Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor

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1 NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer Lnear Algebra Al 22; 9:83 84 Publshed onlne 8 October 2 n Wley Onlne Lbrary (wleyonlnelbrarycom) OI: 2/nla822 Lnear convergence of an algorthm for comutng the largest egenvalue of a nonnegatve tensor Lng Zhang and Lqun Q 2, *, eartment of Mathematcal Scences, Tsnghua Unversty, Bejng 84, Chna 2 eartment of Aled Mathematcs, The Hong Kong Polytechnc Unversty, Hong Kong SUMMARY An teratve method for fndng the largest egenvalue of a nonnegatve tensor was roosed by Ng, Q, and Zhou n 29 In ths aer, we establsh an exlct lnear convergence rate of the Ng Q Zhou method for essentally ostve tensors Numercal results are gven to demonstrate lnear convergence of the Ng Q Zhou algorthm for essentally ostve tensors Coyrght 2 John Wley & Sons, Ltd Receved 25 Setember 2; Revsed 7 August 2; Acceted 28 August 2 KEY WORS: nonnegatve tensor; essentally ostve tensor; egenvalue; teratve method; convergence INTROUCTION Consder an m-order n-dmensonal tensoraconsstng of n m entres n the real feld <: A a m, a m 2 <, 6, : : :, m 6 n An m-order n-dmensonal tensor A s called nonnegatve (or, resectvely, ostve) f a m > (or, resectvely, a m > ) A tensoras called reducble, f there exsts a nonemty roer ndex subset J f, 2, : : :, ng such that a m, 8 2 J, 8 2, : : :, m 62 J If A s not reducble, then we say that A s rreducble Ths defnton was used n [ 6] In [7, 8], ths roerty s called ndecomosable To an n-dmensonal column vector x x I x 2 I : : : I x n /, real or comlex, and any comlex number, we defne n-dmensonal column vectors Ax and x Œ : Ax a 2 ::: m x 2 : : : x m A, x Œ W xį 2,:::, m 66n 66n Let C be the comlex feld A ar, x/ 2 C C n nfg/ s called an egenvalue egenvector ar of A, f they satsfy: Ax x Œ () *Corresondence to: Lqun Q, eartment of Aled Mathematcs, The Hong Kong Polytechnc Unversty, Hong Kong E-mal: maqlq@olyueduhk Coyrght 2 John Wley & Sons, Ltd

2 LINEAR CONVERGENCE FOR FINING THE LARGEST EIGENVALUE 83 Ths defnton was ntroduced by Q [9] when m s even and A s symmetrc and extended to the general case n [] Indeendently, Lm [] gave such a defnton but restrcted x and to be real Unlke matrces, the egenvalue roblem for tensors s nonlnear Recently, the largest egenvalue roblem for a nonnegatve tensor has attracted much attenton because t has many mortant alcatons such as multlnear agerank [], hyergrahs [2], hgher-order Markov chans [4, 3], and ostve defnteness of a multvarate form [3] Chang, Pearson, and Zhang [] generalzed the Perron Frobenus theorem from nonnegatve matrces to rreducble nonnegatve tensors Yang and Yang [6] generalzed the weak Perron Frobenus theorem to general nonnegatve tensors Bulò and Pelllo [2] gave new bounds on the clque number of grahs on the bass of sectral hyergrah theory The calculaton of these new bounds reles on fndng the largest egenvalue of a f, g nonnegatve tensor Ng, Q, and Zhou [4] roosed an teratve method for fndng the largest egenvalue of an rreducble nonnegatve tensor, whch s an extenson of the Collatz method [4] for calculatng the sectral radus of an rreducble nonnegatve matrx Pearson [5] ntroduced the noton of essentally ostve tensors and roved that the unque ostve egenvalue s real geometrcally smle when the tensor s essentally ostve wth even order Here, real geometrcally smle means that the corresondng real egenvector s unque u to a scalng constant Lu, Zhou, and Ibrahm [3] modfed the Ng Q Zhou method such that the modfed algorthm s always convergent for fndng the largest egenvalue of an rreducble nonnegatve tensor Chang, Pearson, and Zhang [2] ntroduced rmtve tensors An essentally ostve tensor s a rmtve tensor, and a rmtve tensor s an rreducble nonnegatve tensor but not vce versa Chang, Pearson, and Zhang [2] establshed convergence of the Ng Q Zhou method for rmtve tensors Fredland, Gaubert, and Han [7] onted out that the Perron Frobenus theorem for nonnegatve tensors has a very close lnk wth the Perron Frobenus theorem for homogeneous monotone mas, ntated by Nussbaum [5] and further studed by Gaubert and Gunawardena [8] Frendland, Gaubert, and Han [7] ntroduced weakly rreducble nonnegatve tensors and establshed the Perron Frobenus theorem for them enote < n C fx 2 <n W x > g and < n CC fx 2 <n W x > g A ma F W < n C! <n C s called a homogeneous monotone ma f Ftx/ tfx/ for any x 2 < n C and any ostve number t and f Fx/ 6 Fy/ for any x, y 2 < n C wth x 6 y For an m-order n-dmensonal tensor A, defne F W < n C! <n C by Fx/ Ax Œ for any x 2 < n C Then F s a homogeneous monotone ma Hence, n < C < n C, egenvalues and egenvectors of nonnegatve tensors, dscussed here, fall n the class of egenvalues and egenvectors of homogeneous monotone mas Egenvalues and egenvectors of nonnegatve tensors are defned ncc n nfg/ In Corollary 43 of [7], Frendland, Gaubert, and Han showed how to extend the Perron Frobenus result for a nonnegatve tensor from < C < n C toc Cn nfg/ In ths aer, we establsh an exlct lnear convergence rate of the Ng Q Zhou method for essentally ostve tensors Such a result has not aeared n the lterature of egenvalues of homogeneous monotone mas and nonnegatve tensors Ths aer s organzed as follows In Secton 2, we recall some relmnary results In Secton 3, we rewrte the Ng Q Zhou method n the srt of the earler work of Hall and Porschng [6, 7] An exlct lnear convergence rate of the Ng Q Zhou method s establshed for essentally ostve tensors, wth a secfed startng ont, n Secton 4 Fnally, n Secton 5, we reort some numercal results Note that Chang, Pearson, and Zhang [2] showed that lnear convergence rate dd not hold for some rmtve but not essentally ostve tensors for the Ng Q Zhou method Ths shows that our result s shar 2 PRELIMINARIES Frst, we state the Perron Frobenus theorem for nonnegatve tensors gven n [, Theore4] and the mnmax theorem for rreducble nonnegatve tensors gven n [, Theorem 42] These two results, as we stated n the ntroducton, may be derved from the earler results on homogeneous monotone mas Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

3 832 L ZHANG AN L QI Theorem 2 If A s an rreducble nonnegatve tensor of order m and dmenson n, then there exst > and x 2 < n CC such that Ax x Œ Moreover, f s an egenvalue wth a nonnegatve egenvector, then If s an egenvalue of A, then jj 6 Theorem 22 Assume that A s an rreducble nonnegatve tensor of order m and dmenson n Then mn max x2< n 66n CC Ax x max mn x2< n 66n CC Ax / x, where s the unque ostve egenvalue corresondng to a ostve egenvector On the bass of Theorem 22, the Ng Q Zhou method resented n [4] works as follows Choose x / 2 < n CC and let y/ A x / For k,, 2, : : :, comute x kc/ yk/ Œ, yk/ Œ y kc/ A x kc/, kc mn x kc/ > y kc/ x kc/, kc max x kc/ > y kc/ x kc/ (2) It s shown n [4] that the obtaned sequences f k g and f k g converge to some numbers and, resectvely, and we have 6 6, where s the largest egenvalue of A, defned n Theorem 2 If, that s, the ga s zero, then both the sequences f k g and f k g converge to However, a ostve ga may haen, whch can be seen n an examle gven n [4] That examle s rreducble but not rmtve Chang, Pearson, and Zhang [2] establshed convergence of the Ng Q Zhou method for rmtve tensors and gave an examle, whch s rmtve but not essentally ostve, such that lnear convergence fals for ths examle wth the Ng Q Zhou method We thus ntend to establsh an exlct lnear convergence rate of the Ng Q Zhou method for essentally ostve tensors For ths urose, we need the followng defnton gven n [5] efnton 2 A nonnegatve m-order n-dmensonal tensor A s essentally ostve f Ax 2 < n CC for any nonzero x 2 < n C By Theorem 32 and efnton 2 n [5], t s easy to obtan the followng result Prooston 2 A nonnegatve m-order n-dmensonal tensor A s essentally ostve f and only f a j :::j > for, j 2 f, 2, : : :, ng For the remander of ths aer, denote I f 2, : : :, m / j 2, : : :, m 2 f, : : :, ngg Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

4 LINEAR CONVERGENCE FOR FINING THE LARGEST EIGENVALUE ALGORITHM We frst gve ncluson bounds for the largest egenvalue of an rreducble nonnegatve tensor The followng lemma was gven n [, Lemma 22] Lemma 3 If a nonnegatve tensor A of order m and dmenson n s rreducble, then R W 2,:::, m a 2 ::: m >,, 2, : : :, n On the bass of Theorem 22 and Lemma 3, we obtan the followng ncluson bounds Prooston 3 Let A be an rreducble nonnegatve tensor of order m and dmenson n, and let be the largest egenvalue of A Then mn M 6 6 max M, 66n 66n where M R 2,:::, m m a 2 ::: m R m 2 R m,, 2, : : :, n efne an n-dmensonal column vector R W R / 66n, where R s defned n Lemma 3 By Lemma 3, the vector R Π2 < n CC Takng x RΠnto the equaltes n Theorem 22, we mmedately get the lower and uer bounds Let For, 2, : : :, n, we have R R R 2,:::, m R max 66n R, a 2 ::: m 6 M 6 R R R mn 66n R (3) 2,:::, m a 2 ::: m R Ths shows that the lower and uer bounds gven n Prooston 3 are better than the bounds n [6, Lemma 56] On the bass of Prooston 3 and n the srt of the earler work of Hall and Porschng [6,7], we rewrte the Ng Q Zhou method as follows Algorthm 3 Ste Let A / A and S / np 2,:::, m " > be a suffcently small number and set k W Ste Set A kc/ a kc/ ::: m, S kc/ a 2 ::: m for, : : :, n Let the accurate tolerance 2,:::, m a kc/ 2 ::: m Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

5 834 L ZHANG AN L QI Ste 2 Let for, : : :, n, where a kc/ 2 ::: m 2 ::: m 2 m k max S kc/, k mn S kc/ 66n 66n (4) If k k < ", sto Outut the maxmal egenvalue 2 k C k / Otherwse, set k W k C and go to Ste Algorthm 3 s well defned By Lemma 3, for, : : :, n, S / R > and there exsts at least one subndex array 2, : : :, m / 2 I such that a 2 ::: m > Hence, a / 2 ::: m > and S / > By nducton, we have 2 ::: m > and > for k 2, 3, : : : We now rove that Algorthm 3 s actually the Ng Q Zhou method wth a secfed startng ont Prooston 32 Let A be an rreducble nonnegatve tensor of order m and dmenson n Then Algorthm 3 s just the Ng Q Zhou method wth the startng ont x / I I : : : I / 2 < n For k,, : : :, defne an n-dmensonal column vector ky x k/ A S j / j 66n BecauseAs rreducble, by Lemma 3, x k/ 2 < n CC for k,, : : : Set y k/ W xk/ ΠBy the defnton ofa k/ n Ste, we obtan for 6 6 n, A y k/ y k/ x k/ 2,:::, m, k,, : : : (5) xk/ Π2,:::, m a / 2 ::: m 2,:::, m a 2 ::: m x k/ 2 x k/ m Qk j S j / 2 2 ::: m Qk j S j / m Q k j S j / 2 m S kc/, (6) whch concludes that k s just kc n (2) for k,, 2, : : : Ths also holds for the sequence f k g Hence, we conclude the statement of ths rooston Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

6 LINEAR CONVERGENCE FOR FINING THE LARGEST EIGENVALUE 835 Numercal results reorted n [3,4] mly that Algorthm 3 s effcent In artcular, Lu, Zhou, and Ibrahm [3] aled a modfcaton of Algorthm 3 to study the ostve defnteness of a multvarate form Testng ostve defnteness of a multvarate form s an mortant roblem n the stablty study of nonlnear autonomous systems va Lyaunov s drect method n automatc control Researchers n automatc control studed the condtons of such ostve defnteness ntensvely [9] For n > 3 and m > 4, ths roblem s a hard roblem n mathematcs There are only a few methods to answer the queston, and these methods are comutatonally exensve when n > 3 [3] Numercal results n [3] show the method of usng the largest egenvalue of nonnegatve tensors s effectve n some cases 4 LINEAR CONVERGENCE By Prooston 32, t suffces to establsh an exlct lnear convergence rate of Algorthm 3 n the case of essentally ostve tensors By some straghtforward comutatons, we mmedately obtan the followng two roostons Prooston 4 For k,, 2, : : :, S kc/ 2,:::, m For, 2, : : :, n, by (4), we have S kc/ Prooston 42 For k,, 2, : : :, 2,:::, m 2 ::: m a kc/ 2 ::: m 2 m 2,:::, m 2 ::: m ::: a :::,, 2, : : :, n j :::j ak/ j ::: a j :::j a j :::, For, 2, : : :, n, t follows from (4) that S / ::: ak ::: k / k / S S k /,, 2, : : :, n 2 m, j 2 f, 2, : : :, ng For, j 2 f, 2, : : :, ng, accordng to (4), by a drect comutaton, k / a ::: a ::: j :::j ak/ j ::: ak / j :::j a k / j :::j k / S j k / S k / k / S j ::: k / S j a ak / j ::: a j :::j a j ::: Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

7 836 L ZHANG AN L QI LetAbe an rreducble nonnegatve tensor of order m dmenson n By Theorem 2, there exsts a ostve egenvalue of A, whch s the largest n modulus among all the egenvalues of A Algorthm 3 yelds two sequences of lower and uer bounds for ths largest egenvalue Theorem 4 Let A be an rreducble nonnegatve tensor of order m and dmenson n and be ts largest egenvalue Assume that f k g and f k g are two sequences generated by Algorthm 3 Then R k k R For k,, : : :, let y k/ be defned by (5) Takng the vector y k/ nto the two equaltes n Theorem 22, we obtan from (6), We now rove for any k >, k mn S kc/ 6 6 max S kc/ k, k,, : : : 66n 66n k 6 kc and kc 6 k We assume, wthout loss of generalty, that kc S kc2/ f, 2, : : :, ng We have by Prooston 4, and kc S kc2/ q where, q 2 kc S kc2/ S kc/ 2,:::, m > mn S kc/ 66n k a kc/ 2 ::: m S kc/ 2 S kc/ m Smlarly, we can rove that kc 6 k Ths, together wth (3), comletes our roof Theorem 4 ndcates that f k g and f k g converge, and the sequences f k k g s nonnegatve and monotoncally decreasng Hence, f k k g has a lmt We now show that n the case of essentally ostve tensors, f k k g lnearly converges to zero wth an exlct convergence rate Ths establshes an exlct lnear converge rate of the Ng Q Zhou method for essentally ostve tensors Theorem 42 Let A be a nonnegatve tensor of order m and dmenson n If A s essentally ostve, then k k 6 k k, k, 2, : : :, (7) where W ˇ 2, /, (8) R ˇ W mn,j 2f,2,:::,ng a j :::j, R max 66n R Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

8 LINEAR CONVERGENCE FOR FINING THE LARGEST EIGENVALUE 837 and R a 2 ::: m 2,:::, m By Prooston 2, the nonnegatve tensor A s rreducble Wthout loss of generalty, assume that k S kc/ and k kc/ Then by Prooston 4, we have k k S kc/ kc/ 2 ::: m S k/ 2,:::, m! q 2 ::: m k/ 2 m (9) efne Ik/ ( 2, : : :, m / 2 I ˇ 2 ::: m ) > ak/ q 2 ::: m k/ By the defnton of, for,, n, we have 2,:::, m /2Ik/ 2 ::: m C 2 ::: m 2,:::, m /2InIk/ Lettng and q and combnng these two equaltes, we have 2 ::: m! q 2 ::: m S k/ k/ 2,:::, m /2Ik/ 2,:::, m /2InIk/ 2 ::: m! q 2 ::: m () k/ Combnng (9), (), and the defntons of k and k, by Theorem 4, we obtan k k 2,:::, m /2Ik/ C 2,:::, m /2InIk/ 6 k 2,:::, m /2Ik/ 2 ::: m 2 ::: m 2 ::: m k k k 2,:::, m q 2 ::: m k/ 2 m! q 2 ::: m k/ 2 m! q 2 ::: m q 2 ::: m 2,:::, m /2InIk/ 6 k k C 2 6 k k k C 2 R C k q 2 ::: m q 2 ::: m C 2,:::, m /2InIk/! 2 ::: m q 2 ::: m 2,:::, m /2Ik/ k/ AA! q 2 ::: m k/, () Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

9 838 L ZHANG AN L QI where 2,:::, m /2InIk/ 2 ::: m, 2 2,:::, m /2Ik/ Because A s essentally ostve, we have from Prooston 2, Because A s rreducble, from (3) and (2), we have q 2 ::: m a j :::j >,, j 2 f, 2, : : :, ng (2) < ˇ < R, (3) where ˇ s defned n the statement of the theorem We now consder the subndex array, : : :, / 2 I If, : : :, / 2 InIk/, then the summaton must nclude a k/ ::: If, : : :, / 2 Ik/, there are two ossbltes: () q, : : :, q/ 2 Ik/ In ths case, the summaton 2 must nclude q q:::q (2) q, : : :, q/ 2 InIk/ In ths case, the summaton must nclude q:::q, and the summaton 2 must nclude q ::: From the dscusson, by Prooston 42, we can obtan n o C 2 > mn a k/ :::, ak/ q q:::q, ak/ q:::q C ak/ q ::: > mn a :::, a q q:::q, 2 a q:::q a q ::: Take > ˇ (4) W ˇ R Then < < follows from (3) Combnng () and (4), we obtan (7) for k, 2, : : : Theorem 42 acheved the target of ths aer, namely, to establsh an exlct lnear convergence rate of the Ng Q Zhou method for essentally ostve tensors In the followng corollary, we gve an exlct asymtotc estmate on the number of teratons Corollary 4 Let and R, R be defned by (8) and (3), resectvely Let " > be the suffcently small number gven n Algorthm 3 IfAs essentally ostve, then Algorthm 3 termnates n at most teratons wth 2 3 " log K R R 6 log / K K < " By (7) n Theorem 42, we have for k, 2, : : :, It follows from (5) and 2, / that log log K/ K log / < log / 7 C (5) k k 6 kr R/ (6) " R R log / " log, R R Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

10 LINEAR CONVERGENCE FOR FINING THE LARGEST EIGENVALUE 839 whch yelds K < " R R Ths, together wth (6), mles K K 6 KR R/ < " Ths comletes the roof 5 NUMERICAL EPERIMENTS To demonstrate the lnear convergence of the Ng Q Zhou algorthm (Algorthm 3 ) for essentally ostve tensors, we made numercal exerments on some numercal examles wth the sto rule " 8 We consder the followng fve classes of nonnegatve tensors of order m 3 and dmenson n A W A jj j for, j, 2, : : :, n, and zero elsewhere A2 W A3 W A2 jj A3 jj j for, j, 2, : : :, n, and zero elsewhere for, j, 2, : : :, n, and zero elsewhere Table I Numercal results for Algorthm 3 Tensor menson Eg No Iter Rato (Max, Mn) Rato A , 79, 79 (46, 79) ,, (5, ) e 4, 36e 4, 36e 4 (367e 4, 36e 4) A , 346, 346 (3327, 2736) , 22, 22 (2448, 86) , 76, 76 (2246, 466) A , 667, 667 (78, 472) , 323, 323 (54, 323) , 64, 64 (38, 64) A , 3796, 38 (5635, 3333) , 2588, 2858 (4823, 3) , 36, 33 (447, 877) 5 x n=5 2 5 n=3 5 n= Fgure The curves of rato fora Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

11 84 L ZHANG AN L QI A4 W A4 jj for j 2, : : :, n, A4 jj for 2, : : :, n, A5 W j, : : :, n C, and zero elsewhere A5 22 4, A5 2, and zero elsewhere Note that A, A2, and A3 are all essentally ostve A4 s rmtve but not essentally ostve A5 s rreducble but not rmtve and not essentally ostve We aly Algorthm 3 to fnd the 35 3 n= n=3 n= Fgure 2 The curves of rato fora2 8 6 n= n=3 n= Fgure 3 The curves of rato fora n=5 n= n= Fgure 4 The curves of rato fora4 Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla

12 LINEAR CONVERGENCE FOR FINING THE LARGEST EIGENVALUE 84 largest egenvalues of all the fve tensors wth dfferent dmensons n 5, 3, 6 We summarze the numercal results n the followng table, where NoIter denotes the number of teratons, Eg denotes the largest egenvalue, and Rato denotes the rato of kc kc to k k at the last three teratons, (Max, Mn) Rato denotes the maxmum rato and the mnmum rato We also draw the curve of the convergence rato to descrbe the lnear convergence n the followng fgures For tensor A5, the Ng Q Zhou algorthm s dvergent From Table I and Fgures 3, we see that the Ng Q Zhou algorthm s lnearly convergent for tensors A, A2, and A3 From Table I and Fgure 4, we see that the Ng Q Zhou algorthm s also convergent fora4, but t s not lnearly convergent Ths echoes the results n [2] Hence, numercal examles show that the Ng Q Zhou algorthm s lnearly convergent for essentally ostve tensors, and ths result s shar ACKNOWLEGEMENTS We would lke to thank edtor and referees for ther comments We also thank r Y u and Prof Zhengha Huang for ther hel n erformng the numercal exerments The work of Lng Zhang was suorted by the Natonal Natural Scence Foundaton of Chna (Grant No 873), whereas that of Lqun Q was suorted by the Hong Kong Research Grant Councl REFERENCES Chang KC, Pearson K, Zhang T Perron Frobenus theorem for nonnegatve tensors Communcatons n Mathematcal Scences 28; 6: Chang KC, Pearson K, Zhang T Prmtvty the convergence of the NZQ method, and the largest egenvalue for nonnegatve tensors SIAM Journal on Matrx Analyss and Alcatons 2; 32: Lu Y, Zhou G, Ibrahm NF An always convergent algorthm for the largest egenvalue of an rreducble nonnegatve tensor Journal of Comutatonal and Aled Mathematcs 2; 235: Ng M, Q L, Zhou G Fndng the largest egenvalue of a nonnegatve tensor SIAM Journal on Matrx Analyss and Alcatons 29; 3: Pearson K Essentally ostve tensors Internatonal Journal of Algebra 2; 4: Yang YN, Yang QZ Further results for Perron Frobenus theorem for nonnegatve tensors SIAM Journal on Matrx Analyss and Alcatons 2; 3: Fredland S, Gaubert S, Han L Perron Frobenus theorem for nonnegatve multlnear forms and extensons To aear n: Lnear Algebra and Its Alcatons 8 Gaubert S, Gunawardena J The Perron Frobenus theorem for homogeneous, monotone functons Transactons of the Amercan Mathematcal Socety 24; 356: Q L Egenvalues of a real suersymmetrc tensor Journal of Symbolc Comutaton 25; 4: Lm LH Sngular values and egenvalues of tensors: a varatonal aroach In Proceedngs of the IEEE Internatonal Worksho on Comutatonal Advances n Mult-Sensor Addatve Processng (CAMSAP 5), Vol IEEE Comuter Socety Press: Pscataway, NJ, 25; Lm LH Multlnear agerank: measurng hgher order connectvty n lnked objects, July 25 The Internet: Today and Tomorrow 2 Bulò SR, Pelllo M New bounds on the clque number of grahs based on sectral hyergrah theory In Learnng and Intellgent Otmzaton, Stützle T (ed) Srnger Verlag: Berln, 29; L W, Ng M Exstence and unqueness of statonary robablty vector of a transton robablty tensor March 2 eartment of Mathematcs, The Hong Kong Batst Unversty 4 Collatz L Enschlessungssatz für de charakterstschen Zahlen von Matrzen Mathematk Zetschrft 942; 48: Nussbaum R Convexty and log convexty for the sectral radus Lnear Algebra and Its Alcatons 986; 73: Hall C, Porschng T Comutng the maxmal egenvalue and egenvector of a ostve matrx SIAM Journal on Numercal Analyss 968; 5: Hall C, Porschng T Comutng the maxmal egenvalue and egenvector of a nonnegatve rreducble matrx SIAM Journal on Numercal Analyss 968; 5: Coyrght 2 John Wley & Sons, Ltd Numer Lnear Algebra Al 22; 9:83 84 OI: 2/nla