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1 DEIM Forum 2015 E ,,,, 1. Freeman [6] 2 [1, 8, 9, 16] Brandes [4] n m O(n 2 + nm) [2, 5, 7, 18, 21] [10 13] 2 Bergamini [3] 1 Bergamini [3] [15, 20] 1.6M 44M Brandes [4]

2 n O(n 2 + nm) O(nm + n 2 log n) 2 [2, 5, 7, 18] Brandes Pich [5] Geisberger [7] Brandes Pich Bader [2] Riondato [18] 2 k [21] DAG 2. 2 [10 13] Lee [13] Brandes [4] Kas [11] Ramalingam [17] Lee [13] 10 Green [10] Kourtellis [12] [10] 2 Bergamini [3] Riondato [18] Riondato Bergamini [3] O(m + n) G = (V, E) V n E m G 2 u, v V dist G(u, v) u v dist G(u, v) = s t s, t 1 P s,t P s,t = {v V dist G(s, v) + dist G(v, t) = dist G(s, t)} P s,t e E s P s,t (DAG) P s,t DAG s, t DAG 3. 2 s, t (v) s, t v s = v t = v (v) = 0 G (v) v C B(v) = v V 3. 3 [21] G = (V, E) H Algorithm 1 M M (s, t) H s, t v P st v v (v) { } (v, (v) ) v P st V (e) e w e (v) e v v / V (e) w e (v) = 0

3. 1 G = (V, E) V n E m G 2 u, v V dist G(u, v) u v dist G(u, v) = s t s, t 1 P s,t P s,t = {v V dist G(s, v) + dist G(v, t) = dist G(s, t)} P s,t e E s P s,t (DAG) P s,t

3 Algorithm 1 [21] procedure BuildHypergraph(G, M) H for i = 1 to M do (s, t) { } e = (v, (v) ) v P st H e v H w H(v) e H we(v) C B(v) C B(v) = n2 wh(v) M C B(v) C B(v) w H(v) Lemma 1. v V E H[w H(v)] = M n 2 C B(v) Proof. (s, t) w H(v) (v) w H(v) 1 n 2 C B(v) C B(v) Hoeffding Lemma 2 (Hoeffding ). X 1, X 2,..., X n [0, 1] X = 1 n n i=1 Xi Pr[ X E[X] > = t] < = 2 exp( 2t 2 n) Hoeffding C B (v) C B (v) C B (v) ϵ C B (v) ϵn 2 δ Lemma 3. ϵ, δ M = O( 1 log 1 ) Pr[ ϵ 2 CB (v) C δ B (v) > = ϵn 2 ] < = δ Proof. Lemma 1 E[ C B (v)] = C B (v) C B (v) [0, 1] M Hoeffding Pr[ C B(v) C B(v) > = ϵn 2 ] = Pr[ n2 n2 wh(v) M M E[wH(v)] > = ϵn2 ] = Pr[ 1 M wh(v) 1 M E[wH(v)] > = ϵ] < = 2 exp( 2ϵ 2 M) M = 1 2ϵ 2 log 2 δ = O( 1 ϵ 2 log 1 δ ) ϵ, δ Pr[ C B(v) C B(v) > = ϵn 2 ] < = δ (BFS) O(n + m) 1 Lemma 3 M = O( 1 log 1 ) Pr[ CB(v) C ϵ 2 δ B(v) > = ϵn 2 ] < = δ O( n+m ϵ 2 4. log 1 δ ) [21] M 4. 1 (u, v) (u, v) 2 M M 2 (s, t) x x x s, t DAG P st v s d sv (v) 3 H st = { (v, d sv, (v) ) v P st } s, t v s d sv s V (H st) s, t H st

.., X n [0, 1] X = 1 n n i=1 Xi Pr[ X E[X] > = t] < = 2 exp( 2t 2 n) Hoeffding C B (v) C B (v) C B (v) ϵ C B (v) ϵn 2 δ Lemma 3. ϵ, δ M = O( 1 log 1 ) Pr[ ϵ 2 CB (v) C δ B (v) > = ϵn 2 ] < = δ Proof.

4 s d s v v B s = {(v, dist G(s, v)) dist G(s, v) < = d s} t d t v v B t = {(v, dist G(t, v)) dist G(t, v) < = d t} d s, d t x d s + d t + 1 = dist G(s, t) x B s, B t v d w B v = {(w, dist G(v, w)) dist G(v, w) < = d} v d R(B v) d V (B v) v d w V (B v) dist Bv (w) dist G(v, w) w / V (B v) dist Bv (w) = 2 B s, B t 3 (H st, B s, B t) G 4. 2 G = (V, E) x (s, t) 3 (H st, B s, B t) s, t 2 B s, B t B s, B t 0 2 BFS V (B s) V (B t) s, t B s, B t 1 B s dist G(s, v) = R(B s) v w B B {(w, R(B s) + 1) dist G(s, w) = R(B s)} B t s, t BFS (s, t) 3 (,, ) s, t B s, B t s, t DAG Algorithm 2 V (B s) V (B t) S S 2 B s, B t P st S v B s, B t dist G(s, v) + dist G(v, t) = R(B s) + R(B t) = dist G(s, t) S P s,t S B s, B t s, t BFS s t S P st B s S P s B t S P t P st = P s P t S P st R(B s) + R(B ) + 1 = dist G(s, t) x B s, B t B s, B t 1 B s, B t P st BFS H st P st P H st Algorithm 3 s BFS P v s d sv s σ sv P v d tv σ tv v σ svσ vt v s, t { } σ svσ vt H st = (v, d sv, σ svσ vt ) v P st Algorithm 2 DAG procedure TwoBallsToDAG(B s, B t) S V (B s) V (B t) P S Q s S FIFO while not Q s.empty() do v = Q s.pop() for each (v, w) E s.t. w / P do if dist Bs (w) + 1 = dist Bs (v) then P P {w} Q s.push(w) t B t return P P st s B s Algorithm 3 P st P P (s, t) procedure ComputeHyperedge(s, t, P ) s P BFS v P d sv, σ sv t P BFS v P d vt, σ vt e for each v P do if d sv + d vt = d st then return e 4. 3 e e (v, d sv, σ svσ vt ) v P st 3 (H st, B s, B t) (u, v) u v (u, v), (v, u) (u, v) 2 B s, B t H st Algorithm 5 Tretyakov [19] s (u, v) B s Algorithm 4 dist Bs (u) + 1 < dist Bs (v) s v (u, v) dist Bs (v) dist Bs (u) + 1 v

2 G = (V, E) x (s, t) 3 (H st, B s, B t) s, t 2 B s, B t B s, B t 0 2 BFS V (B s) V (B t) s, t B s, B t 1 B s dist G(s, v) = R(B s) v w B B {(w, R(B s) + 1) dist G(s, w) = R(B s)} B t s, t BFS (s, t)

5 v w s dist Bs (v) + 1 < dist Bs (w) dist Bs (w) w Algorithm 4 BFS s R(B s) B s B t B s Algorithm 4 (u, v) B s procedure BallInsertEdge(B s, u, v) if dist Bs (u) < R(B s ) then Q FIFO if dist Bs (v) > dist Bs (u) + 1 then dist Bs (v) dist Bs (u) + 1 Q.push(v) while not Q.empty() do v Q.pop() if dist Bs (v) = R(B s ) then continue; for each (v, w) E do if dist Bs (w) > dist Bs (v) + 1 then dist Bs (w) dist Bs (v) + 1 Q.push(w) (u, v) u, v B u, B v u BFS B u 0 1 V (B s ) V (B u ) = B u B v u, v G s, t (u, v) s, t B u, B v R(B u ) R(B v ) B s, B t x R(B u ) R(B v ) u, v B s, B t 1 u v V (B s ) V (B t ) ( 1a) u, v Lemma 4. u, v s, t u, v x B u, B v V (B s ) V (B u ) =,V (B v ) V (B t ) = Proof. x B u, B v V (B s ) V (B u ) = V (B v ) V (B t ) = V (B s ) V (B u ) = B u B v u v v u s t s t R B s R B t R B s R B t B s B t B s B t (a) u, v V (B s) V (B t) (b) u V (B s) 1 s, t dist G (s, u) + dist G (u, v) + dist G (v, t) > = dist G (s, u) R(B t ) > R(B s ) + x R(B t ) = dist G (s, t) u, v s, t V (B v ) V (B t ) = R(B u ), R(B v ) x 2 u V (B s ) ( 1b) dist G (s, u) dist Bs (s, u) u, v Lemma 5. u, v s, t v d st dist Bs (u) R(B t ) 1 B v V (B v ) V (B t ) = R(B v ) d st dist Bs (u) R(B t ) 1 u V (B s ) R(B u ) 0 3 v V (B t ) (2) R(B u ) d st dist Bt (v) R(B s ) 1 R(B v ) 0 (2),(3) (2) B u, B v V (B u ) V (B s ) = V (B v ) V (B t ) = (u, v) s, t DAG P = V (H st ) P su P vt ComputeHyperedge s, t, P H st DAG (u, v) 3 (H st, B s, B t ) B s, B t (u, v) B s dist Bs (u) + 1 = dist Bs (v) B s s BFS B v

pop() if dist Bs (v) = R(B s ) then continue; for each (v, w) E do if dist Bs (w) > dist Bs (v) + 1 then dist Bs (w) dist Bs (v) + 1 Q.

6 Algorithm 5 (u, v) procedure UpdateHyperedge(u, v) B u {(u, 0)}, B v {(v, 0)} d u 0, d v 0 B u, B v if u / V (B s ) V (B t ) and v / V (B s ) V (B t ) then d u x, d v x else if u V (B s) then d v d st dist Bs (u) R(B t) 1 else if v V (B t ) then d u d st dist Bt (v) R(B s ) 1 while V (B s) V (B u) = and R(B u) < d u do B u 1 while V (B v) V (B t) = and R(B v) < d v do B v 1 if V (B s ) V (B u ) = or V (B v ) V (B t ) = then P V (H st ) P P TwoBallsToDAG(B s, B u) P P TwoBallsToDAG(B v, B t) H st ComputeHyperedge(s, t, P ) P st Tretyakov [19] H st (u, v) H st {u, v} V (H st) d su + 1 = d sv (u, v) 2 1 s, t, t (u) = 1 (v) = 1 H st s, t BFS (H st, B s, B t) 2 s, t s, t DAG V (H st) H st ComputeHyperedge(s, t, V (H st)) Stanford Network Analysis Project (SNAP) [14] 1 1 d D 10,000 n m d D GrQc 5,242 28, Enron 36, , Epinions1 75, , Slashdot ,360 1,092, Pokec 1,632,803 44,603, Stanford 281,903 3,985, NotreDame 325,729 2,235, Google 875,713 8,644, BerkStan 685,230 13,298, Patents 3,774,768 33,037, C++11 Google Sparse Hash 1 gcc O3 Linux (CPU: Intel(R) Xeon(R) CPU E GHz: 512GB) V (B s ) + V (B t ) 1 (s, t) B s, B t V (H st ) x x = 0 x = x x = , ,000 99% 0.05n 2 1

= or V (B v ) V (B t ) = then P V (H st ) P P TwoBallsToDAG(B s, B u) P P TwoBallsToDAG(B v, B t) H st ComputeHyperedge(s, t, P ) P st Tretyakov [19] H st (u, v) H st {u, v} V (H st) d su + 1 = d sv

7 Stanford BerkStan NotreDame Google 2 V (B s) + V (B t) x = 0 x = 1 x = 2 V (H st ) GrQc Enron Epinions Slashdot Pokec 1, Stanford 6,807 3,454 1, NotreDame 3,026 1, Google 2, BerkStan 25,677 12,707 4, Patents 5,680 1, x = 0 Pokec BerkStan 3 x x 6. [1] A. Abbasi, L. Hossain, and L. Leydesdorff. Betweenness centrality as a driver of preferential attachment in the evolution of research collaboration networks. Journal of Informetrics, 6(3): , [2] D. A. Bader, S. Kintali, K. Madduri, and M. Mihail. Approximating Betweenness Centrality. In WAW, pp , [3] E. Bergamini, H. Meyerhenke, and C. L. Staudt. Approximating Betweenness Centrality in Large Evolving Networks. In ALENEX, pp , [4] U. Brandes. A Faster Algorithm for Betweenness Centrality. J. Math. Sociol., 25(2): , [5] U. Brandes and C. Pich. Centrality Estimation in Large Networks. Int. J. Bifurcat. Chaos, 17(07): , [6] L. C. Freeman. A Set of Measures of Centrality Based on Betweenness. Sociometry, 40(1):35 41, [7] R. Geisberger, P. Sanders, and D. Schultes. Better Approximation of Betweenness Centrality. In ALENEX, pp , [8] M. Girvan and M. E. J. Newman. Community structure in social and biological networks. P. Natl. Acad. Sci., 99(12):7821 6, [9] K.-I. Goh, E. Oh, B. Kahng, and D. Kim. Betweenness centrality correlation in social networks. Phys. Rev. E, 67:017101, [10] O. Green, R. McColl, and D. A. Bader. A fast algorithm for streaming betweenness centrality. In SocialCom/PASSAT, pp , [11] M. Kas, M. Wachs, K. M. Carley, and L. R. Carley. Incremental Algorithm for Updating Betweenness Centrality in Dynamically Growing Networks. In ASONAM, pp , [12] N. Kourtellis, G. D. F. Morales, and F. Bonchi. Scalable Online Betweenness Centrality in Evolving Graphs. CoRR, abs/ , [13] M.-J. Lee, J. Lee, J. Y. Park, R. H. Choi, and C.-W. Chung. QUBE: a Quick algorithm for Updating BEtweenness centrality. In WWW, pp , [14] J. Leskovec and A. Krevl. SNAP Datasets: Stanford large network dataset collection. data, June [15] S. Milgram. The Small World Problem. Psychology today, 2(1):60 67, [16] M. E. J. Newman and M. Girvan. Finding and evaluating community structure in networks. Phys. Rev. E, 69:026113, [17] G. Ramalingam and T. W. Reps. On the Computational Complexity of Incremental Algorithms. University of Wisconsin-Madison, Computer Sciences Department, [18] M. Riondato and E. M. Kornaropoulos. Fast Approximation of Betweenness Centrality Through Sampling. In WSDM, pp , [19] K. Tretyakov, A. Armas-cervantes, L. García-ba nuelos, and M. Dumas. Fast Fully Dynamic Landmark-based Estimation of Shortest Path Distances in Very Large Graphs Categories and Subject Descriptors. In CIKM, pp , [20] D. J. Watts and S. H. Strogatz. Collective dynamics of small-world networks. Nature, 393(6684): , 1998.

Hossain, and L. Leydesdorff. Betweenness centrality as a driver of preferential attachment in the evolution of research collaboration networks. Journal of Informetrics, 6(3):403 412, 2012. [2] D. A.

8 3 (ms) (s) x = 0 x = 1 x = 2 x = 0 x = 1 x = 2 GrQc Enron , ,557 Epinions , , ,748 Slashdot ,291 1, , ,867 Pokec ,931 12,149 1,652 34, ,892 Stanford , NotreDame , ,013 Google , , ,445 BerkStan , , , , , Patents ,513 3,238 2,948 13, ,270 [21] Y. Yoshida. Almost Linear-Time Algorithms for Adaptive Betweenness Centrality using Hypergraph Sketches. In KDD, pp , 2014.

5 348.6 256.3 118.2 1,401.0 28.2 414 66 90 195 16 814 NotreDame 9.9 3.5 287.7 69.1 107.3 612.3 9.7 2,784 34 143 92 16 1,013 Google 8.3 4.2 11.7 19.7 1.1 193.0 0.

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