The (Non-perfect) Mathematics of Trust

@vaidehijoshi

1735

Professor Roman Sznajder

but seemed to me worthy of attention

This question is so banal,

in that [neither] geometry,

nor algebra,

nor even the art of counting

was sufficient to solve it.

"

sufficient

—Leonard Euler

MAA Euler Archive

Vaidehi's Super Fast And Exceedingly Friendly Guide to Graph Theory

™

(no PhD required!)

nodes or vertices

edges

undirected

directed

ordered pairs

(x, y)

(V, E)

(x, y)

G = (V, E)

a set of vertices

a set of edges

G = (V, E)

a

b

c

 V = { a, b, c } 
 E = { {a, b}, {b, c}, {c, a} }

G = (V, E)

a

b

c

 V = { a, b, c } 
 E = { (a, b), (b, c), (c, a) }

degree

how many vertices are adjacent to this one?

a

b

c

d

e

d = 2

e = 1

a = 4

path

getting from an origin to a destination

simple

without

without repeating vertices

a

b

c

simple path

Eulerian path

All of the vertices with a non-zero degree must be connected

Two vertices must have an odd degree

All of the vertices in the connected graph must be of an even degree

OR

2.

1.

2.

Euler proved something impossible by proving what was possible.

was

"Geometry of Position"

Graph theory

Euler's problem

graph theory is all over computer science

The Knight's Tour

Wikimedia Foundation

Cayley Graphs

Jaap's Puzzle Page

Google's PageRank Algorithm

Wikimedia Foundation

graph theory

is all over the web

literally is the web

is all over the web

website a

website b

website c

website d

website e

website f

Euler's problem

computer
science

bioinformatics

sequencing the

human genome

2001

de Bruijn graph

Cold Spring Harbor Laboratory Press

Euler's problem

computer
science

genome sequencing

Euler's problem

computer
science

genome sequencing

Euler's problem

computer
science

2001

Nicholas Christakis

We’re all embedded in massive social networks that have massive influences on our well-being.

"

Harvard University

our connections can influence each other in positive or negative ways

humans are connected by edges

genome sequencing

Euler's problem

computer
science

genome sequencing

Euler's problem

computer
science

sociology

6 degrees

of Kevin Bacon

genome sequencing

Euler's problem

computer
science

sociology

genome sequencing

Euler's problem

computer
science

Kevin Bacon?

sociology

But it's not just Kevin Bacon!

directed graphs

undirected graphs

genome sequencing

Euler's problem

computer
science

Kevin Bacon?

sociology

sharing
economy

genome sequencing

Euler's problem

computer
science

Kevin Bacon?

sociology

sharing
economy

genome sequencing

Euler's problem

computer
science

Kevin Bacon?

sociology

trust

sharing
economy

the sharing economy leverages our interconnectedness

how we relate to one another

how we navigate our careers

People are more likely to be referred for jobs by their 2nd and 3rd degree connections.

Connections matter most in technology-related industries.

Men typically have larger professional networks than women, but women’s connections appear to be higher quality when it comes to job-hopping.

how we choose our government

Fake accounts based in Russia purchased more than $100,000 worth of ads on divisive issues in the run-up to the 2016 presidential election.

$10 and 15 minutes

20,658 employees

2 billion monthly users

1 employee for every 97,000 users

It can be very easy to trust the algorithms that automate our world.

But the mathematics behind them can't be trusted blindly.

Trust is a (very) hard math problem.

to pick up where the math ends,

We need humans to intervene,

to innovate better approaches to the really hard problems.

to evaluate whether the math is fair to everyone,

That's exactly what Euler did.

Whatever happened to Königsberg?