1. Inferring Networks of Diffusion and Influence
Manuel Gomez Rodriguez1,2 Jure Leskovec1 Andreas Krause3
1 Stanford University 2 MPI for Biological Cybernetics 3 California Institute of Technology

2. Hidden and implicit networks
Many social or information networks are implicit or hard to observe:
Hidden/hard-to-reach populations:
Network of needle sharing between drug injection users
Implicit connections:
Network of information propagation in online news media
But we can observe results of the processes taking place on such (invisible) networks:
Virus propagation:
Drug users get sick, and we observe when they see the doctor
Information networks:
We observe when media sites mention information
Question: Can we infer the hidden networks?

3. Inferring the Network
There is a directed social network over which diffusions take place: a a a b b b d d c c c e e
But we do not observe the edges of the network
We only see the time when a node gets infected:
Cascade c1: (a, 1), (c, 2), (b, 6), (e, 9)
Cascade c2: (c, 1), (a, 4), (b, 5), (d, 8)
Task: inferring the underlying network

4. Examples and Applications
Word of mouth & Viral marketing
Virus propagation
Viruses propagate through the network
We only observe when people get sick
But NOT who infected whom
Recommendations and influence propagate
We only observe when people buy products
But NOT who influenced whom
Process
We observe
It’s hidden
Can we infer the underlying network?

5. Our Problem Formulation
Plan for the talk:
Define a continuous time model of diffusion
Define the likelihood of the observed cascades given a network
Show how to efficiently compute the likelihood of cascades
Show how to efficiently find a graph G that maximizes the likelihood
Note:
There is a super-exponential number of graphs, O(NN*N)
Our method finds a near-optimal graph in O(N2)!

6. Cascade Diffusion Model
Continuous time cascade diffusion model:
Cascade c reaches node u at tu
and spreads to u’s neighbors:
With probability β cascade propagates along edge (u, v)
and we determine the infection time of node v
tv = tu + Δ
e.g.: Δ ~ Exponential or Power-law tu tb tc Δ1 Δ2 a u u b b b c c c d
We assume each node v has only one parent!

7. Likelihood of a Single Cascade
Probability that cascade c propagates from node u to node v is:
Pc(u, v)  P(tv - tu) with tv > tu
Since not all nodes get infected by the diffusion process, we introduce the external influence node m: Pc(m, v) = ε m
Prob. that cascade c propagates in a tree pattern T: ε ε ε b d e a c
Tree pattern T on cascade c: (a, 1), (b, 2), (c, 4), (e, 8)

8. Finding the Diffusion Network
Good news
Computing P(c|G) is tractable:
Bad news
We actually want to search over graphs:
There is a super- exponential number of graphs!
There are many possible propagation trees that are consistent with the observed data:
c: (a, 1), (c, 2), (b, 3), (e, 4)
Even though there are O(nn) possible propagation trees.
Matrix Tree Theorem can compute this in O(n3)! b d e a c b d e a c b d e a c
Need to consider all possible propagation trees T supported by the graph G:
Likelihood of a set of cascades C:
Want to find a graph:

9. An Alternative Formulation
We consider only the most likely tree
Maximum log-likelihood for a cascade c under a graph G:
Log-likelihood of G given a set of cascades C:
The problem is still intractable (NP-hard)
But we present an algorithm that finds near-optimal networks in O(N2)

10. Max Directed Spanning Tree
Given a cascade c,
What is the most likely propagation tree?
where
A maximum directed spanning tree (MDST):
The sub-graph of G induced by the nodes in the cascade c is a DAG
Because edges point forward in time
For each node, just picks an in-edge of max-weight:
Greedy parent selection of each node gives globally optimal tree!

11. Objective function is Submodular
Theorem:
Log-likelihood Fc(G) of cascade c is monotonic, and submodular in the edges of the graph G
Gain of adding an edge to a “small” graph
Gain of adding an edge to a “large“ graph
Fc(A  {e}) – Fc (A) ≥ Fc (B  {e}) – Fc (B)
A  B  VxV
Proof:
We use greedy parent selection and submodularity definition
Log-likelihood FC(G) is a sum of submodular functions, then it is submodular too

12. Finding the Diffusion Graph
Use the greedy hill-climbing to maximize FC(G):
For i=1…k:
At every step, pick the edge that maximizes the marginal improvement
1. Approximation guarantee (≈ 0.63 of OPT)
2. Tight on-line bounds on the solution quality
3. Speed-ups:
Lazy evaluation (by submodularity)
Localized update (by the structure of the problem)
Benefits:
Marginal gains a a b
: 20 b c b
: 17
: 18 d b
: 1
: 2
: 4 d e b
: 3
: 1
: 5 c a c
: 15 b c
: 8
: 6 b d
: 16 e c d
: 7
: 8 e d
: 10
: 8 b e
: 7 d e
: 13

13. Experimental Setup We validate our method on: Synthetic data Real data
Generate a graph G on k edges
Generate cascades
Record node infection times
Reconstruct G
Real data
MemeTracker: 172m news articles
Aug ’08 – Sept ‘09
343m textual phrases (quotes)
How many edges of G can we find?
Precision-Recall
Break-even point
How well do we optimize the likelihood Fc(G)?
How fast is the algorithm?
How many cascades do we need?

14. Pick k strongest edges:
Small Synthetic Example
Small synthetic network:
True network
Baseline network
Our method
Pick k strongest edges: 14

15. Synthetic Networks
1024 node hierarchical Kronecker exponential transmission model
1000 node Forest Fire (α = 1.1) power law transmission model
Performance does not depend on the network structure:
Synthetic Networks: Forest Fire, Kronecker, etc.
Transmission time distribution: Exponential, Power Law
Break-even point of > 90%

16. How good is our graph?
We achieve ≈ 90 % of the best possible network!

17. How many cascades do we need?
With 2x as many infections as edges, the break-even point is already !

18. Running Time
Lazy evaluation and localized updates speed up 2 orders of magnitude!
Can infer a networks of 10k nodes in several hours

19. Real Data: Information diffusion
MemeTracker dataset:
172m news articles from Aug ’08 to Sept ‘09
343m textual phrases (quotes)
Want to infer the network of information diffusion
We use the hyperlinks between sites to generate the edges of a ground truth G
From the MemeTracker dataset, we have the timestamps of:
1. cascades of hyperlinks:
time when a site creates a link
2. cascades of (MemeTracker) textual phrases:
time when site mentions the information e a c f e a c f

20. Real Network: Performance
500 node hyperlink network using hyperlinks cascades
500 node hyperlink network using MemeTracker cascades
Break-even points of 50% for hyperlinks cascades and 30% for MemeTracker cascades!

21. Diffusion Network
5,000 news sites:
Blogs
Mainstream media

22. Diffusion Network (small part)
Blogs
Mainstream media

23. Networks and Processes
We infer hidden networks based on diffusion data (timestamps)
Problem formulation in a maximum likelihood framework
NP-hard problem to solve exactly
We develop an approximation algorithm that:
It is efficient -> It runs in O(N2)
It is invariant to the structure of the underlying network
It gives a sub-optimal network with tight bound
Future work:
Learn both the network and the diffusion model
Applications to other domains: biology, neuroscience, etc.

24. Thanks!
For more (Code & Data):