DNA above

CGA

GAC ACT

CTA

TAT

ATA TAA

AAG

AGGGGCGGC

CAT

ATCGAA

TCG

GCA

ATCGACTATAAGGCATCGAA

CREATING THE 2CREATING THE 2--IN 2IN 2--OUT DIGRAPHOUT DIGRAPH

Notice only 3 vertices with more than degree two. We can redraw this graph so it only has vertices with degree 4.

CGA

GAC ACT

CTA

TAT

ATA TAA

AAG

AGGGGCGGC

CAT

ATCGAA

TCG

GCACGA

ATC

TCG

CGATCGATC

EULERIANEULERIAN DIGRAPHSDIGRAPHS

Assumptions (probalisticallyjustified):1st and last fragments don’t match—we connect them at the end.1.No three-way or greater repeats.

(From repeats of length greater than n)

(From sequences of length greater than n without repeats elsewhere)

This leads to Eulerian digraphs

Some simplifications

Final form is a 4-regular Eulariandigraph (two in-arrows and two out-arrows at each vertex

ENUMERATING THE RECONSTRUCTIONSENUMERATING THE RECONSTRUCTIONS

Each Eulerian circuit corresponds to a reconstruction of the DNA strand.

Some graphs have many Eulerian circuits.

Only ONE corresponds to the correct sequencing of the original strand…

ENUMERATION IS POSSIBLEENUMERATION IS POSSIBLE

Given the set of fragments, one can count the number of Eulerian circuits in the resulting graph.This gives the number of possible ‘misconstructions’ of the strand.The number of Eulerian circuits can be found in polynomial time (matrix tree/BEST).The related question of how many sets of DNA data could be ‘misconstructed’ in exactly k ways has been explored using graph polynomials.This question amounts to asking how many Euleriandigraphs with maximum degree 4 have k Eulerian circuits.

RECOGNIZING AND RECOGNIZING AND ANALYZING NETWORKSANALYZING NETWORKS

http://infosthetics.com/archives/facebook_graph2.jpg

http://apps.facebook.com/touchgraph/?ref=appd_my_recent

Social Networks

Jo EllisJo Ellis--MonaghanMonaghan

INFORMATION TRANSFER IN A NETWORKINFORMATION TRANSFER IN A NETWORK

The Small World Phenomenon

Stanley Milgram sent a series of traceable letters from people in the Midwest to one of two destinations in Boston. The letters could be sent only to someone whom the current holder knew by first name. Milgram kept track of the letters and found a median chain length of about six, thus supporting the notion of "six degrees of separation."

http://mathforum.org/mam/04/poster.html

GRAPH DRAWINGGRAPH DRAWING——DISPLAYING DATA DISPLAYING DATA

"Risk network Structure in the early epidemic phase of HIV transmission in Colorado Springs," Sexually Transmitted Infections, 78 (2002). Pp. i159-i163.

THE THE ‘‘SHAPESHAPE’’ OF OF YOUR DATAYOUR DATA

•Stock Ownership (2001 NY Stock Exchange)

•Children’s Social Network

http://mathforum.org/mam/04/poster.html

QUANTIFYING STRUCTUREQUANTIFYING STRUCTURE

MapQuest

JetBlue

Scale FreeScale Free

DistributedDistributed

Num

ber o

f ver

tices

Vertex degree

Num

ber o

f ver

tices

Vertex degree

ROLLING BLACKOUTS IN AUGUST 2003ROLLING BLACKOUTS IN AUGUST 2003

http://encyclopedia.thefreedictionary.com/_/viewer.aspx?path=2/2f/&name=2003-blackout-after.jpg

SOME NETWORKS ARE MORE ROBUST THAN SOME NETWORKS ARE MORE ROBUST THAN OTHERS.OTHERS.

BUT HOW DO WE MEASURE THIS?BUT HOW DO WE MEASURE THIS?

http://www.caida.org/tools/visualization/mapnet/Backbones/

A NETWORK MODELED BY A GRAPHA NETWORK MODELED BY A GRAPH(communication, transportation)

A functional network(can get from any vertex to any other along functioning edges)

A dysfunctional network (vertices s and t can’t

communicate)

s

t

Question: If each edge operates independently with probability p, what is the probability that the whole network is functional?

If an edge is working (this happens with probability p), it’s as thought the two vertices were “touching”—i.e. just contract the edge:

If an edge is not working (this happens with probability 1-p), it might as well not be there—i.e. just delete it:

Thus, if R(G;p) is the reliability of the network G where all edges function with a probability of p, and e is not a bridge nor a loop, then

R(G;p) =(1-p)R(G-e;p) + p R(G/e;p)

DELETION AND CONTRACTION IS A NATURAL DELETION AND CONTRACTION IS A NATURAL REDUCTION FOR NETWORK RELIABILITYREDUCTION FOR NETWORK RELIABILITY

= (1-p)p2 + p(1-p)p + p2+ p (1-p) + p p= (1-p)p2

(1-p) + p

RELIABILITY EXAMPLERELIABILITY EXAMPLE

Note that, if every edge of the network is a bridge*, then R(G;p) = (p)E, where E is the number of edges.

Also note that R(loop;p) = 1

E.g.:

So R(G;p) = 3p2- 2p3 gives the probability that the network is functioning if every edge functions with probability p.

E.g. R(G; .5)=.5625*A bridge is an edge whose removal disconnects the graph.

THE THE ISINGISING/POTTS MODEL/POTTS MODEL

Consider a sheet of metal:

It has the property that at low temperatures it is magnetized, but as the temperature increases, the magnetism “melts away”*.

We would like to model this behavior. We make some simplifying assumptions to do so.

The individual atoms have a “spin”, i.e., they act like little bar magnets, and can either point up (a spin of +1), or down (a spin of –1).

Neighboring atoms with the same spins have an interaction energy, which we will assume is constant.

The atoms are arranged in a regular lattice.

*Mathematicians should NOT attempt this at home…

ONE POSSIBLE STATE OF THE LATTICEONE POSSIBLE STATE OF THE LATTICE

A choice of ‘spin’ at each lattice point.

2q

Ising Model has a choice of two possible

spins at each point

THE ENERGY (HAMILTONIAN) OF THE STATETHE ENERGY (HAMILTONIAN) OF THE STATE

here is ( )H w -10J

A state w with the value of δ marked on each edge.

Endpoints have different spins, so δ is 0.

Endpoints have the same spins, so δ is 1.

0

10

00

00 00

0

0

1

1

1 111

0

11

1

0

0

000 0

0 0 0

0

,0 for 1 for a b

a ba b

The Hamiltonian of a state is the sum of the energies on edges withendpoints having the same spins.

,edges

H J a b

where a and b are the endpoint of an edges, J is the interaction

energy, and

NOT JUST MAGNETISMNOT JUST MAGNETISM——ANY SYSTEM WHERE LOCAL ANY SYSTEM WHERE LOCAL MICROSCALEMICROSCALE INTERACTION DETERMINE THE INTERACTION DETERMINE THE MACROSCALEPROPERTIESMACROSCALEPROPERTIES OF THE SYSTEMOF THE SYSTEM

Colorings of the points with q colors

Cellular models (temp= cellular motility)Healthy cancerous Necrotic

Epidemiology (temp = virulence)

Healthy Contagious Symptomatic Immune Deceased

MORE STATESMORE STATES----SAME HAMILTONIANSAME HAMILTONIAN

The Hamiltonian still measures the overall energy of the a state of a system.

10H J

1

1

1

1

1

1 1

1

1

1 0 0

0

0

0

0

0

0

0

0

0

00

0

The Hamiltonian of a state of a 4X4 lattice with 3 choices of spins (colors) for each element.

,( ) a be d g e s

H w J

(note—qn possible states)

The probability of a particular state S occurring depends on the temperature, T

(or other measure of activity level in the application—e.g. virulence of a disease)

--Boltzmann probability distribution--

PROBABILITY OF A STATEPROBABILITY OF A STATE

all states

exp( ( ))

exp( ( ))

H SP S

HS

S

4

231 where 1.38 10 joules/Kelvin and is the temperature of the system.k TkT

The numerator is easy. The denominator, called the Potts Model Partition Function,

is the interesting (hard) piece.Turns out to be related to that reliability polynomial (!)

EXAMPLEEXAMPLE

all states

exp( ( ))

exp( ( ))

H SP SH

S

S

The Potts model partition function of a square lattice with two possible spins

12exp(2 ) 2exp(4 ) 2J J

4

4H J 2H J 2H J 2H J

2H J 2H J 2H J

2H J 2H J 2H J

2H J 2H J 2H J 4H J

0H

0H

exp(4 )

12exp(2

all r

) 2exp(4 ) 2

edPJ

J J

Minimum Energy States

PROBABILITY OF A STATE OCCURRING DEPENDS ON PROBABILITY OF A STATE OCCURRING DEPENDS ON THE TEMPERATURETHE TEMPERATURE

P(all red, T=0.01) = .50 or 50%

P(all red, T=2.29) = 0.19 or 19%

P(all red, T = 100, 000) = 0.0625 = 1/16

Setting J = k for convenience, so

exp(4/ )12exp(2/ ) 2exp(4/ ) 2

all red TPT T

EFFECT OF TEMPERATUREEFFECT OF TEMPERATURE

Consider two different states A and B, with H(A) < H(B). The relative probability of the two states is:

At high temperatures (i.e., for kT much larger than the energy difference |D|), the system becomes equally likely to be in either of the states A or B - that is, randomness and entropy "win". On the other hand, if the energy difference is much larger than kT (very likely at low temperatures), the system is far more likely to be in the lower energy state.

( ) ( )

all states all states

H A H B

H H

P A e eP B

e e

S S

S S

( )

( ) , where 0.

H A D

kTH B

DkTee e D H A H B

e

ISINGISING MODEL AT DIFFERENT TEMPERATURESMODEL AT DIFFERENT TEMPERATURES

Cold Temperature Hot Temperature

Critical TemperatureHere H is

and energy is

i js s

# of squaresH

movie

applet

Images from http://bartok.ucsc.edu/peter/java/ising/keep/ising.html

MONTE CARLO SIMULATIONSMONTE CARLO SIMULATIONS

?

http://www.pha.jhu.edu/~javalab/potts/potts.html

EPIDEMIOLOGY MODELEPIDEMIOLOGY MODEL

1. You are a node.2. Identify your neighbors (anyone you can reach out and touch).

These are the folks you can infect (there is an edge between youand each of them).

3. If you have an ‘L’ in your name, you are infected. Raise your hand.

4. If you are next to an infected person, think of a number between1 and 10.

5. If your number is ___________, then you are infected. Raise your hand. (END Round 1)

6. Go to step (4).7. If you have been sick for two rounds, fold your arms. You are

now immune.8. QUESTIONS: Will this disease ‘peter out’? Will everyone end

up immune? How many people might be sick at a time?

MONTE CARLO SIMULATIONSMONTE CARLO SIMULATIONS

B (old)

A (change to new)

B (stay old)

Generate a random number r between 0 and 1.

P Ar

P B

P Ar

P B

CAPTURE EFFECT OF TEMPERATURECAPTURE EFFECT OF TEMPERATURE

exp(‘-’/kT) ~ 0exp(‘-’/kT) ~1H(B) < H(A)B is a lower energy

state than A

H(B) > H(A)B is a higher energy

state than A > r, so change to lower energy state.

> r, so change states.

exp(‘+’/kT) ~1exp(‘+’/kT) ~1

< r, so stay in low energy state.

> r, so change states.

Low TempHigh Temp

Given r between 0 and 1, and that , with Bthe current state and A the one we are considering changing to, we have:

expP A H B H AP B kT

This models how tumor growth is influenced by the amount and location of a nutrient.

The energy function is modified by the volume of a cell and the amount of nutrients.

TUMOR MIGRATIONTUMOR MIGRATION

' ' ' '

2( ) ( ) 1 ( ) ( , )

ij i j ij i j Tij i j

H J V Kp i j

7

1

11

1

FOAMSFOAMS

“Foams are of practical importance in applications as diverse as brewing, lubrication, oil recovery, and fire fighting”.

The energy function is modified by the area of a bubble.

Results: Larger bubbles flow faster.There is a critical velocity at which the foam starts to flow uncontrollably

2

{ , }

(1 ) ( )i j n n

i j n

H J a A

9

Y. Jiang, J. Glazier, Foam Drainage: Extended Large-Q Potts Model Simulation

We study foam drainage using the large-Q Potts model... profiles of draining beer foams, whipped cream, and egg white ...

Olympic Foam: http://mathdl.maa.org/mathDL?pa=mathNews&sa=view&newsId=392

A PERSONAL FAVORITEA PERSONAL FAVORITE

http://www.lactamme.polytechnique.fr/Mosaic/images/ISIN.41.16.D/display.html

SOCIOLOGICAL APPLICATIONSOCIOLOGICAL APPLICATION

The Potts model may be used to “examine some of the individual incentives, and perceptions of difference, that can lead collectively to segregation …”.(T. C. Schelling won the 2005 Nobel prize in economics for this work)

Variables: ‘temperature’ = economic factorsPreferences of individualsSize of the neighborhoodsNumber of individuals

8