Analysis of multiview legislative networks with structured matrix factorization: Does Twitter influence translate to the real world?

The Annals of Applie d Statistics 2015, V ol. 9, No. 4, 1950–197 2 DOI: 10.1214 /15-A OAS858 c  Institute of Mathematical Statistics , 2 015 ANAL YSIS OF MUL TIVIEW LEGISLA TIVE NETW ORKS WITH STR UC TURED MA TRIX F A C TORIZA TION: D OES TWITTER INFLUENCE TRANSLA TE TO THE REAL W ORLD? By Sha wn Mankad and Geor ge Micha ilidis Cornel l University and University of Michigan The rise of so cial media p latforms has fundamentally altered the public discourse by pro vidin g easy to use and u b iquitous forums for the exchange of ideas and opinions. Elected officials often use such platforms for communication with th e broader public to disseminate information and engage with their constitu en cies and other p u blic of- ficials. In this wo rk, we inv estigate whether Twitter conversa tions b e- tw een legislators reveal their real-worl d p osition and influ ence by an- alyzing multiple Twitter net works that feature different typ es of link relations b etw een the Members of Parlia ment (MPs) in the United Kingdom an d an identical d ata set for p oliticians within Ireland. W e develop and app ly a matrix factorizatio n technique that allows th e analyst to emphasize nodes with con textual local netw ork structures by sp ecifying net work statistics that gu id e the factorization solution. Levera ging only link relation data, w e find th at important p oliticians in Twitter netw ork s are asso ciated with real-world leadership p osi- tions, and that rankings from t he prop osed met h od are correlated with the n umber of future media headlines. 1. In tro d uction. There is a gro wing literature that attempts to under- stand and exploit so cial n et wo r king platforms for resource optimization and mark eting, as it is a ma jor in terest for p riv ate en terpr ises and p olitical cam- paigns attempting to propagate particular opinions or pro ducts [ NYTimes ( 2011 , 2012 , 2013 )]. An imp ortan t pr ob lem is the identi fication of influential individuals that facilitate comm un ication o ver the n et wo r k. In this p ap er, w e deve lop a mo deling approac h that captures influence from m u ltiple net- w orks that feature d ifferen t link r elations b et wee n the same set of no d es (e.g., Twitter accoun ts). Suc h multiview data are increasingly common due to the complex structur e of man y net working platforms. S p ecifically , w e analyze three d ifferen t t y p es of n et works that are commonly deriv ed from Twitter data, eac h comp osed of either w eighte d or binary links. Received Octob er 2014; revised July 2015. Key wor ds and phr ases. Matrix factorizatio n, net w ork s, influ ence, Twitter. This is an e lectronic r eprint of the original ar ticle published by the Institute o f Mathema tica l Statistics in The Annals of Applie d Statistics , 2015, V ol. 9, No. 4, 1950–1 972 . This reprint differs from the orig inal in pagination and typog raphic detail. 1 2 S. MANKAD AND G. MICHAILIDIS Twitter is a p opular platform with o ve r 270 million activ e acco unt s eac h mon th as of September 2014 [T w itter ( 2014 )]. Twitter allo w s accoun ts to p ost short messages of 140 c h aracters or less, commonly referred to as “t we ets,” th at can b e read by an y visitor. A tw eet that is a cop y of an- other account’ s t weet is called a “ret weet. ” Within a t we et, an account can men tion another accoun t b y referrin g to their accoun t name with the @ sym- b ol as a prefix. Accoun ts also declare the other accoun ts th ey are intereste d in “follo wing,” whic h means the f ollo w er receiv es notification wh enev er a new t weet is p osted by the follo wed accoun t. T hese thr ee directed actions define p olitical Tw itter netw orks that we analyze in this work. The first netw ork is a ret weet netw ork, where links are d irected and w eighte d to d enote the log-num b er of retw eets from one accoun t to another o ve r an inte r v al of time. The second netw ork is also comp osed of directed and w eigh ted lin ks that denote the log-n umb er of men tions one accoun t giv es another. Th e third n et work is constru cted with dir ected b inary links that denote the follo wer and follo wed r elationships b et we en accoun ts. These three netw orks, eac h featuring 416 Mem b ers of Pa rliamen t (MPs) in the Un ited Kin gd om, are drawn in the top panel of Figure 1 , where accoun ts are registered to 172 Cons erv ative MPs, 185 Lab our, 43 Lib eral Demo cr ats, 5 MPs representing the S cottish National Pa rt y (SNP), and 11 MPs b elonging to other p arties. T here are 650 MPs formin g the House of Commons, the lo wer house in the bicameral legislat iv e b o d y for the United Kingdom. Eac h MP is demo cratically elected to r epresen t constituencies for fi v e yea r terms, though often elections are held more frequently w hen P arliament is dissolv ed. The second set of p olitical Twitter netw orks that we analyze are d ra wn in th e b ottom panel of Figure 1 . E ac h net work is comp osed of 348 no des that represent the ac counts of Irish p oliticians and p olitical organizations at all lev els of go vernmen t, includin g the Pr esid en t of the Republic of Ireland, mem b ers of the lo cal and national go ve rnment, and elected r ep resen tativ es for the Europ ean Union. The ra w data for b oth data sets, collected and p ro cessed by Greene and Cunnin gham ( 2013 ), consists of app r o ximately 500,000 tw eets and 40,000 follo w er links f rom late 2012. An empirical pattern observed in these data and also in previous studies [Hub erman, Romero and W u ( 2008 )] is that the follo w er net work is v ery d ense in con trast to the ret wee t and men tions net works. Almost all p oliticians inte ract v ia ret weet ing or men tioning with a smaller num b er of other account s, r elativ e to their follo we r declarations. Moreo v er, users with man y follo wers p ost up dates less ofte n than those with few er follo wers [Hub er m an, Romero and W u ( 2008 )]. Suc h empirical find- ings su ggest that not all links are created equally , and u s ually the follo w er net work is discarded b ecause it d o es not accurately capture patterns of con- v ersation. How ev er, eac h netw ork, including the follo we r netw ork, con tains ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 3 (a) Retw eet netw ork (b) Men tions netw ork (c) F ollo ws netw ork Fig. 1. The top p anel shows networks of UK Memb ers of Parliament and the b ottom p anel sh ows net works of Irish p olitici ans and p oli ti c al or ganizations. No de c ol or and vert ex shap es denote p arty affiliation. The aver age de gr e e f or the UK Re twe et, Mentions and F ol lows network is 9.13, 25.51 and 65. 25, r esp e ctively. The aver age de gr e e for the Irish R etwe et, Mentions and F ol lows network shown in the b ottom r ow i s 5.81, 15.28 and 48.44, r esp e ctively. meaningful inf orm ation, esp ecially since we only consider th e p opulation of p oliticians in a sp ecific legislativ e b o dy ins tead of a broad set of users or ev en the en tire Twitter userbase. Previous researc h has f ound that Twitter and other so cial net working platforms h elp facilitate communicati on b et wee n p oliticians, go ve rnment agencies and the broader pu blic. Golb ec k, Grimes an d Rogers ( 2010 ) fi n d b y text mining t w eets that mem b ers of th e United States Congress emplo y Twitter for primarily t w o purp oses: information dissemination and self pro- motion. T umasjan et al. ( 2010 ) fin d that the n um b er of tw eets from th e general public men tioning a p olitica l party or p olitician is a v alid indicator of p olitical sentimen t and a go o d predictor of federal election r esu lts in Ger- man y . More recent ly , similar results ha ve b een foun d for f ederal electi ons in Australia and the U.S. House of Representat iv es [Unank ard et al. ( 2014 ), McKelv ey , DiGrazia and Ro jas ( 2014 )]. In con trast to these previous works, w e rely only on the link relations, so-called “meta-data,” among p oliticians to measure influence and iden tify conv ersation flo ws with n et wo r k analysis. 4 S. MANKAD AND G. MICHAILIDIS Approac h es that u tilize con tent analysis can face significant c hallenges as- so ciated w ith text and image analysis (accoun ts can p ost a p hoto within a t wee t), suc h as language differences, tone and sentimen t characte rization, and so on. There has b een extensive work on r anking no des on a netw ork by th eir imp ortance primarily motiv ated by searc h on the W orld Wide W eb. W e fin d our prop osed m etho d co mpares fa v orably for ranking p oliticians aga inst t wo seminal works called P ageRank [Pag e et al. ( 1999 )] and HIT S [Hyp erlink- Induced T opic Searc h; Klein b erg ( 1999 )]. Th e id ea b eh in d PageRa n k is to use as a measure of imp ortance an estimate of the probabilit y of reac hing a giv en no d e by randomly follo win g edges. HITS utilizes th e so-called author- it y and hub s cores, whic h are compu ted by the leading eigen vecto r of A T A and AA T , r esp ectiv ely , where A is an adjacency matrix. Our m ain goal of iden tifying influent ial p oliticians is also closely r elated to role identi fication, wh ich aims to assign roles based on local connectivit y patterns. T ypically , role analysis metho ds rely on analyzing ego net w ork s (the u nion of a n o de and its neigh b ors), net w ork statistics or graph-coloring tec hniques [Salter-T o wns hend and Murphy ( 2015 )]. Also note that w hile there ha ve b een many recen t adv ances in communit y detection, includin g the sto c hastic blo c k mo d el, laten t p osition cluster mo dels and others [see Fien b erg ( 2012 ), Salter-T ownshend et al. ( 2012 ) for survey articles], the task in this article is differen t from typical communit y detection, wh ic h aims to extract groups of no d es that feature relativ ely dense within-group connectivit y and sparser b et ween-group connectivit y . Th at said, comm unity detection could help guide a searc h for influ en tial p oliticians. F or in stance, an analyst ma y examine eac h netw ork separately by firs t d isco ve r ing com- m u nities, if unkno wn , then s earc hing for in teresting net w ork stat istic profiles within eac h group . There are in principle man y w ays to com bine comm u nit y detection with n et wo rk statistics for the iden tifi cation of influ en tial no des, (e.g., p oliticians), bu t it r emains unclear whic h is the preferred metho d. In this pap er, w e integrat e b oth steps together to address this issue. The pro- p osed factorization mo del is also able to emphasize no des with int eresting path-related prop erties b y in corp orating no de-lev el statistics that capture these nonlinear relationships, th u s leading to more inte rpretable measures of in fl uence and su bstructure. The main idea is to guide the mappin g of the multiview net works to lo we r-dimensional spaces using structured matrix factorizatio n . Nonnegativ- it y constraint s are also imp osed on the lo w er-d imensional spaces to impro ve data representa tion and structural disco very . Suc h constrain ts ha ve b een p opularized with the n onnegativ e matrix f actorization (NMF) and S emi- NMF, where one or all matrix factors are comp osed of on ly n onnegativ e en tries and ha ve b een shown to b e adv antag eous for data representa tion [Lee and Seu n g ( 1999 ), Ding, Li and Jord an ( 2010 )]. As v alidation, we find ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 5 that imp ortan t p oliticians id en tified using our m o deling appr oac h are as- so ciated with real-w orld leadership p ositions, and that r ankings fr om the prop osed metho d are significant ly correlated with f uture media h eadlines. The consisten t fin dings b et wee n b oth data sets suggest the mo del can b e a relativ ely straigh tforward tec h nique for identifying influ en tial individuals with p olitical Twitter net works from other count ries th at feature different go v ernment structures, and that it can complemen t the p oten tially more in volv ed con tent analysis for related tasks. The n ext sectio n introd u ces the matrix factorization m o del, follo w ed by estimation details in Section 3 . Section 4 summarizes and compares r esults of th e prop osed mo del against alternativ e metho dologies with UK MPs and Irish p oliticians. This article closes with a brief discussion in Section 5 . 2. Structured semi-NMF for influen ce disco very . T h e use of lo w-r ank appro ximations to netw ork r elated matrices follo ws a long line of p r evious w ork. In classical sp ectral la yo ut, the co ordinates of eac h no de are giv en by the Singular V alue Decomp osition (SVD) of th e Laplacian matrix [K oren ( 2005 ), Brandes, Fleisc her and Pupp e ( 2006 )]. Recen tly , there has b een ex- tensiv e in terest in sp ectral clustering [Rohe and Y u ( 2012 ), Rohe, Chatterjee and Y u ( 2011 )], whic h disco vers communit y stru ctur e in th e eigen ve ctors of the Lap lacian matrix. Lo w-rank appr o ximations satisfying differen t constrain ts other than or- thonormalit y are also p opular. F or instance, NMF has b een prop osed for o ve rlapping communit y d etectio n on static [Psorakis et al. ( 2011 ), W ang et al. ( 2011 )] and dyn amic [Lin et al. ( 2008 )] net works. When o v erlaps among communities exist, an adv an tage of NMF o ver sp ectral clustering is that NMF can s till fi nd basis vec tors for eac h communit y , wh ile orthogo- nalit y of S VD mak es it un lik ely that the singular vec tors will corresp ond to eac h of the communities [Xu, Liu and Gong ( 2003 )]. The b asic framew ork for NMF in net wo rk analysis is A ≈ U V T , wh ere A is an adjacency matrix and U, V ∈ R n × K ≥ 0 . W ritten in elemen t form, A ij ≈ U i 1 V j 1 + · · · + U iK V j K , one can easily see that eac h edge of the giv en net work is app ro ximated with a non n egativ e su m. Consequentl y , eac h term in the sum , U ik V j k , represents the con tr ib ution of the k th laten t stru cture (often capturin g communit y structure esp ecially when decomp osing sparse adjacency matrices [Mank ad and Mic hailidis ( 2013b )]) to the edge f rom i to j . Ed ge decomp ositions can b e aggreg ated by no de or one can u se the rows of V to directly determine no de communit y mem b ership. T he factors are foun d by minimizing min U ≥ 0 ,V ≥ 0 k A − U V T k 2 F , 6 S. MANKAD AND G. MICHAILIDIS where k · k F denotes the F r ob enius norm . T h e optimizatio n ca n b e p erformed using gradient-desce n t algorithms for p enalized optimization. Give n that the p rop osed mod el in this article u tilizes n onnegativit y , we follo w a similar algorithmic app roac h to th e NMF literature. Enforcing nonn egativit y on a single matrix factor was fir s t prop osed in Ding, Li and Jord an ( 2010 ) w ith the so-called Semi-NMF to improv e inter- pretabilit y of the resultan t factorizatio n s w ith data of mixed signs. W e uti- lize the flexibilit y of Semi-NMF and extend it to the net w ork setting w ith a structured approac h that incorp orates graph geometry in to the factorizatio n through u s er-sp ecified matrices. In particular, we aim to utilize the man y no de-lev el statistics that h a ve b een p rop osed in the net work literature to guide the factorizat ion solution. Next we in tro duce the mo d el for single- view net works, then extend to multiview net works, follo w ed by estimation pro cedures in the next s ection. 2.1. Singleview networks. Let A denote the a djacency matrix from a sin- gle, giv en netw ork with n no des. W e start with the follo wing graph Struc- tured S emi-NMF mo del of Mank ad and Mic h ailidis ( 2013a ): min Λ , Θ ≥ 0 k A − S ΛΘ T k 2 F , (1) where S ∈ R n × D , Λ ∈ R D × K , and Θ ∈ R n × K ≥ 0 . Note that Θ is nonnegativ ely constrained, but Λ is not, w h ic h is why the m o del fits into the Semi-NMF framew ork . Each factor in the pro duct ΛΘ T is estimated from the data and pro v id es co efficient s for ea c h n o de that represen t the giv en adjacency matrix in terms of S . The S m atrix is comp osed of D n o de-lev el statistics that are sp ecified b y the analyst b efore p er f orming the factorization to emph asize no d es th at driv e influ ence. There is an extensiv e literature in n et work analysis p r o viding p oten tial no de-lev el statistics [Newman ( 2010 )]. I n our an alysis, the S matrix is constru cted using D = 4 n etw ork statistics and has form S i = [clustering co efficien t i , b et weenness i , closeness i , degree i ] , where i = 1 , . . . , n . The clustering c o efficient for a giv en no de quantifies ho w close its neigh b ors are to forming a complete graph [Newman ( 2010 )]. A higher clustering co efficien t will emphasize p oliticia ns that “create b uzz.” Betwe e nness [F reeman ( 1979 )] and closeness [Newman ( 2 010 )] rely on short- est path statistics and captur e imp ortan t links from hub no des. De gr e e , the n u m b er of connections a n o de has obtained, ensures that activ e p oliticians within communities are emphasized in the f actoriza tion. If there are no no d e-sp ecific v alues that are ob v ious to use for S , one can start with man y candidate no de-lev el statistics and searc h for s u bsets that fit the data we ll wh ile m aintaining int erpretabilit y . T his strategy will b e ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 7 discussed further b elow to also show robustness and assess the sp ecification of S in our application. I nstead of searc hing o ver no de-sp ecific statistics, one could also b e tempted to set S = I n × n to b e the identit y m atrix. In this case, the factorizati on is essen tially the standard Semi-NMF f actorization. Our results sh o w th at the Semi-NMF mo del p erforms similarly to classical imp ortance m easur es, lik e P ageRank and HITS, wh ich s hould b e p r eferred due to their more efficien t implemen tations. The pr op osed mo del implies certain connectivit y dynamics that can b e seen w hen equation ( 1 ) is written in element form A ij ≈ ( S Λ) i 1 Θ j 1 + · · · + ( S Λ) iK Θ j K , ( S Λ) ik = S i 1 Λ 1 k + · · · + S iD Λ D k . F or any nod e i , outgoing edges are co ntrolled b y its local top ological c harac- teristics, as measured in S , and ho w comm un ities load onto the statistics in S , giv en in the columns of Λ. When m u ltiplied toget her, S Λ form cen troids in a K -d imensional space that capture the outgoing no d e in fluence from eac h of the comm un ities. T he r eceiving n o de j in an edge is determined by the j th row of Θ , where larger v alues mean the n o de is more lik ely to hav e incoming connections and , hence, greater influ ence. Due to nonn egativit y and the fact that Θ mo dulates incoming connec- tions, we accomplish our ultimate go al of measur in g o v erall infl uence for the i th no de by taking its cumulativ e sum of imp ortance to eac h communit y I i = K X k =1 Θ ik . (2) As illustrated in the s upplementa l article [Mank ad and Mic hailidis ( 2015 )] on a to y example, the S matrix plays a piv otal role in the factorizatio n , and causes I to b e an effectiv e imp ortance measure even with its relativ ely simple definition. Next w e prop ose an extension of th is mod el to the m ultiview setting fou n d in p olitical Twitter net works. 2.2. Multiview networks. Let A m denote the adjacency m atrix from the corresp onding Twitter net w ork, wh ere m = { ret weet , men tions , follo ws } . W e extend the singleview mo del w ith min Λ m , Θ ≥ 0 ,V m ≥ 0 X m k A m − S m Λ m (Θ + V m ) T k 2 F , (3) where S m ∈ R n × D , Λ m ∈ R D × K , and Θ , V m ∈ R n × K ≥ 0 . Θ is common to all m net works to capture g eneral structure and mak es the ob jectiv e fu nction n on- separable, whereas V m rev eals n et wo rk-sp ecific structur e and also implicitly w eights eac h net work according to its imp ortance in the f actoriza tion. 8 S. MANKAD AND G. MICHAILIDIS The S m matrices are defin ed similarly to the singleview case, usin g no de- lev el net wo rk statistics. W e define S m using the same four net work statisti cs for eac h netw ork view. W eigh ted v ersions of the clustering co efficient and de- gree are u tilized for the Ret w eet an d Men tion net wo rks in order to tak e into accoun t the frequency of in teraction b et wee n p oliticians, since the fr equency should help measure the strength of a relatio n ship [Barrat et al. ( 2004 )]. F or instance, a w eigh ted n et wo rk statistic will distinguish b etw een a p olitician that is r et we eted by the same account h u ndreds of times v ers u s retw eeted once. The mo d el do es allo w for d ifferent statistics to b e defi n ed w ith eac h net work view, whic h ma y b e adv ant ageous in other con texts. The fi nal imp ortance measure I can also b e calculated similarly u s ing equation ( 2 ). Since Θ is common to all n et wo rks, the imp ortance measure is a result of integ r ating multiple n et wo rk views in addition to structur ed disco very . 3. Algorithms. The estimation algorithm w e present is an iterativ e one that cycles b etw een optimizing with resp ect to Θ , V m and Λ m with the f ol- lo wing u p dates: Θ = X m A T m S m Λ m (Λ T m S T m S m Λ m ) − 1 , V m = A T m S m Λ m (Λ T m S T m S m Λ m ) − 1 , Λ m = ( S T m S m ) − 1 S T m A m (Θ + V m )((Θ + V m ) T (Θ + V m )) − 1 . The u p dates are based on alternating least squares (ALS) and deriv ed through stand ard argumen ts [K ro onen b erg and de L eeuw ( 1980 )], wh ic h are sh o wn in the supplemental article [Mank ad and Michail idis ( 2015 )]. T ec hnically , b oth Θ and V m require solving nonn egativ ely constrained least squares p roblems, w hic h result in high iteration costs. S o, instead of exactly solving the constrained least squares p r oblem, we follo w a heuristic that solv es for an un constrained solution, then sets any entry less than a u ser- sp ecified constan t to that constan t. Pro jecting to a small constan t instead of zero follo w s the discussion in Gillis and Glineur ( 2008 ) and Kata yama, T ak ahashi and T ak euc h i ( 2013 ) to o v ercome n umerical instabilities th at oc- cur wh en to o man y elemen ts are exactly zero. Theoretical prop erties are difficult to obtain due to the p ro jection step. Y et th is appr oximati on is computationally efficien t, easy to implemen t, and has b een shown to ac hieve high quality s olutions [Berry et al. ( 2007 )]. The algorithm easily scales to net works w ith tens of thousands of no des. F or ev en larger netw orks on the order of millions of no d es, lo w-rank factoriza- tions should b e found usin g recent algorithmic adv ances that exploit parallel computing architec ture [Gem u lla et al. ( 2011 ), Rec h t and R ´ e ( 2013 )]. F or ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 9 our data, we find that the alternativ e least squares algorithm is s traigh tfor- w ard to implemen t and able to reco v er meaningful factorizations in a timely fashion. In the supplementa l article [Mank ad and Mic h ailidis ( 2015 )], w e also dis- cuss an alternativ e up dating ap p roac h for Θ and V m that is similar to the p opular “m u ltiplicativ e up dating” for NMF. While this approac h is also v ery easy to implement, w e fin d the ALS algorithm m ore numerically stable in higher dim en sions. 3.1. Initialization and c onver genc e criteria. An adv an tage of the ALS algorithm is that only Λ m needs to b e initializ ed if the order of the up dates is Θ , V m , Λ m . Moreo ver, recall that Λ m is unconstrained , thus bypassing the difficulties of in itializing the nonn egativ e factors whic h ha ve receiv ed extensiv e fo cus in th e NMF literature. W e find stable results b y initializing Λ m with norm ally distrib uted en tr ies having unit m ean and v ariance. Another imp ortant issue is sp ecifying the rank of the matrices Θ and V m . Ideally , the rank should b e equal to th e num b er of u nderlying communities and can b e ascertained by examining the accuracy of the reconstruction as a function of rank . In pr inciple, one could also apply cross-v alidation pro cedures for matrix factorization [Owen and P erry ( 2009 )], though this ma y b ecome cum b ersome w ith sparse or extremely large-sized net works. W e follo w a strategy similar to using a scree plot to choose the n umb er of comp onen ts to retain in Principal Comp onen t Analysis [Jolliffe ( 1986 )]. T o our knowledge, this rank selec tion approac h has not b een previously p ursued in the cont ext of NMF or Semi-NMF. S ho wn in Figure 2 , w e find that r anks greater than six (roughly the n um b er of und erlying p olitical parties) yield little marginal explanatory p o wer. Eac h subfigure is constructed by plotting the b est fitting facto rization o v er all p ossible net work sta tistic s u bsets of s ize t wo through four. The appropr iate rank of the m atrices Θ and V m is stable across the S m subsets, though there app ears to b e signifi can t impro vemen t when S m is defin ed with at least th r ee of the netw ork s tatistics. W e k eep all four n et wo rk statistics wh en defin ing S m for our analysis. Last, w e discuss conv ergence criteria used for the ALS algorithm. Let O ( i ) denote the v alue of the ob jectiv e fun ction at iteration i . Th en the algo rithm stops when |O ( i ) −O ( i − 1) | O ( i − 1) ≤ ε = 10 − 4 . W e find in all our in v estigations that the algorithm con verges w ithin 50 iterations. ε = 10 − 4 is also used for the pro jection th reshold. 4. Analysis of the p olitical multiview Twitter net works. 4.1. Do es Twitter influenc e tr anslate to the r e al world? Using the b est rank six factorization with S m defined with all four n et wo r k statistics, we 10 S. MANKAD AND G. MICHAILIDIS Fig. 2. The p er c entage of varianc e explaine d [ 100 ∗ (1 − P m k A m − ˆ A m k 2 F / k A m − ˆ µ m k 2 F ) , wher e ˆ µ i s a m atrix fil le d with the av er age value of A m ] for the Structur e d Semi-NMF with differ ent c onstruct ions of S m . Plotte d is the most ac cur ate mo del over thirty trials with r andom i nitializations for Λ m at e ach p ossible sp e cific ation. We use the b est r ank six mo del with f our network statistics c omp osing S m for the final analysis. rank MPs according to the estimated Θ and the imp ortance measure defined in equation ( 2 ). Figure 3 shows the imp ortance scores from the Structur ed S emi-NMF, Semi-NMF, P ageRank and HITS . Page Rank and HITS are computed using the ret w eet n et wo rk, whic h h as b een sh o wn to capture con v ersation dynam- ics b etter than the other net wo r k types [Ch a et al. ( 2010 )]. Not su rprisingly , the d ifferen t imp ortance measures are all p ositiv ely correlated. Accordingly , as sho w n in T able 1 , there is general agreemen t b et wee n Structured Semi-NMF, S emi-NMF and HITS in the top ten imp ortan t MPs. Man y of these MPs h eld leadership p ositions in the coalition or Op p osition cabinets. F or instance, Ed Milib and , leader of the Lab our Part y and of the Opp osition a t the time of w r iting, is prominen tly emphasized in all rankings. T om Wats on w as the Deputy Ch air of the Lab our Part y , and Chuka Umunna is the Shado w Secretary of State for Bu s iness, Innov ation and Skills. The exceptions are R achel R e eves , who b ecame the Sh ado w Secretary of S tate for W ork and Pensions for the Opp osition after the data w as collected, and ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 11 Fig. 3. I mp ortanc e sc or es b ase d on Structur e d Semi-NMF, Semi-NMF ( S m = I n × n ), PageR ank and HITS (Authority Sc or es). PageR ank and HITS ar e b oth c alculate d using the R etwe et network, while the ot her me asur es utilize al l thr e e networks . The r adius of the cir cle indic ates the c ount of futur e newsp ap er he ad lines as me asur e d with L exis–Nexis. The top t en MPs for the metho ds in e ach sc atterplot ar e lab ele d. David Camer on, who is Pri me Minister and in b oldfac e, was not in the top ten for any metho d. David Milib and , wh o held several imp ortan t p ositions in pr evious terms pr ior to data collection. Another commonalit y is that, w ith the exceptio n of P ageRank, every MP in the top ten is fr om the Lab our Pa rt y . Lab our MPs tend to b e estimated as most imp ortant, follo we d b y Conserv ativ e, and then Lib eral Demo crat MPs. The relativ e ranking among parties is consisten t with the data, wh ere Lab our MPs tend to b e the most activ e users in our data. Of the top fi ft y Twitter accounts in terms of n u m b er of retw eets or menti ons, only four are affiliated with another party—the Conser v ativ es. The Lib eral Demo crats are even less activ e, ranked in the hundreds in terms of num b er of ret w eets or m en tions. F or instance, Nic k Cle g g , leader of the Lib eral Democrats and Deput y Prime Minister at the time of wr iting, is t yp ically the top-ranked mem b er of his part y at fort y-nin e with Stru ctured Semi-NMF, fort y with P ageRank, and outside th e top h u ndred with b oth S emi-NMF and HIT S. Activit y in the data set is lik ely associated with longevit y on Twitter. F or instance, David Camer on , P r ime Minister and leader of the Conserv ativ es, is rank ed tw en ty-nine with Structur ed Semi-NMF, sixt y-eight with Semi- NMF, sixteen w ith Pag eRank, and tw o hundred and f orty-t w o with HITS. Cameron joined Twitter j ust as the data w as collected in Octob er 2012, and, th us, m ay ha v e artificially lo w lev els of acti vit y when compared against more recen t data. In spite of these c h allenges, Pa geRank and Structured 12 S. MANKAD AND G. MICHAILIDIS T able 1 MP r ankings and in p ar entheses the p arty and fr e quency that the M P app e ars in futur e he ad lines for Structur e d Semi -NMF, Semi-NMF ( S m = I n × n ), PageR ank and HITS (Auth ority Sc or es). L denotes L ab our, C denote s Conservat ive Rank Structured Semi-NMF Semi-NMF Pa geRank HITS 1 Ed Miliband (L, 2478) Ed Miliband (L, 2478) Ian Austin (L, 3) Mic h ael Dugher (L, 120) 2 Ed Balls (L, 580 ) Ed Balls (L, 580 ) William Hague (C, 771) Ed Miliband (L, 2478) 3 T om W atson (L, 253) Mic h ael Dugher (L, 120 ) Hugo Swire (C, 57) Ed Balls (L, 580) 4 Mic h ael Dugher (L, 120) T om W atson (L, 253) T om W atson (L, 253) Chuk a Umunna (L , 203) 5 Chuk a Umunna (L , 203) Chuk a Umunna (L, 203) Ed Balls (L, 580) Andy Burnham (L, 125) 6 Rachel R eeves (L, 54) Rachel Reeves (L, 54) Mic hael D ugher (L, 120) T om W atson (L, 253) 7 Stella Crea sy (L, 178) Chris Bryan t (L, 164) P at McF adden (L, 1) Rac hel Reeves (L, 54 ) 8 Chris Bryan t (L, 164) Stella Crea sy (L, 178) Ed Miliband (L, 2478) Chris Bry ant (L, 164) 9 T om Harris (L, 113) Luciana Berg er (L, 133) S tella Ceas y (L, 178) Diana Johnson (L, 105) 10 Da vid Miliband (L, 489) Andy Burnham (L, 125) Matthew Hanco ck (C, 32) T om Harris (L, 113) ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 13 Semi-NMF with us e of the S m matrix are able to b o ost these k ey MPs imp ortance, ev en though they in teract via Twitter with their MP colleagues relativ ely inf requen tly . W e h a ve s o far seen anecdotal evidence that man y MPs in leadership p o- sitions are emph asized b y the differen t tec hniques. Next, w e test in a regres- sion setting whether these different measures of Tw itter imp ortance p redict media cov erage, whic h is measured us in g Lexis–Nexis ( www.lexisnexis.com ) searc hes of the num b er of times an MP’s name app ears in headlines from Jan u ary 1, 2013, to Octob er 17, 2013. Th is interv al of time is strictly af- ter the Twitter data w as collected to a void end ogeneit y issues. Because the headline counts were ov erdisp ersed , w e use a quasi-P oisson regression. The mean and v ariance of the r egression h as form E (HeadlineCoun t i ) = exp( α + β I i + γ Cont rols i ) , (4) V ar (HeadlineCoun t i ) = ρ E (HeadlineCount i ) , (5) where ρ ≥ 1 is estimated fr om the data. HeadlineCount is the headline oc- currence frequency , I is d eriv ed using the different imp ortance measuremen t tec hniques, and Controls cont ain the v ariables Age, Gender, Constituency Size, Po litical Part y an d an ind icator v ariable denoting wh ether eac h MP represent s a constituency within the cit y of L on d on. Age is an imp ortant con trol v ariable, since w e exp ect younger MPs to b e more s avvy w ith s o cial media, whic h could affec t their headline co verage . S imilarly , we exp ect MPs with la rger co n stituencies, certain p olitical affiliations or London-b ased MPs to receiv e more media atten tion. Additional discuss ion in th e supplemental article [Mank ad and Mic h ai- lidis ( 2015 )] sho ws the Po isson d istributional assumption app ears more v alid when compared to other distribu tions f or o verdisp ersion, lik e negativ e bino- mial. Moreo ver, the qu asi-Poi sson results featured the smallest ro ot mean squared err or (RMSE) for all sp ecifications that we discuss n ext. In Figure 4 , w e examine the RMSE of the mo d el when using only cont rol v ariables, as w ell as con trol v ariables with ea c h influ ence measure separately . W e fi nd that the mo del using the p rop osed factorization features the lo w est RMSE, esp ecially after remo ving an outlier, Da vid Cameron, w ho receiv ed man y more future headlines than p redicted. As ment ioned ab ov e, Da vid Cameron joined Twitter just as the original data set wa s collect ed, p oten- tially creating an artificially lo w presence on Twitter. T able 1 in the supplemental article [M ank ad and Mic hailidis ( 2015 )] sho ws the fu ll results for the estimated m o del with Str u ctured Semi-NMF, w h ere the corresp onding co efficien t is statistically significan t and p ositiv e as ex- p ected. Sp ecifying S m leads to an imp ortance measure th at is asso ciated with future media headlines ev en when controlli ng for other influ ence mea- sures and demographic information, thus illustrating the imp ortance of guid- ing th e factorizati on solution. 14 S. MANKAD AND G. MICHAILIDIS Fig. 4. R o ot me an squar e d err ors f or the pr e dicte d numb er of he ad li nes using differ ent sp e cific ations of the r e gr ession mo del in e quations ( 4 ) and ( 5 ). “None” r ef ers to including only c ontr ol variables. “PageR ank” r efers to the c ontr ol variables plus the PageR ank i n- fluenc e me asur e, “HITS” r efers to the c ontr ol variables plus the HIT S influenc e me asur e, and so on. 4.2. Identifying imp ortant c onversation flows. Another adv an tage of the prop osed factorizatio n is that it can also b e us ed to extract p oten tially imp ortant con versation flows. W e construct su bgraphs b y k eeping no des in the top q th p ercen tile of P k (Θ + V m ) ik to reco v er structure sp ecific to eac h net work view. The Stru ctured Semi-NMF do es not incorp orate p art y affiliation for the factorizat ion. Y et it resu lts in more inte rpretable subgraph s than the al- ternativ e approac h in Figure 5 of lo oking at high degree no des within eac h part y . Sho w n in Figure 6 , there are denser within and b et wee n p art y connec- tions, and few er isolated no d es. Moreo v er, w ith th e exception of a handfu l of MPs, eac h no de can reac h ev ery other no de on the graphs. Thus, these net works h elp explain the infl uence r ankings f rom the pr evious section by iden tifyin g paths through whic h int eresting con tent flow ed. T racing the flow of con v ersations in th e 95 p ercen tile sub grap h s in Fig- ure 7 , w e see that the Lab our p oliticians tend to retw eet eac h other of- ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 15 (a) Ret w eet n etw ork (b) Men tions netw ork (c) F ollow s netw ork Fig. 5. Subnetworks of UK Memb ers of Parliament chosen by taking the highest de gr e e MPs in e ach p arty, with c olor and vertex shap es denoting p arty affiliation. MPs ar e dr awn in the same p ositi on as in Figur e 1 . ten. Man y of the Lab our MPs, including Stel la Cr e asy , Ed Milib and , Chuka Umunna , R achel R e eves , T om Watson and others, w er e universall y r ank ed as imp ortan t in the p revious section. E d Bal ls from Lab our in teracts d i- rectly with Gr e g Hands of the C onserv ativ e part y , who in tur n forms a m uch Fig. 6. Networks of UK Memb ers of Parliament, with c olor and vertex shap es denoting p arty affil iation. MPs in the top q th p er c entile of P k (Θ + V m ) ik ar e kept and dr awn in the same p osition as in Figur e 1 . 16 S. MANKAD AND G. MICHAILIDIS (a) Ret w eet n etw ork (b) Men tions n etw ork (c) F ollow s netw ork Fig. 7. Sub gr aphs c onstructe d for the UK MPs (top p anel) and Iri sh p olitic ans (b ottom p anel), whose no des ar e in t he top q = 95 p er c entile of P k (Θ + V m ) ik . Gr aphs ar e r e dr awn to optimize vertex lab els. smaller ret wee t clique with fello w Conserv ativ es Matthew Hanc o ck and Mike F abric ant . Since retw eeting can amount to an endorsement , w hile menti oning allo ws the author to con trol the con tent and sentimen t, there are a greater n umb er of cross-part y mentions edges. F or instance, David Camer on is mentio n ed often and follo w ed b y Lab our MPs, elev ating his imp ortance on those sp ecific net works, but is nev er ret weete d. This illustrates the v alue of utilizing all three typ es of net works for measuring imp ortance. 4.3. Analys is of Twitter networks fr om the Irish p olitic al spher e. W e pro- duce comparable, though less pronounced results with similar Twitter net- w ork d ata from the Irish p olitical scene fr om late 2012. W e organize the r aw data again pro vided in Greene and Cunningham ( 2013 ) in to the same three Twitter netw orks, eac h con taining 348 no d es that represen t the accoun ts of Irish p oliticians and p olitical organizations. The d ata con tains p oliticia ns from all lev els of go vernmen t, includin g th e Pr esiden t of the Republic of Ireland, mem b ers of the lo cal and n ational gov ernmen t, and elected repre- sen tativ es for the E u rop ean Union. ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 17 A ma jorit y of accoun ts b elong to member s of the Irish national p arlia- men t, which is also a b icameral legislativ e b o dy with elections held at least once ev ery five years us ing a system [Coakley and Gallag her ( 2005 )]. Th e lo we r house (D´ ail ´ Eireann) is the principal house in the Irish system an d con- tains 166 elected mem b ers, the senate (Seanad ´ Eireann) con tains a mixture of 60 app ointe d and elected members. Th er e are multiple p olitical p arties in the d ata: 33 Fianna F´ ail, 127 Fine Gael, 6 Green, 20 In dep end en t, 68 Lab our, 22 Sinn F´ ein and 8 Others. Appro ximately 60 Twitter accoun ts are registered to p olitical parties, for example, “Fine Gael O ffi cial,” “Lab our W omen,” etc. After sp ecifying S m as b efore and setting K = 7 (c h osen in a similar fashion), w e p lot th e imp ortance scores in Figure 8 and list the top ten ac- coun ts in T able 2 from the S tructured S emi-NMF, Semi-NMF, P ageRank and HITS. In contrast to the British MP dynamics, p olitical organizations seem to play a m u c h more imp ortan t role in online conv ersations within the Ir ish p olitical sp here, as there is b road agreemen t among th e different imp ortance measures that part y organizatio n accoun ts are h ighly rank ed, suc h as Fine Gael O ffic i al , Y oung Fine Gael , and The L ab our Party . S ome p oliticians are also universall y rank ed as imp ortan t. Michael D Higgi ns , the President at the time of wr iting, is ranke d elev en th u n der the S tructured Semi-NMF, thirteen th u nder P ageRank and in the top ten for all other m eth- Fig. 8. I mp ortanc e sc or es b ase d on Structur e d Semi-NMF, Semi-NMF ( S m = I n × n ), PageR ank and HITS (Authority Sc or es) ar e b oth c alculate d using the Re twe et network. The r adius of the cir cle indi c ates c ount of futur e newsp ap er he ad lines as m e asur e d with L exis–Nexis. The top ten Irish p olitici ans for the metho ds in e ach sc atterplot ar e l ab ele d. Michael Higgins, Pr esident, is b oldfac e d. 18 S. MANKAD AND G. MICHAILIDIS T able 2 Irish p olitician r ankings and in p ar entheses the p arty and fr e quency that the p olitician app e ars in futur e he ad l ines f or Structur e d Semi-NMF, Semi -NMF ( S m = I n × n ), PageR ank and HITS (Auth ority Sc or es). L denotes L ab our, FG denotes Fi ne Gael, I nd denotes Indep endent and SF denote s Sinn F´ ein. Ther e ar e no p ar enthetic al he ad l i ne c ounts or p arty names for p olitic al or ganizations Rank Structured Semi-NMF Semi-NMF Pa geRank HITS 1 Fine Gael Official The Labour Part y Fine Gael Official Fine Ga el Official 2 Y oung Fine Gael Ao dh´ an ´ O R ´ ıord´ ain (L, 1) Fianna F´ ail Y oung Fine Gael 3 Enda Kenny (FG, 166) Fine Gael Official The Lab our Pa rty The Labour Part y 4 Lucinda Creigh ton (F G, 20) Jillian v an T urnhout (Ind, 0) Sinn F ´ ein Simon Harris (FG , 4) 5 Jillian v an T urnhout (Ind, 0) Mic h ael D Higgins (L, 25) Jillia n va n T urnhout (Ind, 0) Ao d h´ an ´ O R ´ ıord´ ain (L, 1) 6 The Labour Part y Ciar a Conw ay (L, 0) Ao d h´ an ´ O R ´ ıord´ ain (L, 1) Jillia n va n T urnhout (Ind, 0) 7 Jerry Buttimer (FG, 2) Simon Harris (FG , 4) Y oung Fine Gael F rances Fitzgerald (FG, 7) 8 Simon Harris (F G, 4) John Gilro y (L, 3) Dermot Lo oney (Ind , 0) Mic h ael D Higgins (L, 25) 9 Simon Co veney (FG, 10) Dermot Looney (Ind, 0) Simon H arris (FG, 4) Jerry Buttimer (FG, 2) 10 P asc h al Donoho e (FG, 4) Jerry Buttimer (FG , 2) Matt Carth y (SF, 0) Dermot Looney (Ind, 0) ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 19 o ds. Jil lian v an T urnhout is an app ointe d mem b er of the Seanad ´ Eireann and is consistently ranke d highly by the differen t influence measur es. Lik e- wise, Jerry Buttimer is a mem b er of the D´ ail ´ Eireann and form er ly of the Seanad ´ Eireann, and Simon H arris was elected to the D´ ail ´ Eireann in 2011 as its yo u ngest memb er. There are k ey differences, ho wev er, among th e v arious imp ortance mea- sures. De rmot L o oney is ranke d in the top ten for Semi-NMF, P ageRank a nd HITS, b ut nineteenth und er Structured Semi-NMF. He seems to b e ranked higher than one ma y exp ect, since Lo oney was part of a lo cal gov er n men t and serv ed as ma y or of the South Dublin Count y Council. Lucinda Cr eighton is rank ed fourth for the Str u ctured S emi-NMF, but is not in the top ten for other imp ortance measures. A t the time of data co llection, Creigh ton serv ed as Minister for E urop ean Affairs rep resen ting I reland in negotiations on Ireland’s EU/IMF bailout and the hosting of Ireland’s presidency of the Eu- rop ean Union. W e also see that Enda K enny , an Irish Fine Gael p olitician who has b een the T aoiseac h (pr im e minister) since Marc h 2011, is ranked in the top ten only under th e Structured Semi-NMF approac h. He is r ank ed fortieth with Semi-NMF, thirty-fourth with P ageRank and s ev ent y-second with HITS . The larger differences b et ween the Stru ctured Semi-NMF and other im- p ortance measures when compared to the UK MP resu lts can b e explained b y the s parser in put net w orks , as sh o wn in Figure 9 , whic h increase the effect of the S m matrices. Figure 7 sho w s the conv ersation d ynamics that help explain w hy certain accoun ts are r ank ed h ighly with the stru ctured approac h. F or instance, we see that Jil lian van T urnhout , an In dep end ent, tends to b e ret w eeted or menti oned b y Fianna F´ ail organizations in add ition to Fine Gael, Lab our and other In dep end en t p oliticians. Accoun ts w ithin the Lab our part y also form their o w n cli que, cen tered around Michael D Higgins and the official Lab our party accoun t. Finally , w e test whether these d ifferen t m easur es of T witter imp ortance predict media co verag e with the s ame quasi-Poi sson mo del as in equa- tions ( 4 ) and ( 5 ). Headline o ccurren ce f requency from Jan u ary 1, 2013, to Octob er 17, 2013, is again measured using Lexis–Nexis searc h es, I is deriv ed using th e differen t imp ortance measurement tec hniqu es, and Con - trols conta in s the v ariables Age, Gend er, P oliticia n Type (local, presiden tial, D´ ail ´ Eireann, Seanad ´ Eireann, Eur op ean Union), Constituency and P olit- ical P art y . Since the d ata conta in s p oliticians in lo cal go vernmen t, wh ere, for example, exact constituency size is not easily defin ed for council mem- b ers, w e include a fixed effect for every unique elec toral district or area. The 134 uniqu e areas are identified usin g a num b er of online sour ces, includ - ing official p art y and candidate website s, newsp ap er articles and election results p osted on h ttps ://ele ctionsireland.org/ . P art y organizatio n account s are remov ed when estimating the r egression mo d el. 20 S. MANKAD AND G. MICHAILIDIS Fig. 9. Networks of Irish p olitici ans, with c olor and vertex shap es denoting p arty affil- iation. Politicians in the top q th p er c entile of P k (Θ + V m ) ik ar e kept and dr awn i n the same p osition as in Fi gur e 1 . T able 2 in the supplemental article [M ank ad and Mic hailidis ( 2015 )] sho ws the Structured Semi-NMF measure is again a s tatistically significan t predic- tor for headline co verage rate, after con tr olling for all other v ariables, and Figure 4 shows again that the prop osed appr oac h results in an influ ence measure that improv es forecasting accuracy relativ e to alternativ e m o del sp ecifications. 5. Conclusion. The Structured Semi-NMF p erforms b est in b oth data sets, th ough the improv emen t wa s only sligh t in the Irish cont ext. The o v erall results were drive n by utilizing all three t yp es of n et wo rks for measurin g imp ortance and sp ecifying the S m matrices to b o ost imp ortant p oliticians with particular t yp es of link ages. One p otenti al issue w ith the analysis is that Lexis–Nexis co v erage of non- US media and, in particular, the Irish media app ears to b e imp erfect. How- ev er, even with p o or cov erage, as long as it is representa tive of the ov erall media landscap e, then the rep orted results will b e meaningful. W e are also una w are of other to ols that can b e us ed for suc h searc hes. An other issue is that p oliticians m a y app ear in headlines that reference their office, f or example, “the president .” A more comprehensive newsp ap er h eadline coun t is difficult to ascertain, bu t could in f uture w ork provide further v alidation of the results presente d here. ANAL YZING MUL TIVIEW NETWORKS WITH MA TRIX F ACT ORIZA TION 21 Giv en that b oth data sets are exclusiv ely link meta-data, our find ings supp ort the notion that the significant c hallenges asso ciated with con ten t analysis can often b e complimen ted or a voided with netw ork analysis to ols for tasks lik e identifying individuals influentia l w ith in so cial netw orking plat- forms. W e b eliev e this is partly explained b y the restrictio n of the p opu lation to p oliticia ns and cl osely related orga nizations, which ensures to some extent that the unob s erv ed conte n t is b oth homogeneous and relev an t. A related pr oblem of identifying emergence of key individ u als, comm u- nities or trend s b ased on net work data requires data collected ov er time. Smo othing strategies, su c h as in Mank ad and Michai lidis ( 2013b ), should b e useful to extend the giv en mo d el for n et wo rk time-series. W e b eliev e the prop osed model ca n b e useful for applicatio n s in marke tin g and e- commerce, where data is collected on ecosystems that are close to a steady state. Oth - erwise, as w e sa w with Da vid Cameron, the mo del can misc haracterize the imp ortance of k ey individuals. S p ecific q u estions relati ng to path prop erties, suc h as information diffusion [Romero, Meeder and Kleinberg ( 2011 )] or the spread of epidemics [Chew and Eysenbac h ( 2010 )], like ly require additional metho ds an d tec h n iques sp ecific to those su btopics. There also has b een recen t w ork on a related problem w h en no de f ea- tures are measured along with net w ork data [ F osdick and Hoff ( 2013 , 2014 ), Y ang, McAuley and Lesk ov ec ( 2013 )]. F or instance, one ma y ha v e access to demographic information or topics and themes of eac h accoun t’s tw eets as in Greene, O’Callaghan and Cun ningham ( 2012 ). While it app ears the pro- p osed mod el could b e us efu l in this setting, using external cov ariates on th e no des to construct S m lik ely raises additional issues th at requir e care, such as v ariables b eing a v ailable for some, but not all no d es. In this work, the no de-lev el statistics are “in tern ally” calculated directly from th e n et wo rk and, thus, will alwa ys co ve r the full n et wo rk. A strength of the Str u ctured Semi-NMF mo d el is th at it encompasses differen t t yp es of links (w eigh ted and binary ), in tegrates information f rom m u ltiple net works and allo ws the analyst to utilize cont extual kn owledge ab out the give n netw ork ed system. The method dep ends up on the analyst c ho osing appropriate, con text-sp ecific no de-leve l statistics. As s u c h, the al- ternating least squares algorithm pr o vides opp ortun ities for additional regu- larizatio n in situations where the S m matrices are high dimensional or when there are no no de-sp ecific v alues that are ob vious to use. 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