MDSR Proposition and Evaluation of MDSR Method for Core Analysis of Multiple Directed Graphs Shoko KATO Kazumi SAITO Kazuhiro KAZAMA Tetsuji SATOH MDSR Multiple-Directed-Spectral-Relaxation MDSR 2 Twitter ( bot) ( null ) 3 k-core 2 MDSR In this paper, we propose the MDSR method for a problem of extracting core portions of a multiple directed network. The MDSR method extracts plural core portions by repeating the following two steps; quantizing elements of the left- and right-eigenvectors of an adjacency matrix of a network to binary ones as an indicator of extracting core portion, and removing links of the extracted one. By calculating the left- and right-eigenvectors, the MDSR method extracts 445.ka10@gmail.com k-saito@u-shizuoka-ken.ac.jp kazama@sys.wakayama-u.ac.jp satoh@ce.slis.tsukuba.ac.jp pairs of asymmetric node sets which have complementary roles, i.e., initial and terminal nodes. In our experiments using a reply network on Twitter, we demonstrate that the MDSR method uncover the following three types of users; 1) users who send tweets frequently (e.g., bots), 2) users who receive tweets frequently (e.g., null ), 3) small groups who send/receive tweets frequently each other. We also show that some communities were overlapped ones. Furthermore, we show that such communities were hard to be automatically found by two methods, which were constructed by straightforwardly extending the conventional k-core method. 1. Twitter Facebook [1, 2] [3] MDSR (Multiple- Directed-Spectral-Relaxation) Twitter [4] 2 1 [5, 6, 7] 1 1 ( ) [8, 9, 10, 11] MDSR MDSR SR [11] SR MDSR HITS [12] PageRank [13] MDSR 2 MDSR [4] 1 Vol. 14-J, Article No. 1
2. 2. 1 [14, 15] MDSR HITS [12] k-core [10] 2. 2 HITS Kleinberg HITS V = {1,, N} A G = (V, E) (i, j) A(i, j) A(i, j) i V j V A(i, j) A( j, i) A(i, i) = 0 v( j) = i j u(i) = i A(i, j)u(i) u(i) = j i v( j) = j A( j, i)v( j) v = Au Hub u = A T v Authority 2. 3 k-core k-core 2 G i V d + (i) = j>0 A(i, j) d (i) = j>0 A( j, i) d + (i) d (i) k S C(k) k-csc d + (i) d (i) k WC(k) k-cwc k k-csc V S C(k) V E S C(k) E S C(k) = (V S C(k), E S C(k) ) V S C(k) = {i : d + (i) k, d (i) k}, E S C(k) = {e i, j : i, j V S C(k) }. (1) k-cwc V WC(k) V E WC(k) E WC(k) = (V WC(k), E WC(k) ) V WC(k) = {i : d + (i) + d (i) k}, E WC(k) = {e i, j : i, j V WC(k) }. (2) S C(k) WC(k) k-csc k-cwc k k-csc G = (V, E) (V S C(k), E S C(k) ) A1. V S C(k) = V, E S C(k) = E ; A2. P = {i : d + (i) < k or d (i) < k} P ; A3. P = (V S C(k), E S C(k) ) ; V S C(k) = V S C(k) P, E S C(k) = E S C(k) {e i, j : i P, j P} A2. P k-csc (V WC(k), E WC(k) ) P = {i : d + (i) + d (i) < k} A2. k k-csc max i {d + (i)} max i {d (i)} S C(k + 1) S C(k) k k-csc B1. V S C(1) = V, E S C(1) = E, k = 2 ; B2. (V S C(k 1), E S C(k 1) ) (V S C(k), E S C(k) ) ; B3. V S C(k) = ; k = k + 1 B2. max i {d + (i) + d (i)} k-cwc 2 k-csc k-cwc 3. MDSR MDSR SR [11] Web MSDR 2 SR MDSR 2.2 HITS Authority Hub Twitter Hub Authority, 2 Vol. 14-J, Article No. 1
. 3. 1 G V = {1,, N} A (i, j) A(i, j) i j A(i, i) = 0 A(i, j) 2 W V X V G(W, X) = 1 A(i, j). (3) W X W W MDSR W X (3) W X N MDSR 3. 2 W N q i W q(i) = 1 q(i) = 0 X N r (3) G(q, r) = i W j X rt Aq qt qr T r. (4) r T r q G(q, r) A q r q r 2.2 Authority u = A T v Hub v = Au q r E1. t = 1, q (0) = (1,, 1) T ; E2. q = A T Aq (t 1), q (t) = q/ max i q(i) ; E3. max i q (t) (i) q (t 1) (i) < ϵ ; E4. t = t + 1 E2 ϵ q = q (t) r = Aq A q (0) q E2 0 q (t) (i) 1 1 L q L q N O(N) 3. 3 q r q r q S = [s(1),, s(n)] s(i) i q (s(i)) q (s(i + 1)) tie-break E(0) E(0) (5) E(0) = (q(i) 1 N q( j)) 2 = j=1 q(i) 2 1 N ( q(i)) 2. (5) S = [s(1),, s(n)] m W(m) N m E(m) (7) E(m) = = m mj=1 q(s( j))) 2 (q(s(i)) m N j=m+1 q(s( j)))2 + (q(s(i)) N m i=m+1 q(i) 2 1 m m ( q(s(i))) 2 1 N m ( i=m+1 (6) q(s(i))) 2 MDSR (7) E(m) m W(m ) m (q(s(1)),, q(s(n))) (y(0), y(1),..., y(n)) y(0) = 0, i y(i) = q(s( j)) = y(i 1) + q(s(i)) (i = 1,, N). (7) j=1 E(m) E(m) = q(i) 2 1 1 m y(m)2 N m (y(n) y(m))2. (8) F1. q s(i) ; F2. (q(s(1)),, q(s(n))) (y(1),..., y(n)) (7) ; F3. E(1),, E(N 1) (8) ; F4. m = arg max m E(m) W(m ) ; F1 O(N log N) F2 (7) N F3 E(1),, E(N 1) q(i) (8) O(N) 3 Vol. 14-J, Article No. 1
1: WX 1 2: WX 2 3. 4 T T G1. t = 1 T ; G2. E1 E4 q m ; G3. q F1 F4 W k (m ) ; G4. r F1 F4 X k (n ) ; G5. i W k (m ), j X k (n ) A(i, j) = 0 T T (W 1, X 1 ),, (W T, X T ) 4. Twitter MDSR k-csc CWC MDSR MDSR t (W t, X t ) WX t W t X t T = 100 1 t 10 4. 1 2012 3 14 2013 3 14 Twitter [16, 17] @screen name @screen name 1: MDSR 10 WX t W t X t accounts in W accounts in X WX 1 1 9 8 1 8 2...6, hir 1...4 WX 2 1 5 null gat, yuu,... WX 3 1 1 113 e21 WX 4 7 2 toa 1...4, mik,... toa 1,2 WX 5 1 1 Ten Key WX 6 4 1 kyo 1...4 Ya WX 7 8 1 ziz, dr8,... dq WX 8 1 1 pos S c WX 9 3 1 Sox, car, Ara mom WX 10 1 19 null chi, miy,... 11,500,369 1,649,048,139 12.8 4. 2 1 2 WX 1, WX 2 2 1 WX 1 2 t 3 2 2 WX 2 t 3 W t t = 2 W 2 t = 10 W 10 4.3 4. 3 MDSR MDSR MDSR 1 1 t 10 WX t Wt Wt Xt abc 1 1 WX 1 WX 4 WX 6 MDSR WX 2 WX 10 X 2 X 10 null Twitter null @null 4 Vol. 14-J, Article No. 1
2: k-csc 10 S C r k V S C(k) accounts WX t S C 1 35184 2 toa 1,2 WX 4 S C 2 22388 2 sen, get - S C 3 22010 2 twi, non - S C 4 18640 2 ant, hha - S C 5 10745 2 ats, Not - S C 6 10352 2 rks, gom - S C 7 9948 2 sou, 287 - S C 8 9599 3 AOI, bot 1,2 - S C 9 9423 2 U S, h t - S C 10 9124 2 God, eru - 3: k-cwc 10 WC r k V WC(k) accounts WX t 3: W t X t null X t t 11 100 W t X t 3 W t X t W t X t W t X t (1,1) (2,2) WX t 1 t 100 W t X t W t X t (1) W t X t (2) W t > X t (3) W t < X t 3 3 (1) (2) X t reply bot (3) W t = 1 W t null YouTube 1 YouTube null YouTube Twitter @YouTube W t X t 2 MDSR 4. 4 k-csc k-cwc MDSR k-csc k-cwc MDSR 2 3 k 10 r S C r WC r V S C(k) 1 http://www.youtube.com/ WC 1 110052 2 113, e21 WX 3 WC 2 91941 2 Ten, Key WX 5 WC 3 76086 3 toa 1...3 WX 4 WC 4 73394 2 pos, S c WX 8 WC 5 66623 2 null, gat WX 2 WC 6 65458 4 toa 1...3, mik WX 4 WC 7 58978 2 8 1, hir 1 WX 1 WC 8 58384 5 toa 1...4, mik WX 4 WC 9 57740 3 8 1, hir 1,2 WX 1 WC 10 52342 2 Sox, mom WX 9 S C(k) V WC(k) MDSR WX t 2 k-csc 2 3 k- CWC 10 MDSR WC 3 WC 6 WC 6 WC 8 WX 4 WC 7 WC 9 WX 1 k-cwc k k k MDSR k-csc k-cwc MDSR W t X t MDSR 5. MDSR 2 5 Vol. 14-J, Article No. 1
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