Analysis of Temporal Interactions with Link Streams and Stream Graphs

Transcription

1 Anlysis of Temporl Intertions with n Strem Grphs, Tiphine Vir, Clémene Mgnien ltpy@ LIP6 CNRS n Soronne Université Pris, Frne 1/23

2 intertions over time ,,, n for 10 time units time 2/23

3 intertions over time time,,, n for 10 time units lwys present, leves from 4 to 5, present from 4 to 9, from 1 to 3 2/23

4 intertions over time time,,, n for 10 time units lwys present, leves from 4 to 5, present from 4 to 9, from 1 to 3 n intert from 1 to 3 n from 7 to 8; n from 6 to 9; n from 2 to 3. 2/23

5 intertions over time time,,, n for 10 time units lwys present, leves from 4 to 5, present from 4 to 9, from 1 to 3 n intert from 1 to 3 n from 7 to 8; n from 6 to 9; n from 2 to 3. e.g., soil intertions, network trffi, money trnsfers, hemil retions, et. 2/23

6 intertions over time time,,, n for 10 time units lwys present, leves from 4 to 5, present from 4 to 9, from 1 to 3 n intert from 1 to 3 n from 7 to 8; n from 6 to 9; n from 2 to 3. e.g., soil intertions, network trffi, money trnsfers, hemil retions, et. how to esrie suh t? 2/23

7 struture or ynmis e f time grph theory network siene struture signl nlysis, time series ynmis 3/23

8 struture n ynmis? e f time time slies grph sequene 3/23

9 struture n ynmis? e f time grph theory network siene struture signl nlysis, time series ynmis time slies grph sequene informtion loss wht slies? grph sequenes? 3/23

10 struture n ynmis e f time MAG / temporl grphs TVG lossless ut grph-oriente + -ho properties (mostly pth-relte) + ontt sequenes + reltionl event moels /23

11 struture n ynmis e f time MAG / temporl grphs TVG lossless ut grph-oriente + -ho properties (mostly pth-relte) + ontt sequenes + reltionl event moels /23

12 wht we propose e f el with the strem iretly strem grphs n link strems time grph theory network siene signl nlysis, time series wnte fetures: simple n intuitive, omprehensive, time-noe onsistent, generlizes grphs/signl 5/23

13 wht we propose e f el with the strem iretly strem grphs n link strems time grph theory network siene signl nlysis, time series wnte fetures: simple n intuitive, omprehensive, time-noe onsistent, generlizes grphs/signl 5/23

14 grph-equivlent strems strem with no ynmis: noes lwys present, either lwys or never linke e grph 6/23

15 grph-equivlent strems strem with no ynmis: noes lwys present, either lwys or never linke e grph strem properties = grph properties generlizes grph theory 6/23

16 our pproh very reful generliztion of the most si onepts strem grphs n link strems numers of noes n links lusters n inue su-strems ensity n pths uliing loks for higher-level onepts neighorhoo n egrees lustering oeffiient etweenness entrlity mny others + ensure onsisteny with grph theory + ensure lssil reltions re preserve 7/23

17 efinition of strem grphs Grph G = (V, E) with E V V uv E u n v re linke Strem grph S = (T, V, W, E) T : time intervl, V : noe set W T V, E T V V (t, v) W v is present t time t T v = {t, (t, v) W } (t, uv) E u n v re linke t time t T uv = {t, (t, uv) E} (t, uv) E requires (t, u) W n (t, v) W i.e. T uv T u T v 8/23

18 efinition of strem grphs Grph G = (V, E) with E V V uv E u n v re linke Strem grph S = (T, V, W, E) T : time intervl, V : noe set W T V, E T V V (t, v) W v is present t time t T v = {t, (t, v) W } (t, uv) E u n v re linke t time t T uv = {t, (t, uv) E} (t, uv) E requires (t, u) W n (t, v) W i.e. T uv T u T v 8/23

19 efinition of strem grphs Grph G = (V, E) with E V V uv E u n v re linke Strem grph S = (T, V, W, E) T : time intervl, V : noe set W T V, E T V V (t, v) W v is present t time t T v = {t, (t, v) W } (t, uv) E u n v re linke t time t T uv = {t, (t, uv) E} (t, uv) E requires (t, u) W n (t, v) W i.e. T uv T u T v 8/23

20 efinition of strem grphs Grph G = (V, E) with E V V uv E u n v re linke Strem grph S = (T, V, W, E) T : time intervl, V : noe set W T V, E T V V (t, v) W v is present t time t T v = {t, (t, v) W } (t, uv) E u n v re linke t time t T uv = {t, (t, uv) E} (t, uv) E requires (t, u) W n (t, v) W i.e. T uv T u T v 8/23

21 efinition of strem grphs Grph G = (V, E) with E V V uv E u n v re linke Strem grph S = (T, V, W, E) T : time intervl, V : noe set W T V, E T V V (t, v) W v is present t time t T v = {t, (t, v) W } (t, uv) E u n v re linke t time t T uv = {t, (t, uv) E} (t, uv) E requires (t, u) W n (t, v) W i.e. T uv T u T v 8/23

22 n exmple time T = [0, 10] V = {,,, } W = T {} ([0, 4] [5, 10]) {} [4, 9] {} [1, 3] {} T = T T = [0, 4] [5, 10] T = [4, 9] T = [1, 3] E = ([1, 3] [7, 8]) {} [6, 9] {} [2, 3] {} T = [1, 3] [7, 8] T = [6, 9] T = [2, 3] T = 9/23

23 few remrks works with... isrete time, ontinuous time, instntneous intertions or with urtions, irete, weighte, iprtite... if v, T v = T then S L = (T, V, E) is link strem if u, v, T uv {T, } then S G = (V, E) is grph-equivlent strem 10/23

24 size of strem grph How mny noes? How mny links? T = 10 T = 2 time 11/23

25 size of strem grph How mny noes? How mny links? T = 10 T = 2 time n = v V T v T n = T 10 + T 10 + T 10 + T 10 = = 2.6 noes 11/23

26 size of strem grph How mny noes? How mny links? T = 10 T = 2 time n = T 10 + T 10 + T m = T 10 + T n = v V m = uv V V 10 + T T 10 T v T T uv T = = 2.6 noes = = 0.7 links 11/23

27 lusters, su-strems Cluster in G = (V, E): suset of V. Cluster in S = (T, V, W, E): suset of W T V time C = [0, 2] {} ([0, 2] [6, 10]) {} [4, 8] {} S(C) su-strem inue y C S(C) = (T, V, C, E C ) properties of (su-strems inue y) lusters 12/23

28 lusters, su-strems Cluster in G = (V, E): suset of V. Cluster in S = (T, V, W, E): suset of W T V time C = [0, 2] {} ([0, 2] [6, 10]) {} [4, 8] {} S(C) su-strem inue y C S(C) = (T, V, C, E C ) properties of (su-strems inue y) lusters 12/23

29 neighorhoo in G = (V, E): N(v) = {u, uv E} in S = (T, V, W, E): N(v) = {(t, u), (t, uv) E} N() = ([2, 3] [5, 10]) {} [5.5, 9] {} N(v) is luster 13/23

30 in G n in S: (v) is the size of N(v) egree N() = ([2, 3] [5, 10]) {} [5.5, 9] {} () = [2,3] [5,10] 10 + [5.5,9] 10 = = 0.95 egree istriution, verge egree, et if grph-equivlent strem then grph egree reltion with n n m 14/23

31 ensity in G: pro two rnom noes re linke δ(g) = n links n possile links = 2 m n (n 1) in S: pro two rnom noes re linke t rnom time instnt δ(s) = n links n possile links = uv V V Tuv uv V V Tu Tv? rnom 15/23

32 ensity in G: pro two rnom noes re linke δ(g) = n links n possile links = 2 m n (n 1) in S: pro two rnom noes re linke t rnom time instnt δ(s) = n links n possile links = uv V V Tuv uv V V Tu Tv?? rnom rnom if grph-equivlent strem then grph ensity reltion with n, m, n verge egree 15/23

33 liques in S: su-strem of ensity 1 in G: su-grph of ensity 1 ll noes re linke together ll noes intert ll the time time 16/23

34 lustering oeffiient in G n in S: ensity of the neighorhoo (v) = δ(n(v)) N() = ([2, 3] [5, 10]) {} [5.5, 9] {} 17/23

35 lustering oeffiient in G n in S: ensity of the neighorhoo (v) = δ(n(v)) N() = ([2, 3] [5, 10]) {} [5.5, 9] {} () = δ(n()) = [6,9] [5.5,9] = /23

36 pths in G: from to : (, ), (, ), (, ) length: 3 shortest pths in S: from (1, ) to (9, ): (2,, ), (3,, ), (7.5,, ), (8,, ) length: 4 urtion: 6 shortest pths fstest pths 18/23

37 pths in G: from to : (, ), (, ), (, ) length: 3 shortest pths in S: from (1, ) to (9, ): (2,, ), (3,, ), (7.5,, ), (8,, ) length: 4 urtion: 6 shortest pths fstest pths 18/23

38 etweenness entrlity in G: (v) = frtion of shortest pths from ny u to ny w in V tht involve v shortest pths in S: (t, v) = frtion of shortest fstest pths from ny (i, u) to ny (j, w) in W tht involve (t, v) 19/23

39 etweenness entrlity in G: (v) = frtion of shortest pths from ny u to ny w in V tht involve v shortest pths in S: (t, v) = frtion of shortest fstest pths from ny (i, u) to ny (j, w) in W tht involve (t, v) shortest fstest pths 19/23

40 A B C mny other onepts e time time time e f e u v g u v w u x v y w u v w time time time 0 1 time u v u v w u x v y w time = e f g h time 0 1 time time time rxiv preprint 20/23

41 reltions vs intertions grph/networks = reltions (like frienship) ynmi grphs/networks = evolution of reltions (like new friens) strem grphs / link strems = intertions (like fe-to-fe ontts) intertions = tres/reliztion of reltions? link strems = tres of grphs/networks? reltions = onsequenes of intertions? grphs/networks = tres of link strems? 21/23

42 onlusion we provie lnguge (set of onepts) tht: mkes it esy to el with intertion tres, omines struture n ynmis in onsistent wy, generlizes grphs / networks ; signls / time series? meets lssil n new lgorithmi hllenges, opens new perspetives for t nlysis, lrifies the interply intertions reltions. stuies in progress: internet trffi, finnil trnstions, moility/ontts, miling-lists, sles, et. 22/23

43 lls for ppers speil issues of interntionl journls Theoretil Computer Siene (TCS) : moels n lgorithms Computer Networks : methos n se stuies eline: July 15th 23/23