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IPAM Intelligent Extraction of Information from Graphs & High Dimensional Data July 26, 2005. Random Dot Product Graphs. Ed Scheinerman Applied Mathematics & Statistics Johns Hopkins University. Coconspirators. Libby Beer John Conroy (IDA) Paul Hand (Columbia) Miro Kraetzl (DSTO)

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random dot product graphs

IPAM

Intelligent Extraction of Information from

Graphs & High Dimensional Data

July 26, 2005

Random Dot Product Graphs

Ed Scheinerman

Applied Mathematics & Statistics

Johns Hopkins University

coconspirators
Coconspirators
  • Libby Beer
  • John Conroy (IDA)
  • Paul Hand (Columbia)
  • Miro Kraetzl (DSTO)
  • Christine Nickel
  • Carey Priebe
  • Kim Tucker
  • Stephen Young (Georgia Tech)
overview
Overview
  • Mathematical context
  • Modeling networks
  • Random dot product model
  • The inverse problem
graphs i have loved
Graphs I Have Loved
  • Interval graphs & intersection graphs
  • Random graphs
  • Random intersection graphs
  • Threshold graphs & dot product graphs
intersection graphs
Intersection Graphs

{1}

{1}

{1,2}

{2}

random graphs
Random Graphs

Erdös-Rényi style…

p

1 – p

Randomness is “in” the edges. Vertices are “dumb” placeholders.

random intersection graphs
Random Intersection Graphs
  • Assign random sets to vertices.
  • Two vertices are adjacent iff their sets intersect.
  • Randomness is “in” the vertices.
  • Edges reflect relationships between vertices.
threshold graphs
Threshold Graphs

0.5

0.6

0.3

0.8

dot product graphs
Dot Product Graphs

[1 0]

[2 0]

[0 1]

[1 1]

Fractional intersection graphs

physical networks
Physical Networks

Internet

Telephone

Power grid

Local area network

social networks

Alice

Bob

Social Networks

2003-4-10

B

A

example email at hp
Example: Email at HP
  • 485 employees
  • 185,000 emails
  • Social network (who emails whom) identified 7 “communities”, validated by interviews with employees.
properties of social networks
Properties of Social Networks
  • Clustering
  • Low diameter
  • Power law
properties of social networks18
Properties of Social Networks
  • Clustering
  • Low diameter
  • Power law

b

a

c

properties of social networks19
Properties of Social Networks
  • Clustering
  • Low diameter
  • Power law

“Six degrees of separation”

properties of social networks20
Properties of Social Networks
  • Clustering
  • Low diameter
  • Power law

Degree Histogram

log N(d)

log d

degree histogram example 1
Degree Histogram Example 1

2838 vertices

Number of vertices

degree

degree histogram example 2
Degree Histogram Example 2

16142 vertices

Number of vertices

degree

random graph models

Random Graph Models

Goal: Simple and realistic random graph models of social networks.

erd s r nyi
Erdös-Rényi?
  • Low diameter!
  • No clustering: P[a~c]=P[a~c|a~b~c].
  • No power-law degree distribution.

Not a good model.

model by fan chung et al
Model by Fan Chung et al

Consider only those graphs with

with all such graphs equally likely.

people as vectors
People as Vectors

Sports

Politics

Movies

Graph theory

shared interests
Shared Interests

Alice and Bob are more likely to communicate when they have more shared interests.

whence the vectors
Whence the Vectors?
  • Vectors are given in advance.
  • Vectors chosen (iid) from some distribution.
random dot product graphs ii
Random Dot Product Graphs, II
  • Step 1: Pick the vectors
    • Given by fiat.
    • Chosen from iid a distribution.
  • Step 2: For all i<j
    • Let p=f(xi•xj).
    • Insert an edge from i to j with probability p.
megageneralization
Megageneralization
  • Generalization of:
    • Intersection graphs (ordinary & random)
    • Threshold graphs
    • Dot product graphs
    • Erdös-Rényi random graphs
  • Randomness is “in” both the vertices and the edges.
  • P[a~b] independent of P[c~d] when a,b,c,d are distinct.
isolated vertices
Isolated Vertices

Thus, the graph is not connected, but…

mostly connected
“Mostly” Connected

“Giant” connected component

A “few” isolated vertices

diameter 6 proof outline

Isolated

Attached

Attachedpair

Diameter = 2

Diameter ≤ 6 Proof Outline
graphs to vectors

Graphs to Vectors

The Inverse Problem

given graphs find vectors
Given Graphs, Find Vectors
  • Given: A graph, or a series of graphs, on a common vertex set.
  • Problem: Find vectors to assign to vertices that “best” model the graph(s).
maximum likelihood method
Maximum Likelihood Method
  • Feasible in dimension 1. Awful d>1.
  • Nice results for f(t) = t / (1+t).
convergence55
Convergence

diagonal entries

iteration

convergence56
Convergence

diagonal entries

iteration

convergence57
Convergence

diagonal entries

iteration

convergence58
Convergence

diagonal entries

iteration

convergence59
Convergence

diagonal entries

iteration

convergence60
Convergence

diagonal entries

iteration

applications

Applications

Network Change/Anomaly Detection

Clustering

synthetic lethality graphs
Synthetic Lethality Graphs
  • Vertices are genes in yeast
  • Edge between u and v iff
    • Deleting one of u or v does not kill, but
    • Deleting both is lethal.
sl graph status
SL Graph Status
  • Yeast has about 6000 genes.
  • Full graph known on 126 “query” genes (about 1300 edges).
  • Partial graph known on 1000 “library” genes.
random dot product graphs71
Random Dot Product Graphs
  • Extension to higher dimension
    • Cube
    • Unit ball intersect positive orthant
  • Small world measures: clustering coefficient
  • Other random graph properties
vector estimation
Vector Estimation
  • MLE method
    • Computationally efficient?
    • More useful?
  • Eigenvalue method
    • Understand convergence
    • Prove that it globally minimizes
    • Extension to missing data
  • Validate against real data
network evolution
Network Evolution
  • Communication influences interests:
rapid generation
Rapid Generation
  • Can we generate a sparse random dot product graph with n vertices and m edges in time O(n+m)?
  • Partial answer: Yes, but.