emotion expression is a psychological behavior with the purpose of communicating affective states between different individuals .this behavior could be either verbal or nonverbal , including text , voices , faces and bodies  . in the epoch of the internet ,tremendous developments of online social media provide abundant innovative and powerful means of information exchange , bringing unprecedented richness and diversity to the ways of emotional expression .  among these new forms , _emoticon _ is getting more and more popular with the rapid growth of weibo , a twitter - like service in china . in general , an emoticon is a kind of non - verbal sentiment language represented by a vivid image , which differs itself from a _ smiley _ formed by combining punctuation marks as used in twitter .indeed , users seem to show increasing interests in expressing sentiments through around 2,000 emoticons designed by weibo , ranging from facial expressions like laughing or bursting into tears , body languages like applause or hug , to some icons like the sun , christmas trees or birthday cakes .this is not unusual , since tweets are very short and context - absent for the 140-word limitation , while emoticons could easily convey specific emotions in a vivid and personalized way .it has been found that both smileys and emoticons are strongly related with typical sentiment words , and could serve as convincing indicators of different emotions  .tossell et al .confirm that emoticon usage is contextual , and people use more negative emoticons than positive ones  . researchers also realize that emoticons and smileys could be effective features for texts to improve the precision of sentiment analysis  , or be treated as sentiment labels to avoid intensive labor costs for preparing training samples  .  as a speical social behavior ,emoticon usage is shaped by cultural and social factors .the pattern of emoticon usage in short - message texting is investigated and the discrepancies between male and female users are revealed  .for instance , female users send more messages containing emoticons , while male users use a wider range of emoticons .schnoebelen et al .distinguish users by whether they use nose smiley `` :-) '' or non - nose smiley `` :) '' , and demonstrate that the variants correspond to different types of users , tweeting with different vocabularies and writing styles  .park et al . point out that emoticons are socio - cultural norms and their meanings could be affected by the identity of the speaker  .the above research , though being very interesting , is mainly focused on emoticons rather than emoticon users .who they are , what they are talking about , how they behave online and why they adopt emoticons these human - centric problems are actually very interesting to sociologists and perhaps marketers , who are always seeking for potential consumers.meanwhile , different from expensive but spatio - temporally limited surveys in tradition  , digital traces created by interactions with technology indeed offers a new probe to collective human behavior and these new data are fuelling the rapid development of computational social science  .these indeed motivates our study in this paper , which takes ambivalent emoticons in a tweet as latent clue for tracing the unusual behaviors of a speical group called `` ambivalent users '' .more specifically , we manually select 79 positive and 36 negative emoticons with unambiguous polarities to label the sentiments of tweets ; that is , tweets containing only positive ( negative ) emoticons are defined as positive ( negative ) tweets .surprisingly , we find many _ambivalent tweets _ containing both positive and negative emoticons , indicating inconsistent emotions  .the _ ambivalent users _ are then defined as the users who have published more than 30 ambivalent tweets in 2012 .the _ senior ambivalent users _ are defined accordingly as a group of special ambivalent users who have published more than 50 ambivalent tweets in 2012 . as contrast , the _ ordinary users _ are the users who have never published any ambivalent tweet in 2012 . after excluding abnormal users like verified celebrities , organizations , inactive users ( post less than 12 tweets in 2012 or have less than 65 followers ) and spam accounts , we finally have 1,069 ambivalent users , among whom 357 are senior users , and the ordinary users total 46,245 .a first study of demographics shows that ambivalent expression is more popular in weibo than in twitter , and the female occupies a significantly higher proportion of ambivalent users .this indeed validates the conventional theory in psychology that people from east show significantly stronger association of positive and negative in emotion expression than the west , especially the female  .further explorations on the behaviors of ambivalent users disclose more interesting patterns .specifically , ambivalent users like to express ambivalence in midnights or at weekends , focus much more on topics of entertainment and sports , actively mention their close friends in tweets and gain more replies rather than reposts than ordinary users , and frequently use positive terms while expressing negative mood .the above observations suggest the subtle link between users mood status and the usage of online social media , which has become a very hot topic in recent years and attracted increasing research interests  .specifically , based on the theoreies of emotion management in sociology , we point out that ambivalent users unusual behaviors can be well explained from the view of _ self regulation _ against negative feelings .moreover , this self - regulation behavior of ambivalent users seems to be conscious , which makes our finding distinct from existing studies mostly on unconscious behaviors .finally , as a natural conjecture , we testify the self - correcting effect of shopping on ambivalent users negative mood using the tweets posted around the singles day in 2012 .this indicates that ambivalent users like compulsive users are ideal targets for online marketing .we first observe ambivalent expression on weibo .it is interesting that nearly 1.9% of the emotional tweets on weibo are ambivalent , but it reduces to 1.1% for twitter , indicating stronger inclination of chinese weibo users for ambivanent expression ( see _ methods _ ) .this may owe to the extraordinary vividness and ease - of - use of the built - in emoticon modular of weibo , and the linguistic differences between chinese and english may also contribute .we then focus on the ambivalent users on weibo .it is interesting that ambivalent users generally showed more passive mood by publishing significantly more negative tweets than ordinary users ( the negative - tweet ratios are 27.1% against 18.8% for ambivalent and ordinary users , respectively , in 2012 ) .moreover , the female occupies 79.0% of total ambivalent users , but only 42.8% of the ordinary users . among themwe find a verified community named _ weibo lady _ , whose members are most active females passionate about sharing their personal lives . in this community ,82.8% of the users ever posted ambivalent tweets , which is consistent with the theory that people from the east show a stronger association between positive and negative affect than the west , especially the female  .we further observe the posting behavior of ambivalent users .[ fig : tweet_time_pattern](a ) shows the daily pattern , where ambivalent tweets seem more noticeable in the midnight from 11 pm to 2 am .similar situation happens for the weekly pattern in fig .[ fig : tweet_time_pattern](b ) , where significantly more ambivalent tweets are posted at the weekend , especially on sunday . since midnights and weekends are usually the leisure time for individuals , we could conjecture that ambivalent users are apt to express mixed feelings when relaxed .we further explore the topic preference of ambivalent users .it has been reported that the variation of social behavioral indexes , like `` the number of posts '' , could be employed to detect hot topics or external events in social media  .accordingly , we trace the time series of positive and negative tweets , respectively , and highlight the peaks with clear event semantics ( see _ methods _ ) , as shown in fig . [fig : events ] .  fig .[ fig : events](a ) suggests that , in terms of positive emotion , besides the festivals such as spring festival , valentine s day , and moon cake festival , ambivalent users enjoy talking more about some niche topics of sports and entertainment , like day to speak out love ( _ j _ ) , a singer , xin liu s birthday ( _ l _ ) , european cup competitions ( _ l _ and _ m _ ) , and olympic games ( _ n _ ) .the variations of negative sentiment in fig .[ fig : events](b ) further validate this point ; that is , besides the natural disasters with universal concerns , ambivalent users pay more attention to topics like the rescue of dogs ( _ b _ ) , the competitions in european cup ( _ d _ ) , the final episode of a tv play ( _ f _ ) and the final of voice of china ( _ h _ ) .these unveil that ambivalent users are indeed more sensitive and emotional than the ordinary ones .the extracted topic words in tables [ tab : pos_event ] and [ tab : neg_event ] provide more details ( see _ methods _ ) .it is interesting that ambivalent tweet is an important channel for ambivalent users to socialize online . to illustrate, we take senior ambivalent users as well as ordinary users for comparative study .simple statistics show that ambivalent users are a group of people who are socially more active online , e.g. , having more fans ( 2506 versus 735 ) and bi - friends ( 270 versus 194 ) and posting more tweets ( 1890 versus 336 ) , than the ordinary users .  more interestingly, ambivalent users have a subtle communication mode . to illustrate this , we trace the `` @ '' behavior on all the tweets they posted in 2012 ( a weibo user can use the `` @ '' symbol in her tweet to inform someone deliberately , which could be regarded as a way for more `` private '' communication on weibo ) . fig .[ fig : mention_pattern ] shows that ambivalent users are generally more apt to mention a friend in an ambivalent tweet .further exploration reveals that the friends mentioned in ambivalent tweets share averagely 20 common followers with the ambivalent users , but the number reduces to 12 for the friends mentioned in positive tweets and even 7 in negative tweets .since to share more friends indicates a stronger social tie between two users  , the above implies that ambivalent users prefer to interact with their closer friends via ambivalent tweets .in other words , we can touch close ties on weibo by simply checking who is mentioned by whom in ambivalent tweets .the reactions from the friends of ambivalent users are also very interesting .to illustrate this , we observe the _ replies _ and _ reposts _ of the tweets posted by senior ambivalent users in 2012 .we find that ambivalent tweets are generally more apt to attract _ replies _ a tweet containing dialectic emotions has 36 replies on average , whereas a positive ( negative ) tweet only has 34 ( 23 ) replies .this contradicts with the _ reposts _case , where a positive tweet elicits more reposts than an ambivalent tweet ( 63 versus 53 ) .since a reply is a more private feedback than a repost , the above implies that ambivalent users are easier to gain private feedbacks via ambivalent tweets . in a nutshell, we can conclude that ambivalent users on weibo depend heavily on ambivalent tweets to interact with their close friends in a relatively private way .we here demonstrate that posting ambivalent tweets can lead to sentiment shift and thus help to ease negative emotion .this is particularly important for ambivalent users , who seem generally more passive than ordinary users .this may also uncover the root of ambivalent expression on social media .  to this end, we collect all the ambivalent tweets posted in 2012 and analyze the sentiment shift indicated by ambivalent emoticons within a tweet .simple statistics reveal that there are 60,293 tweets shifting from negative emotion to positive one , but only 39,457 in reverse ( we abandon tweets with ambiguous sentiment shift ). the `` negative positive '' pattern of emotional shift implies that weibo indeed can ease negative mental state of ambivalent users .we also try `` perceiving '' subtle emotional shifts from the words adopted in ambivalent tweets . to this end, we collect all the ambivalent tweets posted in 2012 and split them into short clauses , which are then tagged as positive ( negative ) clauses if they only contain positive ( negative ) emoticons .the frequent terms are then extracted from the positive ( negative ) clauses , and are compared with the terms used in purely positive ( negative ) tweets ( see _ methods _ ) .it is interesting that the sentiment shift in ambivalent tweets in the term level is surprisingly evident .specifically , the ambivalent users tend to use some special terms in negative ( positive ) clauses that are actually more frequently occurred in positive ( negative ) tweets .this contradiction is particularly evident for the case of negative clauses , implying the more positive inclination of ambivalent tweets .this agrees with the mainstream pattern of emotion shift : negative positive , and implies that posting ambivalent tweets indeed can help weibo users realize self - correcting .we will have a more detailed discussion below .the sentiment shift of ambivalent tweets gives an important clue that online social media might affect or even manipulate users mood status , which indeed has attracted specialized interests from researchers of various domains .for example , it has been found that socialising , information - seeking , and entertainment in online social media significantly influence user s positive emotion  , persistent emotional expressions for individual users and channels are revealed in online chat rooms  , and positive terms in news feed may trigger online friends to express similar feelings  . from the dark side , however , disordered online social networking use leads to difficulties with emotion regulation  , and daily time spent on social networking is related to depression  .these pilot studies , though being very interesting , have not touched the speical group of ambivalent users hidden inside online social media . nor did they provide adequate explanations to the abnormal behavioral patterns of these users . in particular , the sentiment shift from negative to positive implies that ambivalent users `` actively '' leverage the weibo platform to realize self - correcting , which has clear contrast to the `` unconscious '' emotional influences discussed in the above - mentioned literature .nevertheless , we can find some evidence for the self - regulation of negative emotions from earlier sociological research . indeed , based on offline survey studies , these research reveals that people could tune their moods through music , social interactions , enjoyable activities , shopping , religion , distraction  and cognitive reappraisal  .these results , though being widely accepted in the field of sociology , have hardly been tested in an online environment .therefore , it would be very interesting if our findings on ambivalent users in weibo could resonate with these offline findings .indeed , as illustrated below , we can really explain the complicated behavioral patterns of ambivalent users from a sentiment self - regulation perspective .first , as can be found in sect .ii.a , ambivalent users are apt to express passive moods and post ambivalent tweets in the midnights or at the weekends .this justifies that ambivalent users objectively have the need for self regulation of negative emotions , espeically when they are not occupied by work .hence , we can conjecture that publishing tweets should be an ideal way to distract them from negative mentality , especially the ambivalent tweets that can help them realize cognitive reappraisal . to understand the last point , recall the finding in sect . ii.cthat ambivalent users demonstrate an unusual sentiment - shift behavior , most evidently from negative mood to positive mood .this pattern implies that ambivalent users try to reduce their passive feelings through building new cognitions to the context , including opinions , attitudes , sentiments , etc .second , as illustrated in sect .ii.b , ambivalent users are particularly fond of topics like entertainment and sports .these enjoyable activities could arouse relaxed feelings and thus might liberate them from depression or stress .moreover , the special social interaction patterns of ambivalent users given in sect .further hint the existence of self regulation .that is , ambivalent users prefer to mention their close friends in tweets and get encouraging feedbacks from those friends .this coincides with previous findings from surveys that supportive interactions with friends produce positive affect  and positive feedbacks on the profile enhance social self - esteem and well - being in online social networking  .more interestingly , our findings suggest that interacting with close friends through more private actions like `` @ '' and `` reply '' in online social networking can help boost self regulation .this subtle detail exhibits the possible uniqueness of self regulation theory in an online environment , which can not been touched in traditional offline studies .axis , the unit is a day and 0 stands for the shopping date .( a ) positive tweets .( b ) negative tweets . ]  in a nutshell , the self regulation theory in sociology indeed well explain the distinct behaviors of ambivalent users in weibo , and social media big data even enable us to find more distinct details in an online environment .it is also noteworthy that the self regulation of ambivalent users seems to be conscious , which makes our finding an important supplement to existing online studies . along this way, one interesting corollary according to the self regulation theory is that ambivalent users would like to use shopping to self - correct negative mentality .we test this point by finding the shopping behaviors of ambivalent users around the singles day ( or doule eleven , a famous day for promotion sales in china advocated by taobao inc . ) in 2012 and observing their emotional shifts ( see _ methods _ ) .it is interesting that similar to compulsive buyers  , ambivalent users experience an evident increment in positive mood after shopping .specifically , around 20.5% of ambivalent users post shopping tweets during the promotion sales while the ratio for ordinary users is just 7.8% .moreover , as shown in fig.[fig : shopping_tweet_time_pattern ] , in time window of one day before and after shopping , the fraction of positive tweets posted by ambivalent users grows from 65.2% to 71.5% ( negative tweets drop from 30.2% to 24.9% ) , while the value for ordinary users seems dropped slightly .this reversed shift of sentiments suggests that ambivalent users indeed regulate their negative feelings actively through shopping , which makes them ideal targets for online marketing .in this paper , we take ambivalent expression in weibo as an important clue for tracing unusual online behaviors of ambivalent users .our findings first confirm traditional sociological studies by showing that the ambivalent expression is preferred by the east , especially the female .further investigation reveal various interesting behaviors of ambivalent users , including the topic preference in entertainment and sports events , the increasing communication desire when relaxed , and the private contact to close friends via @ symbol in ambivalent tweets as well as the replies as feedback .moreover , we find a clear negative to positive mode in sentiment shift in ambivalent tweets , which guides us to find reasonable explanations to the above behaviors via the conventional theory of emotion management in sociology . as a result, we conclude that ambivalent expression is an effective way to self - regulate negative feelings in online social media , which is an important supplement to traditional offline sociological studies .extended studies on the self - correcting effect of shopping suggest that our findings could shed light on applications like online marketing .this study inevitably has limitations .first , for the online users investigated , the demographics other than gender , like age , occupation and geo - distribution , are not well discussed for data limitation , and their correlations with ambivalence expression thus remain unclear .second , because of missing a complete online following graph , this study is not embedded into the context of a social network , which otherwise might help us structurally characterize the detailed mechanism of emotional interplay between different users .therefore , further explorations focusing on the above points would be interesting directions in future work ._ weibo data ._ weibo is a twitter - like service in china , which has accumulated more than 500 million registered users since founded in 2009 .we select 137,981 users by breadth - first search starting from several weibo - verified seeds and then crawl each user s profile page .we collect totally 68 million tweets posted by these users in 2012 , among which 22.7% are emoticonized tweets . in this paper, we only conduct the experiment on the data set of 2012 , because from april of 2013 , alibaba has been a strategic cooperative partner of weibo and weibo therefore experiences a sudden growth in marketing tweets , which might badly contaminate the analysis .we select emoticons appeared more than 1,000 times and manually label them as positive or negative according to their images and descriptive words .to ensure the consistency of an emoticon s sentiment orientation , we check the top-200 frequently used keywords extracted from the 1,000 tweets containing that emoticon .the results show that many popular emoticons are strongly and consistently bonded with one specific emotion .finally , we have 115 verified emoticons , among which 79 are positive and 36 are negative .they appear in more than 84.3% of the emoticonized tweets , which consist of 312,456 ambivalent tweets , 13,430,096 positive tweets and still 2,858,413 negative ones .our data sets can be freely downloaded from www.datatang.com/data/47207 or http://pan.baidu.com/s/1mg67cbm .  _ twitter data . _for the comparison purpose , 467 million twitter posts published by 20 million users are employed in this study  , covering about 30% of all public tweets posted in a 7-month period from june 1 , 2009 to dec .31 , 2009 .we use 15 positive smileys ( i.e. , :) " , : d " , `` = d '' , `` =) '' , `` :] '' , `` = ] '' , `` :-) '' , `` : -d '' , `` :-] '' , ;) " ,`` ; d '' , `` ; ] '' , `` ;-) '' , `` ; -d '' and `` ; - ] '' ) and 9 negative smileys ( i.e. , `` :( '' , `` =( '' , `` : [ '' , `` = [ '' , `` :-( '' , `` : - [ '' , `` : ( '' ,`` : [ '' and``d : '' ) , which are also adopted in  for polarity classification . finally , we find 34 million tweets containing at least one smiley , among which 28 million are positive , 6 million are negative , and only 385,145 are ambivalent.  _ computation of topic preference ._ we draw everyday tweets published in 2012 by ambivalent and ordinary users , respectively , for topic inference .let be the number of tweets published by user at day with emotion ( either positive or negative ) .let be the average number of tweets published daily by .then for a group of users , the emotional intensity of at day is given by .we trace continuously along the year 2012 , with being the group of ambivalent and ordinary users and being the positive and negative emotions , respectively , to find emotional peaks that might indicate real events .we finally obtain 16 and 8 peaks with clear event semantics from the top-20 peaks in positive and negative emotion lines , respectively , as shown in fig .[ fig : events ] .the topics discussed by ambivalent users with extracted frequent terms are reported in table [ tab : pos_event ] and table [ tab : neg_event ] .+ * f : jul 27 , 2012 , witness insecurity * + ending , witness insecurity , mr xu , tvb , die + * g : aug 7 , 2012 , xiang liu got injured in the olympic games * + xiang liu , china , london , olympic games , cry , fail down , referee , hero , fighting + * h : sep 30 , 2012 , the final of voice of china * + voice of china , mochou wu , ads , zhiwen jin , shady deal , bo liang , sherry chang huei - mei +  _ computation of emotional shift . _we take all the emoticonized tweets containing at least one of the 115 verified emoticons to study the emotional shift inside a clause of ambivaent tweets .we first cut ambivalent tweets into short clauses by punctuation characters or whitespaces , and label the clauses containing only positive emoticons as positive clauses .we then extract the top-2,000 frequent keywords from the positive clauses , denoted as .similarly , we extract top-2,000 keywords from purely positive tweets and form the keyword set .the different set with then indicates the distinctive keywords used in ambivalent tweets to express positive feeling , which are subject to the emotional orientation test below .that is , for any keyword , we compute the occurrence frequencies of in postive and negative tweets , denoted as and , respectively .let denote the postive intensity of and denote the negative intensity .it is interesting to find that , indicating that the positive clauses of ambivalent tweets actually convey more negative feelings than purely positive tweets .we also find this subtle mismatch from the negative clauses of ambivalent tweets .let denote the distinctive keywords used in ambivalent tweets to express negative feelings with , we have , indicating the stronger positive feelings in ambivalent tweets negative clauses .  _ computation of shopping behaviors ._ we collect all the tweets published by ambivalent users from nov . 4 to nov . 18 , 2012 for identifying their shopping behaviors around the singles day .the tweets containing keywords like `` buy '' , `` shopping '' and `` taobao '' are then selected as candidates and subject to manual labeling for finding shopping tweets depicting users real shopping behaviors . for each shopping tweet, we trace back to its author s tweeting history and calculate the fraction of positive and negative tweets , respectively , in the specified time window before and after shopping .the results of corresponding ambivalent or ordinary users are then averaged to get the final shift patterns .note that we here neglect neutral tweets without emoticons .statistical significance of sentiment shift is also testified by randomly shuffling the posting time of each user s tweets , after which the sentiment shift produced by the shopping behavior disappears .this work was partially supported by national natural science foundation of china ( 71322104 , 71171007 , 71471009 ) , national center for international joint research on e - business information processing ( 2013b01035 ) , national high technology research and development program of china ( 863 program ) ( ss2014aa012303 ) , foundation for the author of national excellent doctoral dissertation of pr china ( 201189 ) , and program for new century excellent talents in university ( ncet-11 - 0778 ) .thanks the fund of the state key laboratory of software development environment ( sklsde-2015zx-28 ) .

Abstract: recent decades have witnessed online social media being a big - data window for quantificationally testifying conventional social theories and exploring much detailed human behavioral patterns . in this paper , by tracing the emoticon use in weibo , a group of hidden `` ambivalent users '' are disclosed for frequently posting ambivalent tweets containing both positive and negative emotions . further investigation reveals that this ambivalent expression could be a novel indicator of many unusual social behaviors . for instance , ambivalent users with the female as the majority like to make a sound in midnights or at weekends . they mention their close friends frequently in ambivalent tweets , which attract more replies and thus serve as a more private communication way . ambivalent users also respond differently to public affairs from others and demonstrate more interests in entertainment and sports events . moreover , the sentiment shift of words adopted in ambivalent tweets is more evident than usual and exhibits a clear `` negative to positive '' pattern . the above observations , though being promiscuous seemingly , actually point to the self regulation of negative mood in weibo , which could find its base from the emotion management theories in sociology but makes an interesting extension to the online environment . finally , as an interesting corollary , ambivalent users are found connected with compulsive buyers and turn out to be perfect targets for online marketing .


in computational mechanics , the complexity of a process that generates a single time series is defined as the least amount of information required for a maximally accurate statistical description of the series .this definition classifies random processes , as well as simple periodic ones , as having low complexity . in a 2004 article ,shalizi et al.extended the definition to spatially extended dynamical systems and introduced an algorithm for measuring the complexity of a discrete system given time series data for all components .applying the algorithm to 2d cyclic cellular automata ( cca ) confirmed that it classified cca rules generating fixed states or incoherent local oscillations as having low complexity and cases that produce turbulent spiral waves as having high complexity .shalizi s complexity measure , , is defined as the amount of information stored in local causal states , where a causal state is an equivalence class of all past configurations that give rise to the same distribution of future outcomes .a set of causal states can be discerned from time series data for all elements in the system . is obtained by measuring correlations between truncated past and future light cones associated with each element in the system .the formalism developed by shalizi et al .is intriguing due to its principled application of information theoretic concepts to the analysis of complex dynamical systems .it is not immediately clear , however , how the formalism can be applied to random or heterogeneous networks .we show here that a natural extension requires distinguishing between averages over nodes and averages over time , and also that the relevant quantities can be computed analytically for certain classes of random boolean networks .our primary goal is to clarify the meaning of shalizi s complexity measure . in doing so ,we propose a new measure of the information processing occurring at a given node , which gives a new statistic for distinguishing the roles played by different nodes in a complex network .we show that this measure depends upon global features of the dynamics and , in particular , that it is not simply related to the sensitivity of the network and is not maximized in critical networks . along the way , we also present a conjecture concerning the dynamics of boolean networks consisting only of parity functions .we study two complexity measures that differ only in the choice of the ensembles of spacetime points used for averaging the local complexity .one approach considers the ensemble of all spatial points at the same time instant , which corresponds to shalizi s . in computing ,all nodes are given equal weight in determining the probabilities of observing different states .shalizi et al .used to investigate self - organization of cellular automata , which are logically and topologically uniform networks of discrete , interacting elements .a second approach is to assign a complexity to each individual element by averaging over time .the system average of these individual complexities is denoted . for a homogeneous system in which all elements have statistically indistinguishable time series ( a regular lattice with rules that lead to turbulent dynamics , for example ) , and are the same .we suggest that is the one that is most informative for spatially inhomogeneous systems .  to develop conceptual insights into the behavior of the quantities and , we study their dependence on the parameters specifying different ensembles of random boolean networks ( rbns ) .rbns were first studied by kauffman as toy models of gene regulatory networks .they have garnered much attention in the last few decades , and features such as steady - state bias , sensitivity to perturbations , attractor length and mutual information between nodes have been extensively investigated .we show here that for any given distribution of logic functions , the value of for a rbn can be analytically calculated as a function of the bias and is not simply related to the sensitivity . on the other hand , is always near zero for sensitivity values , where the network is in the ordered or critical regime , and is also zero for the highest possible value , where the network dynamics is strongly chaotic .thus reflects the intuitive notion that systems with short periodic cycles or apparently random behavior should both have low complexity .the maximum of for rbns occurs somewhere in the disordered regime .we find also that the amount of information processed by any given individual depends on global properties of the network dynamics as well as the logic functions of the node in question and others in its neighborhood .section ii defines the two measures and in detail .section iii describes an implementation of these definitions in the context of rbns and presents theoretical and numerical results on the complexity of a certain class of rbns .the relation between network complexity and sensitivity , and the relation between individual nodes complexity and role in determining the network dynamics is also discussed .we conclude with some general remarks and suggestions for future research .the grassberger - crutchfield - young statistical complexity is defined as the least amount of information about the past trajectory required for optimal prediction of future trajectory , given time series data for a single variable  .this measure is calculated from time series data alone , without reference to the physical laws that govern the dynamical processes .shalizi et al .extended the concept to processes with spatial extent  .we summarize the basic ideas here . for details and information theoretic support of these ideas , see ref . .  given a field that varies over space and time in a system where information propagates at a maximum speed of , the past light cone of a space - time point consists of all space - time points where events can influence .the future light cone consists of all points that can be influenced by . and denote the configurations of the field in the past and futre light cones , respectively : note that each element of consists of three quantities : a field value , a time lapse , and a _ relative _ position .thus ( or ) is exactly the same for two distinct space - time points with identical past ( or future ) light cones .each space - time point is associated with one and one .an ensemble of such pairs defines a probability distribution for future light cone configurations conditioned on past configurations .a causal state is defined as a set of past light cone configurations that have the same distribution of future configurations .all instances of a given causal state predict the same distribution of future light cone configurations :   by definition , is a sufficient local statistic ; knowing the causal state at a given point provides the same predictive power as knowing the exact past light cone configuration . is also a _ minimal _ sufficient statistic  , meaning that the sufficient statistic contains the least amount of information among all statistics that have the same predictive power : \leqslant h[\eta(l^{-})]},\ ] ] where is a sufficient statistic and =-\sum_{i}p(x = x_{i})\log_{2}p(x = x_{i}) ] is the least amount of information for optimal prediction of the future dynamics  , which is taken to be the relevant measure of a system s complexity .we use the shorthand notation }.\ ] ]  the value of depends upon the choice of the ensemble of space - time points used to determine the causal states .we consider two choices .for the first , denoted and studied by shalizi et al . for homogeneous systems , causal states are determined at any given time by considering the ensemble of spatial locations in the system at that time .in other words , causal states are defined as in eq .[ eq : causalstates ] with the restriction that is determined from the set of past and future light cone pairs present at time .this approach allows one to speak of the complexity of a system as a function of time , which may exhibit transient dynamics .systems that exhibit a spontaneous increase in have been described by shalizi et al . as going through a self - organization process . for present purposes, we use to refer to the complexity after transients have decayed .alternatively , for the causal states at a given spatial point are defined using the ensemble of light cone pairs observed at different times . in other words , causal states are defined as in eq .[ eq : causalstates ] with the restriction that is determined from the set of past and future light cones pairs occurring at different times at . receives negligible weight from transients , and in practice we compute it by taking data only after transients have relaxed . because is associated with a particular element of the system , it provides information about the role that element plays in the dynamical evolution of the system . averaging over all , we obtain a global complexity measure .  in practice , estimating the complexity requires restricting the depths of the light cone configurations to a manageable size .for the boolean networks discussed below , we will show that it is sufficient to consider a single update step in the past and a single step in the future .we study and in synchronously updated rbns to determine how ( or whether ) they are related to well - understood measures of the dynamics , such as the sensitivity of the network or the overall bias in the values of the binary variables . in a synchronous rbn , at each time step each node is updated based on a logic function that is applied to the current values of its input nodes : where is a positive integer . here is a quenched boolean logic function for node , chosen randomly from some a weighted distribution over the logic functions with the appropriate number of inputs , and the quenched choice of nodes that act as inputs to are randomly selected , independently for each , from the full set of nodes , with each given equal weight .the s are the binary values of the inputs to .an important feature of rbn s is that the density of feedback or feedforward loops of any finite size goes to zero as the network size goes to infinity  ; the local structure around each node is `` tree - like . ''the number of inputs and outputs per node , which may or may not be the same for all nodes , and the distribution of logic functions are the parameters that characterize an ensemble of synchronous rbns . from these parameters , two global measures ,the bias and the sensitivity , can be calculated analytically  .the bias is the fraction of nodes with value 1 after transients have decayed .let be an input vector of length , and be the fixed number of input per node for a network , then the bias map of the network is given by where denotes expectation taken over distribution of logic functions and is the number of 1s in . the fixed point or cyclic behavior of the biasis determined by the bias map .the sensitivity is the average rate of increase of the hamming distance between two state space trajectories that initially differ at a single node : where and denotes the xor function .the sensitivity distinguishes qualitatively different network behaviors that have been termed ordered ( ) , disordered ( ) , and critical ( )  .  for the present study, we fix the number of inputs and outputs of the nodes such that the light cones of all nodes have the same shape .we study the simplest nontrivial case , in which all nodes have exactly two inputs and two outputs .for these networks , the greatest possible sensitivity is .  ref . outlined an algorithm for computationally distinguishing the causal states of cellular automata , and we can implement similar procedures to distinguish the causal states of our rbns . because nodes in rbns are not assigned spatial positions , the definition of and requires some technical modifications . in the calculation of , of node at timestep is defined as where is the shortest distance between nodes and ( the minimum number of links that must be traversed to get from node to node ) , and the speed of propagation of information is now 1 . by this definition , different nodes that have the same distance from a given node are deemed indistinguishable for the purpose of identifying a light cone configuration .although the update rules ( eq . [ eq : rbn ] ) contain indices for node inputs , the calculation of statistical complexity is defined without reference to the underlying physical laws and hence without the knowledge of which input is which at any given node .  for calculating the complexity of an individual node , however , different inputs and outputs are distinguishable . when comparing two past or future light cone configurations of the same node _i _ at different time steps , one can keep track of which input is which and observe that when the two inputs differ , the future light cone configuration depends upon which input value is 1 .consequently , is defined as     in measuring or for a rbn in the limit of large system size , as opposed to a regular lattice , it is sufficient to restrict the computation to light cones of depth one .the the lack of memory in the update rules ensures that all the relevant information for determining the state of a node is determined by the configuration one time step in the past , and the lack of short loops in the random graph ensures that all information propagating from a given past light cone to a given future light cone must pass through the single node under consideration .[ fig : rbn ] illustrates the latter point . in a regular lattice, a node s past light cone can influence its future light cone through multiple paths , whereas in an rbn , the chance of finding such a path is vanishingly small in the limit of large system size .thus , even though the definition of statistical complexity relies on arbitrarily deep light cones , the results for depth 1 become exact for rbns with system size .this fact enables us to calculate and , in some cases , analytically in the large system limit . is calculated from depth-1 light cones as follows . for a node with two inputs and , there are only three possible past light cone configurations : , , and or ( note we have omitted the relative time and distance entries in writing the light cone configurations because they are now trivial after we restrict the light cone depth to 1 ) , occurring with probability , , and , respectively , and yielding probabilities for the reference node being on of , , and respectively .as discussed above , the probability of observing a future light cone configuration , , depends only on the state of the reference node .thus , if each of the three possible s yields a unique probability for the reference node to be on or off , which in turn yields a unique distribution of s , then each is itself a causal state and we have   modifications to eq .[ eq : cmu ] are required if the above assumptions do not hold .for example , if any two of , and are equal , then there would be less than three causal states .the following scenario is also possible .if the state of a reference node is , the probability that an output is 1 is and if , the probability that an output is 1 is in the case of an accidental degeneracy , the distribution of is independent of and therefore independent of , in which case all three possible collapse to a single causal state , yielding .[ fig : cmu ] shows the comparison between the analytical calculation of and simulation results for ensembles of networks with mixtures of two logic functions .the horizontal axis denotes the fraction of nodes that get assigned the indicated boolean function . the ensemble in fig .[ fig : cmu](a ) satisfies the requirements for eq .[ eq : cmu ] .the ensemble in fig .[ fig : cmu](b ) only has two causal states because == , and =1 , where is the fraction of xor nodes in a xor - on network .the first two terms in eq .[ eq : cmu ] collaps into one for such an ensemble , and the corresponding is given by from eq .[ eq : bias ] we can obtain the fixed - state bias for a xor - on network to be which is monotonically decreasing from to for  ] because monotonically decreases in  ] .we obtain an approximate measure of by taking and considering for . a study of the xor - and and if - and ensembles shows that the correlation coefficient between and in the disordered regime ranges between 0.4 to 0.7 for network size n=1000 .this suggests an interpretation of : it is a measure of importance of the logical processing performed at node for determining the global dynamics .a low value of means that the computations done by node can be effectively simulated by a random number generator .we note that correlations associated with the existence of two paths in the network from one node to another can cause frozen nodes to have nonzero .the correlation arises when a path of unfrozen nodes from some node passes through the frozen node and then immediately to an unfrozen node , while another path of unfrozen nodes of the same length goes from to without passing through .let be the input to along the first path .then and may be correlated due to the common source at node , in spite of the fact that node does not pass on any information .( see fig .[ fig : twopaths ] . ) in the large system limit , the fraction of nodes affected by this type of correlation vanishes .  :node is frozen , but its past and future light cones are correlated because nodes and receive input from the same node through chains of unfrozen nodes that differ in length by exactly two links . ]  because is necessarily zero for any frozen node , the existence of multiple paths in a finite system causes some nodes with high to have low .we further note that itself is not a perfect measure of dynamical importance of a given node s activity because the average euclidean distance between the bias vectors may be small even though the sequence of states is substantially different . in fact , we observe that saturates at low values .comparing with more precise measures of dynamical importance would be an interesting topic for future research .in extending the formalism of shalizi s complexity measure to random boolean networks , a distinction must be made between determining causal states by averaging over nodes at a given time step versus averaging over time for each node separately .the two methods are equivalent for systems described by the same input - output function at every node of a uniform lattice , but yield different results for systems that are spatially inhomogeneous either because the input - output functions are different for different nodes or because the network topology is not a regular lattice .the networks we have studied have both types of inhomogeneity .we find that , obtained from statistics at a single time step , can be calculated analytically for rbn s , and that it is not simply related to sensitivity to small perturbations .the maximum of can occur in the ordered , disordered , or critical regime , depending on the details of the probability distribution chosen for the boolean rules in the network . is directly related to the steady state ( or long - term oscillatory ) bias in the node values . , obtained from statistics compiled over time for individual nodes , is zero ( in the large system limit ) everywhere in the ordered and at the limiting value of sensitivity in the disordered regime , so it is maximized somewhere in the disordered regime .the value at an individual node , , may be interpreted as a measure of the role that node in determining the global dynamics .nodes that can be replaced by random number generators without substantially altering the global dynamics ( and hence perform no essential information processing ) tend to have lower .this last finding may be helpful in characterizing the behavior of social or biological systems and the individual agents or components that comprise them .interestingly , the s depend on the global features of the network that determine its attractors , not just on the local logic functions and topology . in other words, the identification of important players in a network requires global information about the network , not just a characterization of each individual s behavior .              c. r. shalizi , optimal nonlinear prediction of random fields on networks , in _ discrete models for complex systems , dmcs03 _ , edited by m. morvan and .rmila , volume ab of _ dmtcs proceedings _ , pp .1130 , discrete mathematics and theoretical computer science , 2003 .

Abstract: we study two measures of the complexity of heterogeneous extended systems , taking random boolean networks as prototypical cases . a measure defined by shalizi et al . for cellular automata , based on a criterion for optimal statistical prediction [ shalizi et al . , phys . rev . lett . 93 , 118701 ( 2004 ) ] , does not distinguish between the spatial inhomogeneity of the ordered phase and the dynamical inhomogeneity of the disordered phase . a modification in which complexities of individual nodes are calculated yields vanishing complexity values for networks in the ordered and critical regimes and for highly disordered networks , peaking somewhere in the disordered regime . individual nodes with high complexity are the ones that pass the most information from the past to the future , a quantity that depends in a nontrivial way on both the boolean function of a given node and its location within the network .


low - density parity - check convolutional ( ldpcc ) codes have been shown to be capable of achieving capacity - approaching performance with iterative message - passing decoding .the excellent iterative decoding thresholds that these codes display has been attributed to the _ threshold saturation _effect .in addition to good threshold performance , it can also be shown that the minimum free distance typical of most members of these ldpcc code ensembles grows linearly with the constraint length as the constraint length tends to infinity , i.e. , they are _ asymptotically good _ . a large free distance growth rate indicates that codes randomly drawn from the ensemble should have a low error floor under maximum likelihood ( ml ) decoding .generalized ldpc ( gldpc ) block codes were first proposed by tanner and have been shown to possess many desirable features , such as large minimum distance and good iterative decoding performance . following this construction , more complicated constraints than a single parity - check ( spc ) constraint are permitted .in other words , a constraint node with inputs can represent an arbitrary linear block code .in , lentmaier and fettweis showed that ensembles of generalized terminated ldpcc codes , called generalized spatially - coupled ldpc codes ( gsc - ldpc ) codes , constructed from a protograph have better iterative decoding thresholds than their block code counterparts , and that , for large termination lengths , their thresholds coincide with the maximum a - posteriori ( map ) decoding threshold of the underlying gldpc block code ensemble .  in this paper , using weight enumerator evaluation techniques presented by abu - surra , divsalar , and ryan , we study the asymptotic weight spectrum of gsc - ldpc code ensembles .we show , using a -regular gldpc block code with hamming code constraints as an example , that the corresponding gsc - ldpc code ensembles are asymptotically good and have large minimum distance growth rates . as the termination length increases, we obtain a family of codes with capacity approaching iterative decoding thresholds and declining minimum distance growth rates .however , since these are convolutional codes , a more appropriate distance measure for assessing the ml decoding performance of such code ensembles is the _ free distance_ growth rate of the associated ensemble of periodically time - varying generalized ldpc convolutional ( gldpcc ) codes .consequently , in the final part of the paper , we show that the terminated gsc - ldpc code ensembles can be used to obtain an upper bound on the free distance growth rate of ensembles of periodically time - varying gldpcc codes .the free distance growth rate can also be bounded below by using ensembles of tail - biting gldpcc codes using a similar technique to one previously presented for ldpcc code ensembles with spc constraints . by comparing and evaluating these boundswe find that , for a sufficiently large period , the bounds coincide , giving us exact values for the gldpcc code ensemble free distance growth rates .a protograph is a small bipartite graph that connects a set of variable nodes to a set of generalized constraint nodes by a set of edges . in a protograph - based gldpc code, each constraint node can represent an arbitrary block code of length .figure [ fig:27ham ] displays the protograph of a -regular gldpc block code .-regular gldpc block code .the white circles represent generalized constraint nodes and the black circles represent variable nodes .the labels on the edges indicate the corresponding columns of the parity check matrix of the generalized constraint code.,width=230 ]  a protograph can be represented by means of an bi - adjacency matrix , which is called the _ base matrix _ of the protograph .the entry in row and column of is equal to the number of edges that connect nodes and .the base matrix of the protograph in fig .[ fig:27ham ] is given by .\label{hammingbase}\ ] ]  in order to construct ensembles of protograph - based gldpc codes , a protograph can be interpreted as a template for the tanner graph of a derived code , which can be obtained by a copy - and - permute operation .the protograph is lifted by replicating each node times and the edges are permuted among these replicated nodes in such a way that the structure of the original graph is preserved .allowing the permutations to vary over all possible choices results in an ensemble of gldpc block codes .an unterminated gldpcc code can be described by a _convolutional protograph _ with base matrix }=\left [ \begin{array}{cccccc } \mathbf{b}_{0 } &  & \\ \mathbf{b}_{1 } & \mathbf{b}_{0 } &  \vspace{-2.2mm}\\ \vdots & \mathbf{b}_{1 } & \vspace{-2mm}\ddots \\ \mathbf{b}_{m_s } & \vdots & \ddots\\  & \mathbf{b}_{m_s } & \vspace{-2mm}\\  & & \ddots\\ \end{array}\right],\ ] ] where denotes the syndrome former memory of the convolutional code and the _ component base matrices _ , , represent the edge connections from the variable nodes at time to the ( generalized ) constraint nodes at time .an ensemble of ( in general ) time - varying gldpcc codes can then be formed from } ] , we can form a periodically time - varying -fold graph cover with period by choosing , for the submatrices in the first columns of } ] such that , .an ensemble of periodically time - varying gldpcc codes with period , rate , and decoding constraint length can then be derived by letting the permutation matrices used to form , for , vary over the choices of an permutation matrix .suppose that we start the convolutional code with parity - check matrix defined in at time and terminate it after time instants .the resulting finite - length base matrix is then given by }=\left [ \begin{array}{ccc } \mathbf{b}_0 & & \\ \vdots & \ddots &  \\\mathbf{b}_{m_s } &  & \mathbf{b}_0 \\ & \ddots & \vdots\\ & & \mathbf{b}_{m_s } \end{array}\right]_{(l+m_s)b_c \times lb_v}. \vspace{-1mm}\ ] ] the matrix } ] by lifting with some factor has columns and rows , where denotes the number of parity - checks of the constraint code .parity - checks. this assumption can be relaxed in general . ]it follows that the rate of the gsc - ldpc code is equal to for some that accounts for a slight rate increase due to the shortened constraint nodes .if has full rank , the rate increase parameter is given by .however , shortened constraint codes at the ends of the graph can cause a reduced rank for , which slightly increases . in this case , and depends on both the particular constraint code chosen and the degree of shortening .as , the rate converges to the rate of the underlying gldpc block code with base matrix .the generalized convolutional base matrix } ] are removed and added to the first rows to form the tail - biting parity - check matrix with tail - biting termination factor : . ] in such a way preserves the design rate of the ensemble , i.e. , , and we see that has exactly the same degree distribution as the original block code base matrix .in , abu - surra , divsalar , and ryan presented a technique to calculate the average weight enumerator and asymptotic spectral shape function for protograph - based gldpc code ensembles .the spectral shape function can be used to test if an ensemble is _ asymptotically good _, i.e. , if the minimum distance typical of most members of the ensemble is at least as large as , where is the _ minimum distance growth rate _ of the ensemble and is the block length .consider the protograph with generalized constraint nodes shown in fig .[ fig:27ham ] . if we suppose the constraint nodes to be hamming codes with parity - check matrix ,\ ] ] then the resulting ensemble has design rate , is asymptotically good , and has growth rate  .we will construct the base matrix of a gsc - ldpc code ensemble using ( [ convbase ] ) and component base matrices ( [ b1 ] ) and ( [ b2 ] ) .the resulting protograph is shown in fig .[ fig : scprot ] .we use the hamming code with parity - check matrix for the generalized constraint nodes .the numbers on the edges of the protograph in fig .[ fig : scprot ] indicate which columns of ( or shortened version of ) the nodes are connected to .after termination , the resulting ensemble corresponds to a gsc - ldpc code ensemble .the design rate of the ensemble is given as note that in this example because the two leftmost ( shortened ) constraint nodes in fig .[ fig : scprot ] correspond to shortened codes with rate , i.e. , the number of parity - checks in these two constraint nodes is , while all of the other constraint nodes have parity - checks .these ensembles were shown to have thresholds numerically indistinguishable from capacity as in .the evaluation of the asymptotic weight enumerators for gsc - ldpc codes is complex , since the conjecture regarding simplification of the numerical evaluation proposed in can not immediately be applied to these ensembles .this conjecture relies on grouping together nodes of the same type and optimizing them together .in the gsc - ldpc case , nodes from different time instants must be optimized separately , even if they are of the same type . fig .[ fig : termgrowth ] shows the asymptotic spectral shape functions for the gsc - ldpc code ensembles with termination factors and . also shownare the asymptotic spectral shape functions for `` random '' codes of corresponding rate calculated using ( see ) where .we observe that the gsc - ldpc code ensembles are asymptotically good and have large minimum distance growth rates .this indicates that a long code based on this protograph has , with probability near one , a large minimum distance . as increases , the design rate increases and the minimum distance growth rate decreasesthis behavior is the same as was observed in the spc case .in fig . [ fig : termgrowth ] we saw that the minimum distance growth rates of gsc - ldpc codes decrease as the termination factor increases .however , since gsc - ldpc codes are terminated gldpcc codes , a more appropriate distance measure for assessing the ml decoding performance of such codes is the _ free distance _ growth rate of the gldpcc ensemble . in this section ,we first calculate the minimum distance growth rates for gtb - ldpc code ensembles and show that for sufficiently large termination factors , the growth rates coincide with those calculated for the gsc - ldpc code ensembles in section [ sec : dist ] .we then show that the growth rates of the gtb - ldpc code ensembles and gsc - ldpc code ensembles can be used to obtain lower and upper bounds on the free distance growth rate of the gldpcc code ensemble , respectively .we now consider terminating the protograph in fig .[ fig : scprot ] as a gtb - ldpc code with termination factor . unlike the previous termination technique , this results in a -regular protograph with design rate for all .the minimum distance growth rates of the gtb - ldpc code ensembles are presented in fig .[ fig : termtbgrowth ] alongside those corresponding to the gsc - ldpc code ensembles .we observe that the growth rates remain constant at ( the growth rate of the original gldpc block code ensemble ) for and then begin to decay to zero as . also , as a result of the convolutional structure , we observe that the gtb - ldpc and gsc - ldpc growth rates coincide for .this is the same behavior that we observed for tb - ldpc and sc - ldpc codes with spc constraints .consider an ensemble of periodically time - varying gldpcc codes with rate and period constructed from a convolutional protograph with base matrix }} ] ( see ( [ termbase ] ) ) with termination factor .consider a rate unterminated , periodically time - varying gldpcc code ensemble with memory , decoding constraint length , and period derived from }} ] in the gsc - ldpc ( terminated convolutional ) code can also be viewed as a codeword }=[\begin{array}{cccccc } x_0 & x_1 & \cdots & x_{l n b_v-1}&0&\cdots\end{array}] ] .similarly , using a similar argument to that presented in , we have where is the minimum distance growth rate of the gtb - ldpc code ensemble with tail - biting termination factor and base matrix .the free distance growth rate that we bound from above using ( [ ub ] ) is , by definition , an existence - type lower bound on the free distance of most members of the ensemble , i.e. , with high probability a randomly chosen code from the ensemble has minimum free distance at least as large as as .note that the free distance growth rate may also be calculated with respect to the encoding constraint length , which corresponds to the maximum number of transmitted symbols that can be affected by a single nonzero block of information digits . as a result of normalizing by the decoding constraint length , it is possible to have free distance growth rates larger than .for further details , see .   as an example, we consider once more the -regular gldpcc code ensemble with memory and rate depicted in fig .[ fig : scprot ] . for this case , we calculate the upper bound on the free distance growth rate of the periodically time - varying gldpcc code ensemble as using ( [ ub ] ) for termination factors . fig .[ fig : termtbgrowth ] displays the minimum distance growth rates of the gsc - ldpc code ensembles defined by }$ ] for that were calculated using the technique proposed in and the associated upper bounds on the gldpcc code ensemble growth rates for .also shown are the minimum distance growth rates of the gtb - ldpc code ensembles defined by base matrix for and the associated lower bounds on the gldpcc code ensemble growth rates for calculated using ( [ lb ] ) .we observe that the calculated gtb - ldpc code ensemble minimum distance growth rates remain constant for and then start to decrease as the termination factor grows , tending to zero as tends to infinity .correspondingly , as exceeds , the lower bound calculated for levels off at .the calculated gsc - ldpc code ensemble minimum distance growth rates are larger for small values of ( where the rate loss is larger ) and decrease monotonically to zero as . using ( [ ub ] ) to obtain an upper bound on we observe that , for , the upper and lower bounds coincide , indicating that , for these values of the period , , significantly larger than the underlying gldpc block code minimum distance growth rate .in addition , we note that , at the point where the upper and lower bounds on coincide , the minimum distance growth rates for both termination methods also coincide .recall that the gtb - ldpc code ensembles all have rate , wheras the rate of the gsc - ldpc code ensembles is a function of the termination factor given by ( [ ratesc ] ) .this general technique can be used to bound the free distance growth rate above and below for any regular or irregular periodically time - varying protograph - based gldpcc code ensemble .  while large free distance growth rates are indicative of good ml decoding performance , when predicting the iterative decoding performance of a code ensemble in the high snr region other graphical objects such as trapping sets , pseudocodewords , absorbing sets , etc ., come into effect .based on results from the spc case , we would expect gsc - ldpc codes with large minimum / free distance growth rates to also have large trapping set growth rates , indicating good iterative decoding performance in the high snr region .gsc - ldpc codes constructed from a protograph are known to have better iterative decoding thresholds than their block code counterparts , and , for large termination lengths , their thresholds coincide with the map decoding threshold of the underlying gldpc block code ensemble . in this paper, we used an asymptotic weight enumerator analysis to show that gsc - ldpc code ensembles are also asymptotically good .we saw , using a -regular gldpc code as an example , that the corresponding gsc - ldpc code ensembles have large minimum distance growth rates for all computed values of .this indicates that long codes chosen from these ensembles have , with probability near one , large minimum distances as well as excellent iterative decoding thresholds .finally , we obtained asymptotic minimum distance growth rates for gtb - ldpc code ensembles and showed that the growth rates of gtb - ldpc and gsc - ldpc code ensembles can be used to obtain lower and upper bounds , respectively , on the free distance growth rate of the associated periodically time - varying gldpcc code ensemble .this work was partially supported by nsf grant ccf-1161754 .a. e. pusane , r. smarandache , p. o. vontobel , and d. j. costello , jr ., `` deriving good ldpc convolutional codes from ldpc block codes , '' _ ieee trans .inf . theory _ , vol .57 , no . 2 ,835857 , feb .m. lentmaier , a. sridharan , d. j. costello , jr . , and k. sh .zigangirov , `` iterative decoding threshold analysis for ldpc convolutional codes , '' _ ieee trans .inf.n theory _ ,56 , no .10 , pp . 52745289 , oct .m. lentmaier , g. p. fettweis , k. sh .zigangirov , and d. j. costello , jr ., `` approaching capacity with asymptotically regular ldpc codes , '' in _ proc .inf . theory and app .workshop _ , san diego , ca , feb .s. kudekar , t. j. richardson , and r. l. urbanke , `` threshold saturation via spatial coupling : why convolutional ldpc ensembles perform so well over the bec , '' _ ieee trans .inf . theory _ ,57 , no . 2 ,803834 , feb . 2011 .a. sridharan , d. truhachev , m. lentmaier , d. j. costello , jr . , and k. sh .zigangirov , `` distance bounds for an ensemble of ldpc convolutional codes , '' _ ieee trans .inf . theory _53 , no . 12 , pp .45374555 , dec . 2007 .d. g. m. mitchell , a. e. pusane , and d. j. costello , jr ., `` minimum distance and trapping set analysis of protograph - based ldpc convolutional codes , '' _ ieee trans .inf . theory _ ,59 , no . 1 ,254281 , jan . 2013 .d. truhachev , k. sh .zigangirov , and d. j. costello , jr ., `` distance bounds for periodically time - varying and tail - biting ldpc convolutional codes , '' _ ieee trans .inf . theory _ ,56 , no . 9 , pp . 43014308 , 2010 .m. lentmaier , d. g. m. mitchell , g. p. fettweis , and d. j. costello , jr ., `` asymptotically regular ldpc codes with linear distance growth and thresholds close to capacity , '' in _ proc .inf . theory and app .workshop _ , san diego , ca , feb .

Abstract: families of generalized spatially - coupled low - density parity - check ( gsc - ldpc ) code ensembles can be formed by terminating protograph - based generalized ldpc convolutional ( gldpcc ) codes . it has previously been shown that ensembles of gsc - ldpc codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts , and that , for large termination lengths , their thresholds coincide with the maximum a - posteriori ( map ) decoding threshold of the underlying generalized ldpc block code ensemble . here we show that , in addition to their excellent iterative decoding thresholds , ensembles of gsc - ldpc codes are asymptotically good and have large minimum distance growth rates .