Diffusion-Aware Sampling and Estimation in Information Diffusion Networks By: Motahareh Eslami...

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

[email protected]

4th September, 2012

mailto:[email protected]

2 DMLP2P Live Video Streaming DMLDML2

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


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



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



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



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«


Related WorkRelated Work

Complete Data

Partial Data

8 DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation

8

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

9 DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation

9

Partial Data

Problem FormulationProblem Formulation Basic Notations and Definitions Problem Definition

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DMLP2P Live Video Streaming DMLDMLDiffusion-Aware Sampling and Estimation

11

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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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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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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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.

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Experimental EvaluationExperimental 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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25

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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27

Performance Evaluation

Figure 3: Link Attendance characteristic evaluation in different sampling rates

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28

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

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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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