hierarchical organisation is a widespread phenomenon in nature and society .signs of hierarchy were recorded in various animal flocks , in social interactions , in urban planning , in ecological systems and in evolution . since a natural representation of hierarchiesis given by directed acyclic graphs , hierarchical organisation became a very relevant concept also in complex network theory .the association of tags to various on - line contents have became ubiquitous , as various tags may indicate the topic of news - portal feeds and blog post , the genre of films or music records on file sharing portals , or the kind of goods offered in web stores .these tags usually serve as keywords , providing a rough description of the given entity , helping the users in a fast decision whether the given article , film , etc .is of interest or not .keywords , categories , classes , etc . are also used in e.g. , library classification systems and biological classification for helping the search and browsing amongst a large number of objects . in the latter casesthe involved entities are categorised hierarchically , with a set of narrower or broader categories building up a tree - like structure composed of `` is a subcategory of '' type relations .in contrast , the tags appearing in on - line platforms are usually free words chosen by the author or owner of the given object , and tags are almost never organized into a pre - defined hierarchy of categories and sub - categories . furthermore , in many tagging systems like flickr , citeulike or delicious the tagging process is collaborative , as in principle an unlimited number of users can tag photos , web pages , etc . ,with free words . in order to highlight this collaborative nature ,the arising set of free tags and associated objects are often referred to as folksonomies .since the tagging actions involve user - tag - object triplets , a natural representations of these systems is given by hypergraphs , where the hyperedges connect more than two nodes together .an interesting problem related to free tagging is to extract a hierarchy between the tags based on their co - occurrences on the tagged items .the basic motivation is that the way users think about objects presumably has some built in hierarchy , e.g. , `` pigeon '' is usually considered as a special case of `` bird '' . revealing the hidden hierarchy between tags in a folksonomy or in a tagging system in general can significantly help broadening or narrowing the scope of search in the system , give recommendation about yet unvisited objects to the user , or help the categorisation of newly appearing objects .here we apply an improved version of a recent tag hierarchy extraction method to keywords associated to on - line articles , collected from the portals of spiegel online , the guardian , the new york times and the australian .the obtained hierarchies show very interesting differences , indicating that the methods for choosing keywords are based on rather different principles in the studied journals .the structure of the paper is as follows : in , the principles of hierarchy construction and comparison are shortly described . in the empirical datasetsare presented , in we compare the obtained hierarchies with each other , and in results are discussed .we employ an upgraded version of a recent method , which is based on two assumptions :  * tags positioned high in the hierarchy also have high centrality values in the tag - tag coappearance graph , * parent - child pairs coappear more frequently than expected from pure chance .according to the first assumption , the algorithm orders the tags by their centrality , then , for each tag ( which become child ) the parent candidates are collected .all tags with higher centrality are parent candidates of the child tag .candidate parents are assigned a score , indicating the probability of the observed number of co - occurrences according to a random null - model . using the second assumption ,the final parent is the candidate with the highest score sum , where the sum runs over all descendants of the child tag .note , that the algorithm builds up the hierarchy bottom up , starting from the leaves with lowest centrality .the full detailed description of the currently used version of the algorithm involving a couple of improvements is given in the supplementary information .hierarchies are frequently represented by directed acyclic graphs ( dags ) , in which directed cycles are forbidden .however , children are allowed to have more than one parent in general . for simplicity ,we have restricted the number of parents to one in the present analysis .a natural idea for comparing two dags is to compare the hierarchical relations , i.e. , the sets of ancestor - descendant relationships .here we adopt the approach proposed in ref . , defining a similarity measure based on mutual information .we note that mutual information plays a central role also in the comparison method introduced in for the related , but separate problem of comparing hierarchical community structures , ( where only the lowest - level nodes in dag actually exist in the input data - set ) .the dag similarity measure we use can be formulated as follows  where and are two dags , having and tags from which are common , and is the set of descendants of in dag .equation [ eq : infsym ] is 0 for independent dags and 1 for identical ones .  a further very closely related similarity measure that turned out to be useful in previous studies is given by the linearised mutual information ( lmi ) , based on the fraction of links that have to be rewired in a randomisation procedure on leading to a hierarchy with the same nmi when compared to as the .the formal definition of this measure is given as follows .let denote the average nmi obtained for a fraction of randomly rewired links , . by projecting the nmi of the empirical case , , to the axis using this function as receive the fraction of randomly chosen links to be rewired in the empirical case for obtaining a randomized hierarchy with the same nmi .based on that we define the linearised mutual information , ( lmi ) as this quality measure corresponds to the fraction of unchanged links in a random link rewiring process , resulting in a hierarchy with the same nmi as the empirical value .( the reason for calling it `` linearised '' is that equation [ e : lin ] is actually projecting to the linear curve ) .we analyse four tagged datasets , obtained from online news portals .they contain tagged news items , covering a more than 2 years long time window , in the same period .the four sources are : spiegel online , the guardian , the new york times and the australian .there are a few observations which hold for all four datasets .for example , very long tags exist , more like headlines ( `` ` muntazer al - zaidi : the iraqi shoe thrower ` '' ) .some of the tags form `` frozen '' cliques in the coappearance network , where each member of such a clique appear only together with the other members of the clique , e.g. , ` haiti ` and ` haiti earthquake disaster 2010 ` , ` diana ` and ` princess of wales ` .since members of a large clique have large centrality values , such tags will be placed to unwanted high positions by the first step of the hierarchy construction algorithm .therefore we have considered such `` frozen '' cliques as single tags , which fits better to the assumed usage of tags .some concepts are represented by two or more tags , where the same idea is expressed with different , but synonymous words , e.g. , ` art ` and ` arts ` .these were left as observed , unless explicitly stated otherwise .another problem is posed by the occurrence of very rare tags , that are usually names .  in order to avoid misleading results due to the above observed problems, we have prefiltered the tags by requiring that each tag pair in the coappearance network has to occur on at least news items . the case corresponds to skipping the prefiltering .we set to its optimal value for each dataset by keeping the number of tags as high as possible and minimizing the number of misleading tags described above .finally we note , that temporarily important topics can produce unexpected co - occurrences ( e.g. , `` ` japan ` '' - `` ` fukushima nuclear catastrophe ` '' - `` ` nuclear power ` '' ) .the dataset is from april 2011 to january 2013 .it contains 4802 news items and 388 tags . for the pre - filtering , minimum 1 common news item for each tag pair ( i.e. , no filtering ) seems to be a good tradeoff between noise reduction and info loss .the dataset looks very well organised ( e.g. , there are only 400 tags , general tags are used consistently , and there are only a few duplicated tags , long tags or `` frozen '' cliques ) .the dataset is from november 2009 to january 2013 , containing 55835 news items and 6797 tags .pre - filtering needs minimum 3 news items ( removes 2530 tags and 61 news items ) .here we found several ad hoc tags ( mostly names ) , that were used only once or a handful of times .we found synonymous tags , e.g. , `` ` middle east and north africa ` '' and `` ` middle east ` '' , that will appear as two local roots of two branches in the dag .these branches correspond to the same topic , thus divide the related tags between them .the dataset reaches from november 2010 to january 2013 .it contains 35736 news items and 23009 tags .cliques are a huge problem here .there are 2902 ones , collapsing them removes about 6000 tags .several cliques appear on numerous objects , therefore the minimum news item - filtering does not solve the problem automatically . cliques also reach very large sizes : there is a 809-tag clique ( may contain much more characters than a news item itself ) ; after the minimum news items filtering , the largest one still consists of 44 tags as follows from the definition of cliques , these tags appear strictly together on each object . for the pre - filtering ,minimum 5 news items were required , leaving finally 2981 tags ( out of 23009 ) .news items were much less affected , 31184 out of 35736 remained .data is from december 2009 to january 2013 .it contains 31501 news items and 79054 tags thus , there are much more tags than news items .cliques are present , but have only 1 - 2 objects , so it is not a serious problem , the pre - filtering can solve it .multiple synonyms occur on the same object very often e.g. , `` ` economist_paul_samuelson ` '' `` ` paul_a._samuelson ` '' `` ` paul_samuelson ` '' .another example is the set of synonyms for barack obama , which are : ` barack_obama ` , ` barack_obama ` , ` obama ` , ` president_barack_obama ` , ` president_barack_obama ` , ` president_obama ` , ` us_president_barack_obama ` , ` us_president_barack ` , ` us_president_barack_obama ` , ` barack_obama ` .pre - filtering with minimum 5 news items leaves 1673 tags out of 79504 .the news items are reduced from 31501 to 10550 .the tags have relatively few objects , and not only due to the large number of very infrequent tags , e.g. , even the prime minister has only 900 objects .although there are very general tags like ` community ` , ` committee ` or ` claim ` , most of tags are very specific , almost tailored for one object , e.g. , ` rebels_storm_gaddafi_compound ` .we analysed the tag hierarchies obtained from an improved version of `` algorithm b '' published in ref . ; a brief description of the idea of the method can be found in , the full details of the used algorithm are given in the supplementary information .in first we summarise the most important properties of the individual hierarchies corresponding to the different news portals , which is followed by the pairwise comparisons in .finally , in we examine the overall quality of the hierarchies from different aspects .   *spiegel online * the constructed dag consists of 1 connected component .most of the tags are under 3 branches : `` ` world ` '' , `` ` europe ` '' , `` ` germany ` '' .a visualisation of the dag is shown on fig .[ f : s_overw ] .the spiegel dag seems to be somewhat concerned with immigrants and integration , they have a branch containing of the tags , similarly to australian s , and in contrast to and of guardian and nyt ( note that the latest data come from january 2013 , well before the beginning of the recent migrant crisis ) .  * the guardian * the overall structure of the dag is quite well organised , the top 2 - 3 levels are very impressive .the dag consists of four similarly - sized connected components : `` ` uk news ` '' , `` ` world news ` '' , `` ` culture ` '' , `` ` sport ` '' , although the tags `` ` world news ` '' and `` ` uk news ` '' are in isolated components , they are not completely mutually exclusive , e.g. , both of them appear on the news items of `` defence policy '' . note that while the components top tags correspond well to the menu items on the journal s website , they are placed totally automatically by the dag construction algorithm .visualisation is omitted due to the relatively large size of the dag , however , a smaller sample is shown in the supplementary information .    * the new york times * here we found numerous duplicated branches in the constructed dag ( e.g. , for research , television , education , medicine , defence and military forces ) .this indicates that for these topics , two distinct sets of tags were used in parallel .the dag is much less organised than that of the spiegel and of the guardian .there are 31 isolated components , most of them correspond to one theme ( e.g. `` ` baseball ` '' ) .the sizes of the components varies from 898 to 2 , and there is a continuous range of them from the 2nd largest one ( 274 tags ) down .there are no very general categories .although a number of large related components exists ( under the tags `` ` basketball ` '' , `` ` baseball ` '' , `` ` football ` '' ) , these components are not collected under a general `` ` sport ` '' tag .it seems as if there were no demand for using general tags .note that there is a tag called `` ` sports ` '' , however , it appears only on 5 news items , and it is negligible .a technical consequence is that the dag construction algorithm does not always select the most general tags as roots , because they lack the important connections to other components . instead, one of the more specific tags can be selected for a central position , for example , `` middle east and north africa unrest ( 2010- ) '' for foreign affairs , or `` ` european sovereign debt crisis ( 2010- ) ` '' for europe - related tags . in other words ,the centrality no longer correlates only with the generality for the top tags .some lower - level branches end up at unexpected places , e.g. , ` environment ` under ` iran ` .superfluous levels appears , for example , ` international relations ` under ` united states international relations ` .  *the australian * the dag looks disorganised overall .there are about 1900 components for the 79504 tags without the pre - filtering , and about 300 components for the min .5 news items - filtered 1673 .there are no macroscopic components , the largest one s size is just 3480 ( out of 79504 tags ) and 165 ( out of 1673 tags ) , which is less than 10% of the total nodes .even the existing components look more like just bunches of more or less associated tags than small hierarchical structures .  in general , the top of the constructed dags are much better than the bottom .this is no surprise - there is much more information for the construction algorithm at the top of the dag .we carried out a pairwise comparison between the journals from the point of view of their content organisation .since the audience and the interests of the journals are different , the list of tags appearing on the articles was unique for each news portal . therefore , before actually comparing the tag hierarchies , first we needed to create a common tag set for each pair of journals . in a number of cases , finding the corresponding tag pairs went beyond a simple string matching andwas based on semantic matching , e.g. , `` ` fossil fuels ` '' ( guardian ) was matched with `` ` oil ( petroleum ) and gasoline ` '' ( nyt ) .the size of the reduced common tag sets were 252 ( spiegel - guardian ) , 217 ( spiegel - nyt ) , 985 ( guardian - nyt ) , 93 ( australian - spiegel ) , 278 ( australian - guardian ) , 274 ( australian - nyt ) .the reduced hierarchies were obtained by keeping only the common tags in the original dags and erasing the rest of the tags . in most casesthis resulted in deletion of leafs , sub - branches , or lower parts of sub - branches from the original hierarchies .however , a small number of times this procedure erased a tag higher in a given branch while keeping other tags lower in the same branch , therefore distorting the original dag structure in a radical way .to ensure as much similarity to the original hierarchies as possible , under these circumstances the ancestors standing higher in the branch were also kept , despite that they were not part of the common tag set , ( see the si for more details ) .the reduced dags can be found in the si .  for each pair of journalswe have computed the linearised information similarity measure described in between the reduced dags , the obtained values are shown in fig .[ f : table ] . according to the resultsspiegel and guardian provide the largest similarity measure , which is also supported by a number of identical or almost identical sub - branches between the two dags , as shown in fig .[ f : sg ] . herethe background colouring of the sub - branches indicate the similarity to the corresponding ( most similar ) sub - branch in the other dag .the spiegel , the guardian and the new york times have an overall similar structure , as fig . [f : table ] shows , opposed to the australian , which is dissimilar to all of them . still , there are some differences between the first three journals .the guardian , compared to the spiegel , has a level of intermediately - sized branches , e.g. , ` law ` or ` society ` in ` uk news ` .this level is missing from the dag of spiegel .their global dag structures are shown in fig .[ f : sg ] .   to the given branch and the most similar branch from the other hierarchy .note that sub - branches on all hierarchical levels have their own colour . ]meanwhile , the new york times has interestingly no ` world ` tag , and foreign countries are separated into 4 different branches , in 3 components ( see the si for more details ) .although the linearised information similarity between the guardian and the new york times is somewhat lower , they also have a few quite similar branches ; a prominent example is shown in fig .[ f : gn ] .     according to the results presented in the previous sections the tag hierarchies obtained for the studied journals show strong differences . herewe examine to what extent does their overall structure follow a few simple intuitive requirements that can be formulated for a well organised tag hierarchy .  * correlations with google news .* one of the basic properties of a well organised hierarchy is that frequent , more general tags are expected to be higher compared to rare , specific tags . in order to examine the obtained hierarchies from this perspective we compared the centrality score of the tags in the tag co - occurrence network ( determining their position in the hierarchy ) with their number of hits provided by google news . for each pair of tags with a significant number of co -occurrence we checked whether the difference between their centrality score and the difference between their number of hits in google news have the same or the opposite sign . if the signs of the differences match for the majority of the tag pairs , then we can assume that the structure of the hierarchy is consistent with word frequencies of english news texts around the world .  in table[ t : inversions ] we show the relative frequency of the cases , where the differences have the opposite sign , calculated for tag pairs co - appearing in statistically significant numbers . if tags are assigned to articles absolutely at random , the result would correspond to a inversion rate , i.e. , half of the coappearing tag pairs would have similar centrality and frequency ordering ..ratios of inversions between centralities and real - world occurrence frequencies , calculated for tag pairs coappearing in statistically significant numbers .totally random case corresponds to . [ cols="^,^",options="header " , ]   spiegel s is dominated by a single contribution .the global root , ` international ` has a branch containing almost the whole dag under ` news ` .most of comes from small branches , although there are a few exceptions . in the guardian dag ,dwarf sub - branches are common , due to the huge size of the components which dwarf several branches , as well as to nearly star - shaped branches , sometimes containing hundreds of leaf - tags ( e.g. , ` film ` , ` music ` ) . for the nyt ,contrary to the guardian , is much larger than .two important reasons are misplacing a number of branches and letting less general tags getting high centralities . since the australian dag has quite limited structure inside the numerous small components , and are not very informative measures here .however , the tiny components seem to be well balanced .  further analysis of the dags can be found in the si .we studied the hierarchy of keywords associated to news articles in four different on - line news portals .the datasets contain various artefacts , such as long and complex keywords , `` frozen '' cliques of exclusively coappearing tags , synonyms or very rare and specific tags . nonetheless , it was possible for the construction method to obtain very reasonable dags from the data .the identification of frozen cliques might also be applied by disambiguation techniques , to identify cliques of equivalent semantic meaning , used in the field of natural language processing .the constructed dags suggest that the tags appearing in the different news portals are organised to different degrees .our analysis revealed that guardian has an extra intermediate level of organisation at certain locations .a further very interesting result is that the number of connected components in the dags conveys information about the extent of organisation in the data : the spiegel and guardian have components and are quite organised , the new york times has a few dozen components and breaks the world into independent pieces , and the australian has components which are barely informative at all .a similar picture was emerging from the comparison between the frequencies of tags in google news and their centrality score in the tag - tag co - appearance graphs .the correlation was quite strong in case of the spiegel and the guardian , medium for the new york times , and almost equivalent to the totally random case for the australian .a more detailed characterisation of the dags can be obtained by quantifying the extents of too large and too small sub - branches .although being a geometry - based analysis , it can also identify problems with tag functions , like a non - comprehensive set of intermediate - level branches in the guardian , or misplaced branches in the new york times.  in summary , the following picture is arising from the different analyses we carried out : the spiegel and guardian datasets are quite well - organised , the new york times is significantly less but still has relevant hierarchical structure , and the australian is close to being random , from a hierarchical point of view .the consistency of the results is encouraging , and suggests that the measures used are useful in the quantification and comparison of datasets from the aspect of hierarchical organisation .10 goessmann , c. , hemelrijk , c. & huber , r. the formation and maintenance of crayfish hierarchies : behavioral and self - structuring properties . * 48 , * 418428 ( 2000 ) .  nagy , m. , kos , z. , biro , d. & vicsek , t. hierarchical group dynamics in pigeon flocks .* 464 , * 890893 ( 2010 ) .nagy , m. et al .context - dependent hierarchies in pigeons .* 110 , * 1304913054 ( 2013 ) .fushing , h. , mcassey , m. p. , beisner , b. & mccowan , b. ranking network of captive rhesus macaque society : a sophisticated corporative kingdom . * 6, * e17817 ( 2011 ) .guimer , r. , danon , l. , daz - guilera , a. , giralt , f. & arenas , a. self - similar community structure in a network of human interactions .* 68 , * 065103 ( 2003 ) .pollner , p. , palla , g. & vicsek , t. preferential attachment of communities : the same principle , but a higher level . * 73 , * 478484 ( 2006 ) .valverde , s. & sol , r. v. self - organization versus hierarchy in open - source social networks .* 76 , * 046118 ( 2007 ) .krugman , p. r. confronting the mystery of urban hierarchy .* 10 , * 399418 ( 1996 ) .batty , m. & longley , p. ( academic , 1994 ) .hirata , h. & ulanowicz , r. information theoretical analysis of the aggregation and hierarchical structure of ecological networks .* 116 , * 321341 ( 1985 ) .wickens , j. & ulanowicz , r. on quantifying hierarchical connections in ecology .* 11 , * 369378 ( 1988 ) .eldredge , n. ( oxford university press , 1985 ) .mcshea , d. w. the hierarchical structure of organisms .* 27 , * 405423 ( 2001 ) .ravasz , e. , somera , a. l. , mongru , d. a. , oltvai , z. n. & barabsi , a .-hierarchical organization of modularity in metabolic networks .* 297 , * 15511555 ( 2002 ) .trusina , a. , maslov , s. , minnhagen , p. & sneppen , k. hierarchy measures in complex networks .* 92 , * 178702 ( 2004 ) .clauset , a. , moore , c. & newman , m. e. j. hierarchical structure and the prediction of missing links in networks .* 453 , * 98101 ( 2008 ) .pumain , d. , vol . 3 _ methodos series _ ( springer , 2006 ) .corominas - murtra , b. , rodrguez - caso , c. , goi , j. & sol , r. v. measuring the hierarchy of feedforward networks .* 21 , * 016108 ( 2011 ) .mones , e. , vicsek , l. & vicsek , t. hierarchy measure for complex networks .* 7 , * e33799 ( 2012 ) .corominas - murtra , b. , goi , j. , sol , r. v. & rodrguez - caso , c. on the origins of hierarchy in complex networks .* 110 , * 1331613321 ( 2013 ) .kaiser , m. , hilgetag , c. c. & ktter , r. hierarchy and dynamics of neural networks .* 4 , * 112 ( 2010 ) .ma , h. w. , buer , j. & zeng , a. p. hierarchical sructure and modules in the escherichia coli transcriptional regulatory network revealed by a new top - down approach .* 5 , * 199 ( 2004 ) .mika , p. ontologies are us : a unified model of social networks and semantics in _ the semantic web iswc 2005 _ , vol .3729 ( eds gil , y. , motta , e. , benjamins , v. r. & musen , m. a. ) in _ lect .notes comput ._ , 522536 ( springer , 2005 ) .spyns , p. , moor , a. d. , vandenbussche , j. & meersman , r. from folksologies to ontologies : how the twain meet in _ on the move to meaningful internet systems 2006 : coopis , doa , gada , and odbase _ , vol .4275 ( eds meersman , r & tari , z. ) in _ lect .notes comput ._ , 738755 ( springer , 2006 ) .voss , j. tagging , folksonomy & co - renaissance of manual indexing ?_ arxiv preprint arxiv : cs/0701072v2 _ ( 2007 ) .tibly , g. , pollner , p. , vicsek , t. & palla , g. ontologies and tag - statistics .* 14 , * 053009 ( 2012 ) .cattuto , c. , loreto , v. & pietronero , l. semiotic dynamics and collaborative tagging .* 104 , * 14611464 ( 2007 ) .lambiotte r. & ausloos , m. collaborative tagging as a tripartite network in _ computational science iccs 2006 _ , vol .3993 ( eds alexandrov , v. n. , van albada , g. d. , sloot , p. m. a. & dongarra , j. ) in _ lect .notes comput ._ , 11141117 ( springer , 2006 ) .cattuto , c. , barrat , a. , baldassarri , a. , schehr , g. & loreto , v. collective dynamics of social annotation .* 106 , * 1051110515 ( 2009 ) .ghoshal , g. , zlati , v. , caldarelli , g. & newman , m. e. j. random hypergraphs and their applications .* 79 , * 066118 ( 2009 ) .zlati , v. , ghoshal , g. & caldarelli , g. hypergraph topological quantities for tagged social networks .* 80 , * 036118 ( 2009 ) .floeck , f. , putzke , j. , steinfels , s. , fischbach , k. & schoder , d. imitation and quality of tags in social bookmarking systems - collective intelligence leading to folksonomies in _ advances in intelligent and soft computing _ , vol .76 ( eds bastiaens , t. j. , bauml , u. & krmer , b. j. ) ch . on collective intelligence , 7591 ( springer , 2011 ) .  lu , l. et al .recommender systems .* 519 , * 149 ( 2012 ) .heymann , p. and garcia - molina , h. , _ collaborative creation of communal hierarchical taxonomies in social tagging systems .stanford infolab technical report ._ ( 2006 ) available at : ` http://ilpubs.stanford.edu:8090/775/?auth=basic . `( accessed : 18th january 2012 )  plangprasopchok , a. , lerman , k. & getoor , l. a probabilistic approach for learning folksonomies from structured data . in _fourth acm international conference on web search and data mining ( wsdm ) _ , 555564 ( 2011 ) .schmitz , p. inducing ontology from flickr tags .in _ proceedings of collaborative web tagging workshop at the 15th international conference on world wide web ( www ) _ ( 2006 ) .van damme , c. , hepp , m. & siorpaes , k. folksontology : an integrated approach for turning folksonomies into ontologies . * 2 , * 5770 ( 2007 ) .tibly , g. , pollner , p. , vicsek , t. & palla , g. extracting tag hierarchies .* 8 , * e84133 ( 2013 ) .juszczyszyn , k. , kazienko , p. & katarzyna , m. personalized ontology - based recommender systems for multimedia objects in _ studies in computational intelligence _ , vol . 289 ( eds hkansson , a. , hartung , r. & nguyen , n. ) ch .agent and multi - agent technology for internet and enterprise systems , 275292 ( springer , 2010 ) .fattore , m. , grassi , r. & arcagni , a. measuring structural dissimilarity between finite partial orders in _ multi - indicator systems and modelling in partial order _( eds brggemann , r. , carlsen , l. & wittmann , j. ) , 6984 ( springer , 2013 ) .brandenburg , f. j. , gleiner , a. & hofmeier , a. the nearest neighbor spearman footrule distance for bucket , interval , and partial orders .* 26 , * 310332 ( 2013 ) .palla , g. , tibly , g. , mones , e. , pollner , p. & vicsek , t. hierarchical networks of scientific journals .* 1 , * 15016 ( 2015 ) .tibly , g. , pollner , p. & palla , g. partial order similarity based on mutual information ._ arxiv preprint arxiv:1601.05922 _ ( 2016 ) .tessone , c. j. , perotti , j. i. & caldarelli , g. hierarchical mutual information for the comparison of hierarchical community structures in complex networks . * 92 , * 062825 ( 2015 ) .financial support of the hungarian national science fund ( otka k105447 ) is acknowledged .gt , pp and gp conceived and designed the experiments .dsr provided the datasets .gt analysed the data .gt and gp wrote the manuscript .all authors reviewed the manuscript .* competing interests * : the authors declare no competing financial interests .

Abstract: the tagging of on - line content with informative keywords is a widespread phenomenon from scientific article repositories through blogs to on - line news portals . in most of the cases , the tags on a given item are free words chosen by the authors independently . therefore , relations among keywords in a collection of news items is unknown . however , in most cases the topics and concepts described by these keywords are forming a latent hierarchy , with the more general topics and categories at the top , and more specialised ones at the bottom . here we apply a recent , cooccurrence - based tag hierarchy extraction method to sets of keywords obtained from four different on - line news portals . the resulting hierarchies show substantial differences not just in the topics rendered as important ( being at the top of the hierarchy ) or of less interest ( categorised low in the hierarchy ) , but also in the underlying network structure . this reveals discrepancies between the plausible keyword association frameworks in the studied news portals .


in recent years , most industrialized societies have started to see the limits of the growth of urban traffic .the traffic demand in metropolitan areas has largely exceeded the vehicular capacity and we face the problem of increasing pollution and growing frequency of accidents .typical remedies such as the expansion of road way systems or road improvement do not work well anymore . as a consequence ,many metropolitan areas seek to solve this problem by constructing additional roadway , but this is undesirable due to political and environment objections or impractical because of the high costs of construction .thus , the consistent management of large , distributed man - made transportation systems has become more and more important for planning and prediction of traffic .computer simulations as a means of evaluating , planning , and controlling traffic systems have gained considerable importance . lately , more and more work has been done to consider the dynamical aspects of the traffic system .specially , the micro - simulation work has taken great advantage of models developed in classical research areas such as physics , mathematics , and computer science .computer models can be used to simulate the influence of governmental actions like road - pricing or building of new highways , , , in solving the traffic problems . moreover, output data of traffic models can be used as an input to climate models for calculating traffic - caused air pollution and the dependency of this pollution on time - limits .  for this reason , in the last few yearsthere has been considerable interest in the investigation of traffic flow using methods and models from statistical physics , especially the cellular automata ( ca ) models have allowed the large - scale simulations of large traffic networks faster than real time .these kind of models are so - called _ particle hopping models_. recently , particle hopping models using ca has emerged as a very promising alternative to describe the traffic flow , , , .cellular automata are alternatives to differential equations in an attempt to model transportation systems .ca s are dynamical systems in which space and time are discrete .a cellular automaton consists of a regular grid of cells , each of which can be in one of a finite number of possible states , updated synchronously in discrete time steps according to a local , identical interaction rule .the state of a cell is determined by the previous states of the surrounding neighborhood of cells .thus , ca models have the desirable capacity to capture micro - level dynamics and relate these to macro - level behavior .also they are capable of representing individual vehicle interactions and relating these interactions to macroscopic traffic flow metrics , such as throughput , time travel , vehicle speed , etc .in addition , ca models are suitable to represent both single and multi - lane traffic in an easy form , which is particularly crucial for the modeling of highways .the technical difference between car - following and ca models for traffic flow is that in the latter , space and time are discrete , whereas in the car - following models , they are continuous . simulations of car - following models use discrete time but the space is continuous .the objective of this paper is to explore a new minimal ca model for the high - speed simulation of microscopic traffic alternatives .we use particle hopping models as a starting point for this investigation because their highly discrete nature reduces the number of free parameters . here, we present a new extension of a ca model for a single - lane highway on a ring built on the pioneering work of nagel ( ns ) and his colleges .the ns model has been applied to the project transims on transportation simulation .this model consider four rules to update the state of the road involving braking , acceleration , stochastic driver reaction , and the car movement .the ns model decelerates a vehicle if its distance from the vehicle in the front is less than its current speed but does not consider the speed of the vehicle ahead . by including a parameter of safe distance we determine the distance that one vehicle must maintain with respect to the vehicle in the front by taking their speeds into account .the knowledge of this speed is a property of some kind of automated vehicles .we analyze the relationship between the flow and the density , the so - called fundamental diagram , and we find that the new model works better under the condition of high traffic density . by analyzing the variance of speed and its relationship with the flow , we show that the behavior of this variance can have important consequences for the design and control of traffic flow . we think that knowledge of this relationship can be very useful for analyzing _ automated system _ especially for vehicles equipped with infrared sensors to determine the distance and the velocity from other vehicle . recently such vehicles have been used to study an automated traffic . in the following we will refer as automated vehicles to this kind of vehicles .the paper is organized as follows . in sect .2 , we give a brief description of our model which modifies the existing ns model by an inclusion of a safety criterion . in section 3 ,the results of computer simulations from the new model are presented , by analyzing fundamental diagrams for different values of the safety parameter . in this section, we also show that a suitable safety factor produces a lesser speed variance .finally , sect .4 concludes with a short summary and discussion of our findings .our model is a probabilistic cellular automaton .it consists of cars moving on a one - dimensional lattice of cells with periodic boundary conditions ( the number of vehicles is conserved ) .space and time are discrete and hence also the velocities .each cell is either empty , or it is occupied by just one vehicle ( see fig .1 ) with a discrete velocity with the same limt speed . in the present paper ,velocity ranges are from the limit speed .this limit speed may be different depending of the kind of vehicle under consideration , trucks , cars , etc .however , here will use the same maximum velocity for all the vehicles , then we consider only one type of car .the velocity is equivalent to the number of sites that a vehicle advances in one time step provided that there are no obstacles ahead .vehicles only move in one direction .the length of a cell is around  , which is interpreted as the length of a vehicle plus the distance between vehicles in a jam , but can be suitably adjusted with respect to the problem under consideration .one time step ( ) lasts 1 s , which is of the order of the reaction time of humans .then , means km / h in real units .let and denote the current velocity and position respectively of the vehicle and and be the velocity and position respectively of the vehicle ahead ( preceding vehicle ) in a fixed time .we denote by , the distance ( number of empty cells ) in front of the vehicle in the position . in the context of ca models proper unitsare often omitted due to the given discretization of space and time .thus , the proper units for the model would be : =]number of cells per time step , =$]number of time steps , etc . for that reason we will use instead of , because .we have the following set of rules , which are applied in _ parallel _ ( simultaneously ) for all cars :  s1 : acceleration ( ) .if , the velocity of the car is increased by one , i.e. ,    this rule simulates that all the drivers like to reach the maximum velocity   s2 : randomization ( ) . if , the velocity of the car  is decreased randomly by one unity with probability , i.e. ,   with probability   this rule represents all the random situations , such as the random driver behavior or the random highway physical state .s3 : deceleration ( ) .if , the velocity of the car  is reduced to ( , i.e. , the new velocity of the vehicle is :   .the term represents the driving schemes respecting the safety distance ( gap ) and this distance is determined by the safe distance parameter the physical meaning of will be discussed latter .s4 : vehicle movement ( ) .each car is moving forward according to its new velocity determined in steps 1 - 3 , i.e. ,    the rules allow to obtain the new velocities . then the vehicles are moved ( thus , one has to divide the update in two parts , the first part is to obtain the new velocities while the second part is the movement of cars .step 1 reflects the general tendency of the drivers to drive as fast as possible without crossing the maximum speed limit .the randomization in step 2 takes into account the different behavioral patterns of the individual drivers , specially , nondeterministic acceleration as well as overreaction while slowing down .this is important for the spontaneous formation of traffic jam .step 3 is intended to avoid collision between the cars .we use a parallel updating scheme since it takes into account the reaction time and can lead to a chain of overreactions .suppose , a car slows down due to the randomization step , and if the density is large enough then this might force the following cars to brake in the deceleration step .in addition , if is larger than zero , it might brake even further in step 2 .the main modification to the ns model takes place in step 3 . to determine consistently for all vehicles ,step 3 must be iterated ( times .we consider the distance between the vehicles and and their coresponding velocities .the knowledge of the velocity of the vehicle ahead is incorporated through the safety parameter .this parameter may take values between 0 and 1 .if takes its maximum value , the speed of the vehicle ahead is not considered in the deceleration process ( such as the ns model , which has been compared succesfully with standard traffic flow in germany ) . on the contrary , when the speed of the vehicle ahead is considered without restrictions , i.e. , without establishing a safe distance .this fact is regarding those automated vehicles which can obtain information from the vehicles in front of them .thus , the safe distance parameter may allow us to define the required degree of automation for the system .this model is a minimal model in the sense that all the four steps are necessary to reproduce the basic features of real traffic . however , additional rules may be needed to capture more complex situations .thus , the parameters of the model are the following : the limit speed , the braking parameter , the global density and the parameter of safe distance in our case , takes the following values this latter parameter determines the safety distances between two vehicles depending of the gap between them and their speeds .thus , if is smaller the restrictions of safety for the system are lesser and the safety is lower .before going on , we would like to describe our standard simulation set up for the following observations .we simulated a system of length sites with closed boundary conditions , ( i.e. traffic was running in a loop ) for a given value of and .we start with random initial conditions . cars are randomly distributed on the lane around the complete loop with an initial speed taking a discrete random value between 0 and .since the system is closed , the average density remains constant with time .next , we update the individual car velocities and positions in accordance with the rules of the model .each density is simulated for time steps , of which the first half were discarded to let transients die out and for the system to reach its asymptotic steady state .later , we start to extract data to make the statistical analysis .  for each simulation , we establish a value for parameter by taking into account the desired degree of safe distance among vehicles for example , the case of is equivalent to taking into account the distance between two cars and the exact velocity of the vehicle ahead , i. e. , we allow that a vehicle may be behind the other one with the same speed without the existence of a safe distance between them to prevent accidents .  a convenient way to investigate the model is to draw a diagram of flow versus density , the so - called fundamental diagram .it is a smooth curve with a well - defined maximum at a certain density , by analyzing the fundamental diagram , we found that for a given the flux exhibits a maximum at the density  that decreases when the range of interaction of the vehicles increases and the length of this interaction is determined by the safety parameter . in particular , we found that for the flow and the magnitude of decrease with increasing . on the contrary , for all the magnitude of decreases with decreasing and the flow increases .moreover , smaller is , i.e. as the safety factor tends to zero , the longer the length of interaction of the vehicles is and the flow decreases more rapidly ( see fig .2 ) .  on the other hand , for and we can distinguish two different dynamical regions of traffic flow .flow is linearly increasing with increase in traffic density ( laminar flow , free - flow ) ; beyond the flow becomes linearly decreasing in density ( back traveling start - top waves ) . in the free - flow regionall vehicles can move with high speed close to the speed limit and the average speed converges to the maximum velocity .however , beyond a certain density , a further increase of the density leads to a more rapid decay of flow due to the strong length correlation among vehicles and jams emerge ( as in fig .2 ) . nevertheless ,for we distinguish an additional mixed flow region where the flow increases with increasing but the vehicles do not move with maximum velocity , the speeds of vehicles start decreasing  by analyzing the relationship between velocity and density we observe that the velocity decreases more rapidly with decreasing ( see fig .3 ) . as a consequencethe vehicles move rather slowly in this region as compared to the free - flow region .the flow takes higher values increasing the highway vehicular capacity due to formation of small platoons of vehicles with similar velocities to avoid the propagation of jams .   on the other hand , for all increasing not only leads to lesser flow but also lowers ( see fig 4a ) .in addition , we found that for small and , the flow has the same behavior , i.e. the system is insensitive to the braking ( see fig 4b ) .we now examine the higher moments of the velocity distribution . in particular , we examine the speed variance and its behavior . for all values of , during both free - flow and congested phases ( where the average speed tends to the maximum speed ) the speed variance is negligible . in particular , for values of such that there does not exist the region of mixed flow , i.e.  or , the speed variance starts to increse immediately after the free - flow phase .this variance continues increasing with density until the variance reaches its maximum value , which occurs after the free - flow phase ( as shown in the fig . 5 for ) .on the other hand , for values we observe a local maximum of the speed variance after the free - flow phase , but during the mixed flow phase we note a local minimum in the variance before the maximum flow is reached ( see fig . 6 for ) ; the speed variance newly starts to increase and reaches its maximum value after the density of maximum flow .  by comparing the maximum flow and the maximum speed variance for and ( fig . 5 and 6 respectly )we observe that the maximum flow for is higher than for , although the maximum speed variance at is 50% smaller than .the behavior observed is a consequence of the safety factor we introduced , i.e , if the safety factor is higher , there exists a smaller propagation of fluctuactions and prevent variations of speeds by the formation of platoons or clusters of vehicles.  in general for all values of the density where the speed increases until it is maximum value occurs after the maximum flow density , ( the goal of most traffic planners ) .high speed variance means that different vehicles in the system have widely varying speeds .it also means that a vehicle would experience frequent speed changes per trip through out the system . in turn , this results in higher trip travel time variance . in reality, this could also increase the probability of traffic accidents .thus it seems reasonable to attempt to steer traffic away from the density immediately after the density of maximum flow .in this paper , we have introduced and investigated a statistical model capable of describing the traffic of automated vehicles accurately , considering a safety factor for the distance between two vehicles .we modified an existing ca model to capture interesting characteristics of traffic flow . here , we studied the addition of a safe distance parameter , to the nagel - shreckenberg model to consider the anticipation of a driver .this modification allows the two vehicles to be as near as possible by respecting a safe distance , i.e. , a driver decides his new velocity by taking into account the velocity from the driver ahead and the distance that exists between them .we think that a suitable safety factor is important to study the behavior of traffic flow and increasing the vehicular capacity . by the inclusion of the consideration of the velocity of the vehicle ahead in the process of deceleration by taking into account a safe distance ( i.e., the driver knows the velocity of the preceding vehicle ) we can obtain different correlations among vehicles that can increase the capacity of vehicles and decrease the maximum flow .a decrease in the capacity implies an increase in the maximum flow or the speed variance reaches its maximum value after the region of mixed flow ; whereas for safety factors very big or null ( there does not exist distance of safety ) this variance reaches its maximum value inmediately after the region of free - flow . thus , the definition of a suitable safety factor may increase the vehicular capacity .  finally , our approach here was to search for minimal sets of rules which reproduce certain macroscopic facts .the advantage of this model is that relations between rules and macroscopic behavior can be more easily identified ; and we also obtain higher computational speed .thus , the particle hopping models are a good tool to simulate a big scale traffic flow .the authors thank to dr .vivechana agerwal and dr .mariano lpez de haro for the critical reading of the paper .this work was partially supported by dgapa unam under project in103100 .blue , v. , bonetto , f. and embrechts , m. ( 1996 ) a cellular automata of vehicle self organization and nonlinear speed transitions ._ proceedings of transportation reserach board annual meeting _ , washington , dc .

Abstract: in recent years the modelling of traffic flow using methods from statistical physics , especially cellular automata models have allowed simulations of large traffic networks faster than real time . in this paper , we study a probabilistic cellular automaton model for microsimulations of traffic flow of automated vehicles in highways . this model describes single - lane traffic flow on a ring . we study the equilibrium properties by including a parameter of safe distance in the model and calculate the so - called fundamental diagrams ( flow vs. density graph ) considering parallel dynamics . this is done numerically by computer simulations of the model and by means of an analysis of speed variance .   traffic flow models microsimulation cellular automata simulation modeling


the study of systems composed of a large number of degrees of freedom was endowed a definitive means of reasoning with the introduction of the differential stochastic dynamical framework about a century ago .specifically , differential stochastic dynamics allows one to describe the time evolution of a given observable as a function of macroscopic variables of the system as well as non - deterministic ones .the latter are expressed in terms of noise(s ) , reflecting the microscopic features of the system , whose univocal description is beyond the bounds of possibility  . despite the impossibility of a deterministic description of the evolution of the observable, one can successfully obtain time dependent statistical details .undeniably , einstein s and bachelier s grounbreaking works , respectively on brownian motion and stock price movements , are outstanding examples of the relevance of differential stochastic dynamics in diverse fields .the inference of a differential stochastic process is generically made from statistical features such as statistical moments , correlation functions and probability density functions built from a time series of measurements . in particular , a well established theoretical background for identifying the stochastic dynamics exists for the important class of processes following the markov property  .however , in practice , several hindrances arise . specifically , besides matters related to the finite size of data sets  and to the direct error of the measurements ( associated with the quality of the equipments and/or their calibration  ) , the sample rating ( the spell between logged measurements ) plays a crucial role in the determination of actual underlying stochastic process . as a matter of fact, we are generally left a set of snapshots reproducing a fraction of the events that occurred in the continuous time process from which the dynamics is due to be determined . in any case, it would be important to compare the acquisition interval with the characteristic times of the process to determine whether the kramers - moyal ( km ) coefficients estimates are trustworthy . in this respect ,another difficulty is that the characteristic times are not always known beforehand , although they can be estimated , e.g. , through the computation of auto - correlation functions .moreover , when the sampling interval is found to be too long compared with the characteristic timescales , for suitably uncovering the process , it is not always possible to upgrade it , particularly for historical data . for such cases , it is therefore of primary importance a careful analysis of the impact of the time interval of data sampling on the observable finite - time km coefficients , specially as one approaches the independence limit .  for a markovian timeseries ,one can obtain the evolution equation for the conditioned probability density ( pdf ) by computing the km coefficients , with ^{k}\rangle \ , , \notag \\ & = & \frac{1}{k!\tau } \sum_{j=0}^{k}\left ( \begin{array}{c } k\cr j \end{array } \right ) \langle x^{j}\rangle ( -x_{0})^{k - j}\ , , \end{aligned}\ ] ] where we have already assumed that the coefficients are ( at least locally ) stationary ( for non - stationary data - sets see ref . ) . aiming to simplify the notation, we denote the statistical averages conditioned to the initial value : for usual stationary unconditioned averages .  in practice ,only the finite - time estimates can be directly computed , with limited by the minimal time interval , , of data acquisition . furthermore , when the sampling interval is larger than the correlation time , i.e. , the pdf becomes unconditioned and under stationarity , one gets , ^{k }  \notag \\ & \equiv & \frac{1}{k!\tau } \langle \lbrack x(\tau ) -x_{0}]^{k}\rangle_u \ , , \notag \\ & = & \frac{1}{k!\tau } \sum_{j=0}^{k}\left ( \begin{array}{c } k\cr j \end{array } \right ) \langle x^{j}\rangle _ { u}(-x_{0})^{k - j}\,.\end{aligned}\ ] ] notice that , in eq .( [ dk_independence ] ) the stationary averages are unconditioned , hence do not depend on , differently from eq .( [ dtilde ] ) .therefore , eq . ( [ dk_independence ] ) represents a -order polynomial in , no matter how complex the intrinsic coefficients are . in practical applications ,this feature introduces a complete uncertainty on the actual form of the intrinsic coefficients , in the absence of a model _ a priori _ .  in this manuscriptwe obtain the finite - time estimates for several standard processes described by it - langevin stochastic differential equations , where is a standard wiener process . in previous work , this task was accomplished for the particular class of processes with and , by means of it - taylor expansions .herein , we will consider a larger class of processes following two different approaches .the first one , based on the solution of the evolution equation of the raw statistical moments , represents a simplification regarding the whole process of appraising the finite - time coefficients when the drift is linear .second , we employ the fokker - planck adjoint operator technique that is useful for processes with nonlinear drifts , for instance .furthermore , we will analyze the extreme instances of no - fluctuations and independence . the deterministic limit ( absence of fluctuations )is relevant in stochastic dynamical systems since it provides a reference on the functional behavior of quantities that can be perturbed by augmenting the intensity ( broadness ) of the noise . in the meanwhile ,the independence limit sets the bounds expected for large acquisition interval and/or large noise intensity .although it is not always possible a direct inversion of the finite - time estimates to extract the intrinsic ones , the expressions herein obtained can be contrasted against the observed km coefficients , allowing the identification of the underlying process as well as the associated parameter values .given the fokker - planck equation ( fpe ) for +\partial _{ xx}[d_{2}p]\ , ,  \label{fpe}\ ] ] the evolution equation for a mean value can be obtained by multiplying both sides of eq .( [ fpe ] ) times the quantity to be averaged and integrating by parts with suitable boundary conditions ( vanishing at the boundaries ) . in this way, the equations for the moments have the form if is linear and is at most quadratic , the equations for the moments can be successively solved from the lowest order  . however , a hierarchy of equations depending on higher order moments generally arises . in such cases ,one can still resort to approximate techniques such as hierarchy truncation , substitution of terms , etc .alternatively , the conditional averages can be computed by means of the short - time solution expansion  , which leads to ^{n}q(x)_{|x_{0}}\,\tau ^{n}/n!\ , ,  \label{adjoint}\ ] ] where is the ( backwards ) fokker - planck adjoint operator ( see also  ) .  in the deterministic case , it is enough to solve eq .( [ eqmn ] ) for the first moment , which becomes where the superindex stands for `` deterministic '' and whence we obtain the full range of values , .then , the deterministic part of , with , is let us note that the deterministic are non - null functions , although they vanish in the limit except for . nevertheless , we should bear in mind that for not too small the deterministic part can dominate over the noise contribution .this represents another drawback one may face in a practical application .in fact , if the noise is small , the deterministic contribution to may screen the information on the noise .let us first consider the case of linear drift , with assuring the existence of a stationary solution . as a matter of fact , a raft of phenomena are described by an exponential relaxation , which is concomitant with a parabolic drift potential . from eq .( [ eqmn ] ) one has whose solution is assuming the initial condition and defining . hence , according to eq .( [ dtilde ] ) , that in the limit recovers .the opposite limit of independence yields in accordance with eq .( [ dk_independence ] ) .it is noteworthy that eq .( [ d1linear ] ) is independent of the particular form of .this is due to the fact that in ( and only in ) the linear case , eq .( [ eqm1 ] ) coincides with its deterministic version , eq .( [ determ ] ) , hence . for other forms of the deterministic part of the stochastic differential equation ,the derivative of with respect to the time is equal to more complex expressions which depend on higher - order moments and/or powers of the first conditional moment . despite the slope evolution with ,it is worth noting that the finite- drift preserves the linear form of the intrinsic .equation ( [ m1 ] ) also allows one to estimate the conditional two - time covariance function given by  , that exponentially decays towards the long - time average .still one has to average over initial conditions to find , where is usually the steady pdf .however , any choice of will not alter the exponential character of the correlations since the integration is not carried out over time .in addition , the correlations are only ruled by the drift and do not depend on the noise amplitude .it is straightforward to understand that this quirk comes to pass because of the functional properties of the conditional average .   if together with drift linearity the noise intensity is second - order polynomial , with parameters such that , then each evolution eq .( [ eqmn ] ) is linearly coupled to those of lower order only . in this casethe equations can be solved one after another . in particular , for the second moment , one has , thus , we just need to solve a non - homogeneous 1st order differential equation of the form whose solution , given the initial condition , is   therefore , with the identifications ,  and , following eqs .( [ eqm2 ] ) and ( [ m1 ] ) , and the initial condition .namely , where and .the finite- second km coefficient can be obtained by means of eq .( [ dtilde ] ) with and given by eqs .( [ m1 ] ) and ( [ m2 ] ) , respectively .explicitly , we find \,x_{0}^{2 }  \notag \\ & & + 2\left [ \frac{b+b}{a-2a } ( z - z^2w ) + \frac{b}{a } ( z-1 ) \right ] \,x_{0 } \notag \\ & & + \frac{b(b+b)(a z^2w -2(a - a ) z + a-2a)}{a(a - a)(a-2a ) }  \notag \\ & & + \frac{c}{a - a}(1-z^2w ) \biggr ) \,.\end{aligned}\ ] ]  when , , while for , eq . ( [ dk_independence ] ) is verified .these results permit one to embrace many fundamental models such as ornstein - uhlenbeck , feller and harmonic drift with additive-(linear)multiplicative noise .also in finance , in most well - known models of volatility ( ) , the drift is linear and , with 0 ( ornstein - uhlenbeck process ) , 1 ( square - root , feller or cox - ingersoll ross model ) and 2 ( hull and white model ) .let us also recall that for the linear drift , its finite - time expressions are also linear in .similarly , with linear drift and the dependence of on is quadratic . as a matter of fact , a similar scenario holds for -order finite - time km coefficients . in other words ,the -order km coefficients are polynomials of order , whose coefficients depend on . thus , it is not simple to nimbly rescue the value of just by identifying the polynomial order of the observed coefficients .  for the quadratic noise intensity , the evolution equations of higher order conditional moments , from eq .( [ eqmn ] ) , are where ; \\ b_{n } & = & n\left [ \left ( n-1\right ) b+b\right ] ; \\ c_{n } & = & n\left ( n-1\right ) c.\end{aligned}\ ] ]  high - order km coefficients are particularly relevant as well . in other words , for the present class of markovian processes the pawula theorem  implies that they must vanish . in the absence of any other reason to discard the validity of chapman - kolmogorov approach , if the finite - time conditional higher - moments are non - null , it is important to probe whether it is the outcome of a finite - time effect .let us restrain our calculations to the explicit formulae of the third and fourth coefficients in two important cases : _ (i ) _ the reversion to a null value in the absence of coupling between additive and ( linear ) multiplicative white noises ( and ) and _ ( ii ) _ the feller process .\(i ) for the additive - multiplicative process with and , notice that eq .( [ eqmn ] ) with has the same form of eq .( [ ode ] ) , through the identification , , and , and the initial condition , yielding    then , according to eq .( [ dtilde ] ) , one has where the averaged quantities are given by eqs .( [ m3 ] ) , ( [ m2 ] ) and ( [ m1 ] ) , respectively .  for the fourth moment , eq .( [ eqmn ] ) with has the same form of eq .( [ ode ] ) , through the identification , , and , together with the initial condition , yielding thus , is obtained by substitution of the averaged quantities eqs .( [ m4 ] ) , ( [ m3 ] ) , ( [ m2 ] ) and ( [ m1 ] ) .the present results generalize those previously obtained for the particular case ( with ) and in  .   on first , second and fourth finite - time km coefficients for the process , with , and .the values of used are indicated on the figure .artificial timeseries were generated by means of the euler algorithm with and data points of the timeseries were considered for the computation of km coefficients in each case .symbols correspond to numerical estimates and the associated colored solid lines to the theoretical expressions for finite - time coefficients given by eqs .( [ d1linear ] ) , ( [ d2linear ] ) and ( [ d4feller ] ) .black solid lines correspond to the intrinsic coefficients , and and black dashed lines ( practically coinciding with the curves for ) , to the independence limit forms given by eq .( [ dk_independence ] ) with , scaledwidth=45.0% ]  \(ii ) for the feller process ( ) , by successive integration of eqs .( [ ode ] ) and using ( [ dtilde ] ) we find , and + b(b+b ) } { a^2(1-z)^2}x_0 ^ 2  \notag \\ & & -4\frac{(b+b)(b+2b)(b-(b+3b)z)}{a^3(1-z)}x_0  \notag \\ & & \left.+ \frac{b(b+b)(b+2b)(b+3b)}{a^4 } \right).\end{aligned}\ ] ]  . in all cases and the different values of are indicated on the figure .black solid lines correspond to the intrinsic coefficients , black dashed lines to the independence limit forms at and black dotted lines to the deterministic forms ( eq . ( [ dk_determ ] ) ) at .symbols correspond to numerical estimates and the associated colored ( thin ) solid lines to the theoretical expressions for finite - time coefficients given by eqs .( [ d1linear ] ) and ( [ d2linear ] ) . in the lower panelthe inset shows the same data in log - linear scales ., scaledwidth=45.0% ]  considering that for the feller process , then the km coefficients can be rewritten in terms of the unconditioned raw moments , which can be straightforwardly estimated from data , as follows ,        and where .  in fig .[ fig : feller ] , we exemplify the behavior of , and for the feller process ( that is , or also ) as the independence limit is approached .notice the variety of behaviors that can be observed even for not too large .the slope of the linear changes with taking the values , while soon becomes quadratic with .also largely departs from the intrinsic null value .for all the coefficients , the independence limit is practically attained at ( ) .figure [ fig : fellerdet ] illustrates the impact of the intensity of noise on the finite - time estimates .noise intensity does not affect which coincides with the deterministic straight line . largely disagrees with the true form , departing towards the independence limit , for large noise intensity , and towards the deterministic limit for small noise .of course , as one approaches the deterministic ( no - fluctuations ) limit the range of shrinks around the deterministic ( equilibrium ) value .when the drift is linear , whatever the noise intensity , the parameters and can be unraveled by a linear fit to the observed : the value of the slope allows one to determine the parameter , once is known , while the abscissa at which vanishes corresponds to .that is , if the observed drift can be considered linear in good approximation , a direct inversion of to obtain the intrinsic is possible .this will also facilitate the obtention of the functional form ( and parameter values ) of .   for , or more generally for non - quadratic , the finite - time are also non quadratic in .the equation for the second moment , following eq .( [ eqmn ] ) , can not be solved straightforwardly in this case .let us consider the especial case with ( known as 3/2-model ) used as a volatility model as well . for small noise intensity , one can find the correction to the deterministic part of by means of expansion ( [ adjoint ] ) , taking . by identifying the general form of the coefficients of , with the aid of algebraic manipulation programs , we obtain   then and on that account, which in the limit tends to as expected .as in the case of linear drift and non - quadratic noise , if the drift is nonlinear , the equations for the moments can not be solved successively from the lowest order .moreover , the useful independence of the evolution equation for on higher order statistical moments , which are noise dependent , fails .consequently , the existence of a nonlinear drift introduces a troublesome relation between and the noise as well as a dependence of the correlation function on the same noise .however , we can still use the calculations at the deterministic limit to represent the upper bound which is quite reliable for the cases presenting small noise intensity .explicitly , if the contribution of diffusion is neglected , the evolution is almost deterministic .accordingly , we will solve the deterministic equation for the first moment whereas for higher order moments we will take into account the lowest order correction due to noise .let us consider a process with cubic drift  also called bernoulli oscillator , where .when the system represents a stochastic motion in a bistable potential .the statistical moments of this stochastic system were early studied within a variational approach context  . however , this method lands up introducing an extra ( fitting ) parameter which we adamantly want to avoid , since it would increase the level of uncertainty of the results .figure [ fig : cubic ] exhibits the first , second and fourth finite - time km moments , obtained for different values of the sampling interval .notice that , with increasing , tends to a linear form that can erroneously lead to assume that the drift is linear .however , under this assumption , the inversion of should give incongruous results for , thus leading to discard the linearity of .for small noise intensity , the deterministic expression fitted ( with two fitting parameters and ) to the observed allows to recover .notice also the complex influence of on the higher order coefficients , leading to forms with two minima , in contrast with the linear drift case .   on the first , second and fourth finite - time km coefficients , for the process defined by eq .( [ sdecubic ] ) , with , and .the values of are indicated on the figure .symbols correspond to numerical estimates .black solid lines correspond to the intrinsic coefficients , and , while black dashed lines to the independence limit forms at ., scaledwidth=45.0% ]  ) . in all cases , and the different values used are indicated on the figure .black solid lines correspond to the intrinsic coefficients , black dashed lines to the independence limit at , and black dotted lines to the deterministic forms ( [ dk_determ ] ) at .symbols correspond to numerical estimates and the associated colored lines to the approximations for small noise intensity given by eqs .( [ d1low ] ) and ( [ d2low ] ) , respectively . in the lower panelthe inset shows the same data in log - linear scales . for the smallest noise intensity twodifferent initial conditions were used to fill both potential wells.,scaledwidth=45.0% ]  by integration of eq .( [ determ ] ) , one obtains the deterministic expression , that does not preserve the original simple cubic form underscoring the unique character of the linear drift . in the limit ,one recovers the linear case studied above .the case where is a perturbative parameter may be of interest for systems in the vicinity of a phase transition , such as in refs . . in the limit , eq .( [ zeroth ] ) becomes , independent of the signal of , for short , is equal to . for large values of ,the picture depends on whether is positive or negative . in the former case , which corresponds to a single well potential , defined by a straight line , which corresponds to the independence limit . meanwhile , in the latter case , which entails bi - stable potentials , one obtains , in this case , for large values of , one can describe the limits corresponding to large values of , and for small values of the same quantity , -1}{\tau } x_{0}-\frac{b\exp \left [ 3|a|\tau \right ] } { 2\,|a|\,\tau } x_{0}^{3}.\ ] ] as a result , we can verify that the polynomial dependence of is preserved for the central region that dwindles as the sampling rate increases resulting in the straight line limit ( [ anegx0big ] ) , which describes the full dependence ( beyond relaxation scale ) situation .  in the generic case ( with ) ,the solution to eq .( [ determ ] ) is , which includes eq .( [ purecubic ] ) as a particular case ( for ) and also the linear case with ( for ) . in the latter instanceit is provable that the exponential functional dependence is recovered .for all the cases and in the limit , one recovers whereas in the opposite limit , one gets .  for small fluctuations ,higher order finite- km moments are dominated by the deterministic part given by eq .( [ dk_determ ] ) with further corrections dependent on the noise .  for the cubic drift , in the presence of small fluctuations , such that , with , and . then , where is given by eq .( [ zeroth ] ) .similarly , for the second moment we found the first correction to the deterministic part with then , finally which verifies that tends to as .figure [ fig : cubicdet ] shows the impact of the intensity of noise on the finite - time estimates .in contrast to the linear drift case , here the noise influences , in such a way that as the intensity of noise increases , approaches the independence limit even for small sampling rates . this could be attributed to the decorrelating role of noise ( it is worth regarding that already depends on the noise for non - linear potentials ) .similarly to the linear case , largely disagrees with the true form , departing towards the independence limit , for large noise intensity , and towards the deterministic limit for small noise .notice that if the intensity of noise is too small , one of the wells of the potential may become inaccessible .precisely , let us focus on the insets in the lower panel of fig .[ fig : cubicdet ] .when one is in the no - fluctuation regime the system is characterized by three fixed points , namely one unstable ( at , which prevails if and only if is the initial condition ) and two stable points ( at ) .minimal noise is able to make the observable , which we can envisage as a virtual particle , move off the unstable point to one of the wells , each one defined by one of the remaining extremes ( minima ) , but it gives a very little rate of transition between the regions of stability . as the noise intensity augments , the rate of transition increases as well and we are able to detect the finite values of the km moments in the vicinity of the unstable point .this can be adduced by computing the first non - vanishing eigenvalue , , when the fpe is transformed into a schrodinger equation . in first approximation ,the eigenvalue of this bistable potential , is given by , for the values presented in fig .[ fig : cubicdet ] , goes from ( for ) to ( for ) .in this manuscript we obtained analytical results about the impact of the sampling rate on the km coefficients directly computed from timeseries .we analyzed stochastic processes subject to a linear drift , which already comprise a huge variety of processes .we managed to compute exact expressions for the evolution of conditional moments , from which we obtained the finite - time km coefficients and correlation functions .the first moment and the linear correlation function are both independent of the noise intensity .we also analyzed standard nonlinear drift cases , where the scenery exhibits a sharp change .the moments are now ruled by a cascade of differential equations which hampers the obtention of exact analytical solutions .nonetheless , approximate expressions valid for small noise intensity were achieved by means of the adjoint operator approach . differently from the linear case, the intensity of the noise affects every conditional moment and the correlation function as well .we also showed that by increasing the noise intensity the functional dependence of the km heads towards the independence regime , which is characterized by a polynomial of the same order .this fact has notorious consequences in time series analysis , since one can be brought onto a situation where independent - like coefficients are obtained , but one is unable to assign this finding to either an improper sample rating or a strong noise intensity .  for sampling rates smaller than the relaxation time , in a steady state approach , our results permit one to sieve the set of processes giving raise to the heuristic stationary probability density function in order to select the appropriate markovian differential stochastic process and consequently to determine the respective parameter values .furthermore , it enables the judgement of the validity of the markovian proposal by supplying precise estimates of higher order km moments .otherwise , if the relaxation time is shorter than the sampling rate , the measured km coefficients will tend to a polynomial of order matching the order of the moment .this functional form is independent of the primary stochastic process , hence introducing uncertainty in the recognition of the process .uncertainties might be hedged by inspecting probability densities and correlation functions as well as other measures of dependence such as the relative ( kullback - leibler ) entropy .ca is grateful to brazilian agencies faperj and cnpq for partial financial support and smdq acknowledges the warm hospitality of puc - rio and the financial support of the national institute of science and technology for complex systems during the early stage of this work .r. friedrich and j. peinke , _ physica d _ * 102 * , 147 ( 1997 ) ; ch .renner , j. peinke and r. friedrich , _ j. fluid .mech . _ * 433 * ( 2001 ) , 383 ; c. anteneodo , e.e .ferrero , s.a .cannas , _ j. stat_ , p07026 ( 2010 ) .

Abstract: we analyze the impact of the sampling interval on the estimation of kramers - moyal coefficients . we obtain the finite - time expressions of these coefficients for several standard processes . we also analyze extreme situations such as the independence and no - fluctuation limits that constitute useful references . our results aim at aiding the proper extraction of information in data - driven analysis .