Diffusion on networks: modelling the spread of innovations and customer churnover social networks .

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Diffusion on networks: Modelling the spread of innovations and customer churn over social networks

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1 Introduction

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2 Modelling concept

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Innovating nodes

2 1

3

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3 Model estimation

3.1 The data

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t

Not innovated Innovated

t67

A neighbour of subscriber 1 innovates

Another neighbour of subscriber 1 innovates

•

Individuals can be in two states: not innovated and innovated

•

No re-entry permitted

t15 t32

Subscriber 1 innovates

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3.2 Estimation method

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4 Model results

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(6)

*96 32D:4 9JA@E96D:D E92E H6 H@F=5 =:<6 E@ E6DE :D H96E96C ?6:893@FCD @7 E96 7@42= ?@56 92G6 2? :?7=F6?46 @? E96 49FC?:?8 564:D:@? @7 E96 7@42= ?@56 ?F>36C @7 :?56A6?56?E G2C:23=6D 2C6 FD6 7@C E6DE:?8 E9:D 9JA@E96D:DNumber of infected neighbours:D 2 4@F?E6C

@7 E96 ?F>36C @7 49FC?6CD E92E 2C6 5:C64E=J 4@??64E65 E@ E96 7@42= ?@56 2?5% of infected neighbours5:G:56D E9:D 4@F?E6C 3J E96 E@E2= ?F>36C @7 ?6:893@FCD D E96 ?6EH@C< :D 2 H6:89E65 8C2A9 E96C6 :D :?7@C>2E:@? @? E96 DEC6?8E9 @7 E96 =:?< 36EH66? EH@ ?@56D -6 H@F=5 6IA64E E92E DF3D4C:36CD H9@ :?E6C24E >@C6 H:E9 6249 @E96C 2=D@ 92G6 2 DEC@?86C :?7=F6?46 @? 6249 @E96C *9:D 67764E :D 42AEFC65 3Jweights of infected neighbours2?5% of infected neighbours (weighted) FCE96C>@C6 H6 4@?;64EFC6 E92E DF3D4C:36CD E92E 92G6 2 9:89 568C66 46?EC2=:EJ 92G6 2 9:896C AC@323:=:EJ @7 49FC?:?8 *96 C62D@? 369:?5 E9:D 4@F=5 36 E92E DF3D4C:36CD H9@ E2=< E@ 2 =@E @7 @E96C DF3D4C:36CD 92G6 >@C6 64@?@>:4 :?46?E:G6D E@ A6C:@5:42==J 6G2=F2E6 H96E96C E96 4FCC6?E >@3:=6 A9@?6 @A6C2E@C DE:== :D E96 36DE 2=E6C?2E:G6 log(degree centrality unweighted) H9:49 4@F?ED E96 ?F>36C @7

?6:893@FCD 2?5 degree centrality (weighted) H9:49 :D E96 DF> @7 2== =:?< H6:89ED

@C:8:?2E:?8 :? E96 7@42= ?@56 % of off-net neighboursE2<6D :?E@ 244@F?E E92E E96C6 2C6 AC:46 5:776C6?46 36EH66? @? 2?5 @77?6E 42==D 2?5 :E :D E96C67@C6 36?67:4:2= 7@C DF3D4C:36CD E@ 36 @? E96 D2>6 ?6EH@C< 2D E96 A6@A=6 E96J H2?E E@ 42== ? E96 D2>6 G2:?

weights of on-net and not-infected neighbours D9@F=5 8:G6 2? :?5:42E:@? H96E96C

?6:893@FCD E92E 2C6 @? E96 D2>6 ?6EH@C< 2?5 ?@E :?764E65 AF== 324< 2 DF3D4C:36C 7C@>

49FC?:?8 :?2==J H6 :?4=F56 2?ageG2C:23=6 2?5 5F>>:6D 7@C H96E96C E96 DF3D4C:36C :D male@C 2prepaid4FDE@>6C

*23=6 D9@HD E96 C68C6DD:@? C6DF=ED *96 D64@?5 C@H D9@HD E96 A2C2>6E6C 6DE:>2E6D 2?5 E96 D:8?:7:42?46 =6G6= *96 A2C2>6E6C 6DE:>2E6D 92G6 E96 6IA64E65 D:8?D 2?5 2C6 9:89=J D:8?:7:42?E 2E E96 =6G6= *96 E9:C5 C@H D9@HD E96 >2C8:?2= 67764ED @7 E96 G2C:23=6D 2?5 E96 7@FCE9 C@H E96 2G6C286 G2=F6 @7 E96 G2C:23=6D @C 6I2>A=6 @=56C DF3D4C:36CD 2C6 D:8?:7:42?E=J =6DD =:<6=J E@ 49FC? E92? J@F?86C DF3D4C:36CD 2?5 @?6 6IEC2 J62C @7 286 :?4C62D6D E96 E:>6 E@ 49FC? 3J 23@FE 52JD

*96 >@DE :>A@CE2?E G2C:23=6D FD65 E@ E6DE H96E96C 49FC?:?8 :D 4@?E28:@FD 2C6 E96 7:CDE 7@FC D 42? 36 D66? 7C@> *23=6 49FC? :D DEC@?8=J :?7=F6?465 3J E96 564:D:@? @7

?6:893@FCD E@ 49FC? ? @E96C H@C5D 49FC? :D 4@?E28:@FD 2?5 DAC625D @G6C 2 ?6EH@C<

&?6 255:E:@?2= ?6:893@FC 564C62D6D E96 E:>6 E@ 49FC? 3J 23@FE 52JD D 42? 36 D66?

:? E96 =2DE 4@=F>? E96 2G6C286 G2=F6 @7 E9:D G2C:23=6 :D ;FDE >62?:?8 E92E >@DE @7 E96 A6@A=6 5@?OE 92G6 4@?E24E H:E9 @E96C 49FC?6CD *9:D A2CE=J C67=64ED E96 C6=2E:G6=J D9@CE A6C:@5 7@C H9:49 E96C6 :D 52E2 2G2:=23=6 2E E96 368:??:?8 @7 E96 @3D6CG2E:@? A6C:@5 E9:D G2C:23=6 :D K6C@ 7@C 6G6CJ DF3D4C:36C @H6G6C E96C6 2C6 2 76H DF3D4C:36CD H9@ 92G6 2

=@E @7 49FC?65 ?6:893@FCD 2?5 2C6 DEC@?8=J :?7=F6?465 3J ?6:893@FC9@@5 49FC? %@E6 E92E EH@ 7C:6?5D @7 2? 286?E @7E6? 2=D@ 92G6 2 C6=2E:@?D9:A 2?5 2 ?6:893@FCOD :?7=F6?46 @?

E96 286?E >:89E 36 >F=E:A=:65 3J :?5:C64E =:?<D

(7)

6A ,2C:23=6time to failure '2C2>6E6C DE:>2E6D 92?86 @7

@?6 F?:E @7 I

X

?F>36C @7 :?764E65 ?6:893@FCD

@7 :?764E65 ?6:893@FCD

H6:89ED @7 :?764E65 ?6:893@FCD

@7 :?764E65 ?6:893@FCD

H6:89E65

=@8568C66 46?EC2=:EJ F?H6:89E65

568C66 46?EC2=:EJ H6:89E65

@7 @77?6E ?6:893@FCD

H6:89ED @7 @??6E 2?5 ?@E:?764E65

?6:893@FCD

286

>2=6

AC6A2:5 4FDE@>6C

4@?DE2?E

*23=6 (68C6DD:@? C6DF=ED 7C@> DFCG:G2= 2?2=JD:D >@56=

A2CE 7C@> E96 ?F>36C @7 49FC?6CD E@ H9@> 2 DF3D4C:36C 92D 4@?E24E :E :D 2=D@ :>A@CE2?E H96E96C E96 7C6BF6?4J @7 4@?E24E :D 9:89 @C =@H *96 9:896C E96 :?E6C24E:@? 7C6BF6?4J 36EH66? 2 DF3D4C:36C 2?5 2 49FC?6C E96 >@C6 =:<6=J E92E D96 :D 2=D@ 49FC?:?8 *9:D >2<6D D6?D6 2D 2 9:89 :?E6C24E:@? 7C6BF6?4J :D 2? :?5:42E@C @7 2 4=@D6C C6=2E:@?D9:A 2?5 4@?E28:@? D9@F=5 36 9:896C 36EH66? D@4:2==J 4=@D6 DF3D4C:36CD

:?2==J 2 9:89 A6C46?E286 @7 ?6:893@FCD H9@ FD6 2 5:776C6?E ?6EH@C< :D 2? :?5:42E@C @7 9:896C 49FC? AC@323:=:EJ D AC:46D 7@C @77?6E 42==D 2C6 9:896C E92? AC:46D 7@C @??6E 42==D :E :D >@C6 36?67:4:2= E@ 49FC? :7 2 9:89 A6C46?E286 @7 ?6:893@FCD 2C6 @77?6E H96C62D 2 9:89 A6C46?E286 @7 @??6E ?6:893@FCD D66> E@ 9@=5 2 DF3D4C:36C @? 2 ?6EH@C<

):8?:7:42?E 2E =6G6= ):8?:7:42?E 2E =6G6= ):8?:7:42?E 2E =6G6=

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5 Discussion

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D9@H E92E E9:D DAC625:?8 @44FCD @G6C D@4:2= ?6EH@C<D D 5:D4FDD65 62C=:6C E9:D

>@56==:?8 2AAC@249 92D 2 H:56 2C62 @7 2AA=:423:=:EJ 2?5 42? 36 FD65 E@ >@56= E96 DAC625

@7 :??@G2E:@?D @C ?6H AC24E:46D :? 86?6C2= -:E9 E96 :?4C62D65 2G2:=23:=:EJ @7 6=64EC@?:4 52E2 H6 6IA64E E92E >@C6 2?5 >@C6 DEF5:6D H:== >2<6 FD6 @7 E96D6 @AA@CEF?:E:6D

*96 >2:? 5:776C6?46D 36EH66? E96 >@56==:?8 @7 E96 DAC625 @7 4FDE@>6C 49FC? 2?5 E96 DAC625 @7 :??@G2E:@?D AC@5F4E FAE2<6 :? @FC @A:?:@? 2C6 E96 7@==@H:?8 EH@ :CDE=J 7@C MC62=N :??@G2E:@?D E96C6 :D EJA:42==J @?=J =:>:E65 @C ?@ A2DE 52E2 2G2:=23=6 *9:D >2<6D AC65:4E:@? 2 =@E >@C6 4@>A=:42E65 3FE :? >2?J 42D6D 2 G2C:6EJ @7 :??@G2E:@?D H:== DAC625

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@7 :??@G2E:@?D H:== DAC625 E@ 2 DF3DE2?E:2= A2CE @7 E96 A@AF=2E:@? 88C682E:@? @7 E96

=@42= AC@46DD6D 56D4C:365 :? E9:D A2A6C H:== 36 6G6? >@C6 :>A@CE2?E E96? 2G:?8 D2:5 E92E H6 H@F=5 2C8F6 E92E E96 32D:4 >6E9@5@=@8:42= 2AAC@249 D9@F=5 36 E96 D2>6 2?5 H6 2C6 4FCC6?E=J 24E:G6=J C6D62C49:?8 E@ 6IE6?5 2?5 E6DE E96 >@56= 7@C E96 DAC625 @7 MC62=N :??@G2E:@?D 2D H6==

? E96 >2C<6E:?8 =:E6C2EFC6 5:776C6?E 492??6=D 42? 36 E96 5@>:?2?E F?56C=J:?8 7@C46 @7 5:77FD:@? $2DD >65:2 492??6=D A=2J 2 G6CJ :>A@CE2?E C@=6 :? E9:D C6DA64E 2?5 25G6CE:D:?8 42>A2:8?D EC25:E:@?2==J EC:65 E@ DAC625 :??@G2E:@?D E9C@F89 2 @?6E@>2?J C6=2E:@?D9:A @H6G6C >2DD >2C<6E:?8 42>A2:8?D 2C6 C682C565 2D AC@3=6>2E:4 2D 4@?DF>6CD 2C6 :?4C62D:?8=J D46AE:42= @7 >2C<6E:?8 @C 25G6CE:D:?8 >6DD286D 0'1 ?E6CA6CD@?2= C6=2E:@?D9:AD 2C6 D66? 2D >@C6 2?5 >@C6 :>A@CE2?E :? convincing 2?

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(9)

References

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0'1 &O6== ) 2?5 ! '2;F?6?The butterfly customer: Capturing the loyalty of today’s elusive consumer E@3:4@4<6 &?E2C:@ -:=6J 2?5 )@?D

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