Visually Intractable Problems

Nathaniel DeanDepartment of Mathematics

Texas State UniversitySan Marcos, Texas 78666 USA

nd17@txstate.edu

Career Options for Underrepresented Groups in Mathematical Sciences,

Minneapolis, MN

March 27, 2010

Models of Human Behavior(social networks, biology, epidemiology)

CallDetail

InternetTraffic

PsychometricBiochemistry

GlobalTerrorismDatabase

MarketBaskets

A variety of massive data sets can be modeled as “large” mathematical structures.

Problem: Extract and catalog interactions to identify “interesting” patterns or collaborative sub-networks. Interactions between genes, proteins, terrorists,

physical contacts, neurons, etc.

Mathematical & ComputationalModeling Cycle

Data from theReal World

Math & ComputerModels

New View ofthe World

Mathematical &ComputationalResults

VerifyExplain

Interpret

OrganizeSimplify

Analyze

Mathematics → Super Abilities

Ecomonics

Biology

Puzzles

Games

Logic

Sociology

Financial Markets

Medicine

Computing

Linguistics

Physics

Engineering

Disease

Graph Model

A graph consists nodes and edges.

The nodes model entities. The edge set models a

binary relationship on the nodes.

Edges may be weighted, reflecting similarities/dissimilarities between nodes.

Graph Drawing• Find an aesthetic layout of the graph

that clearly conveys its structure.• Assign a location for each node so

that the resulting drawing is “nice”.• Example: Protein Interaction Data (

file)V = {1,2,3,4,5,6}

E = {(1,2),(2,3),(1,4),

(1,5),(3,4),(3,5), (4,5),(4,6),(5,6)}

Input (data) Output (drawing)

Clustering Reveals the Macro Structure of Data

dense sub-graph dense sub-graph

dense sub-graphsparse sub-graph

Communitiesof interest?

dense sub-graph

a

deg( b ) = 4

deg( c ) = 4 deg( f ) = 3 deg( g ) = 4

b

g f e

c ddeg( d ) = 1

deg( e ) = 0

Degree of a Vertex

= the number of edges incident with it.

deg( a ) = 2

CountriesRegions

States Counties

Towns Subdivisions

Blocks Lots

Buildings

Hierarchies (geography, families, companies)

Work on Large Graphs & Hierarchies

Show demo

Are some graphs too complicated to understand?

The Algebraic School (end of 19th century)George Boole and others, Algebraic structure of

formulas, Boolean algebra

The Mathematical School (early 20th century)

• The Hilbert program: formalization of all of mathematics with a proof of consistency

• Godel’s Incompleteness TheoremAny axiomatization that includes arithmetic there is a sentence neither provable nor disprovable.

• Church-Turing thesis (computability)Defined what it means to compute.

A Brief History of Logic

Forms of Intractability

PSPACE, NP-hardness

Computability

Undecidability

Incompleteness

Incomprehensibility

Ulam’s Lattice Point ConjectureIn any partition of the integer lattice points into uniformly bounded sets there exists a set that is adjacent to at least six other sets.- Joseph Hammer Unsolved Problems Concerning Lattice Points, 1977

Ulam’s Lattice Point ConjectureIn any partition of the integer lattice points into uniformly bounded sets there exists a set that is adjacent to at least six other sets.- Joseph Hammer Unsolved Problems Concerning Lattice Points, 1977

Not Adjacent

Adjacent

Adjacent

Ulam’s Lattice Point ConjectureIn any partition of the integer lattice points into uniformly bounded sets there exists a set that is adjacent to at least six other sets.- Joseph Hammer Unsolved Problems Concerning Lattice Points, 1977

Not Uniformly Bounded

Ulam’s Lattice Point ConjectureIn any partition of the integer lattice points into uniformly bounded sets there exists a set that is adjacent to at least six other sets.- Joseph Hammer Unsolved Problems Concerning Lattice Points, 1977

Homomorphism

• If xy E(G), then f(x)f(y) E(H) or f(x) = f(y) and

• If ab E(H), then there exists x,y V(G) such that f(x) = a, f(y) = b, and xy E(G).

A surjective map f: V(G) V(H) of G onto H where

Homomorphism f: V(G) V(H)

Homomorphism f: V(G) V(H)

Homomorphism f2: V(G) V(H)

A homomorph H of G is a uniformly bounded homomorph if for some integer m every vertex x of H satisfies

.|)(| 1 mxf

Ulam number u(G) = min {(H): H is a uniformly bounded homomorph of G}.

• u(G) (G).• H is a homomorph of G u(H) u(G).• F G u(F) u(G).

An infinite tree which is locally finite must contain an infinite path.

N

Konig’s Infinity LemmaHierarchical Structure

Konig’s Infinity Lemma

Proof Idea:Since there are finitely many branches, at least one of them must have an infinite subtreeGo in that direction.

Konig’s Infinity Lemma

Proof Idea:

Find an infinite branch of the tree.

Go in that direction.

Proof Idea:

Find an infinite branch of the tree.

Go in that direction.

Konig’s Infinity Lemma

Proof Idea:

Find an infinite branch of the tree.

Go in that direction.

Konig’s Infinity Lemma

Konig’s Infinity Lemma

Proof Idea:

Find an infinite branch of the tree.

Go in that direction.

Konig: An infinite tree which is locally finite contains an infinite path.

Corollary: Every finite homomorph of contains as a subgraph.

NN

Corollary: 2)( Nu

If G has a good drawing in a strip, then u(G) 2.

L

Shrinking each cell to a vertex yields a homomorph isomorphic to a collection of paths.

If G has a good drawing in the plane, then u(G) 6.

Shrinking each cell to a vertex yields a homomorph isomorphic to a subgraph of the triangular grid.

L

(not a good drawing )

u(G) 3 every drawing of G in any strip [0,N] x R is incomprehensible.

u(G) 7 every drawing of G in the plane is incomprehensible.

• d + such that, for any integer N, there is a region of diameter ≤ d containing ≥ N verticesOR

• Edges are arbitrarily long.

G has no good drawing ≡ G is incomprehensible.

Open ProblemsMathematics → Super Abilities

Disease

Health Care

Family

Hunger

Politics

War

Violence

Medicine

Poverty

Emotions

Survival

Disease

Love Happiness

Feelings

Success