dne: a method for extracting cascaded diffusion networks from social networks

DNE: A Method for Extracting Cascaded Diffusion Networks from Social Networks

DNE: A Method for Extracting Cascaded Diffusion Networks from Social Networks

By: Yousef Naderimnaderi@gmail.com

Authors: Motahhare EslamiHamidReza Rabiee

Mostafa Salehi

2011, October, 9th

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Outline Introduction Problem Definition Problem Importance Related Work

Link Prediction Network Completion Network Inference

Proposed Method: “DNE” Experimental Evaluation Contribution Future Work References

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Introduction

Diffusion and Cascading behavior: A process by which information, viruses, ideas

and new behavior spread over the network.

Figure 1: An E-mail Recommendation Network[1]

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Introduction

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An Example of Diffusion Process

Figure 2: Diffusion process over information networks

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The network that diffusion takes place on it is usually unknown and unobserved.

we only observe the times of infection not the one who causes it. So Who-infects-whom?

Figure 3: The diffusion network extraction problem[3]

Diffusion Network Extraction: Problem Definition

Diffusion Network Extraction

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Problem Importance

Related WorkRelated Work

A few work had been done…

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Related Work

[7] for the first time tries to reconstruct epidemic trees of a disease propagation and estimates the sickness outbreak history.

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Proposed Method

Graph G with |V|=n and |E|=eC: The set of propagating cascades over G

Nc members A time vector:

Goal: Finding the diffusion network which is generated by propagating cascades over G.

The only information which is available is infection times.

1 2{ , ,..., }c nT t t t

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Cascade transmission Model

Information Propagation models Threshold model Cascade model

• Independent cascade model Pc(u, v) P(tv - tu)

tv > tu Δ = tv - tu

( )P e

1( )P

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Initial Graph Construction

As cascade c propagates over G, it remains a path of information Sorting its time vector as

Constructing initial graph Gc by considering all probable links attending in diffusion process: Each (i,j) which ti<tj

Assumptions At each moment, only one node can get infected. As each node can infect more than one node but each

node only have one parent, we consider the state transitions from infected node to infecting node.

1 2{ , ,..., }i i in ct t t

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An Example of Initial Graph

Figure5(a)General Initial Graph

(b)Initial Graph of cascade 1 from Figure 2

Figure 4: Initial graph construction

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Random walk Markov Model

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Hitting Time

Hij: The expected number of steps before node j is visited, if we start from node i [23].

Intuitively as Hij increases, the probability of direct infection transmission from i to j will decrease.

A recursive relation for calculating Hij in a strongly connected graph[36,37]

Being strongly connected is necessary for having irreducibility condition.

Calculating this equation needs stationary distribution[23].

ijH 0

1

0

1n

ik kjk

p H

i j

i j

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Hitting Time(cont’d)

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Reaching Time

A new measure based on hitting time RTij: The expected number of steps from node i to j by

“feasible paths”.

As the Reaching time between two nodes’ infection times increases, the probability of infection transmission will decrease.

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Diffusion Network Extraction Problem

Constructing Gtotal

Gtotal =

Defining RT for each edge(i,j) in Gtotal

RTij =

Problem converts to:G’=argmin

cc C

G

cij

c C

RT

( , ) total

iji j G

RT | ' |G m

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Proposed Algorithm: “DNE”

Recursive equation to find RT:1

( 1)j

ij iw wjw i

RT p RT

0iiRT

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Proposed Algorithm: “DNE”

Considering infected nodes instead of infecting ones!

Defining set Sj as all the nodes with lower infection time respect to node j:

As the size of S increases, there will be more candidates to infect j:

Furthermore, the members of S have different priorities to infect j.

1 2{ , ,..., }jj kS n n n

1

| |ijj

PS

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Proposed Algorithm: “DNE”

Considering the order of infection instead of infection times difference.

More independency to cascade transmission modelIntroducing a new parameter named Rank:

Converting the problem to finding m links with least Ranks to construct G’.

ijr 0 i j| | ( )jS j i i j

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Proposed Algorithm: “DNE”For each c C∊

for each (i,j) c∊if (ti < tj)

then Gc G⟵ c (i,j)∪ Sc

j S⟵ cj {i}∪

sort nodes of Gc by infection time.for each (i,j) G∊ c which i < j

rcij | S⟵ c

j | . (j-i)Gtotal G⟵ total (i,j)∪ rij r⟵ ij + rc

ij

Sort edges of Gtotal respect to rij

For i 1 to m⟵G’ G’ {e⟵ ∪ i Gtotal }∊

Return (G’)

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Experimental Evaluation

NetInf[5] for comparison.

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Dataset

Synthetic Networks Forest Fire[38] Kronecker[22] Barabasi-Albert(BA)[39]

Synthetic Network Parameter matrix

a B n e Nc

Forest-Fire [5;0.12;0.1;1;0] 1 0.5 1024 1221 2786

Hierarchical [0.5;0.5;0.5;0.5] 2 0.5 1024 2048 4000

Random(ER) [0.9;0.1;0.1;0.9] 2 0.4 1024 2048 1813

Core-Periphery [0.9;0.5;0.5;0.3] 2 0.1 1024 2048 2350

Barabasi-Albert [2] 2 0.5 1000 2000 4824

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Dataset

Real Networks Co-authorship network[42] Football network[43] President election network[3]

Real Network n e Nc

Co-authorship 1589 2742 6427

Football 115 615 993

President election 1490 19090 5717

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Evaluation Metrics

P2

RF

P R

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Cascade Dependency

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Cascade Dependency

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Extracting Important Diffusion Links

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Extracting Important Diffusion Links

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Running Time

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Contributions

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Future Work…

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