Studying Networks Studying Networks Through an Through an Algorithmic LensAlgorithmic Lens

Michael T. GoodrichUniv. of California, Irvine

A Paradigm ShiftA Paradigm Shift Transitioning from equations to algorithms as

the fundamental model for prediction

Images from wikimedia.org and computerhistory.org

2000

E=mc2

Equations

Algorithms

F=ma

19451700

The Algorithmic LensThe Algorithmic Lens

Image from http://uwnews.org/uweek/uweekarticle.asp?visitsource=uwkmail&articleID=39100

Exposes the computational nature of natural processes and provides a language for their description.

Brings to bear fundamental algorithmic concepts: adversarial and probabilistic models, asymptotic analysis, intractability, computational learning theory, threshold behavior.

Provides a new worldview for many fields.

Two Flavors of the Algorithmic LensTwo Flavors of the Algorithmic Lens

1. Discover inherent combinatoric, topological and geometric properties of various networks that can improve algorithms that operate on such networks.

2. The algorithmic worldview provides new computational insights, models, and metaphors about networks

Image by Argus fin from http://commons.wikimedia.org/wiki/Image:International_E_Road_Network.png, and is in the public domain

Social Sciences and NetworksSocial Sciences and Networks The Web is a powerful laboratory for studying social and

economic systems as computational processes. Insights from algorithmic theory are indispensable for

understanding interactions and relationships that the Internet has spawned.

Images from wikimedia.org, in public domain,under CC license and GNU FDL

Kevin BaconBarack Obama

The World is Not Flat (or Spherical)The World is Not Flat (or Spherical) Road networks are built on the surface of the Earth, which

is like a sphere. Previous researchers have viewed road networks as plane graphs, that is, graphs that are drawn on a sphere without edge crossings. (E.g., see [Chou, 96].)

Image by NASA, is available at http://commons.wikimedia.org/wiki/Image:Europe_terrain.jpg, and is in the public domain

The World is Not Flat (or Spherical)The World is Not Flat (or Spherical) Unfortunately, real road networks are highly non-planar. In particular, a road network with n vertices typically has a

number of edge crossing proportional to

)( nO

Data is from the U.S. TIGER/Line road network database, as provided by the Ninth DIMACS Implementation Challenge

The Geometry of Road NetworksThe Geometry of Road Networks So we want a new geometric way to characterize road

networks that leads to improved algorithms Previous researchers have made assumptions about edge weights,

such as their being Euclidean or having nice distributions (e.g., [Goldberg, 93], [Holzer et al., 05], [Klunder and Post, 06], [Sanders and Schultes, 05], [Sedgewick and Vitter, 86], [Thorup, 99]).

Instead, we took a different approach inspired by the work of [Eppstein et al., 93] and [Miller et al., 95, 97, 98] on separators in disk neighborhood systems.

Image by Pbroks13, from http://commons.wikimedia.org/wiki/Image:Apollonian_circles.svg, used under the GFDL 1.2

Disk Neighborhood SystemsDisk Neighborhood Systems Given a collection of disks in the plane, define a graph G

so that each disk is associated with a vertex in G and there is and edge between v and w if their disks intersect.

Image by David Eppstein, available at http://commons.wikimedia.org/wiki/Image:Unit_disk_graph.svg, in the public domain

For each vertex v, define a disk with radius equal to half the length of the longest road adjacent to v.

The road network is guaranteed to be a subgraph of this Natural Disk Neighborhood System.

The Natural Disk Neighborhood The Natural Disk Neighborhood System for Road NetworksSystem for Road Networks

Data is from the U.S. TIGER/Line road network database, as provided by the Ninth DIMACS Implementation Challenge

Future DirectionsFuture Directions Study additional algorithmic properties of networks,

combining combinatorics, geometry, and topology. Use the algorithm as a part of the statistical models

for social interactions