Diffusion-Aware Sampling and Estimation in Information Diffusion Networks By: Motahareh Eslami...
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Diffusion-Aware Sampling and Estimation in Information Diffusion
Networks
Diffusion-Aware Sampling and Estimation in Information Diffusion
Networks
By: Motahareh Eslami Mehdiabadi, Hamid Reza Rabiee and Mostafa Salehi
4th September, 2012
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Outline Introduction Related Work
Complete Data Partial Data
Problem Formulation Proposed Framework
Sampling Design Estimation Approach
Experimental Evaluation Conclusion References
Diffusion-Aware Sampling and Estimation
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Introduction
Information Diffusion
One of the important topics in large On-line Social Networks (OSN) such as Facebook, Twitter, and YouTube.
Has a latent structure which makes its analysis considerably difficult
Although we usually discover the time of obtaining some information by people, we can not find the source of information easily.
Using partially-observed network data
Diffusion-Aware Sampling and Estimation
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Introduction
Measurement of the network characteristicsSampling
-Collecting the data by a sampling method
- Considering the visiting probabilities of network
elements
Estimation- Using an estimator to
obtain the network characteristics
Diffusion-Aware Sampling and Estimation
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Different Sampling Designs in Diffusion Networks
Considering the sampling approaches to study diffusion behaviors of social networks, apart from their topologies
Figure 1-(a): Using well-known sampling methods such as BFS and RW without considering the behavior of the diffusion process gathering redundant data + losing parts of diffusion data + decrease the performance of these sampling methods
Figure 1-(b): Working with diffusion network directly It is often not feasible to work with this latent network
Diffusion-Aware Sampling and Estimation
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Introduction
A link-tracing based sampling method which utilizes
diffusion process properties to traverse the network more
accurately.Only uses the infection times as local information without
any knowledge about the latent structure of the
diffusion networkExtend the well-known
Hansen-Hurwitz estimator to correct the bias of sampled
data by three efficient estimators of characteristics:
(1) link-based (2) node-based (3) cascade-based
The First attempt to introduce a complete measurement
framework for the diffusion networks
Proposing a Novel Two-Step Framework
(Sampling/ Estimation)»DNS«
Diffusion-Aware Sampling and Estimation
Related WorkRelated Work
Complete Data
Partial Data
8 DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation
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Complete Data The most fundamental approach: Collecting the complete
diffusion data Following the Iraq war petitions in the format of e-mail [11],
[19] studying communication events between faculty and staff of a
university by e-mails [15] tracking the flow of information by extracting short textual
phrases [16]
missing data privacy policiesdiffusion network latent structure
Use sampling methods to obtain
partial data
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Partial Data
Problem FormulationProblem Formulation Basic Notations and Definitions Problem Definition
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DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation
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Basic Notations & Definitions
G = (V, E) The underlying network (n=|V| & m=|E|)
Infection
some diffusible chunks such as information and epidemic diseases which propagate over G
Cascade
each path of infection will build a “cascade”
Diffusion Network
When the cascades spread over the underlying network, the diffusion network G* will be formed.
Gs = (Vs,Es)
the induced sub-graph of G by sampling rate of μ
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DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation
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Diffusion Characteristics Measurement
Element Set T: A set of diffusion network elements such as nodes, links and cascades
L: a finite set of element labels
• Examples of a label: the degree of a node, the weight of a link, or the length of a cascade.
Target function f: T L⇾ : A label le is assigned to each element e T ∊ i.e. f = {(e, le)| e T, l∊ e L∊ }
• Infection as an example
To measure diffusion network characteristics, some aggregative function should be used.
• Average Function
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Problem Definition
Proposing a diffusion-aware measurement
framework
• samples from the underlying network and computes the desired target function f
• computes an estimate of Avg(f) by finding an appropriate estimator M.
Evaluation: Bias Metric
Final Goal • Finding a measurement framework which minimizes the bias ( )⍴
Proposed FrameworkProposed Framework
DNS: Diffusion Network Sampling
1) Sampling Design2) Estimation Approach
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Sampling Design
Existing sampling methods such as BFS & RW do not consider diffusion paths.
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Sampling Design (cont’d)
Each cascade c can be assigned to a time vector tc = {t1, t2, …, tn} which shows the infection times
of nodes by cascade cCT = {t1, t2, …, tNc}: the set of cascades’ time vectors
Waiting time model[1] depends on = tΔ v – tu
Ce: set of cascades which pass over link e
The infection probability of link e
( )P e
Exponential Model
( )
| |cc C
ee
P eP
C
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The Algorithm
moving from a node u, to a neighboring node v, through an outgoing link with the infection probability
Using the infection times (i.e. CT) as local information without any prior knowledge about the latent structure of the diffusion network
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DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation
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Estimation Approach
Correcting the selection bias of a sampling method by re-weighting of the measured values
Hansen-Hurwitz Estimator: elements are weighted inversely proportional to their visiting probability
Xi and п(Xi) are the visited element and its visiting probability on the ith draw of sampling method.
1
01
0
( )
( )ˆ
1( )
ki
i ik
i i
f x
x
x
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DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation
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Estimation Approach
Characteristics
Link-based Node-basedCascade-
based
To use this estimator, the probability of visiting each element in the proposed sampling procedure should be
computed.
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Link-based Characteristics
Gaining some information without having any connection to others for propagation will be not valuable in a network.
Link Attendance: shows the amount of presence in diffusion process for a link
Moving over links with the probability of Pe
п(e) = Pe
Only Using local knowledge to compute the visiting probabilities of the links.
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Node-based Characteristics
Examples: The number of “Seeds” (the beginners of an
infection) [3] and “participation” (the fraction of
users involved in the information diffusion) [12]Modeling the diffusion process
as a Markov random walk over the underlying network G [18]
Where N (u) is the set of node u’ neighbors
Calculating п = {п(0), п(1),…, п(n-1)} needs the global
knowledge of a network (the stationary distribution of the
mentioned Markov chain)Not having the global view of
the network in sampling procedure
Finding the exact value of п is not possible in real systems.
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Cascade-based Characteristics
Cascades: the building blocks of a diffusion network
D
e
p
t
h
o
f a
s
p
rea
d
i
n
g
p
h
e
n
o
m
e
n
o
n
ca
n
b
e
d
eter
m
i
n
e
d
b
y
t
h
e
le
n
gt
h
o
f its
casca
d
es
A cascade visiting probability depends on visiting all of its links:
Calculating п(c) needs having all links of cascade c which is a global knowledge.
23
Experimental EvaluationExperimental Evaluation
• Setup• Dataset• Performance Evaluation•Diffusion Behavior Analysis
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DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation
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Setup
Characteristic: Link-Attendance
Baseline methods: BFS and RW
⍺: Control the speed of cascade transmission
β: control the distance through which a cascade can propagates
Diffusion rate (δ): The fraction of the underlying network G which is covered by the diffusion network G*
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Dataset
Synthetic NetworksReal Networks
Table 1: The network and cascade generation parameters
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Speed of Cascade
The unknown structure of the diffusion network structure unavailability of the cascade speed
Figure 2: Speed of cascade effect of link-attendance bias
0.4 ≤⍺≤ 0.7
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Performance Evaluation
Figure 3: Link Attendance characteristic evaluation in different sampling rates
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Performance Evaluation
Decreasing the bias by 30% in average by the DNS framework in comparison to the sampling design without estimation approach
Table 2: The average performance difference of DNS with BFS, RW & DNS-WoE
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Diffusion Behavior Analysis
Figure 4: Analysis of diffusion rate over sampling frameworks
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DMLP2P Live Video Streaming DMLDMLDiffusion Network Extraction30
Contributions
Proposing a novel sampling design for gathering data from a diffusion network by utilizing the properties of diffusion process.
Pr
oposi
ng t
hree esti
mat
ors f
or c
orrecti
ng t
he
bias
of sa
mple
d
data: li
nk-
base
d,
node-
base
d, a
nd casca
de-
base
d c
haracteristics
Decreasi
ng t
he
bias
of
meas
uri
ng li
nk-
base
d c
haracteristics c
ompare
d t
o t
he
ot
her c
ommon sa
mpli
ng
met
hods
DNS Framewor
k
31
DMLP2P Live Video Streaming DMLDMLDiffusion Network Extraction31
Conclusion & Future Work…
The proposed method outperforms BFS and RW in terms of link-attendance by about 37% and 35%, respectively
The proposed estimator can improve the performance of the sampling design by about 30%
DNS can act very well even in low sampling rates
DNS leads to low bias even in low diffusion rates.
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DMLP2P Live Video Streaming DMLDMLExtracting Network of Information32
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