Time-sensitive Network Inference
in Continuous-Time Diffusion Networks
Emaad Ahmed Manzoor
CS229: Final Presentation
Slide 2
Slide 2 text
Networks
Slide 3
Slide 3 text
Social Networks
Slide 4
Slide 4 text
Epidemic Networks
Individual stick figures from xkcd.com
Slide 5
Slide 5 text
Information Networks
Slide 6
Slide 6 text
Diffusion
Slide 7
Slide 7 text
No content
Slide 8
Slide 8 text
0
Slide 9
Slide 9 text
0
Slide 10
Slide 10 text
0
Slide 11
Slide 11 text
0
Slide 12
Slide 12 text
0 T
Slide 13
Slide 13 text
Diffusion
Formalization
Slide 14
Slide 14 text
0 T
Cascade
Slide 15
Slide 15 text
0 T
Parents &
Children
Slide 16
Slide 16 text
0 T
Infection
Times
t
1
t
2
t
3
t
4
Slide 17
Slide 17 text
0 T
Observation
Limit
t
1
t
2
t
3
t
4
Slide 18
Slide 18 text
0 T
Underlying
Network
t
1
t
2
t
3
t
4
Slide 19
Slide 19 text
0 T
Transmission
Times
0.1
0.7
0.1
0.4 0.7
0.3
0.5
0.8
0.6
0.9
Slide 20
Slide 20 text
f(t
i
) Probability that node i is
infected at time t
i
Slide 21
Slide 21 text
f(t
i
) Probability that node i is
infected at time t
i
f(t
i
|t
j
) Probability that node i is
infected at time t
i
given that node j is
infected at time t
j
Slide 22
Slide 22 text
f(t
i
) Probability that node i is
infected at time t
i
f(t
i
|t
j
) Probability that node i is
infected at time t
i
given that node j is
infected at time t
j
f(t
i
|t
j
) = f
ij
(t
i
- t
j
)
Slide 23
Slide 23 text
Exponential Pairwise Transmission Function
Slide 24
Slide 24 text
Network Inference
Slide 25
Slide 25 text
Set C of cascades
Slide 26
Slide 26 text
Set C of cascades
Each cascade is a set of
observations
Slide 27
Slide 27 text
Set C of cascades
Each cascade is a set of
observations
Each cascade is observed
until a horizon time
Slide 28
Slide 28 text
Set C of cascades
Each cascade is a set of
observations
Each cascade is observed
until a horizon time
Nodes not infected before
this horizon time
Slide 29
Slide 29 text
Set C of cascades
Each cascade is a set of
observations
Each cascade is observed
until a horizon time
Nodes not infected before
this horizon time
Pairwise transmission
functions
Slide 30
Slide 30 text
Set C of cascades
Each cascade is a set of
observations
Each cascade is observed
until a horizon time
Nodes not infected before
this horizon time
Pairwise transmission
functions
Find transmission rates
that maximise the
likelihood of the
observed cascades
Slide 31
Slide 31 text
No content
Slide 32
Slide 32 text
Metrics
Precision
Recall
Slide 33
Slide 33 text
State of the Art
Slide 34
Slide 34 text
Uncovering the temporal dynamics of
diffusion networks
Influence maximisation in continuous-time
diffusion networks
Scalable influence estimation in continuous
time diffusion networks
Rodriguez et al.
ICML '11
Rodriguez et al.
ICML '12
Du et al.
NIPS '13
Slide 35
Slide 35 text
Rodriguez et al.
ICML '11
Uncovering the temporal dynamics of
diffusion networks
1. Define cascade likelihood as the
objective function
2. Since this function is convex, the
problem is a constrained maximisation
problem over transmission rates
Slide 36
Slide 36 text
Rodriguez et al.
ICML '11
Uncovering the temporal dynamics of
diffusion networks
"Our formulation thus does not
depend on the absolute time of
infection of the root node"
"Transmission functions are shift invariant,
and do not depend on the absolute times of
infection of the pair of nodes"
Contribution 1: Formulate a time-dependent
transmission function as a discrete mixture of
distributions.
Slide 43
Slide 43 text
Contribution 1: Formulate a time-dependent
transmission function as a discrete mixture of
distributions.
Slide 44
Slide 44 text
Contribution 1: Formulate a time-dependent
transmission function as a discrete mixture of
distributions.
Slide 45
Slide 45 text
Contribution 2: Model the time-dependent priors
with circular normal distributions
Slide 46
Slide 46 text
Per node
Per edge
Unknowns
How do we fit these from the data?
Slide 47
Slide 47 text
Contribution 3: EM algorithm to fit the unknown
parameters from the data
1. Initialize the state for each node in each
cascade randomly; S
ic
= random(A, S)
2. Estimate and d for every pair of
nodes using convex optimisation (Manuel et
al., 2011).
3. Estimate and using closed-
form maximum-likelihood estimates.
4. Reassign new states S
ic
to nodes in each
cascade
Slide 48
Slide 48 text
Contribution 3: EM algorithm to fit the unknown
parameters from the data
4. Reassign new states S
ic
to nodes in each
cascade
Slide 49
Slide 49 text
Results
Slide 50
Slide 50 text
Synthetic data: 1024 nodes Kronecker
core-periphery (Leskovec, '08), transmission
times and root nodes chosen uniformly at
random, 1000 cascades.
Real data: Memetracker, 1M nodes, >100K
cascades.
Slide 51
Slide 51 text
Future
Slide 52
Slide 52 text
Algorithm:
Continuous states
Remove stationarity assumption
Implementation:
Parallelism
Speed
Experiments:
Real data
New synthetic data