Random Dot Product Graphs Ed Scheinerman Applied Mathematics & Statistics Johns Hopkins University IPAM Intelligent Extraction of Information from Graphs & High Dimensional Data July 26, 2005

Coconspirators Libby Beer John Conroy (IDA) Paul Hand (Columbia) Miro Kraetzl (DSTO) Christine Nickel Carey Priebe Kim Tucker Stephen Young (Georgia Tech)

Overview Mathematical context Modeling networks Random dot product model The inverse problem

Mathematical Context

Graphs I Have Loved Interval graphs & intersection graphs Random graphs Random intersection graphs Threshold graphs & dot product graphs

Interval Graphs

Intersection Graphs {1} {1,2} {2}

Random Graphs Erdös-Rényi style… p1 – p Randomness is “in” the edges. Vertices are “dumb” placeholders.

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

Dot Product Graphs [1 0] [2 0] [1 1] [0 1] Fractional intersection graphs

Communication Networks

Physical Networks Telephone Local area network Power grid Internet

Social Networks Alice Bob A B

Social Network Graphs Vertices (Actors)Edges (Dyads) TelephonesCalls addressesMessages ComputersIP Packets Human beingsAcquaintance AcademiciansCoauthorship

Example: at HP 485 employees 185,000 s Social network (who s whom) identified 7 “communities”, validated by interviews with employees.

Properties of Social Networks Clustering Low diameter Power law

Properties of Social Networks Clustering Low diameter Power law a b c

Properties of Social Networks Clustering Low diameter Power law “Six degrees of separation”

Properties of Social Networks Clustering Low diameter Power law log d log N(d) Degree Histogram

Degree Histogram Example vertices degree Number of vertices

Degree Histogram Example vertices degree Number of vertices

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

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 Consider only those graphs with with all such graphs equally likely.

People as Vectors Sports Politics Movies Graph theory

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

Selecting the Function

Random Dot Product Graphs, I

Generalize Erdös-Rényi

Generalize Intersection Graphs

Whence the Vectors? Vectors are given in advance. Vectors chosen (iid) from some distribution.

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(x i x j ).  Insert an edge from i to j with probability p.

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.

Results in Dimension 1

Probability/Number of Edges

Clustering

Power Law

Power Law Example

Isolated Vertices Thus, the graph is not connected, but…

“Mostly” Connected “Giant” connected component A “few” isolated vertices

Six Degrees of Separation Diameter ≤ 6

Attached Attached pair Diameter ≤ 6 Proof Outline Diameter = 2 Isolated

Diameter ≤ 6 Proof Outline

Graphs to Vectors The Inverse Problem

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 Feasible in dimension 1. Awful d>1. Nice results for f(t) = t / (1+t).

Gram Matrix Approach

Wrong Best Solution

Real Problem

Iterative Algorithm

Convergence

iteration diagonal entries

Convergence iteration diagonal entries

Convergence iteration diagonal entries

Convergence iteration diagonal entries

Convergence iteration diagonal entries

Convergence iteration diagonal entries

Enron example

Applications Network Change/Anomaly Detection Clustering

Change/Anomaly Detection

Graph Clustering

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 Yeast has about 6000 genes. Full graph known on 126 “query” genes (about 1300 edges). Partial graph known on 1000 “library” genes.

What Next?

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 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 Communication influences interests:

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.

The End