Influential Spreaders in Networks with Community Structure

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Influential Spreaders in Networks with Community Structure

Zakariya Ghalmane, Mohammed El Hassouni, Chantal Cherifi, Hocine Cherifi

To cite this version:

Zakariya Ghalmane, Mohammed El Hassouni, Chantal Cherifi, Hocine Cherifi. Influential Spreaders in Networks with Community Structure. International Conference on Complex Systems (ICCS), Jul 2020, Boston (Online), United States. pp.1. �hal-03059257�

Influential Spreaders in Networks with Community Structure

Zakariya Ghalmane & Mohammed El Hassouni Chantal Cherifi Hocine Cherifi

University Mohammed V MOROCCO

Problematic Example: Betweenness α in non-modular networks

Complexity: O(ln)

Betweenness in Modular networks: A = [α _L , α _G ]

Local Component

Ranking the Nodes

Example: Weigthed Modular Betweenness Centrality Ranking

Global Network ^G

^g

Local Network G ^l

1. Split the original network into two networks

University of Burgundy FRANCE

A1 A2 A3

A4 A5 An

Centrality in modular networks?

Hypotheses:

• Community structure available

(Ground-truth or Community detection)

• Non Overlapping community structure

Global Component

4 2

3

6

9 5

1

8

7

¹⁰

11

12

14 15

16

17

13

18 19 20 22 21

G

C

¹

C

²

C

³

4

2 3

6

9 5

1

8

7 ¹⁰

11

12

14 15

16

17

13

C

⁴

18 19 20 22 21

C

¹

C

²

C

3

4

2 3

6

9 5

1

8

7 ¹⁰

11

12

14 15

16

17

13

C

⁴

18 19 20 22 21

G

S

²

S

¹

4

2

6 9

14

16

18 22

7

C

1

4

2 3

5

1

 ¹  ^{* ( ) * ( )}

)

( _{i C L i C G i}

W v v v

l

l   



   

 



 

l i

l i

l

C

v i

C

v i

er

C k v

v k

) (

)

int (









St

k

j jk

i jk i

G

v v

,

) ) (

( 

 







Cl

k

j jk

i jk i

L

v v

,

) ) (

( 

 







G k

j jk

i jk

i

v v

,

) ) (

( 

 

Betweenness α of a node v

ⁱ

^{∈G :}

Strong community structure

LFR (µ=0.1)

Ego-Facebook

Email-Eu-core

3

1

S

¹

4

14

16

22

3 7

1

C ¹

C ²

C ³

4

2 3

6

9 5

1

8

7 ¹⁰

11

12

14 15

16

17

13

C ⁴

18 19 20 22 21

57 G

.

1

 0

 C

44 .

3

 0

 C

21 .

2

 0

 C

42 .

4

 0

 C

 L  _G

 L

 G

LFR (µ=0.7)

Medium community structure strength

LFR (µ=0.4)

2. Using the classical definition:

Compute α _L on G ^L Compute α _G on G ^g

Georgetown

Weak community structure

Results

Combine

University of Lyon 2 FRANCE

Z. Ghalmane, C. Cherifi, H. Cherifi & M. El Hassouni (2019). Centrality in Complex Networks with overlapping Community structure. Scientific Reports, 9(1).