with the downscaling of nanoscale electronic devices , the non - equilibrium green s function ( negf ) method has become the most effective and accurate modeling approach in predicting the electronic transport performance .the self - consistent solution of negf and poisson s equations is capable of accurately modeling coherent as well as decoherent transport by integrating atomic / molecular orbital level physics .an aspect of negf , that currently prevents a broader use , is the large computational cost associated with the method .researchers and analysts have used several approximations to alleviate this cost .examples include the mode - space approximation , which couples 1d negf transport simulation to transverse states from solving 2d schrdinger equation on the cross - section , and the restriction to levels of lower accuracy ( such as tight - binding or effective mass levels ) .although these approximations of negf have successfully predicted the transport characteristics of nanoscale devices , these alternatives remain unable to capture atomic - scale inhomogeneities , such as surface roughness , unintentional doping , and trapped charges ( see , for example , for an illustration of the resulting inaccuracy ) .the modeling of such inhomogeneities requires a full 3d real - space negf simulation .recent studies have performed 3d negf simulations to handle these inhomogeneities but with `` coarse discretization '' ( _ i.e. _ small number of atomic orbitals or small number of grid points ) to control the computational cost . to enhance the predictability of these 3d negf simulations , finer discretizations have to be considered .therefore the computational cost of negf needs to be addressed .the goal of this paper is to present a numerical method that significantly reduces the cost of 3d negf simulations .  during a negf simulation ,the main computational bottleneck is the repeated evaluations of the retarded green s function , , and the lesser green s function , , over a wide range of energy values .physical quantities , such as density of states , carrier density , and transmission coefficients , are obtained from the evaluations of these green s functions .after discretization of the nanoscale device , these repeated evaluations amount to computing entries in the matrices and that approximate , respectively , the retarded and lesser green s functions . recall that the matrix is defined as where is the system hamiltonian matrix and is the identity matrix .the self - energy matrix consists of two parts : the contact self - energy and a self - energy to represent scattering mechanisms .the matrix is defined as where corresponds to the lesser self - energy matrix . to compute diagonal and desired off - diagonal entries of and , the recursive green s function algorithm ( rgf ) has been the algorithm of choice for many years .recently , the hierarchical schur complement ( hsc ) method and the fast inverse using nested dissection ( find ) method have exhibited significant speed - ups over rgf to evaluate the diagonal entries of . both of these methods have a smaller operation count than the rgf method . our hsc - extension to compute diagonal and desired off - diagonal entries of  also demonstrated significant speed - ups over rgf simulations for 2d nanoscale devices . in this paper, the efficiency of the hsc - extension over the rgf method is studied for 3d nanoscale devices .the rest of the paper is organized as follows .section [ sec : algorithm ] gives a brief description of the rgf and hsc - extension approaches and discusses the asymptotic operation counts for both approaches when simulating 3d devices .section [ sec : results ] presents the runtime analysis for a cuboid structure with a regular mesh and three state - of - the - art nanoscale devices : a graphene - boron nitride device with a multilayer hexagonal mesh ; a silicon nanowire structure with 10 localized atomic orbitals ; and a dna molecule sequence with an unstructured atomic connectivity .section [ sec : conc ] provides a brief summary of the numerical experiments .this section briefly describes the rgf method and the hsc - extension approach . then the operation counts for both methods are discussed when simulating cuboid or brick - like nanoscale devices .the rgf algorithm evaluates the specific entries of the matrices and when the system matrix is structured as a block - tridiagonal matrix .the off - diagonal and diagonal blocks of are uniquely associated with layers ( of atoms or of grid points ) orthogonal to the transport direction , usually the -direction .the rgf algorithm is composed of two passes to compute diagonal entries of and two passes to compute diagonal and desired off - diagonal entries of .the two passes to compute entries of ( or of ) are interpreted as follows :  * the first pass marches one layer at a time from _ left to right _ along the -direction and ,recursively , _ folds _ the effect of left layers into the current layer ; * the second pass marches one layer at a time from _ right to left _ along the -direction and , recursively , _ extracts _ the diagonal blocks and the nearest neighbor off - diagonal blocks for the final result .mathematically , when computing diagonal entries of , the first pass amounts to transforming into a block diagonal matrix , where is a lower block - triangular matrix with identity blocks on the diagonal . recall that being a complex symmetric matrix , the block -factorization is a particular form of a block lu - factorization .the second pass in the evaluation of entries for is expressed by the formula ( see for further details ) .starting from the knowledge of the block on the diagonal of for the rightmost layer for the rightmost layer is the corresponding block in the matrix . ], the formula ( [ eqn : takahashi_gr ] ) provides , one layer at a time , the diagonal block and the nearest neighbor off - diagonal blocks when marching from _ right to left _ along the -direction .when computing , the first pass amounts to evaluating the matrix - matrix products , where the matrices and form the block -factorization of .the second pass in the evaluation of entries for is expressed by the formula ( see for further details ) .starting from the knowledge of the block on the diagonal of for the rightmost layer , the formula ( [ eqn : takahashi_gless ] ) provides , one layer at a time , the diagonal block and the nearest neighbor off - diagonal blocks when marching from _ right to left _ along the -direction .   the hsc method and its extension evaluate specific entries of the matrices and when the matrix is symmetric and sparse .in contrast to rgf , the matrix is not required to be block - tridiagonal .the first step of the hsc method and of its extension gathers the grid points ( or atoms ) into arbitrarily - shaped clusters .these clusters are then organized in a binary tree .the ordering of rows of according to this hierarchy of clusters arranges the non - entries of into a block arrow - shaped structure .such choice of ordering allows _ to fold _ and _ to extract _ in any physical direction when following the vertical hierarchy of the binary tree .[ fig : merge ] illustrates a partition for a cuboid device of size and the corresponding binary tree .three levels of separators are depicted in fig .[ fig : merge ] and the colors in the binary tree match the colors in the partition .note that the three separators are orthogonal to three distinct physical directions .the _ fold _ process represents the computation from the lowest level to the top - most level , while the _ extract _ process marches one level at a time from the top - most level to the lowest one .   to cubes of dimension .three levels of separators are colored by red , purple and green respectively .( b ) the multilevel binary tree corresponding to the cuboid decomposition .the three levels of separators are depicted with matching colors .the blue blocks denote the corresponding blue clusters . ]the mathematical interpretation of the two passes for remains unchanged with the stipulation that the rows of , , and are ordered according to the hierarchy of clusters in the binary tree .the first pass for is interpreted as the evaluation of the matrix - matrix products , and the second pass is expressed by the formula where the matrix rows remain ordered according to the hierarchy of clusters in the binary tree .next we provide two pseudo - codes describing , respectively , the hsc method and its extension . on the basis of the hierarchical structure of a binary tree with levels ,let denote the set of all cluster indices such that cluster is an ancestor of cluster .similarly , let denote the set of all cluster indices such that cluster is an descendant of cluster .algorithm [ alg : gr ] describes the computation of _ specific _ entries of ( hsc method ) .+  ' '' ''   + * for * to * do *  all the clusters on level * do * for all in  for all and in  for all and in  and for all in + + * for * to * do *  for all cluster indices in  for all cluster indices in  + computed entries in match corresponding entries in +  ' '' ''   + [ alg : gr ]  algorithm [ alg : gl ] describes the computation of _ specific _ entries of with the hsc - extension . for further details about the hsc method for and about its extension for ,the reader is referred to the references .+  ' '' ''   +  block diagonal + * for * to * do *  all the clusters on level * do * for all and in + + * for * to * do *  for all cluster indices in  for all cluster indices in  + computed entries in match corresponding entries in +  ' '' ''   + [ alg : gl ]  the key element controlling the efficiency of the hsc method and its extension is the definition of clusters in the binary tree . when the binary tree is replaced by a _ degenerate _( or pathological ) tree where each parent node has only one associated child node , the hsc method becomes equivalent to the rgf method ( see , for example , ) .for the hsc algorithm and its extension , the multilevel nested dissection defines the binary tree and the resulting clusters .the _ nested dissection _ approach divides the domain of grid points or atoms into two uncoupled sub - domains ( clusters of grid points / atoms ) with a separator , such that the two disconnected sub - domains are of roughly the same size and the separator is minimized .multilevel nested dissection repeats this division on each sub - domain , reducing the clusters down to a size small enough ( manageable for an inversion of the corresponding block matrix ) , or until the clusters can not be divided by nested dissection . in practice, the metis graph partition library provides the implementation of the multilevel nested dissection .this well - established partition library allows the hsc extension approach to handle any arbitrarily - discretized nanoscale electronic devices .next we discuss the operation counts for the rgf method and for the hsc - extension .consider a cuboid device , covered by a three - dimensional orthogonal mesh with , , and grid points per direction .the discretization of the hamiltonian is obtained via a 7-point stencil .the self - energy matrix is assumed to be represented via a similar 7-point stencil ( for example , with a crude diagonal approximation or with a pml - like approximation ) .the resulting matrix is of dimension , where the -direction is the transport direction .  the rgf approach groups the grid points into disjoint layers , each layer holding grid points .[ fig:3d_opcount_layers ] illustrates these layers for , , and .( the -direction is the transport direction . ) .the colored layers along the -direction show the layered - structure organization of grid points for the rgf approach . ]  by ordering the grid points one layer at a time , the matrix exhibits the block - tridiagonal structure , required by the rgf approach .the operation count for the rgf method on this cuboid device is .the hsc - extension employs a multilevel nested dissection to gather the grid points into a hierarchy of clusters .[ fig:3d_opcount_nd ] illustrates the separators obtained at each level and the eight subdomains .the resulting binary tree is also depicted in fig .[ fig:3d_opcount_nd ] with matching colors for the separators .the operation count for the hsc - extension is derived in the appendix . for the sake of conciseness, the final value is summarized in table [ tab:3d_opcount_hsc ] .  [cols="^,^,^",options="header " , ]   the four different partitionings used for these approaches are depicted on fig .[ fig : dna9 ] .the _ customized _ rgf method with two 2-pairs layers gathers , twice , 2-pairs into one layer .this particular choice of layers indicates the impact of two blocks of dimension on the overall cpu time , _i.e. _ an increase in cpu time .the _ customized _partition for the hsc - extension allows a degenerate sub - tree to avoid any 2-pairs cluster .this _ customized _partition makes the hsc - extension operate on blocks of dimension , at most , .this choice results in a lower cpu time , on par with the original rgf approach . a similar behavior was observed for other dna molecules , where metis gathers 2-pairs into one cluster .these numerical experiments suggest that the hsc - extension , combined with a multilevel nested dissection , is an efficient approach even for smaller but denser matrices .when the graph partitioning is allowed to insert degenerate sub - trees , the performance is comparable with that of the performance of rgf .in this work , we demonstrate the hsc - extension based negf solver as a working methodology for various 3d systems . the cost analysis for hsc - extensionis performed on a cuboid structure .hsc - extension exhibits operation count of when simulating cubic device with dimension , whereas a count is observed for rgf .we also illustrate various asymptotic costs of hsc - extension when the device has an elongated shape ( ) , when the device is flattened ( ) , and when a dense self - energy is used to model the open boundary conditions .the runtime performance of hsc - extension is further investigated for nano - electronic devices of practical interest : graphene - hbn - graphene multilayer heterostructure , silicon nanowire and dna molecule .these devices exhibit distinct atomistic sparsity , indicating different computational efficiency for hsc - extension .the numerical experiments suggest that the hsc - extension exhibits asymptotic runtimes and operation counts proportional to the runtime of the lu - factorization .for all the nano - electronic devices considered , the hsc - extension becomes faster than the rgf method as the device gets larger . a 1,000 speed - upis observed for a graphene - hbn - graphene multilayer device with 40,000 atoms .since the hsc - extension requires less operations than rgf , these speed - ups will increase as the device gets larger .a matlab version of hsc - extension algorithm is available at https://els.comotion.uw.edu/express_license_technologies/glessuw .the authors acknowledge the support by the national science foundation under grant eccs 1231927 .s. r. patil acknowledges the support by ugc - india grant no .f.5 - 50/2014(ic ) .m. p. anantram and j. qi acknowledge the support from the national science foundation under grant no .102781 for the dna part of the paper .to analyze the runtime complexity of hsc - extension , we consider a cuboid device with grid points per direction .first we discuss the case of a cubic mesh , _i.e. _ .the operation count for evaluating diagonal entries in was discussed by lin __ . according to their analysis , the operation count grows as . for the hsc - extension ,the complexity for evaluating diagonal entries of is identical to the complexity for ( as discussed in hetmaniuk _ for two - dimensional devices ) .the overall operation count will grow as .next we consider the case of an elongated device , where the numbers of grid points per direction satisfy .the multilevel nested - dissection will identify subdomains , each discretized with grid points .when evaluating diagonal entries in , the operation count for each cubic subdomains will grow as . the remaining operations will involve dense matrices for the separators of dimension .the algebraic operations for one separator will include dense matrix - matrix multiplications and dense matrix inversions , yielding an asymptotic cost .the number of separators is also .the overall operation count will grow as .finally we consider the case of a flattened device , where the number of grid points per direction satisfy . as discussed by lin _ , this configuration is similar to a two - dimensional problem with grid points .the prefactor will depend on .the hsc algorithm and our extension can proceed as if the device is two - dimensional by replacing scalar algebraic operations with block algebraic operations ( each block being of dimension ) .these block operations will cost .so the overall operation count will grow as .frisch , g.w .trucks , h.b .schlegel , g.e .scuseria , m.a .robb , j.r .cheeseman , g. scalmani , v. barone , b. mennucci , g.a .petersson , h. nakatsuji , m. caricato , x. li , h.p .hratchian , a.f .izmaylov , j. bloino , g. zheng , j.l .sonnenberg , m. hada , m. ehara , k. toyota , r. fukuda , j. hasegawa , m. ishida , t. nakajima , y. honda , o. kitao , h. nakai , t. vreven , j.a .montgomery , jr .peralta , f. ogliaro , m. bearpark , j.j .heyd , e. brothers , k.n .kudin , v.n .staroverov , r. kobayashi , j. normand , k. raghavachari , a. rendell , j.c .burant , s.s .iyengar , j. tomasi , m. cossi , n. rega , j.m .millam , m. klene , j.e .knox , j.b .cross , v. bakken , c. adamo , j. jaramillo , r. gomperts , r.e .stratmann , o. yazyev , a.j .austin , r. cammi , c. pomelli , j.w .ochterski , r.l .martin , k. morokuma , v.g .zakrzewski , g.a .voth , p. salvador , j.j .dannenberg , s. dapprich , a.d .daniels , o. farkas , j.b .foresman , j.v .ortiz , j. cioslowski , d.j . fox .gaussian 09 revision a.1 .gaussian inc .wallingford ct 2009

Abstract: the hierarchical schur complement method ( hsc ) , and the hsc - extension , have significantly accelerated the evaluation of the retarded green s function , particularly the lesser green s function , for two - dimensional nanoscale devices . in this work , the hsc - extension is applied to determine the solution of non - equilibrium green s functions ( negf ) on three - dimensional nanoscale devices . the operation count for the hsc - extension is analyzed for a cuboid device . when a cubic device is discretized with grid points , the state - of - the - art recursive green function ( rgf ) algorithm takes operations , whereas the hsc - extension only requires operations . operation counts and runtimes are also studied for three - dimensional nanoscale devices of practical interest : a graphene - boron nitride - graphene multilayer system , a silicon nanowire , and a dna molecule . the numerical experiments indicate that the cost for the hsc - extension is proportional to the solution of one linear system ( or one lu - factorization ) and that the runtime speed - ups over rgf exceed three orders of magnitude when simulating realistic devices , such as a graphene - boron nitride - graphene multilayer system with 40,000 atoms .


since its founding in 2006 , twitter has become an important platform for news , politics , culture , and more across the globe .twitter , like other social media platforms , empowers new forms of social organization that were once impossible .margetts et al .2016 discuss changing conceptions of membership and organization on social media ; twitter communities and conversations need not be bounded by geography , propinquity , or social hierarchy . as a result, social and political movements have taken to the site as a means of organizing activity both online and offline . in facilitating these movements, twitter simultaneously makes available a data trail never before seen in social research .researchers have embraced these data to create an expanding body of literature on twitter and social media writ large .  yet , this body of literature is only unified in the source of its data ; it remains fractured across many disciplines and fails to establish set procedures for drawing conclusions from these rich datasets .indeed , metareviews of election prediction using twitter have raised significant concerns of this literature s validity .this minireview extends this critical discussion of twitter literature to political action .we selected the reviewed literature in order to sample a variety of topics and methodologies .the sample purposively draws a geographic diversity of papers studying twitter - based political action in europe , the middle east , and the united states . yet ,some gaps certainly remain , including the glaring absence of hashtag activism studies and terrorist propaganda activity , two topics important to political action on twitter that warrant further study .hence we acknowledge that our review is not inclusive in terms of coverage of all the relevant papers in the field .  in reviewing the state of twitter literature on political action, we seek to explain the role of computational social science ( also called social data science ) methodologies in augmenting political scientific and sociological understanding of these phenomena .our minireview is structured as follows .we begin by examining the role of theory , and find that most often authors do not consider the expansive political and social theoretical literature in their analyses of online social phenomena .instead , they provide case studies and methodological developments exclusively for twitter research .we next examine the methodologies of these studies , and , drawing upon ruths et al . , we find that many papers fail to support their choice of methodology within the greater literature .we then examine significant results and discuss implications for further twitter studies of political action .social and political theory serves an important role in making sense of social research by fitting individual studies into larger theoretical frameworks . in this way, individual studies can intelligibly inform future research .alternatively , data analysis without a coherent , defensible theoretical framework serves only to explain a single observation at one point in time .the papers reviewed here fall into three broad categories in their use of theory : no theory , theory - light , and theory - heavy .papers fall into these categories irrespective of methodological or phenomenological focus .papers without theoretical grounding may cursorily cite but fail to engage theoretical texts .beguerisse - dias et al . examine communities and functional roles on twitter during the uk riots of august 2011 .to explain these phenomena , however , they cite no social theory .while the authors offer sophisticated methodical innovations for determining interest communities and individual roles in those communities , they do so without reference to a broader social science literature .some other papers offer cursory theory in their discussions of twitter data .borge - holthoefer et al . investigate political polarization surrounding the events that precipitated egyptian president morsi s removal from power in 2013 .their analysis of changes in `` loudness '' of opposing factions , although quite enlightening , is not grounded in any theoretical model of political action . instead, the authors proceed based on a number of platform - specific assumptions that do not readily permit results to be generalized beyond twitter .the authors suggest their findings contribute to the study of bipolar societies yet do not develop a theoretical model for such applications .the authors do use social theory , however sparingly , in order to contextualize their results , but even here theoretical discussion is lacking .conover et al. study partisan communities and behavior on twitter during the 2010 u.s .midterm elections .they similarly prioritize analytical innovations over theoretical models .the authors analyze behavior , communication , and connectivity between users , but do not seek to explain observed partisan differences .their research yields statistically significant differences between follower and retweet networks of liberal and conservative communities , which begs the question of why they exist ? such explanations could benefit from examining elections literature to develop a general theoretical model of partisan sharing . although the authors do briefly address the 2008 u.s . presidential election, it is only to contrast resulting phenomena , not to offer explanatory theories .in contrast , alvarez et al . explain political action in the spanish 15 m movement using durkheimian theory of collective identity and establish their work on firm basis in collective action literature .yet , while the authors base their methodology in theory , their findings do not directly engage with that theory aside from `` quantifying '' it .a similar fate befalls sampled predictive studies , which draw on theory to produce empirical results , but often fail to engage those results with underlying theory .weng et al . develop a model that predicts viral memes using community structure , based on theoretical insight from contagion theory .the authors find that viral memes spread by simple contagion , in contrast to unsuccessful memes which spread via complex contagion ; still , only the briefest theoretical discussion for this result is offered .garcia - herranz et al . develop a methodological innovation using individual twitter users as sensors for contagious outbreaks based in the `` friendship paradox '' and contagion theory .this mechanism uses network topology as an effective predictor , but does not address the social phenomena that create and sustain that topology .such methodological innovations provide researchers new analytical tools for observational analysis , but these tools remain of dubious explanatory value because they fail to ground methods in theory of the social world .twitter data present an opportunity not simply for analysis of social interactions on the platform but , if done well , these insights hold potential to contribute to new visions of the social world .rigorous data science can generate new theory .coppock et al . are particularly notable in this regard .the authors base their methodological innovation in twitter mobilization inducement on an extensive theoretical literature review , which yields three opposing hypotheses .they assess the political theory of collective action as it applies to twitter via these three hypotheses , and find that the civic voluntarism model is most consistent with their results .likewise , gonzalez - bailon et al . , in their study of protest recruitment dynamics in the spanish 15 m movement , offer both an extensive grounding in social theory and theory - engaging results .the authors findings serve to clarify threshold models of political action and `` collective effervescence '' .  as to the particular theoriesaddressed , the above mentioned papers focus primarily on political action and network theories of diffusion and contagion .important in such topics , but absent from all papers , is discussion of power or hierarchy .twitter operates within numerous contexts , e.g. , the offline influence of particular users and the online influence of those with numerous followers .reconciling methodologies with theories of power promises to provide further insight into political action on twitter .more broadly , a greater focus on theory is needed for twitter analyses to provide externally valid insight into the social world , both online and off .in developing analyses of twitter data , researchers have not drawn on a coherent body of agreed - upon methodologies . rather , methodological choices differ considerably from one paper to another. ruths et al . offers a critique of many common social media analysis practices . drawing from that work as well as our own insights , we examine many of the methodological choices made in our sample papers .we have delineated these choices into several overarching categories : data , filtering , networks and centrality , cascades and communities , experiments , and conjecture .  before addressing the methodological choices outlined above ,we first address several important findings from ref . . today , academic research writ large including social media work and much more is insufficiently transparent .academic journals publish only `` successful '' studies . without publishing methodologies that failed to explain political action phenomena , how is one to weigh the probability that the supposed `` fit '' observed is not due to random chance ?even those papers which address the robustness of their analysis , often stop at a very shallow significance tests using p - value , which is argued to be a flawed practice .similarly , when new methodologies are created , as in , they are justified vis - - vis random baselines and not prior methods .new methods are useful , but are they better than existing tools ?these opacity critiques are fundamental to the current state of twitter scholarship .researchers should be cognizant of these limitations when drawing conclusions from their work and should alter their methodologies to account for these limitations whenever possible .twitter data ultimately comes from the twitter platform .if scholars wish to make claims about the versatility of their methodologies and findings , they must justify their data - collection methods as representative of underlying populations on twitter or elsewhere .this proves a problematic task .the twitter api offers researchers an incredible array of tweet , user , and more data for analysis ; yet , the api acts as a `` blackbox '' filter that may not yield representative data .for example , weng et al . `` randomly '' collect 10 percent of public tweets for one month from the api .not only does the api preclude analysis as to the representativeness of the sample but it too prevents researchers from comparing studies over time , as the api sampling algorithm itself will change .proprietary sampling methods only further exacerbate the opacity problem .in , the authors use a proprietary sampling method to generate their dataset of spanish tweets from spain .the authors of do as well , using twitter4j and tweetmogaz as data sources .other papers do not use a global sampling method , but obtain data in other ways .beguerisse - dias et al . use a list of `` influential twitter users '' published in _ the guardian _ as the starting point for data collection .coppock et al . develop their experimental design in cooperation with the league of conservation voters , and use their twitter followers as test subjects .other papers , including and collect data by following particular hashtags and the users who tweeted them .garcia - herranz et al . twitter data by snowball - sampling from one influential user , paris hilton , as well as all users mentioning trending topics .none of these sampling methods allows authors to make broad claims about the twitter platform and political action in general .the method used in is particularly concerning , as it attempts to collect a large sample to sufficiently model a twitter population , but the choice of method undermines this very goal .a final complication of data in twitter studies regards the publication of that data .once data is collected and analyzed , it is rarely made available for others to replicate these studies the hallmark of good research .the problem here lies with twitter itself ; the terms of use preclude the republication of tweet contents that have been scraped from the site .following data collection , researchers often filter an intractable dataset into a manageable sample .researchers often use filtering to select a coherent sample . language and geography offer clear examples .borge - holthoefer et al . limit their dataset to arabic tweets about egypt .both gonzalez - bailon et al . and alvarez et .al limit their datasets to spanish tweets from spain . to do so , however , both papers use a proprietary filtering process from cierzo development ltd .as addressed above , proprietary methodologies stymie research transparency and replication .filtering can likewise facilitate a narrowing of research focus given a particular sample population .one common means of achieving a relevant dataset is to use hashtags as labels for tweets in which they appear . in authors obtain a sample of protest - related tweets using a list of 70 hashtags affiliated with the spanish 15 m movement .conover et al . filter to a sample of political tweets using a list of political hashtags and , in an excellent technique , allow the list of hashtags to grow based on co - occurring hashtags . in authours go one step further , and query not only hashtags but complete tweet content .arabic tweets were normalized for spelling and filtered by a series of boolean queries with a set of 112 relevant keywords .researchers , after filtering for a relevant sample and topic , may further filter for user attributes .borge - holthoefer et al . restrict their sample to high activity users with more than ten tweets extant in the limited sample .beguerisse - dias et al . limit their dataset to central users , those in the giant component of tweet - connected users .weng et al . limits the data to only reciprocal relationships .conover et al . filter tweets with geo - tags .the authors use a self - reported location field as their data source , despite the fact that someone can put `` the moon '' or anything else as their location .although the authors acknowledge the preliminary status of their analysis and its utility as an illustration of potential data - driven hypotheses , it left us unsatisfied with a lack of methodological rigor that should underlie even the most tentative of filtering claims .authors may choose to filter for no other reason than to obtain a manageable dataset .such decisions need not be arbitrary .garcia - herranz et al . settle on a particular sample size for their analyses , seeking to balance statistical power and the need to keep test and control groups from overlapping in the network .the authors offer an effective defense of their decision , presenting brief analyses of other sample sizes as well .coppock et al . , on the other hand , arbitrarily remove twitter users with more than 5000 followers from their sample because , they argue , these users are `` more likely '' to be influential or organizations , and therefore differ from the rest of the sample .this decision to remove outliers and the arbitrariness of the choice of threshold introduces systematic biases in the results , fundamentally undermining their analyses .these myriad filtering decisions often go insufficiently defended .those who do defend filtering choices often do so without referencing past literature .even sound filtering decisions , however , undermine the general claims researchers can make .this may be one reason most of the studies fail to contribute to social theory beyond their micro case studies .twitter lends itself to fruitful network analyses of both explicit interactions and other derivative relations .conover et al . use three network projections to analyze partisan political behavior during the 2010 u.s .midterm elections : one network sees users connected when mentioned together in a tweet , another where users are linked by retweeting behavior , and , third , the original explicit user follow - ship network .weng et al . also uses three networks mention , retweet , and follow to study meme virality .the authors conduct primary analysis on the follow network and use the other two as robustness tests . in studying protest recruitment to the 15 m movement , gonzalez - bailon et al . make use of two networks , one symmetric ( comprised of reciprocated following relationships ) and one asymmetric to study protest recruitment to the 15 m movement .the authors use these networks to determine the influence of broadcasting users .still other authors use single , traditional follower networks in their analyses .network analyses are all the more powerful when they are combined , as in and . in ,the authors offer another insight when they use network analyses over time with temporally evolving networks in response to events that preceded egyptian president morsi s removal from power .the authors recreate a sequence of networks that evolve over time .this method offers insight into how online activity responds to offline events in egypt , and could be a powerful tool in many other contexts , helping to parse a key question of political action : how groups respond to events and evolve over time .the opposite , to assume a network remains static during a given period , precludes this insight and undermines social analysis . in ,the authors exemplify this pitfall , as a network of protesters being recruited surely saw significant changes during their study s time period .given the fast growing literature on temporal networks , more attention is required in analyzing the dynamics of networked political activities .  beyond decisions of network type and temporality ,authors make important choices in projecting and using twitter networks . does not weight network edges based on number of tweets , and choses to limits the network projection to reciprocal relationships .both decisions fundamentally affect results , and undermine its validity as representing activity on the twitter platform .others , including account for asymmetry in their network projections .in doing network analysis , many researchers use centrality scores as a means to find the most influential users .researchers have developed a number of different definitions and algorithms for centrality .the choice of a specific approach , however , depends on the particular context and research questions . often timesthis choice is not well justified in the given context of online political mobilizations . among the papers considered here ,k - core centrality is the most common choice .while k - core centrality is a very useful tool to find the backbone of the network , it neglects social brokers , or the nodes with high betweenness centrality , relevant features in their own right when studying social behavior .   whether in networks or another form , twitter data yield insight through a multitude of different analytical techniques .one such technique examines tweets as they flow through the network in `` cascades . ''cascades follow a single tweet that is retweeted or similar tweets as they move across a network .the twitter platform makes these analyses difficult , however , as retweets are connected to the original tweet , not the tweet that triggered the retweet .researchers address this pitfall by using temporal sequencing to order and connect tweets or retweets . to achieve meaningful results , studies must sufficiently filter the tweets to establish that sequential tweets are related in content as well as time , which undermines representativeness , as discussed above .another common technique examines tweet content .alvarez et al . analyze their data for its social and sentiment content using semantic and sentiment analytic algorithms that analyzes tweets based on a test set .the authors use this technique to draw conclusions of individual users opinions of the 15 m movement in spain by analyzing up to 200 authored tweets on the topic per user .this technique holds great promise for future studies of political activity , and indeed any activity , on twitter . uses a less sophisticated solution towards a similar goal : they characterize users as either for or against military intervention in egypt .the authors attempt to show changes in opinion , and so can not not rely on comprehensive opinion from a mass of past tweets as done in . instead , uses coded hashtags to indicate users opinions .although this technique allows for discernable changes in opinion , the authors establish a dichotomy that threatens to oversimplify users opinions .community detection is another key analytical tool for twitter researchers . using network topology or node ( user or tweet ) content, researchers can cluster similar nodes and provide insight into social systems on a macroscopic scale .there are a variety of techniques , each with its own set of strengths and weaknesses . uses the infomap algorithm and test the robustness of their results by applying a second community detection technique , link clustering . uses a combination of two techniques , rhaghavan s label propagation method seeded with node labels from newman s leading eigenvector modularity maximization .the authors selected this combination of methods because it `` neatly divides the population ... into two distinct communities . '' yet , the authors fail to defend these observations rigorously in their paper . , on the other hand , effectively defends their decisions in setting resolution parameters for the markov stability method .the authors also use community detection creatively in conjunction with a functional role - determining algorithm to assign `` roles '' to users without a priori assignments of those groups . selects an apt community detection method that corresponds well with their objectives : to follow changes in polarity over time , the authors use label propagation , whereby nodes spread their assigned polarity .this method allows for seeding with nodes of known belief useful in monitoring the progression of the egyptian protests on twitter , as many important actors positions were publicly known . yetthis decision too comes with a cost : the authors program the label propagation to allow for only two polarities : secularist or islamist , even though they acknowledge that a third camp likely existed , namely supporters of deposed hosni mubarak .while community detection is still considered as an open question in network science , both at the definition and algorithmic implementation levels , many papers use one or more of these methods without enough care to make sure that the methods and definitions that they are using in their specific problem is well justified .twitter also lends itself as an experimental platform for researchers to implement controlled studies of social phenomena .in particular , and seek to predict viral memes on twitter using network topology and activation in linked users and communities , respectively . runs two experiments on inducing political behavior on twitter using different types and phrasing of messages . in all cases ,authors necessarily use controls in their experimental context . creates a null distribution of tweets with randomly shuffled timestamps to distinguish the effect of user centrality from user tweeting rate . uses two baseline models to quantify the predictive power of their community - based model .the authors use a random guess and community - blind predictor , against both of which the model is highly statistically significant . , with a true experimental design , offer an extensive discussion of experimental controls on twitter .the platform has inherent limitations for public tweet experiments because there is no effective way to separate experimental and control users given an inherently interconnected network structure .but the authors design their study to use direct messages to selected users as the experimental variable .the authors even tweaked and repeated the study to improve randomization in the control .such a methodology makes an example of a particularly strong experimental twitter paper .as we have seen , twitter provides researchers myriad analytical techniques .methodological choices as to which techniques to use present a fundamental challenge for researchers . they must select and properly defend their choice of methods that both work and fit their theoretical objectives .as we have noted above , there are numerous instances where researchers will do better jobs than others are achieving a methodological fit and defending it in their studies .some researchers may face the temptation to extend analyses to produce exciting results , but do so at the expense of sound methodologies .future twitter research would be well served to stress defensible , rigorous methodologies that are couched within existing theoretical literature from the social sciences , something that is rare today .taken collectively , the reviewed papers offer considerable insight into political activities conducted on the twitter platform , through analyses that examine political action in the abstract and others that offer case studies of concrete political action .these insights particularly address the roles of communities and individual users , connections between such entities , as well as the content they tweet .predictive models take these insights and offer tools for , perhaps , understanding political action in real - time . uses a sensor group of central users to predict virality of content , and extend this predictive sensor beyond twitter to google searches . uses connection topology to predict virality , although the predictive model is not extended to other content . observes viral tweets emerge from randomly distributed seed users , indicating exogenous factors determine the origins of viral content .taken together , these three studies offer an understanding of mass communication on twitter : viral content tends to originate randomly across the platform , reach more central users first , and spread across communities more easily than non - viral content .theoretical explanations of what makes viral content in the first place , however , is lacking in these analyses , and warrants further attention .topology also offers insights into its embedded users . uses topographical analyses to reveal flow based roles , interest communities , and individual vantage points without a priori assignment . assigns political leaning and then examine differences in partisan topologies in communities , tweeting activity , retweeting behavior , and mentions .both approaches offer insight into political behavior using topology , with different strengths .the techniques used in are quite useful when the partisan landscape on a particular issue is unknown ; the approach in greater understanding of known divisions .topology is not the sole determinant of activity , however , and tweet content analyses offer a second means of understanding political activity on twitter . finds that , in the context of the spanish 15 m indignados , tweets with high social and negative content spread in larger cascades .tweet content also readily lends itself to analyses which link twitter with offline phenomena . and that , in 2013 egyptian protests and spanish 15 m protests , respectively , real world events impact tweeting behavior . successfully induces off - twitter behavior using the content of tweets .content analyses offer insight into non - platform - dependent political activity .topology and content are distinct analyses .research that combines the two to answer a single question can yield robust results .several papers attempt this , borge - holthoefer et al . most successfully .the authors use content analysis to classify tweets and users into opinion groups , and then create temporally based retweet networks to follow changes in the activity and composition of those opinion groups .alvarez et al . use content analysis of observed network topological phenomena , e.g. , cascades , to quantify the social and emotional effects of content on sharing outcomes .beguerisse - dias et al . too combine methodologies , although less rigorously : they use word clouds to label topologically derived network communities .  in this vein , many of the above mentioned papers could benefit from incorporating mixed methodologies and drawing on each others analyses .future research papers should seek to emulate the approach in .further use of sentiment analyses from would render even more robust results .additional joint content and topology analyses would be even more useful : would using s central users in communities , i.e. , incorporate weng et al.s methods , result in to more precise virality predictor ? would adding content analysis as used in further improve precision ? if holistic understanding of social phenomena is researchers goal , future efforts should seek to incorporate not one but numerous methodologies in pursuit of that end .the papers considered in this minireview offer several important considerations on the state of twitter research into social phenomena .what was once the arena of solely political scientists and sociologists , political action and social phenomena have now become research topics for computer scientists and social physicists .new disciplines have much to offer social research , as indicated in the methodology review of our sample papers ; yet , these methodologies are often divorced from underlying social theory . thus far, twitter studies offer primarily observational not explanatory analyses .greater dialog between theory and methods , as well as a holistic use of all available methodologies , is needed for data science to truly offer insight into our social world , both on twitter and off it .the authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest .all authors listed , have made substantial , direct and intellectual contribution to the work , and approved it for publication .for providing useful feedback on the original manuscript we thank xxx xxx xxx .

Abstract: [ [ section ] ]  in recent years researchers have gravitated to social media platforms , especially twitter , as fertile ground for empirical analysis of social phenomena . social media provides researchers access to trace data of interactions and discourse that once went unrecorded in the offline world . researchers have sought to use these data to explain social phenomena both particular to social media and applicable to the broader social world . this paper offers a minireview of twitter - based research on political crowd behavior . this literature offers insight into particular social phenomena on twitter , but often fails to use standardized methods that permit interpretation beyond individual studies . moreover , the literature fails to ground methodologies and results in social or political theory , divorcing empirical research from the theory needed to interpret it . rather , papers focus primarily on methodological innovations for social media analyses , but these too often fail to sufficiently demonstrate the validity of such methodologies . this minireview considers a small number of selected papers ; we analyze their ( often lack of ) theoretical approaches , review their methodological innovations , and offer suggestions as to the relevance of their results for political scientists and sociologists .


dielectrophoresis and electrorotation ( er ) offer a unique capability of monitoring the dielectric properties of dispersions of colloids and biological cells . under the action of external fields , these particles exhibit rich fluid - dynamic behaviors as well as various dielectric responses .it is of importance to investigate their frequency - dependent responses to ac electric fields , which yields valuable information on the structural ( maxwell - wagner ) polarization effects .the polarization is characterized by a variety of characteristic frequency - dependent changes known as the dielectric dispersion . in particular , for the -dispersion ( also called the maxwell - wagner dispersion , ranging from khz to mhz ) , it may suffice to focus on the induced dipole moments of the particles . in this work, we will analyze the -dispersion based on the spectral representation theory .  in the last two decades, various experimental tools have been developed to analyze the polarization of biological cells - dielectric spectroscopy , dielectrophoresis and er techniques . among these techniques ,conventional dielectrophoresis and er are usually applied to analyze the frequency dependence of translations and rotations of individual cells in an inhomogeneous and rotating electric field , respectively .moreover , one is able to monitor the cell movements by using automated video analysis as well as light scattering methods . as a matter of fact, the general cause of er is the existence of a phase difference between the field - induced dipole moment and the external rotating field , resulting in a desired torque which causes the cells to rotate .  in the dilute limit ,the er of individual cell can be predicted by ignoring the mutual interaction between the cells .however , the cells may aggregate under the influence of the external field . in this case the brownian motion can be neglected , and the cell system becomes non - dilute even though it is initially dilute . as an initial model , we have recently studied the er of two approaching spherical particles in the presence of a rotating electric field .we showed that when the two particles approach and finally touch , the mutual polarization interaction between the particles leads to a change in the dipole moment of individual particles and hence the er spectrum , as compared to that of isolated particles , via the multiple image method .  in this work , we consider further the effects of cell shape on the er spectrum .in fact , a cell may readily deviate from a perfect spherical shape due to many reasons , e.g. , under a hydrostatic pressure .in addition to a prolate spheroidal shape , there exist cells of oblate spheroidal shape , such as human erythrocytes .regarding the spectral representation approach , it is a rigorous mathematical formalism of the effective dielectric constant of a two - phase composite material .also , the approach was extended to deal with a three - phase material .the spectral approach offers the advantage of the separation of material parameters ( namely the dielectric constant and conductivity ) from the cell structure information , thus simplifying the study . from this approach, one can readily derive the dielectric dispersion spectrum , with the dispersion strength as well as the characteristic frequency being explicitly expressed in terms of the structure parameters and the materials parameters of the cell suspension . in the present work , the analytic expression for the characteristic frequencyis derived with the aid of the spectral representation approach , thus simplifying the fitting of experimental data .we show good agreement with experimental data on human erythrocytes .consider a suspension of spheroidal particles of complex dielectric constant coated with a shell of dispersed in a host medium of , where = + /i2f with being the frequency of the applied field .the depolarization factors of the spheroidal particles is described by a sum rule l_z+2l_xy=1 where and ( ) are the depolarization factors along the - and - ( or - ) axes of the oblate spheroidal particle , respectively . just indicates sphere , while disk .the phenomenon of er is based on the interaction between a rotating electric field and the induced dipole moment .the dipole moment of the particle arises from the induced charges that accumulate at the interface of the particle . incircularly polarized fields , the axis of lowest polarizability , namely of the present particle , should be oriented perpendicular to the plane of field rotation .the angle between and is denoted by , where and is angular velocity of the rotating electric field .the torque acting on the particle is given by the vector cross product between the electric field and the dipole moment , so that only the imaginary part of the dipole moment contributes to the er response . in the steady state , the frequency - dependent rotation speed , which results from the balance between the torque and the viscous drag , is given by , \nonumber \\  & = & -f(\ep_2 , e,\eta)im[\t{b}_{x}]=-f(\ep_2 , e,\eta)im[\t{b}_{y}]\equiv  -f(\ep_2 , e,\eta)im[\t{b}_{xy } ] ,   \end{aligned}\]]where  ] , the angular brackets denote a time average , and is a coefficient which is proportional to the square of magnitude of field , but being inversely proportional to the dynamic viscosity of the host medium . for spherical cells , . regarding ,spin friction expression suffices for spherical particle .however , for spheroidal particles or many particles interacting in a suspension , we must consider the more complicated suspension hydrodynamics . since the angular velocity of the rotating field is much greater than the electrorotation angular velocity , i.e. , , the time averages are just equal to 1/2 . for a single coated spheroidal particle ,the dipole factor is given by _xy= where is the volume ratio of the core to the whole coated spheroid .we are now in a position to represent in the spectral representation . in what follows, we will show that from the spectral representation , we can obtain the analytic expression for the characteristic frequency at which the maximum er velocity occurs .let , and assume , we obtain _ xy = np+ [ bxy ] where denotes the nonresonant part which vanishes in the limit of unshelled spheroidal cells . in eq.([bxy ] ) , the various quantities are given by \}}{x+(1-l_{xy } )  l_{xy}(1-x)^2(1-y)},\nonumber\\ f_1&=&\frac{-x^2y}{3[x+(1-l_{xy})l_{xy}(1-x)^2(1-y)]^2},\nonumber\\ np&=&\frac{[l_{xy}(1-x)+x](1-x)(1-y)}{3[-x-(1-l_{xy } )  l_{xy}(1-x)^2(1-y)]}.\nonumber\end{aligned}\ ] ]  note that we have assumed to be a real number , that is , , which will be justified below .after substituting into eq.([bxy ] ) , we rewrite after simple manipulations _ xy=(np+)+ with and , where the dispersion magnitude and characteristic frequency admit respectively  fig.1(a)-(c ) , the dependence of the pole , characteristic frequency and dispersion magnitude on the parameter are investigated for different particle shape . for clarity, we set from now on .it is evident that both and decreases monotonically as increases ; larger depolarization factor leads to smaller and .however , for increasing , increases concomitantly ; also , increases while increases .  in fig.1(d)-(f ) , we investigate the dependence of the pole , characteristic frequency and dispersion magnitude on for different shell thickness . for ,namely without shell , always .increasing leads to decreasing or .it is evident that thicker shell yields larger pole or as , but smaller as . in the dispersion magnitude plot , is constant for the uncoated particles . in the case of coated particles , increasing yields increasing ; moreover , larger , smaller .  in fig.2 , the dependence of material parameters ( namely , , , , and ) on and is discussed . is weakly dependent on , but strongly dependent on , and .smaller ( or ) , larger .however , larger , larger . on the other hand , is weakly dependent on and .also , it is evident that and play no role in , as eq.([dep ] ) actually shows .note all curves in fig.2(g ) and ( h ) are respectively overlapped . in other words , this predicts that and plays no role in the peak value of the angular velocity ( see fig.4 ) .  in fig.3 ,the dependence of , and on  ] .the effect is small so that it can be neglected .smaller leads to larger ; at the same time , increasing peak value appears .increasing yields decreasing , and the related peak value is unchanged . for increasing , increases concomitantly , but the peak value remains unchanged as well .  in fig.5, the experimental data is extracted from an experiment on human erythrocytes .obviously , there is a good agreement between theory and experiment . during this fitting ,we model the cell as oblate spheroid ( ) .in the present work , we have discussed the effect of oblate spheroidal particle shape on the er spectrum . in reality , there exist many cells in the form of oblate spheroid , such as human erythrocytes .however , there exist only a few theories to discuss the er spectrum of such cells ( e.g. , and references therein ) .our theory is advantageous in that the characteristic frequency at which the maximum rotational velocity occurs , is derived analytically , which simplifies the fitting of experimental data .we have assumed , namely the ratio of the shell to host dielectric constant , to be a real and positive number .in fact , its imaginary part is indeed small .if we had retained the imaginary part of , two peaks would appear , and the conductivity - dominated peak would have occurred at substantially lower frequency .therefore , the neglect of the imaginary part of is to drop the lower frequency peak .moreover , according to our calculations , there are two ( degenerate ) sub - dominant poles associated with in the spectral representation .thus , in the present work , only one capacitance - dominated peak has been shown .  in a recent work , we developed simple equations to describe the er of particles in a suspension from the spectral representation , and obtained a good fitting on a coated - bead er assay by using the coated spherical model . in this work , an extension to oblatespheroidal case has been made .our theory serves as a basis which describes the parameter dependence of the polarization and thereby enhances the applicability of various cell models for the analysis of the polarization mechanisms . in this connection , the shell - spheroidal cell modelmay readily be extended to multi - shell cell model .however , we have shown that the multi - shell nature of the cell may have a minor effect on the er spectrum .we have considered the isolated cell case , which is a valid assumption for low concentration of cells . however , for a higher concentration of cells , we should consider the mutual interaction between cells .when the volume fraction of the suspension becomes large , the particles may aggregate in the plane of the rotating applied field , and the mutual interactions between the suspended particles can modify the spin friction , which is a key to determine the angular velocity of er . for er of two particles , we have successfully applied the spectral representation to deal with the dispersion frequency . however , the determination of the spin friction is still lacking . for two particles , the basic tool is the reflection method , being analogous to the multiple image method in electrostatics , but being only valid for two particles . for more than two particles ,we need a first - principles method , e.g. , the green s function ( oseen tensor ) formulation . for a dilute suspension , however , one may adopt the effective medium theories to capture the effective viscosity of a suspension , hence modifying further the er spectrum .  one may argue that normal erythrocytes may deviate from an oblate spheroidal shape . for cells of non - conventional shapes , one may use a first - principles approach , recently developed to deal with particles of rod shape . as shown in ref. ,the derivation from spheroidal model can be small when the cells are rotating with their long axes along the applied field .  in summary, we have presented a theoretical study of electrorotation based on the spectral representation theory . from the theoretical analysis, we find that cell shape as well as the coating can change the characteristic frequency . by adjusting the cell shape , dielectric properties and the thickness of the coating , it is possible to obtain good agreement between our theoretical predictions and the experimental data on human erythrocytes suspensions .this work was supported by the research grants council of the hong kong sar government under grant cuhk 4245/01p .acknowledges useful conversation with professor g. q. gu .  for a review ,see j. gimsa and d. wachner , biophys .j. * 77 * , 1316 ( 1999 ) .j. gimsa , ann .sci . * 873 * , 287 ( 1999 ) .k. asami , t. hanai and n. koizumi , jpn .* 19 * , 359 ( 1980 ) .g. fuhr , j. gimsa and r. glaser , stud . biophys . * 108 * , 149 ( 1985 ) .j. gimsa , p. marszalek , u. lowe and t. y. tsong , biophys .j. * 73 * , 3309 ( 1991 ) . g. de gasperis , x .- b .wang , j. yang , f. f. becker and p. r. c. gascoyne , meas .* 9 * , 518 ( 1998 ) .j. p. huang , k. w. yu and g. q. gu , phys .e * 65 * , 021401 ( 2002 ) .k. w. yu and jones t. k. wan , comput .. commun . * 129 * , 177 ( 2000 ) .k. asami and t. yamaguchi , annals of biomedical engineering * 27 * , 427 ( 1999 ) .j. p. huang and k. w. yu , j. phys . : condens .matter * 14 * , 1213 ( 2002 ) .d. j. bergman , phys .rep . * 43 * , 379 ( 1978 ) .k. p. yuen and k. w. yu , j. phys . : condens .matter * 9 * , 4669 ( 1997 ) .jun lei , jones t. k. wan , k. w. yu and hong sun , phys .e * 64 * , 012903 ( 2001 ) .j. gimsa , th .schnelle , g. zechel , and r. glaser , biophys .j. * 66 * , 1244 ( 1994 ) .j. gimsa , bioelectrochemistry * 54 * , 23 ( 2001 ) .l. gao , jones t. k. wan , k. w. yu and z. y. li , j. phys . :matter * 12 * , 6825 ( 2000 ) .j. p. huang , k. w. yu , j. lei and h. sun , commun .phys . , in press ( 2002 ). j. happel and h. brenner , _ low reynolds number hydrodynamics _ , ( kluwer academic publishers , dordrecht / boston / london , 1983 ) .t. c. choy , physica a * 221 * , 263 ( 1995 ) .jun lei , jones t. k. wan , k. w. yu and hong sun , j. phys . : condens .matter * 13 * , 3583 ( 2001 ) .

Abstract: we have investigated the effect of particle shape on the eletrorotation ( er ) spectrum of living cells suspensions . in particular , we consider coated oblate spheroidal particles and present a theoretical study of er based on the spectral representation theory . analytic expressions for the characteristic frequency as well as the dispersion strength can be obtained , thus simplifying the fitting of experimental data on oblate spheroidal cells that abound in the literature . from the theoretical analysis , we find that the cell shape , coating as well as material parameters can change the er spectrum . we demonstrate good agreement between our theoretical predictions and experimental data on human erthrocytes suspensions .


let be a random variable on a finite set with probability mass function ( pmf ) given by .suppose that we wish to guess the realization of this random variable by asking questions of the form `` is equal to ? '' , stepping through the elements of , until the answer is `` yes '' ( , ) . if we know the pmf , the best strategy is to guess in the decreasing order of -probabilities .the aim of this paper is to identify good guessing strategies and analyze their performance when the pmf is not completely known . throughout this paper, we will assume that the only information available to the guesser is that the pmf of the source is one among a family of pmfs .  by way of motivation, consider a crypto - system in which alice wishes to send a secret message to bob .the message is encrypted using a private key stream .alice and bob share this private key stream .the key stream is generated using a random and perhaps biased source .the cipher - text is transmitted through a public channel .eve , the eavesdropper , guesses one key stream after another until she arrives at the correct message .eve can guess any number of times , and she knows when she has guessed right .she might know this , for example , when she obtains a meaningful message . from alice s and bob s points of view , how good is their key stream generating source ? in particular , what is the minimum expected number of guesses that eve would need to get to the correct realization ? from eve s point of view , what is her best guessing strategy ?these questions were answered by arikan in and generalized to systems with specified key rate by merhav and arikan in .taking this example a step further , suppose that alice and bob have access to a few sources .how can they utilize these sources to increase the expected number of guesses eve will need ?what is eve s guessing strategy ?we answer these questions in this paper .  when is known , massey and arikan sought to lower bound the minimum expected number of guesses . for a given guessing strategy ,let denote the number of guesses required when .the strategy that minimizes  ] , satisfies \right ] \leq h_{1/2}(p),\ ] ] where is the rnyi entropy of order .bozta obtains a tighter upper bound .  for , arikan also considered minimization of \right)^{1/\rho} ] , grows exponentially with .the minimum growth rate of this quantity ( after normalization by ) is given by the rnyi entropy .this gave an operational significance for the rnyi entropy .in particular , the minimum expected number of guesses grows exponentially with and has a minimum growth rate of .the study of  ] is at least as large as that of the optimal strategy \log \left ( 1 + \ln   .one can only conclude that in source compression with mismatch where the `` nuisance '' term is not but the constant 1 . yet , in the examples in section [ examples ] on guessing we see how to make good use of these bounds .see also the discussion following theorem [ campbellredundancytheorem ] at the end of the next section .campbell in and gave another operational meaning to the rnyi entropy of order . in this sectionwe show that arises as `` inaccuracy '' in this problem as well , when we encode according to a mismatched source . to be consistent with the development in the previous section, we will assume that is coded when the source coder has side information .let and be finite alphabet sets as before .let the true source probabilities be given by the pmf on .we wish to encode each realization of using a variable - length code , given side information .more precisely , let the ( nonnegative ) integer code lengths , satisfy the kraft inequality , the problem is then to choose among those that satisfy the kraft inequality so that the following is minimized : \right),~~ -1 < \rho< \infty , \rho \neq 0,\ ] ] where the expectation is with respect to the pmf . as , this quantity tends to the expected length of the code ,  ] .in particular , . is a concave function of since is concave and the argument is linear in .let be the one - sided derivative of evaluated at ( _ i.e. _ , limit as ) .we can straightforwardly check that with the possibility that the value ( slope at ) may be .we have therefore established that has , is concave and therefore continuous in  ] in raster order ( one row after another ) .every would have figured exactly once in the list , and therefore occurs exactly times in the list .next , prune the list . for each , if there exists an index with and , set .this indicates that the string already figures in the final guessing list .finally remove all s to obtain the desired guessing list , where is the unique position at which occurs in the pruned list .clearly , for every and for every such that , we have .indeed , occurs in the position in the matrix constructed above .it therefore occurs in position and therefore before the position in the unpruned list .it can not be placed any later in the pruned list , and thus .the above observation leads to \leq \frac{1}{\rho } \log \mathbb{e } \left [ g(x^n)^{\rho } \right ] + \log n.\ ] ] the proposition follows from theorem [ guessingredundancytheorem ] , proposition [ guessingavstypeproposition ] , and the bounding .we finally remark that the min - sup redundancy for the finite - state arbitrarily varying source grows linearly with under some circumstances .[ avsconvergence ] for a fixed , let be a pmf on and the corresponding type . let the sequence ( as a function of ) converge to a pmf as . then where .  the second term in the right - hand side of ( [ redundancyequation ] ) , after normalization by , converges to a nonnegative real number as seen below :   we next consider the first term on the right - hand side of ( [ redundancyequation ] ) after normalization , _i.e. _ , , where is given by ( [ minimizeravs ] ) .[ avsconvergencelemma ] for a fixed , let be a pmf on and the corresponding type . let the sequence ( as a function of ) converge to a pmf as .let be the output pmf when the input pmf on is and the channel is .furthermore , let be the limiting output pmf as . then .  as a consequence of this lemma and ( [ secondtermofavs ] ) , we have by the strict concavity of for , and jensen s inequality , we have .this concludes the proof of the theorem ._ remarks _ : if and only if either  for some , or  does not depend on , _ i.e. _ , the state does not affect the source .thus , for all but the trivial finite - state arbitrarily varying sources , the min - sup redundancy grows exponentially with at a rate .this means that the guessing strategy that achieves the min - sup redundancy has an exponential growth rate strictly bigger than that of the best strategy obtained with knowledge of the state sequence .we now prove the rather technical lemma [ avsconvergencelemma ] .\(a ) we first show that .let be the pmf on given by .let denote the -probability of the set . from ( [ minimizeravs ] ) , we may write where ( [ f1 ] ) follows from the observation that for all , ( [ f2 ] ) from ( see proof of ( * ? ? ? * lemma 2.3 ) ) and by enlarging the sum over to all of .  from ( [ f3 ] ) and ( [ conditionalrenyientropy ] ), we have   \(b ) we now show that . for a given pmf on and conditional pmf , let be the induced pmf on and the reverse conditional pmf , _i.e. _ , is the probability of a state given .continuing from ( [ f1 ] ) , we may write where ( [ g1 ] ) follows because and the sum over is restricted to a sum over a type to be chosen later ; ( [ g2 ] ) follows because and the sum over is now restricted over a non - void -shell of , where will be appropriately chosen later .we next observe that for , the following hold : substitution of these inequalities into ( [ g3 ] ) yields } \end{aligned}\ ] ] and therefore \nonumber \\ \label{lowerboundonnormalizedrenyientropyavs } & & - ~ \frac{|\mathbb{x}|(1+\alpha |\mathbb{s}|)}{1-\alpha } \frac{\log ( n+1)}{n}\end{aligned}\ ] ] for any type of sequences and for any such that is a non - void shell for an .clearly , the last term in ( [ lowerboundonnormalizedrenyientropyavs ] ) vanishes as .if we can choose and , we will be done since .we can not do this if is not a type of sequences , or if is not a conditional type given an .but we will show that as , we can get close enough .the following arguments make this idea precise .define and consider .we may restrict our choice of to those that are absolutely continuous with respect to , _i.e. _ , for every . for sufficiently large , we can choose such a that in addition satisfies and .we then have where ( [ h1 ] ) follows from ( * ? ? ?* lemma 2.7 ) . after averaging ,we get this section we look at an interesting geometric property of divergence that makes it behave like squared euclidean distance , analogous to the kullback - leibler divergence . throughout this section ,we assume and .we proceed along the lines of .let and be finite alphabet sets .let denote the set of pmfs on . given a pmf on , the set is called an -sphere ( or ball ) with center and radius .the term `` sphere '' conjures the image of a convex set . that the set is indeed convex needs a proof since is not convex in its arguments .[ ballisconvex ] is a convex set .  let for .for any  ] .the limiting value as , the derivative of with respect to evaluated at , should be . this will give us the necessary condition .note that the derivative evaluated at is a one - sided limit since $ ] .we will first check that this one - sided limit exists .  from ( [ idefinitionrho ] ), can be written as , where is bounded , positive , and lower bounded away from 0 , for every .let and be the derivatives of and evaluated at . clearly , similarly , it is easy to check that with the possibility that it is ( only when and ) .  since we can write ,\end{aligned}\ ] ]it follows that the derivative of exists at and is given by , with the possibility that it might be .however , ( [ limitingifderivative ] ) and imply that consequently , is necessarily finite . in particular , when , we have ascertained that is finite .after substitution of , and we get when , clearly , can not be zero , due to the nonzero assumptions on the other quantities in ( [ simplifiedpythagoreaninequality ] ) .this implies that is finite when as well .an application of ( [ lirelationgeneral ] ) and ( [ idefinitionrho ] ) shows that ( [ simplifiedpythagoreaninequality ] ) and ( [ pythagoreanpropositioninequality ] ) are equivalent .this concludes the proof of the forward implication .the reader will recognize that the basic idea is quite simple : evaluation of a derivative at and a check that it is nonnegative .the technical details above ensure that the case when the derivative of the denominator is infinite is carefully examined .\1 ) : the hypotheses imply that , and are finite . as observed above , ( [ simplifiedpythagoreaninequality ] ) and ( [ pythagoreanpropositioninequality ] ) are equivalent .observe that both sides of ( [ simplifiedpythagoreaninequality ] ) are linear in .this property will be exploited in the proof .clearly , if we set in ( [ pythagoreanpropositioninequality ] ) and ( [ simplifiedpythagoreaninequality ] ) , we have the equalities and a -weighted linear combination of the inequalities ( [ simplifiedpythagoreaninequality ] ) and ( [ trivialpythagoreanequalityexpanded ] ) yields ( [ simplifiedpythagoreaninequality ] ) with replaced by . the equivalence of ( [ pythagoreanpropositioninequality ] ) and ( [ simplifiedpythagoreaninequality ] ) result in this concludes the proof of the first part .\2 ) this follows easily from the first statement . for the forward implication , indeed , ( [ simplifiedpythagoreaninequality ] ) holds for .moreover , ( [ simplifiedpythagoreaninequality ] ) holds when is replaced by . if either of these were a strict inequality , the linear combination of these with the given by ( [ innerpmf ] ) will satisfy ( [ trivialpythagoreanequalityexpanded ] ) with strict inequality replacing the equality , a contradiction .the reverse implication is straightforward .let us now apply proposition [ pythagoreanproposition ] to the -projection of a convex set .for a convex , we call an _ algebraic inner point of _ if for every , there exist and satisfying ( [ innerpmf ] ) .  _( projection theorem ) _ [ l - projection - theorem ] let and a finite set .a pmf is the -projection of on the convex set if and only if every satisfies the -projection is an algebraic inner point of , then every satisfies ( [ pythagoreaninequality ] ) with equality . .this follows easily from proposition [ pythagoreanproposition ] .for the case when not covered by proposition [ pythagoreanproposition ] , ( [ pythagoreaninequality ] ) holds trivially .let , and a pmf be the -projection of on the convex set .if is an algebraic inner point of , then every satisfies ( [ pythagoreaninequality ] ) with equality .  clearly , for any , we have , and therefore .the corollary now follows from the second statement of theorem [ l - projection - theorem ] .  while existence of -projection is guaranteed for certain sets by proposition [ existenceofprojection ] , the following talks about uniqueness of the projection .( _ uniqueness of projection _ ) [ uniquenessofprojection ] let . if the -projection of on the convex set exists , it is unique.  let and be the projections. then where the last inequality follows from theorem [ l - projection - theorem ] .thus , and .analogous to the kullback - leibler divergence case , our next result is the transitivity property .let and be convex sets of pmfs on .let have -projection on and on , and suppose that ( [ pythagoreaninequality ] ) holds with equality for every .then is the -projection of on .the proof is the same as in ( * ? ? ?* theorem 2.3 ) .we repeat it here for completeness .observe that from the equality hypothesis applied to , we have consequently is finite .furthermore , for a , we have where ( [ c2 ] ) follows from theorem [ l - projection - theorem ] applied to , and ( [ c3 ] ) follows from ( [ c1 ] ) .we next compare ( [ c3 ] ) with and cancel to obtain for every .theorem [ l - projection - theorem ] guarantees that is the -projection of on .  as an application of theorem [ l - projection - theorem ]let us characterize the -center of a family .if the -center of a family of pmfs exists , it lies in the closure of the convex hull of the family .  let be the closure of the convex hull of .let be an -center of the family , and , which is at most , the -radius .our first goal is to show that .  by proposition [ existenceofprojection ], has an -projection on , and by proposition [ uniquenessofprojection ] , the projection is unique on . from theorem [ l - projection - theorem ] , for every , we have thus thus , leading to .for the special case when is finite , _i.e. _ , , we found the weight vector such that and .this was done in an explicit fashion in section [ subsubsec : minimizerinconvexhull ] using results on -divergences .we conclude this paper by applying some of our results to guessing of strings of length with letters in .let , , and a pmf on .let denote the pmf of the discrete memoryless source ( dms ) where the -string .theorem [ arikanguessingtheorem ] says that for , the minimum expected number of guesses grows exponentially with ; the growth rate is given by .  if the only information that the guesser has about the source is that , the guesser suffers a penalty ( interchangeably called redundancy ) ; growth rate of the minimum expected number of guesses is larger than that achievable with knowledge of .the increase in growth rate is given by the normalized redundancy , where is the guessing strategy chosen to work for all sources in .this normalized redundancy equals the normalized -radius of , _i.e. _ , , where is given by ( [ minsuplalphaeqn ] ) , to within .  when is a dms , and the pmf on is unknown to the guesser , arikan and merhav shown that guessing strings in the increasing order of their empirical entropies is a universal strategy .their universality result is implied by the fact that the normalized -radius of the family of dmss satisfies .the family of dmss is thus not rich enough from the point of view of guessing .knowledge of the pmf is not needed ; the universal strategy achieves , asymptotically , the minimum growth rate achievable with full knowledge of the source statistics .suppose now that ; we may think of an -string as the outcome of independent coin tosses .suppose further that two biased coins are available . to generate each ,one of the two coins is chosen arbitrarily , and tossed .the outcome of the toss determines .this is a two - state arbitrarily varying source .we may assume .let us assume that as , the fraction of time when the first coin is picked approaches a limit .let us further assume that for each , the receiver knows how many times the first coin was picked , _i.e. _ , it knows the type of the state sequence . if the two coins are not statistically identical , the normalized -radius approaches a strictly positive constant as .this implies that the growth rate in the minimum expected number of guesses for a strategy without full knowledge of source statistics is strictly larger than that achievable with full knowledge of source statistics .we note that in order to maximize the expected number of guesses , the right solution may be to pick one coin , the one with the higher entropy , all the time .the guesser s lack of knowledge of the number of times the first coin is picked results in additional redundancy .however this additional redundancy asymptotically vanishes .the guesser `` stitches '' together the best guessing lists for each type of state sequences .

Abstract: this paper considers the problem of guessing the realization of a finite alphabet source when some side information is provided . the only knowledge the guesser has about the source and the correlated side information is that the joint source is one among a family . a notion of redundancy is first defined and a new divergence quantity that measures this redundancy is identified . this divergence quantity shares the pythagorean property with the kullback - leibler divergence . good guessing strategies that minimize the supremum redundancy ( over the family ) are then identified . the min - sup value measures the richness of the uncertainty set . the min - sup redundancies for two examples - the families of discrete memoryless sources and finite - state arbitrarily varying sources - are then determined .   -divergence , guessing , information geometry , -projection , mismatch , pythagorean identity , redundancy , rnyi entropy , rnyi information divergence , side information