Visually Intractable Problems Nathaniel Dean Department of Mathematics Texas State University San Marcos, Texas USA Career Options for Underrepresented Groups in Mathematical Sciences, Minneapolis, MN March 27, 2010

Models of Human Behavior (social networks, biology, epidemiology) Call Detail Internet Traffic Psychometric Biochemistry Global Terrorism Database Market Baskets 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 & Computational Modeling Cycle Data from the Real World Math & Computer Models New View of the World Mathematical & Computational Results Verify Explain Interpret Organize Simplify 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)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 sparse sub-graph Communities of interest? dense sub-graph

a deg( b ) = 4 deg( c ) = 4 deg( f ) = 3deg( g ) = 4 b g f e c d deg( d ) = 1 deg( e ) = 0 Degree of a Vertex = the number of edges incident with it. deg( a ) = 2

Countries Regions 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 19 th century) George Boole and others, Algebraic structure of formulas, Boolean algebra The Mathematical School (early 20 th century) The Hilbert program: formalization of all of mathematics with a proof of consistency Godel’s Incompleteness Theorem Any 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 PSPACE, NP-hardness Computability Undecidability Incompleteness Incomprehensibility

Ulam’s Lattice Point Conjecture In 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 Conjecture In 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

Ulam’s Lattice Point Conjecture In 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 Conjecture In 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 2 : 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 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. Konig’s Infinity Lemma Hierarchical Structure

  Konig’s Infinity Lemma Proof Idea: Since there are finitely many branches, at least one of them must have an infinite subtree Go 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

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. Corollary:

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

(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 vertices OR Edges are arbitrarily long. G has no good drawing ≡ G is incomprehensible.

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