Numerical Methods for Data Science: Spectral Network Analysis, Part I, Lecture notes of Chemistry

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Numerical Methods for Data Science:

Spectral Network Analysis, Part I

David Bindel

21 June 2019

Department of Computer Science

Cornell University

Lecture plan

Three threads from “lay of the land” to current research:

• Monday: Latent Factor Models
• Wednesday: Scalable Kernel Methods
• Friday: Spectral Network Analysis
  • 1:30-2:30: Network spectra, optimization, and dynamics
  • 3:00-4:00: Network densities of states

Slides posted on web page (linked from my Cornell page).

Networks and Graphs

A graph (network) consists of

• Node (or vertex) set V
• Edges E  V  V
  • Undirected if ( u , v ) 2 E =) ( v , u ) 2 E
• Optional edge weights E 7! R

Can also add node weights or edge/node attributes.

Example Networks: Classic CS

Often small and/or highly structured:

• Finite state automata
• Search trees and DAGs
• Graphical models (correlated random variables)

Mostly not the topic for today.

Example Networks: Citations

http://www.vosviewer.com/

Often directed, some very high-degree nodes, “small world”:

• Web pages, citation networks
• Purchase networks

Other Networks

Lots of others as well!

• Friendship networks
• Interaction networks (phone calls, etc)
• Food webs
• Protein interactions

The Big Questions: Geometric Embedding

Is there an underlying geometry to the network?

The Big Questions: Centrality and Ranking

Who are important players?

The Big Questions

One might ask many more questions:

• Graph alignment: Can we map between similar structures?
• Link prediction: Can we extrapolate the pattern?
• Cascade analysis: How does information spread?

Common approach: map to a linear algebra problem!

From Networks to Matrices

1

2

3

4

5

6

7

























1

1 1

1 1

1 1 1

1 1

1 1

1 1

























Adjacency A ; unweighted is

auv =

1 , ( u , v ) 2 E

0 , otherwise

Degree du =

v

auv is total adjacent edges (edge weight).

Distinguish in/out in directed case.

From Networks to Matrices

1

2

3

4

5

6

7

























1 − 1

− 1 2 − 1

− 1 2 − 1

− 1 3 − 1 − 1

− 1 2 − 1

− 1 2 − 1

− 1 − 1 2

























Laplacian L = G

T G = D A ; unweighted is

l uv

degree d u

, u = v

1 , ( u , v ) 2 E

0 , otherwise

A Bestiary of Matrices

• Adjacency matrix: A
• Laplacian matrix: L = D A
• Unsigned Laplacian: L = D + A
• Random walk matrix: P = AD

1 (or D

1 A )

• Normalized adjacency:

A = D

1 / 2 AD

1 / 2

• Normalized Laplacian:

L = I

A = D

1 / 2 LD

1 / 2

• Modularity matrix: B = A

dd

T

2 n

• Motif adjacency: W = A

2 ⊙ A

Story 1: Dynamics and Diffusion

The Random Walker

Graph with adjacency A ( a ij

denotes edge j to i ),

p ij

= a ij

/ d j

= probability of step j! i

Let w j

( t ) denote probability a walker is at node j at time t ; then

w ( t + 1 ) = Pw ( t ) = P

t

w ( 0 )

Equations for a discrete time Markov chain  power iteration