TEMPORAL NETWORKS ANALYSIS

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Social  and  Technological     Network  Analysis     Lecture  10:  Temporal  Social   Network  Metrics     and  Applica>ons   Dr.  Cecilia  Mascolo    

In  This  Lecture   •  We  will  show  metric  extensions  for  complex   networks  which  keep  >me  into  account.   •  We  will  also  show  how  these  can  be  applied  to   applica>ons.  

Why  Temporal  Social  Network   •  Most  of  the  analysis  we  have  seen  has  been   done  on  aggregated  network  graphs     •  Time  has  not  been  kept  into  account  by  the   metrics   •  Why  does  this  maHer?  

Empirical  Networks   J. Phys. A: Math. Theor. 41 (2008) 224014

F De Vico Fallani et al

Figure 2. (a) Realistic head model for the representative subject. On the right hemisphere of the scalp, the positions of the electrodes are depicted as white little spheres. On the left hemisphere of the cortex, all the cortical regions of interest are displayed and opportunely labelled. The trial-averaged waveforms for a particular subset of areas (7 L, MF L, SM L, CM L, 9 L) are illustrated. (b) Functional networks of the subject in the Beta frequency band during three representative instants (−1 s, onset, +1 s) of the task performance. Dark arrows represent the functional links that persist in all the three instants, while the light arrows represent those flows that changed direction in at least one instant.

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121.811.971.251 )sedoN 41( )591(htlaetS )842(htlaetS )sedoN 2( )sedoN 2( )86(htlaetS )801(htlaetS )171(htlaetS )sedoN 2( )74(htlae)2tS6(htlaetS )69(htlaetS )sedoN 2( )sedoN 2( )s(edoN 2( )sedoN 2 )s e d o N 2( )391(htlaetS )99(htlaetS )151(htlaetS )37(htlaetS )sedoN 2( )621(htlaetS )sedoN 2( )162(htlaetS )sedoN 2( )sedoN 2( 922.812.22.591 )sedoN 2( )sedoN 2( )66(htlae)t3 S2(htlaetS )361(htlaetS )sedoN 42( )sedoN 2)(sedoN 2( )sedoN 2( )77(htlaetS )461(htlaetS )352(htlaetS )332(htlaetS )sedoN 2( )961(htlae)stSedoN 2( )sedoN 2( )41(htlaetS )sedoN 2( )642(htlaetS )sedoN 2( 941.71.36.251 )sedoN 2( 721.812.22.591 e(dhotN )381(htlaetS )96(htlaetS )sedoN 01( ))0s3 lae2t(S )sedoN 32( )18(htlaetS )02(htlaetS )911(htlaetS )91(htlae)stSedoN 2( )sedoN 2( 502.2.36.251 )sedoN 2( )05(htlaetS )sedoN 2( )sedoN 2( )sedoN 2( )sedoN 2( )542(htlaetS )sedoN 91( )36(htlaetS 871.76.031.751 )sedoN 2( )sedoN 2( )sedoN 2( )sedoN 82( )21(htlaetS )2(htlaetS )491(htlaetS 931.691.22.591 902.2.36.251 )sedoN 2( )721(htlaetS )73(htlaetS )s e d o N 2 ( )sedoN 2( )29(htlaetS )sedoN 31( 561.71.36.251 )sedoN 91( )sedoN 2( )sedoN 2( )sedoN )25(62(htlaetS )261(htlaetS )27(htlaetS )sedoN 9( )071(htlaetS )47(htlaetS )sedoN 2( )sedoN 2( )sedoN 2( )sed)o5N 2(h ( tlaetS 6 1 521.71.36.251 )sedoN 2( )93(ht)la )sedoN 2( )251(htlaetS )sedoN 01( 0e 9t(S htlaetS )87(htlaetS )sedoN )041(htlaetS )sedoN 2( )se2d( oN 2( )sedoN 2( 121.71.36.251 )sedoN 2( )822(htlaetS )sedoN 21( )39(htlaetS )sedoN 2( 45.181.441.312 )311(htlaetS 54.791.22.591 )122(htlaetS )sedoN 2( ) s e d o N 2 ( ) s e d o N 2 ( )sedoN 4( )sedoN 2( )04(htlaetS 74.791.22.591 45.891.22.591 )662(htlaetS 902.812.22.591 )sedoN 2( )021(htlaetS )s e d o N 3( )sedoN 3( htl2 1a1(e2tS .812.22.591 )002(htlaetS )sedoN 2( ))s3e7d1o(N )sedoN 2( )89(htlaetS )sedoN 2( )sedoN 3( )sedoN 2( )76(htlaetS )241(h)t2la2e(thStlaetS )sedoN 2( )102(htlaetS 242.891.22).856921(htlaetS se2d(oN 2( )sedoN 2( )sedo)N )sedoN 2( )sedoN )2s(edoN 2( 831.181.441.312 )262(htlaetS 521.74.86.26 )31 )212(htlaetS)841(htlaetS )sedoN )sedoN 2( 42 ( (htlaetS )sedoN 2( )811(htlaetS )sedoN 2( )sedoN 2( )522(htlaetS )s e d o N 2( )97(htlaetS )622(htlaetS )sedoN 2( )062(htlaetS )362(htlaetS )sedoN 2( )17)s(hetdlaoeNtS2( )sedoN 2( )sedoN 2( )581(htlaetS )341(htlaetS )sedoN 2( )441(htlaetS )63(htlaetS )sedoN 2( )732(htlaetS )53(htlaetS )sedoN 2)(001(htlaetS )sedoN 2( )sedoN )620(1(htlaetS )sedoN 2( )sedoN 2( )sedoN 2( )101(htlaetS )16(htlaetS )661(htlaetS )sedoN 2( )5(htlaetS )sedoN 2( )sedoN 2( )sedoN 2( )sedoN 2( )411(htlaetS )sedoN 2( )631(htlaetS )07(htlaetS )sedoN 2( )451(htlaetS )762(htlaetS )sedoN 2( )sedoN 2( )sedoN 2( )28(htlaetS )481(htlaetS )sedoN 2( )sedoN 2(

6.41.452.01 )sedoN 2(

6.9.452.01 )sedoN 2(

6.3.452.01

8.23.042.26

2.74.86.26 )33(htlaetS )sedoN 61( )sedoN 2( )112(htlaetS )081(htlaetS )sedoN 2( )sedoN 2(

2.891.22.591 )sedoN 3(

)802(htlaetS )sedoN 2(

6.8.452.01

)532(htlaetS )sedoN 2(

)121(htlaetS )sedoN 2()061(htlaetS)991(htlaetS )sedoN 2( )sedoN 2( )94(htlaetS )sedoN)022( 2(htlaetS )sedoN 2(

)452(htlaetS )95(htlaetS )sedoN 2( )sedoN 2(

)032(htlaetS )sedoN 2(

1.86.891.56 )sedoN 21( )502(htlaetS )79(htlaetS )sedoN 2( )sedoN 2(

)55(htlaetS )sedoN 2(

521.812.22.591 )sedoN 72( )541(htlaetS )sedoN 2(

902.111.803 23 .1.8 411.802.14

)72(htlaetS )142(htlaetS )sedoN 2( )sedoN 2(

)67(htlaetS )sedoN 2(

)24(htlaetS )sedoN 2(

)781(htlaetS )sedoN 2( )902(htlaetS )sedoN 2(

771.71.36.251 351.71.36.251 )sedoN 7( )sedoN 01)(401(htlaetS )s e d o N 2(

)83(htlaetS )sedoN 2(

)42(htlaetS )sedoN 2(

)71(htlaetS )sedoN 2(

201.291.22.591221.74.86.26 )sedoN 2( )sedoN 2(

)78(htlaetS )sedoN 2(

)942(htlaetS )sedoN 2(

)sedoN 2(

)712(htlaetS )88(htlaetS )sedoN 2( )sedoN 2(

)812(htlaetS )sedoN 2(

)981(htlaetS )sedoN 2(

)49(htlaetS )sedoN 2(

)722(htlaetS )681(htlaetS .71 )se1d6o1N 2.(36.251 )sedoN )9s(edoN 2( )59(htlaetS )sedoN 2( )44(htlaetS

212.042.64 )724.526(htlaetS )sedoN 2 )s1e(doN 2(

)211(htlaetS )sedoN 2(

)82(htlaetS )sedoN 2(

)01(htlaetS)052(htlaetS )sedoN 2( )sedoN 2( )34(htlaetS )sedoN 2(

)11(htlaetS )191(htlaetS )sedoN 2( )19(htlaet)S s e d o N 2( )sedoN 2(

)252(htlaetS )sedoN 2(

)46(htlaetS )sedoN 2( )932(htlaetS )sedoN 2( )152(htlaetS )sedoN 2(

)35(htlaetS )sedoN 2(

)442(htlaetS )sedoN 2(

)291(htlaetS )sedoN 2(

)731(htlaetS )sedoN 2(

202.812.22.591

)51(htlaetS )sedoN 2(

)091(htlaetS )3(htlaetS )sedoN 2)(sedoN 2(

)4(htlaetS )sedoN 2( )342(htlaetS )sedoN 2(

)302(htlaetS )sedoN 2(

)68(htlaetS )stsoH 1(

2.7.452.01 )sedoN 2(

)242(htlaetS )sedoN 2(

the representative Beta frequency band. The overall presence of mutual links in the cortical networks is always higher with respect to random (ρ > 0). However, a different behaviour can be found between the preparation and the execution of the movement. In particular, during the movement preparation the reciprocity of the cortical networks moves from a relative high reciprocal state (ρ > 0.25) to a lower (ρ < 0.17) level as revealed by the negative slope of ρ(t) for −1 < t < 0 s. Instead, during the movement execution the average trend of ρ(t) for 0 < t < 1 s constantly remains in the low reciprocal state reached in proximity of the onset (0.15 < ρ < 0.2). In figure 3(b), the level of reciprocity of all the possible connections within the cortical network is illustrated for the same band and during the entire period of interest. The level of grey codes the number of subjects that actually hold a particular reciprocal link identified by the values at the ordinates. In table 1, the correspondence between the y-values and the bilateral link can be deduced by inspecting the values of the symmetric adjacency matrix. The presence of continuous horizontal lines indicates a sort of ‘persistence’ of particular reciprocal connections which can also remain active during the entire task performance, as for the cingulate motor areas (CM L and CM R) with the ipsi-lateral supplementary motor areas (SM L and SM R), respectively. In such a case, at least three subjects present these persistent )342(htlaetS )sedoN 2(

)631(htlaetS )sedoN 2(

)141(htlaetS )sedoN 2(

37880 JN ,tesremoS ,evirD muirtA 003 ,noitaroproC atemuL 1102 thgirypoC

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slebaL tegraT 11

)EPIR( ertneC noitanidrooC krowteN EPIR

2

)CINIRFA( retneC noitamrofnI krowteN nacirfA

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:detnirP ...02 0040-TMG 30:93:31 42 guA deW

83

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Time  in  networks   •  Timestamps  

–  e.g.  Facebook:  friends  added  and  removed  over   >me  

•  Dura>on  

–  e.g.  Spending  >me  with  friends  

•  Frequency  

–  e.g.  Friends,  colleagues,  strangers  

•  Time-­‐order  

–  e.g.  Timetables  in  public  transport  systems   5  

Temporal  Graph  

t=1  

t=2  

t=3  

Temporal  Graph  

t=1  

t=2  

t=3  

Temporal  Graph  

t=1  

t=2  

• Sta>c   • Shortest  path  (A,G)  =  [A,B,D,E,G]   • Shortest  path  length  (A,G)  =  4  hops  

t=3  

Temporal  Graph  

t=1  

t=2  

• Sta>c   • Shortest  path  (A,G)  =  [A,B,D,E,G]   • Shortest  path  length  (A,G)  =  4  hops   • Temporal   • Shortest  path  (A,G)  =  [A,C,B,D,E,F,G]   • Shortest  path  length  (A,G)  =  6  hops   • Time=3  seconds  

t=3  

Temporal  Measures   •     d      ij          =                  Shortest  Temporal  Path  Dura>on     •     d      ∗          =                  Number  of  Hops  in  shortest  temporal  path   ij

                             

•                             1          Temporal  Efficiency  of  communica>on   Eij =

dij

Temporal  Measures   •  Average  Temporal  Path  Dura>on    

L=

1 N (N −1)



1 N (N −1)



•  Average  Temporal  Path  Hops      

L = ∗

•  Average  Temporal  Efficiency   Eglob =

1 N (N −1)

ij

dij

∗ d ij ij



ij

Eij

Does  it  really  maHer?   •  •  •  • 

Infocom  2005  conference  environment   Bluetooth  coloca>on  scans   5  Minute  Windows   Measure  24  hours  star>ng  12am   Sta$c

Temporal

Day

N



Ac$vity

Contacts  

  L

1

37

25.73

6pm-­‐12pm  

3668  

1.291

0.856

4.090

19h  39m

0.003

2

39

28.31

12am-­‐12pm   8357  

1.269

0.870

4.556

9h  6m

0.024

3

38

22.32   12am-­‐12pm   4217  

1.420

0.798

4.003

10h  32m

0.018

4

39

21.44  

1.444

0.781

4.705

9h  55m

0.013

12am-­‐5pm  

3024  

Eglob

L*

  L

Eglob

Temporal  Centrality  Measures   •  Sta>c  Closeness  and  Betweenness  based  on   sta$c  shortest  paths   •  Reformalise  closeness  and  betweenness  with   temporal  paths:   –  Dura>on   –  Time  Order   –  Frequency  

Temporal  Closeness   W  is  the   number   of   temporal   windows  

� 1 Cih = dhi,j W (N − 1) j�=i∈V

(2 + 2) + (3 + 3 + 3) CA = = 0.867 (3 ∗ (6 − 1)) B   A  

C  

E   F   D  

1

2 Time  

3  

Temporal  Betweenness   •  Using  temporal  path  length  

A  

B  

1  

2  

C   3   Time  

4  

5  

Number  of  temporal   shortest  paths  through  B   for  which  at  >me  4  B   was  carrying  a  message  

6  

Evalua>ng  Centrality   •  Two  perspec>ves:  

–  Seman>c:  known  roles  of  nodes   –  Dynamic  Processes:  mobile  malware  containment  

Enron  in  the  News  

$  

Public  Inves>ga>on   •  •  •  • 

Telephone  logs   Documents   Financials   Emails   -­‐  151  user  mailboxes   -­‐  May  1999  to  Jun  2002   -­‐  250,000  emails   -­‐  NOT  anonymised  

Seman>cs   ID 009 013 017 048 053 054 067 073 075 107 122 127 139 147 150

Role (Unknown) Legal Manager Executive Trader President Vice President Trader Director of Trading Trader Managing Director Manager Director Trader Secretary

Seman>cs   ID 9 13 17 48 53 54 67 73 75 107 122 127 139 147 150

Name Stephanie Panus Marie Heard Mike Grigsby Tana Jones John Lavorato Greg Whalley Sara Shackleton Jeff Dasovich Gerald Nemec Louise Kitchen Sally Beck Kenneth Lay Mary Hain Carol Clair Liz Taylor

Role (Unknown) Legal Manager Executive Trader President Vice President Trader Director of Trading Trader Managing Director Manager Director Trader Secretary

•  Big  bonuses  linked  with  informa>on  mediators  

Mobile  Phone  Malware  

Mobile  Malware  Propaga>on   •  Long  Range  

–  Sms,  mms,  email   –  Can  be  filtered  by  central  service  provider  

•  Short  Range  

–  Bluetooth,  wifi   –  Evades  central  service  provider  

Limita>ons   •  Devices  

–  Resource  constrained  

•  Infrastructure  

–  Limited  bandwidth  

•  Priori>se  Devices  using  SNA  

–  Patch  individual  devices  via  nodes  with  high  Betweenness   –  Flood  patch  via  nodes  with  high  Closeness  

Priority  Patching  Schemes   1.  Tradi>onal  Patching  

➔ Can  we  block  path  of  malware?   ➔ Betweenness    

2.  Opportunis>c  Patching  

➔ Can  we  compete  with  malware?   ➔ Closeness  

Patching  Nodes  

Flood  Network  with  Patch  

Flood  Network  with  Patch   Area  under  Curve   (AUC)  

Peak  Infected  Nodes  

Imax

Complete  patch  >me  

τ

Opportunis>c  Patching  

Malware Start Time

Patch Delay

1.  Finite   Time  

Opportunis>c  Patching   2.  Sta>c  is   Poor  

Malware Start Time

3.   Temporal   is  Best  

Patch Delay

Sta>c  Small  World     •  Graphs  which  both  are  locally  clustered  but   with  small  average  path  length  

–  High  local  clustering  but  long  paths  =>  Lahce   –  Small  average  paths  but  low  clustering  =>  Random  

Temporal  Small  World   •  Does  this  hold  in  >me-­‐varying  graphs   •  Temporal  small  world:  

–   quick  paths  from  one  node  to  another  and   –   have  some  temporal  local  persistence  of  links  

32  

Tes>ng  for     Temporally  Small  World   •  Measure  

–  communica>on  efficiency  

•  Temporal  shortest  path  length  

–  speed  of  change  

•  Temporal  correla>on  coefficient   •  Measure  persistence  of  links  

•  Model  

–  Recreate  a  slowly  changing  and  quickly  changing   temporal  graph   –  Brownian  mo>on  with  prob(jump)   33  

 

Gt if it was already present in graph Gt−1 . To quantify this effect, following Ref. [19] we compute C, the average Coefficient   o f     topological overlap of the neighbor set of a node between Temporal   lustering   two successive graphs in theCsequence: C=

!

Ci 1 Ci = N T −1 i

CA = 1/2

T −1 " t=1

!

aij (t)aij (t + 1) #! ! [ j aij (t)][ j aij (t + 1)] j

(1) We name this metric the temporal-clustering coefficient of G. Node  i   Node  i   A   B   A fundamental concept in graph A   theory B  is that of geodesic, or shortest path. In a static graph, a shortest path nodes i F   and j is defined D   C   E   between D   of minC  as a path E   F   imal length between the two nodes. This is a sequence of adjacent nodes starting at i, ending at j, and visiting t1   t2   the minimum number of nodes. Finally, the distance between node i and node j is set equal to the length of the

te di no of an Fi

th tim gr A (b ita gr wo G, gr es

Temporal  SW  Model   •  N  Random  Walkers  with  Prob  Jumping  Pj   Pj=0.0!

Pj=0.5!

Pj=1.0!

Temporal  SW  Model   •  N  Random  Walkers  with  Prob  Jumping  Pj   Pj=0.0!

Pj=0.5!

Pj=1.0!

Temporal  SW  Model   •  N  Random  Walkers  with  Prob  Jumping  Pj   Pj=0.0!

Pj=0.5!

Pj=1.0!

Temporal  Small  World   •  Graphs  which  evolve  slowly  over  >me  can  s>ll   exhibit  high  communica>on  efficiency  

–  Highly  temporal-­‐clustering  =>  non-­‐jumping  model   –  Low  temporal-­‐delay  =>  fully-­‐jumping  model  

movement. Each time-varying graph has N = 16 nodes, representing cortical regions of interest, and consists in a time sequence of T = 100 directed unweighted graphs, where the directed links represent causal influences between cortical regions (see Ref. [13] for details). We have

Small-­‐world  Behaviour  in  Real  Data   0.1

1

Brain  network   length and temporalraphs produced by the of the probability pj of s we have set N =contacts   100, Bluetooth   produced (INFOCOM’06)   sequences of mporal path length of dashed line.  

C

C rand

L

α β γ δ

0.44 0.40 0.48 0.44

0.18 0.17 0.13 0.17

3.9 (100%) 6.0 (94%) 12.2 (86%) 2.2 (100%)

d1 d2 d3 d4

0.80 0.78 0.81 0.83

0.44 8.84 (61%) 0.35 5.04 (87%) 0.38 9.06 (57%) 0.39 21.42 (15%)

Mar Jun Sep Dec

0.044 0.046 0.046 0.049

0.007 0.006 0.006 0.006

456 380 414 403

Lrand

E

E rand

0.50 0.41 0.39 0.57

0.48 0.45 0.37 0.56

6.00 (65%) 4.01 (88%) 6.76 (59%) 15.55(22%)

0.192 0.293 0.134 0.019

0.209 0.298 0.141 0.028

451 361 415 395

0.000183 0.000047 0.000058 0.000047

0.000210 0.000057 0.000074 0.000059

4.2 3.6 8.7 2.4

(98%) (92%) (89%) (92%)

- We first illustrate (London   network)   in a network model ple motion rules. We lkers which move in TABLE I: Temporal-clustering, characteristic temporal path size D with a fixed length and efficiency for brain cortical networks (subject 1, long-distance jumps and four band frequencies) [13], for the social interaction

Summary   •  We  have  introduced  metrics  for  >me  varying   social  networks     •  We  have  shown  examples  of  use  on  real   networks  

References   •  •  •  •  •  •  • 

Vincenzo  Nicosia,  John  Tang,  Cecilia  Mascolo,  Mirco  Musolesi,  Giovanni  Russo  and  Vito  Latora.   Graph  Metrics  for  Temporal  Networks.  Book  Chapter  in  PeHer  Home  and  Jari  Saramaki  (Editors).   Temporal  Networks.  Springer.  2013.     John  Tang,  Ilias  Leon>adis,  Salvatore  Scellato,  Vincenzo  Nicosia,  Cecilia  Mascolo,  Mirco  Musolesi   and  Vito  Latora.  Applica$ons  of  Temporal  Graph  Metrics  to  Real-­‐World  Networks.  Book  Chapter   in  PeHer  Holme  and  Jari  Saramaki  (Editors).  Temporal  Networks.  Springer.  2013.     J.  Tang,  S.  Scellato,  M.  Musolesi,  C.  Mascolo  and  V.  Latora.    Small-­‐world  behavior  in  $me-­‐varying   graph    In  Physical  Review  E.  Vol.  81  (5),  055101.  May  2010.     J.  Tang,  M.  Musolesi,  C.  Mascolo,  V.  Latora,  V.  Nicosia.  Analysing  Informa$on  Flows  and  Key   Mediators  through  Temporal  Centrality  Metrics.  In  Proc.  of  the  3rd  Workshop  on  Social  Network   Systems  (SNS  2010).  Apr  2010.     J.  Tang,  M.  Musolesi,  C.  Mascolo  and  V.  Latora.  Temporal  Distance  Metrics  for  Social  Network   Analysis.  In  Proc,  of  the  2nd  ACM  SIGCOMM  Workshop  on  Online  Social  Networks  (WOSN09).  Aug   2009.   J.  Tang,  C.  Mascolo,  M.  Musolesi,  V.  Latora.  Exploi$ng  Temporal  Complex  Network  Metrics  in   Mobile  Malware  Containment.  In  Proc.  of  the  IEEE  12th  Interna>onal  Symposium  on  a  World  of   Wireless,  Mobile  and  Mul>media  Networks  (WoWMoM2011).  Jun  2011.   V.  Nicosia,  J.  Tang,  M.  Musolesi,  G.  Russo,  C.  Mascolo,  V.  Latora.    Components  in  $me-­‐varying   graphs.  In  AIP  Chaos.  Vol.22  Issue  2.  2012.