On the Convexity of Latent Social Network Inference

1
On the Convexity of Latent Social Network
Inference

• Seth A. Myers, Jure Leskovec
• NIPS 2010

2
Introduction

• Impossible to collect explicit social network
  data in real-world.
• Drug injection users
• Whole network must be inferred from underlying
  observations.
• Cell-phone call or e-mail interaction
• Select threshold t

k
j
i
m
l
3
Goal

• Consider a diffusion process of a contagion
• Observation
• Infection time of nodes
• Reconstruct the underlying social network along
  the edges of which the contagion diffused

t10
k
j
t3
i
m
l
t1
t8
t15
4
Problem Formulation

•  

5
Problem Formulation

•  

6
Maximum Likelihood Formulation

•  

7
Maximum Likelihood Formulation

• Diagonal of A is zero
• Node cannot infect itself
• N(N-1) variables
• Divided by N independent subproblems
• Each problem has N-1 variables
• Observing that the incoming edges to a node can
  be inferred independently of the incoming edges
  of any other node
• Reduce variables

8
Towards the Convex Problem

• Finding the global optimal MLE for A is difficult
• Derive a convex optimization problem
• Guarantees convergence to a globally optimal
  solution
• Highly optimized convex programming method

9
Towards the Convex Problem
10
Towards the Convex Problem
11
Network Sparsity

•  

12
Implementation

•  

13
Experiments(synthetic data)
14
Experiments(real data)