Download Diffusion-Aware Sampling and Estimation

Diffusion-Aware Sampling and Estimation in
Information Diffusion Networks
By: Motahareh Eslami Mehdiabadi, Hamid Reza Rabiee
and Mostafa Salehi
[email protected]
4th September, 2012
Outline
 Introduction
 Related Work
 Complete Data
 Partial Data
 Problem Formulation
 Proposed Framework
 Sampling Design
 Estimation Approach
 Experimental Evaluation
 Conclusion
 References
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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
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Introduction
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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
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Figure 1-(b): Working with diffusion
network directly
It is often not
feasible to work with this latent network
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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 network
Proposing a Novel TwoStep Framework
(Sampling/ Estimation)
»DNS«
Extend 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
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Related Work
Complete Data
Partial Data
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
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privacy policies
diffusion network
latent structure
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Use sampling methods
to obtain partial data
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Partial Data
Common Sampling Methods
Breadth-First Search (BFS)
Random Walk (RW)
Using these sampling methods without any attention to diffusion paths
results in some redundant data which are not
related to the diffusion process.
Removing these unnecessary data decreases
the efficiency of sampling methods
No previous work considers diffusion process in the sampling strategy.
This paper presents a diffusion-aware
sampling and estimation methods
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The first study to introduce a complete
measurement framework for diffusion
networks.
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Problem Formulation
 Basic Notations and Definitions
 Problem Definition
Basic Notations & Definitions
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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, le ∊ L}
• Infection as an example
To measure diffusion network characteristics, some
aggregative function should be used.
• Average Function
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Problem Definition
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Proposed Framework
DNS: Diffusion Network Sampling
1) Sampling Design
2) Estimation Approach
Sampling Design
 Existing sampling methods such as BFS & RW do
not consider diffusion paths.
Computing the probability of spreading infections over the links of
underlying network
Directing the sampling design toward diffusion paths without any prior
knowledge about the diffusion network structures.
Covering a greater part of unknown diffusion network
Decreasing the redundant data such as nodes and links which do not
attend the diffusion process
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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 c
CT = {t1, t2, …, tNc}: the set of cascades’ time vectors
Waiting time model[1] depends on Δ= tv – tu
P( )  e



Exponential Model
Ce: set of cascades which pass over link e
Pc (e)

c

C
The infection probability of link e Pe 
| Ce |
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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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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
k 1
f ( xi )
probability

i  0  ( xi )
ˆ  k 1
1

i  0  ( xi )
Xi and п(Xi) are the visited element and its visiting
probability on the ith draw of sampling method.
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Estimation Approach
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
Depth of a spreading phenomenon can be determined
by the length of its cascades
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.
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Experimental Evaluation
• Setup
• Dataset
• Performance Evaluation
•Diffusion Behavior Analysis
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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 Networks
 Real 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
0.4 ≤⍺≤ 0.7
Figure 2: Speed of cascade effect of link-attendance bias
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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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Contributions
Proposing a novel sampling design for gathering
data from a diffusion network by utilizing the
properties of diffusion process.
DNS
Framework
Proposing three estimators for correcting the
bias of sampled data:
link-based, node-based, and cascade-based
characteristics
Decreasing the bias of measuring link-based
characteristics compared to the other common
sampling methods
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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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References
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Q&A