MDSR. Proposition and Evaluation of MDSR Method for Core Analysis of Multiple Directed Graphs

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1 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 [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 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) 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

4 1: WX 1 2: WX 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 = t Twitter [16, name 1: MDSR 10 WX t W t X t accounts in W accounts in X WX , hir WX null gat, yuu,... WX e21 WX toa 1...4, mik,... toa 1,2 WX Ten Key WX kyo Ya WX ziz, dr8,... dq WX pos S c WX Sox, car, Ara mom WX null chi, miy,... 11,500,369 1,649,048, WX 1, WX WX 1 2 t WX 2 t 3 W t t = 2 W 2 t = 10 W 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 4 Vol. 14-J, Article No. 1

5 2: k-csc 10 S C r k V S C(k) accounts WX t S C toa 1,2 WX 4 S C sen, get - S C twi, non - S C ant, hha - S C ats, Not - S C rks, gom - S C sou, S C AOI, bot 1,2 - S C U S, h t - S C God, eru - 3: k-cwc 10 WC r k V WC(k) accounts WX t 3: W t X t null X t t 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 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 WC , e21 WX 3 WC Ten, Key WX 5 WC toa WX 4 WC pos, S c WX 8 WC null, gat WX 2 WC toa 1...3, mik WX 4 WC , hir 1 WX 1 WC toa 1...4, mik WX 4 WC , hir 1,2 WX 1 WC 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

6 Twitter 1) 2) 3) 3 k-core 2 MDSR Twitter MDSR - 2 blog MDSR [ ] JSPS [ ] [1] Huberman, B. A., Romero, D. M., Wu, F.: Social networks that matter: Twitter under the microscope, First Monday 14, (1.5). (2009) [2] Kwak, H., Lee, C., Park, H., Moon, S.: What is Twitter, a social network or a news media?, In: Proceedings of WWW 10. pp (2010) [3] Newman, M. E. J., Park, J.: Why social networks are different from other types of networks, Physical Review E, 68(3), (2003) [4] Kato, S., Saito. K., Kazama, K., Satoh, T.: MDSR: An Eigenvector Approach to Core Analysis of Multiple Directed Graphs, In PRICAI 2014: Trends in Artificial Intelligence. Springer International Publishing, pp (2014) [5] Shi, R., Malik, J.: Normalized cuts and image segmentation, IEEE Trans. PAMI, 22(8), pp (2000) [6] Flake, G. W., Lawrence, S., Giles, C. L.: Efficient identification of Web communities, In: Proceedings of SIGKDD 00. pp , (2000) [7] Girvan, M., Newman, M. E. J.: Community structure in social and biological networks, In: Proceedings of the National Academy of Sciences of the United States of America, 99, pp (2002) [8] Palla, G., Derenyi, I., Farkas, I., Vicsek, T.: Uncovering the overlapping community structure of complex networks in nature and society, Nature, Vol.435, pp (2005) [9] Saito, K., Yamada, T., Kazama, K.: Extracting Communities from Complex Networks by the k- Dense Method, IEICE Transactions, Vol.E91-A, No.11, pp (2008) [10] Seidman, S. B.: Network structure and minimum degree, Social Networks, Vol.5, No.3, pp (1983) [11] Saito, K., Ueda, N.: Filtering Search Engine Spam based on Anomaly Detection Approach, In: Proceedings of the KDD2005 Workshop on Data Mining Methods for Anomaly Detection, pp (2005) [12] Kleinberg, J.: Authoritative sources in a hyperlinked environment, Journal of the ACM, Vol.46, No.5, (1999) [13] Brin, S., Page, L.: The anatomy of a large scale hypertextual Web search engine, In: Proceedings of WWW 98, pp (1998) [14] Leicht, E. A., Newman, M. E.: Community structure in directed networks, Physical review letters, 100(11), (2008) [15] Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A.: The architecture of complex weighted networks, Proceedings of the National Academy of Sciences of the United States of America, 101(11), pp (2004). [16] Yamamoto, S., Satoh, T.: Two Phase Extraction Method for Extracting Real Life Tweets using LDA, In: Proceedings of APWeb 13, pp (2013) [17] Yamaguchi,Y., Yamamoto, S. and Satoh, T.: Behavior analysis methods for Twitter users based on transitions in posting activities, Journal of Web Information Systems, Vol.10, No. 4, pp , Emerald (2014) Shoko KATO 2015 ( ) Kazumi SAITO 1985 ( ) 1998 ( ) Kazuhiro KAZAMA 1988 ( ) 2005 ( ) Web Web ACM Tetsuji SATOH ( ) ACM 6 Vol. 14-J, Article No. 1

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DEIM Forum 2012 D2-1 606 8501 150 0002 2-15-1 28F E-mail: {tsukuda,ohshima,tanaka}@dl.kuis.kyoto-u.ac.jp, {miyamamoto,hiwasaki}@d-itlab.co.jp 1 Wikipedia Wikipedia HITS 1. Web Web Web 1 3 Wikipedia 2 Web