despite the importance of software quality , the management of software quality is still an immature discipline in software engineering research and practice .research work has gone in many directions and produced a variety of interesting results . however , there is still no commonly agreed way for quality management .the practice varies strongly from a concentration on testing to a large - scale quality management process .one main problem is that many of the tools and methods work rather isolated .hence , quality is tackled on many levels without a combined strategy .what is missing is a clear integration of these single efforts .one prerequisite for such an integration is a quality management sub - process in the overall development process .the process defines the roles , activities and artefacts and how these work together .a second prerequisite is a clear quality model that defines quality for the software to be developed .the term _ quality model _ is used in various contexts but we use it here to denote a model that breaks down the complex concept _ quality _ and thereby makes it more concrete .these concrete descriptions of quality could then be used to construct , assess and predict software quality .current quality models such as the iso 9126 have widely acknowledged problems . especially as a basis for assessment and prediction , the defined `` -ilities '' are too abstract .a clear transition to measurements is therefore difficult in practice .hence , quantitative quality assessment and prediction is usually done without direct use of such a quality model .this , in turn , leads to isolated solutions in quality management .we use the previously proposed activity - based quality models ( abqm ) as a basis for quality assessment and prediction .they provide a clear structure of quality and detailed information about quality - influences .activity - based quality models have proven useful in practice to structure quality and to generate corresponding guidelines and checklists . in this paper , we add a systematic transition from abqms to bayesian networks in order to enhance their assessment and prediction capabilities .a 4-step approach is defined that generates a bayesian network using an activity - based quality model and an assessment or prediction goal .the approach is demonstrated in a proof of concept .we first motivate and introduce activity - based quality models in section [ sec : quality_models ] . in section[ sec : approach ] the 4-step approach for systematically constructing a bayesian network from an activity - based quality model is proposed .the approach is then demonstrated in a proof of concept in section [ sec : proof ] using publicly available data from a nasa project .related work is discussed in section [ sec : related ] and final conclusions are given in section [ sec : conclusions ] ,quality models describe in a structured way the meaning of a software s quality .we introduce the use of general quality models and how the modelling of activities and facts helps to define quality more precisely .if quality requirements are handled at all in a software project , quality models will be an integral part of it .the quality model describes what is meant by _ quality _ and refines this concept in a structured way . in practice, this is often reduced to metrics such as _ number of defects _ or high - level descriptions as given by the iso 9126 .  in general , there are two main uses of quality models in a software project : ( 1 ) as a basis for defining quality requirements and ( 2 ) for relating quality assurance techniques and measurements to the quality requirements .the first is commonly done by constraining well - known quality attributes ( reliability , maintainability , ) as defined in a quality model . in practice , this is often reduced to simple statements such as `` the system shall be easily maintainable . ''the second use is often not explicitly considered . however , certain metrics are nearly always measured such as the number of faults in the system detected by inspections and testing .the relationship between these measures and quality attributes remains unclear .the reason lies in the lack of practical means to define metrics for these high - level quality attributes .hence , more structure and more detail is needed in quality models to integrate them closely in the development process .it has been proposed to use activity - based quality models ( abqm ) in order to address the shortcomings of existing quality models .the idea is to avoid to use high - level `` -ilities '' for defining quality but to break it down into detailed facts and their influence on activities performed on and with the system . in was shown for maintainability , in for usability .in addition to information about the characteristics of a system , the model contains further important facts about the process , the team and the environment and their respective influence on maintenance activities such as , , or .for example , redundant methods in the source code , also called clones , exhibit a negative influence on modifications of the system because changes to clones have to be performed in several places in the source code .  for abqms , an explicit meta - model or structure model was defined in order to characterise the quality model elements and their relationships .four elements of the meta - model are most important : , , and .an can be any thing , animate or inanimate , that can have an influence on the software s quality , e.g. the source code of a method or the involved testers .these entities are characterised by attributes such as or .the combination of an entity and an attribute is called a _fact_. we use the notation for a fact . for the example of code clones, we can write .these facts are assessable either by automatic measurement or by manual review . if possible , we define applicable metrics for measuring the facts inside the abqm .  an influence of a fact is then specified by an .we concentrate on the influences on activities , i.e. anything that is done with the system .for example , or are high - level activities . the impact on an can be positive or negative .we complete the code clone example by adding the impact on : .this means that if a system entity exhibits the attribute it will have a negative impact on the activity , i.e. changing the method .a further example is the following tuple that describes consistent identifiers : it means that identifiers that can be shown to be consistent have a positive influence on the modification activity of the maintainer of the system . in the model itself, we document more information such as textual descriptions , sources , assessment descriptions and so on .however , the short notation captures the essential relationships .the model does not only contain the impacts of facts on activities but also the relationships among these .facts as well as activities are organised in hierarchies .a top - level activity has sub - activities such as , or .these examples are depicted in figure [ fig : quality_matrix ] . in realistic quality models ,they are then further refined .for example , maintenance can have sub - activities such as and .because facts are a composite of an _ entity _ and an _ attribute _ , the organisation in a hierarchy is straightforward .hierarchical relationships between entities do usually already exist .the top - level in figure [ fig : quality_matrix ] is the of the software development project .it contains , for example , the , its and the development .again , these entities need to be further refined .for example , the system consists of the source code as well as the executable .all entities can be described with attributes such as the of the .the two hierarchies , the fact tree and the activity tree , together with the impacts of the facts on the activities can then be visualised using a matrix as in figure [ fig : quality_matrix ] .the fact tree is shown on the left , the activity tree on the top .the impacts are depicted by entries in the matrix where a `` + '' denotes a positive and a `` - '' a negative impact .the associations between facts in the fact tree can have two different meanings . either an entity is a part or a kind of its super - entity . along the inheritance associations ,parts and attributes are inherited .hence , it allows a more compact description and prevents omissions in the model .for example , naming conventions usually are valid for all identifiers no matter whether they are class names , file names , or variable names .having defined all these entries in the abqm , we can define which activities we want to support and which influencing facts need to be defined . in terms of the above example, if we want to support the activity , we know that we need to inspect the identifiers for their consistency . there exists a prototype tool to define this kind of large and detailed quality models . besides the easier creation and modification of the model , this has also the advantage that we can automate certain quality assurance tasks .for example , by using the tool we can automatically generate customised review guidelines for specific views .although activity - based quality models have proven to be useful in practice , there is no systematic approach for using such quality models in measuring .hence , there are no quantitative assessments and predictions possible so far .we propose an approach that can be used for systematically deriving assessment and prediction models from a activity - based quality model .the general aim of the approach is to provide quality managers with a systematic method to derive assessments and predictions from an activity - based quality model . in the abqm , there are definitions of what quality means w.r.t .different situations , artefacts , and considered activities . currently , we give a textual description in the quality model how a fact could be assessed . for example , the fact described by contains the following assessment description : `` this fact can be assessed manually or semi - automatically .for the automatic assessments there are tools such as clonedetective or ccfinder to detect redundant parts of the source code . ''this information is interesting for quality assurance planning but can not be directly used for an overall assessment , let alone prediction .moreover , as the basic principle of activity - based quality models is that the most important question in quality is how well activities can be performed on and with the system , not only facts but also activities should be assessed and predicted from the knowledge of facts and impacts . currently , only the qualitative statement whether an impact is positive or negative is made .this is suitable for rough assessments only . for more comprehensive and precise assessments of the current state and prediction of future states ,a more sophisticated approach is needed .it has to systematically help to use the given relationships and enrich them with quantitative information .  as most facts and especially the relationships between facts and activities have an associated uncertainty , statistical methods are needed . usually we can not determine the exact relationship but can derive an uncertain range .also values measured can be uncertain , e.g. values from expert opinion .moreover , a statistical method is needed that can directly model the dependencies of different factors from the quality model .we identified bayesian networks as most suitable for that task .bayesian networks , also known as bayesian belief nets or belief networks , are a modelling technique for causal relationship based on bayesian inference .they are represented as a directed acyclic graph ( dag ) with nodes for uncertain variables and edges for directed relationships between the variables .this graph models all the relationships abstractly .  for each node or variablethere is a corresponding _ node probability table _these tables define the relationships and the uncertainty of these variables .the variables are usually discrete with a fixed number of states . for each state, the probability that the variable is in this state is given .if there are parent nodes , i.e. a node that influences the current node , these probabilities are defined in dependence on the states of these parents .an example is shown in table [ tab : example_npt ] . therethe variable is with a probability of 60% in the state _ true _ if both parents are in the state _ low _ , and with 45% in _ true _ if the first parent is in _ high _ and the second is in_ low_.  .an example npt for a variable with two states and two parents [ tab : example_npt ] [ cols="<,>,>,>,>,>,>",options="header " , ]the process of building a bayesian network contains the identification of interesting variables that shall be modelled , representing them as nodes , constructing the topology and constructing the npts .each of these steps is important and non - trivial .first , the identification of _ interesting _ variables includes the assumption that the model builder can decide on some basis what is interesting . in many cases ,this is not clear beforehand .one possibility is to include many variables and use sensitivity analysis to remove insignificant variables .second , the creation of the topology uses the assumption that the model builder can decide on the dependence and independence of the identified variables . in the process of building the bayesian network , especially for independence assumptions ( i.e. missing edges in the graph )detailed justifications should be given .third , the problem of constructing npts is widely acknowledged in the literature .it involves defining quantitative relationships between variables .there are various possible methods for this quantification such as a probability wheel or regression from empirically collected data .all methods have their pros and cons .it is important to note that each of these steps is important and errors in each of these steps can have a large effect on the outcome .bayesian networks and the corresponding tool support make it easy to build models and get quantitative results .however , one needs to be aware that many assumptions are embedded in a bayesian network that need to be validated before it can be trusted .we propose a four - step approach for building a bayesian network as assessment and prediction model derived systematically from an activity - based quality model .the resulting bayesian network contains three types of nodes :  * _ activity nodes _ that represent activities from the quality model * _ fact nodes _ that represent facts from the quality model * _ indicator nodes _ that represent metrics for activities or facts  we need four steps to derive these nodes from the information of the abqm .first , we identify the relevant activities with indicators based on the assessment or prediction goal .second , influences by sub - activities and facts are identified .this step is repeated recursively for sub - activities .the resulting facts together with their impacts are modelled .third , suitable indicators for the facts are added .fourth , the node probability tables ( npt ) are defined to reflect the quantitative relationships . having that , the bayesian network can be used for simulation by setting values for any of the nodes .however , we first describe the four steps in more detail .the first step is a goal - based derivation of relevant activities and their indicators .we use gqm to structure that derivation .we first define the assessment or prediction _ goal _, for example , optimal maintenance planning or optimisation of the security assurance .the goal shows the relevant activities , such as _ maintenance _ or _attack_. this is refined by stating _questions _ that need to be answered to reach that goal . for example , will the maintenance effort increase over the next year or how often will there be a successful attack in the next year ?finally , we derive _ metrics _ or indicators that allow a measurement to answer the question . in the examples , that can be effort change trend or number of harmful attacks over the next year .  in the second step ,we use the quality model to identify the other factors that are related to the identified activities .there are two possibilities : ( 1 ) there are sub - activities of the identified activities and ( 2 ) there are impacts from facts to the identified activities .we repeat this recursively for the sub - activities until all facts that have an impact on the activities sub - tree below the identified activity are collected . for each activitywe immediately see the impacts and hence the corresponding facts .all activities and facts identified this way are modelled as nodes in the bayesian network .we add edges from sub - activities to super - activities and from facts to activities on which they have an impact .figure [ fig : qm2bbn ] gives an abstract overview of that mapping from the quality model to the bayesian network .    in the third step ,we add additional nodes as indicators for each fact and activity node that we want a measurement for . the indicator for our relevant activitywas defined in the first step .hence , we can add additional indicators for sub - activities if needed . in any case , there need to be at least one indicator for each fact that is modelled .there might be a precise description in the quality model already .otherwise , we need to derive our own metric or use an existing one from the literature .the edges are directed from the activity and fact nodes to the indicators , i.e. the indicators are dependent on the facts and activities .an indicator is only an expression of the underlying factor it describes .a main advantage of using a abqms as a basis for the bayesian network is that it prescribes its topology .one of the main points of such quality models is to qualitatively describe the relationships between different factors that are relevant for software quality .we rely on that and assume that all dependencies have been modelled and that all other factors are independent .on the one hand , this constrains the validity of the results of the bayesian network by the validity of the abqm . on the other hand, it frees the network builder from reasoning about independence and dependence .finally , the fourth step , enriches the bayesian network with quantitative information .this includes defining node states as well as filling the npt for each node .the activity and fact nodes are usually modelled as _ ranked _ nodes , i.e. in a ordinal scale .the most common example is low , medium , and high .this has advantages in evaluation and aggregation .the evaluation is easier as not precise numbers have to be determined but coarse - grained levels .these levels actually reflect much more the high uncertainty in the data .also in aggregating over nodes ( up the hierarchy in the activity tree ) coarse - grained ranked data is more simple to handle . for only a few levels in ordinal scale ,it is easier to define a aggregation specification than for continuous data . to define the npt, we use an approach proposed by fenton , neil and galan caballero .the basic idea is to formalise the behaviour observed with experts that have to estimate npts .they usually estimate the central tendency or some extreme values based on the influencing nodes .the remaining cells of the table are then filled accordingly .this is similar to linear regression where a normal distribution is used to model the uncertainty .we use the doubly truncated normal distribution that is only defined in the $ ] region .it allows to model a variety of shapes depending on the mean and variation .it allows , for example , to model the npt of a node by a weighted mean over the influencing nodes .the node states of indicator nodes depend on the scale of the indicator used .this often will be continuous or discrete interval states such as lines of codes in intervals of a hundred .the npts of the indicator nodes are then defined using either common industry distributions or information from company - internal measurements .for example , typical loc distributions can be accumulated over time .the influence of the activity or fact node , respectively , it belongs to can be modelled in at least two ways : ( 1 ) partitioned expressions and ( 2 ) arithmetic expressions .the latter describes a direct arithmetical relationship from the level in the activity or fact node to the indicator . using a partitioned expression ,the additional uncertainty can be expressed by defining probability distributions for each level of the activity or fact node .a main feature of bayesian networks is their capability to simulate different scenarios .having built the bayesian network based on the activity - based quality model , we can ask `` what if ? '' questions .these questions are formulated in scenarios that can be simulated and compared .a scenario involves adding additional information to the model , more specifically , adding an observation for a node .this way , the uncertainty is removed and the consequences for the other nodes can be calculated .interestingly , in a bayesian network it is possible to do forward as well as backward inference , i.e. information can be added to any node and the effect is calculated in any direction of the graph .a first straightforward scenario is to add the currently measured values for the fact indicator nodes .this will drive the calculation up to the activities and the activity indicator node .this indicator node then shows the probability distribution for its value , i.e. the value of the activity .then changes can be made to the fact indicator nodes in other scenarios to reflect possible changes .their effect on the activity can be predicted .another interesting scenario is to set a desired value for the activity indicator and let the network calculate the most probable explanation in the fact indicators .it shows the values that should be reached in order to fulfil the goal .the above presented approach can be used in many different contexts and to answer various assessment and prediction questions .we provide a proof of concept here that applies the approach to a publicly available data set .currently , we can not provide a full validation of the assessment and prediction approach because this would involve measuring a large number of facts from the quality model in order to take full advantage from the knowledge contained in it .this data is not available in public data sets .measuring this at a company will need time and effort .only then a sensible analysis of the predictive validity and comparisons with other prediction models are possible .nevertheless , we provide a proof of concept application of the approach based on a small extract of our quality model for maintainability for which there is publicly available data .it demonstrates the basic principles of the approach on a real data set .this way , we can analyse whether the approach is feasible in an almost realistic setting . the predictive validity in the proof of conceptcan give an indication of the usefulness of the approach .there should at least be an improvement over industry average values .the proof of concept comprises the prediction of the maintainability of a system .the system under analysis was developed by nasa in the project cm1 for which the data has been publicised .the system itself is an instrument in a space craft developed in c. various metrics , especially the mccabe and halstead metrics , have been collected in this project .  as the quality model, we use the activity - based quality model for maintainability from .it contains a complete activity tree for maintenance as well as about 200 facts with an impact on those activities .the cm1 data set does not contain data for all of these facts but we choose a small set of facts and corresponding activities to predict maintainability .the choice was therefore guided by the availability of the data .the actual maintainability can be judged for cm1 because the effort for several changes has been documented .we will use this as our surrogate measure .  for modelling the bayesian network, we use the tool agenarisk .it provides a complete tool environment for bayesian networks including the usage of expressions for describing the npt and sensitivity analysis .the first step in our assessment and prediction approach is to identify the relevant activities and corresponding indicators ( metrics ) in a gqm - like procedure .we want to analyse maintainability and we assume that the quality manager is interested in the question of how high the maintenance efforts will be in the future .this information helps in planning the maintenance team .hence , we formulate the * goal * as `` planning of future maintenance efforts '' .we can directly identify the activity in that goal . to further operationalise that, we define the corresponding * question * as `` what will be the maintenance effort per change request ? '' .this information , probably together with a prediction of the yearly number of change requests , would give a good basis for maintenance planning .however , we concentrate only on the question about the effort per change request .this leads us straightforwardly to the * metric * `` average effort per change request '' .hence , an analysis of the mean is needed .  in the second step , we already start building the bayesian network .we look at the maintainability model and find no impacts directly on that should be considered here .however , we find 10 sub - activities including the following 3 that we will further analyse : , and .all three have impacts from facts but having in mind the available data , we ignore these and use the further sub - activities , and with its child .we create nodes for these activities and connect them with edges corresponding to their hierarchy. they can be seen in figure [ fig : screenshot ] in the box `` activities '' .we only include 3 impacts on each of the lowest level activities chosen so far because we can find corresponding data in the cm1 data set .these facts together with their impacts are :  *  + the size of a module has an impact on reading the code of the module .in essence , the larger the module , the longer it takes to read it . *  + an implementation is regular if it does not use unnecessarily nested branches .this complex structure would render coverage by tests more difficult .*  + comments need to appropriately describe the code it is associated with .the facts are added as nodes in the bayesian network . in figure[ fig : screenshot ] they are contained in the box `` facts '' .the impacts are included as the arrows from the facts to the activities .the indicators that are identified in the fourth step of the approach are taken from the available data of the data set .we only use 1 indicator per fact although we are aware that each fact has more aspects that should be covered . for the _ extent of modules _ , we use the indicator _ average module size _ given in loc .the _ regularity of the implementation _ is indicated by the _average cyclomatic complexity_. this is not a particularly good indicator as it only gives a number for the decision points in the implementation .a manual review can far better decide whether the implementation is regular .however , we have no access to review results .a similar reasoning holds for the indicator _ comment ratio _ for the fact _ appropriateness of comments_. the proportion of the comments in relation to the other code is only of minor importance in comparison with the semantic appropriateness .however , we do not have access to such a semantic judgement .the indicators can be found in figure [ fig : screenshot ] on the right - hand side in the box `` indicators '' .a difficult problem in general with bayesian networks is the definition of the node probability tables ( npt ) .various method can be used to define these npts that we will not describe here in detail . for the approach, we can simplify the problem to 2 cases : ( 1 ) activities and facts npts and ( 2 ) indicator npts .the npts for the activities and facts can , unless there is additional knowledge , be assumed as uniformly distributed .the impacts are only modelled as direct influence . here , a special method for ranked nodes can be applied that simplifies the task . for the indicatornpts either empirically investigated distributions of the company or industry average distribution should be used .this forms the distribution of the indicator values under uncertainty without any observations .  for the average effort of a change ,we refer to that gave an a mean defect removal cost of 27.4 person - hours with minimum 3.9 and maximum 66.6 .although a change does not always have to be a defect removal , it is precise enough for the proof of concept . for the distributions of the other indicators ,no published distributions were available .hence , our own expert opinion was used as a basis .figure [ fig : screenshot ] shows two scenarios .the green values describe the scenario in which no observations have been made , i.e. the general scenario .the blue values are for the scenario _ measured values _ in which for the three fact indicators the real measured values are set as observations .more specifically , the comment ratio is set to .2517 , the average cyclomatic complexity to 5.18 and the average module size to 33.47 loc .this has effects on the rating of the facts in turn on the activities . the most interesting value , however , is the average change effort . in the general scenario ,this variable has a mean value of 27 with a standard deviation of 12.1 .the additional information about the measured values shifts the distribution to the left .the mean decreases to 15.9 with standard deviation 8.5 .this is still not the really measured value of 6 but much closer than the industry standard of 27 .hence , it is an improvement in comparison to just using standard numbers .the reason for the large difference between the prediction and the real value is probably three - fold .first , the distribution from might not be appropriate for the nasa environment in which the effort for a change request seems significantly smaller than average .second , it might also be the case that the data in the cm1 data set does not use exactly the same measures as in .the degree to which additional efforts for re - inspection and re - testing are included could vary .third , several more factors than the 3 considered might have an influence on the maintenance effort .the quality model contains many explanations in terms of facts that should be investigated .  nevertheless , the point of the proof of concept was not mainly to show predictive validity but to investigate the general feasibility .we saw that it is possible with reasonable effort to build a bayesian model .the 4-step approach gives direct guidance for most of the network building .only setting up the npts is still a challenge .there are usually several possibilities how a relationship can be expressed and with how much uncertainty it should be afflicted .this still needs expert opinion and experimentation .nevertheless , agenarisk provides very good tool support to find easier ways to define an npt .as our abqms can get very large with a few hundreds of model elements , it remains to be evaluated whether the approach scales when the quality model is fully mapped to a bayesian network .probably , a selection of a subset of the quality model is necessary first .furthermore , it is important to note that a more in - depth validation of a resulting bayesian network is necessary in order to ensure that all parts , topology , node states , and npts , represent the interdependencies of the quality factors good enough so that a valid statement about the quality of the software system can be made .this is not covered by this proof - of - concept but has to be the next step .the basic idea to use bayesian networks for assessing and predicting software quality has been developed mainly by fenton , neil and littlewood .they introduced bayesian networks as a useful tool and applied it in various contexts related to software quality .in they formulate a critique on current defect prediction models and suggest to use bayesian networks .other researchers also used bayesian networks for software quality prediction similarly   the work closest to the approach proposed in this paper is .they discuss quality models such as the iso 9126 and their problems such as the undefinedness of the relationships in such a model .they aim at solving these problems by defining bayesian networks for quality attributes directly .our work differentiates in using a defined structure for quality models that contain far more details as common quality models .this structure and detail allows a straightforward derivation of a bayesian network from the quality model .this has the advantage that the basic quality model can also be used for other purposes then prediction such as the specification of quality requirements .beaver , schiavone and berrios also used a bayesian network to predict software quality including diverse factors such as team skill and process maturity . in his thesis , beaver even compared the approach to neural networks and least squares regressions that both were outperformed by the bayesian network .however , they did not rely on a structured quality model as in our approach .a high goal in software quality management is the reliable quantitative assessment and prediction of software quality .many efforts in building assessment and prediction models have given various insights in the usefulness but also the constraints of such models .however , these models have not been tightly integrated into other quality management activities .activity - based quality models have proven in practice to be a solid foundation for defining quality on a detailed level .however , quantitative analyses have not been directly possible so far .bayesian networks have been shown to provide promising results in quality predictions . because of that and their clear structuring that can straightforwardly reflect the structure of activity - based quality models , a 4-step approach for transferring activity - based quality models to bayesian networks was proposed .it allows to systematically construct a bayesian network that uses the knowledge encoded in the quality model to provide information about a given assessment or prediction goal . in the terminology of , we use a _ quality definition model _ ( the abqm ) and enrich it with a _ quality assessment _ and _ quality prediction model _ ( the bn ) .although not fully validated , we demonstrated the approach in a case study using real nasa project data .the proof of concept showed the applicability of the approach on such a project and even could improve the prediction in comparison to industry standard values .the prediction was not very accurate but that can have various reasons such as a different effort distribution at nasa or the influence of more factors that have not been considered because there was no data available .the use of bayesian networks opens many possibilities .most interestingly , after building a large bayesian network , a sensitivity analysis of that network can be performed .this can answer the practically very relevant question which of the factors are the most important ones .it would allow to reduce the measurement efforts significantly by concentrating on these most influential facts .we plan to apply this approach in future case studies at our industrial partners in order to further validate the approach .a comprehensive analysis of the predictive validity is necessary to judge the usefulness of the approach and to compare it with other means for assessment and prediction .this work has partially been supported by the german federal ministry of education and research ( bmbf ) in the project quamoco ( 01 is 08023b ) .j. m. beaver , g. a. schiavone , and j. s. berrios .predicting software suitability using a bayesian belief network . in _ proc .fourth international conference on machine learning and applications _ , pages 8288 .ieee computer society , 2005 .f. deissenboeck , e. juergens , k. lochmann , and s. wagner .software quality models : purposes , usage scenarios and requirements . in _ proc .7th international workshop on software quality ( wosq 09)_. ieee computer society , 2009 .f. deissenboeck , s. wagner , m. pizka , s. teuchert , and j .- f .girard . an activity - based quality model for maintainability . in _ proc .23rd international conference on software maintenance ( icsm 07)_. ieee computer society press , 2007 .

Abstract: assessing and predicting the complex concept of software quality is still challenging in practice as well as research . activity - based quality models break down this complex concept into more concrete definitions , more precisely facts about the system , process and environment and their impact on activities performed on and with the system . however , these models lack an operationalisation that allows to use them in assessment and prediction of quality . bayesian networks ( bn ) have been shown to be a viable means for assessment and prediction incorporating variables with uncertainty . this paper describes how activity - based quality models can be used to derive bn models for quality assessment and prediction . the proposed approach is demonstrated in a proof of concept using publicly available data .   [ software quality assurance ( sqa ) ]


common understanding identifies a _ network _ as a set of items , called _ nodes _ ( or vertices ) , with connections between them , called _ links _ ( or edges ) . since many systems in the real wordtake the form of networks ( also called _ graphs _ in much of the mathematical literature ) , they are extensively studied in many branches of science , like , for instance , social , technological , biological and physical science . at the beginning , the study of networks was one of the fundamental topics in discrete mathematics : euler is ascribed as the first providing a true proof in the theory of networks by its solution of the knigsberg problem in 1735 .recently , also thanks to the availability of computers , the study of networks moved from the analysis of single small graphs and the properties of individual nodes and links within such graphs to the consideration of large - scale statistical properties of graphs . thus , statistical methods became a prominent tool to quantify the degree of organization ( _ complexity _ ) of large networks .a typical approach in statistical mechanics of complex networks is the statistical ensemble .such an approach is a natural extension of erds - rnyi ideas .it has been performed through two basic ideas : the configuration space weight and the functional weight .the first one is proportional to the uniform probability measure on the configuration space which accounts for the way to uniformly chose graphs in the configuration space . whereas , functional weights depend on the network topologies and are chosen in order to address the statistical mechanics approach to networks different from the random graphs , which have some typical structures , like the _ small world _property , the power - law degree distribution , the correlation of node degrees , to name the most frequently addressed . during the last decade ,several works have been inspired by this approach .recently , the techniques of statistical mechanics were complemented by new topological methods : a network is encoded through a simplicial complex which can be considered as a combinatorial version of a topological space whose properties can be studied from combinatorial , topological or algebraic points of view .thus , regarding the mentioned topological aspects , different measures of simplicial complexes and of networks stemming from simplicial complexes can be defined .this provides a link between topological properties of simplicial complexes and statistical mechanics of networks from which simplicial complexes were constructed .  in this workwe consider a geometric entropy which is inspired by microcanonical entropy of statistical mechanics and stems from information geometry ( ig ) .in particular , a riemannian manifold ( differentiable object ) is associated to a network ( discrete object ) and the complexity measure is the logarithm of the volume of the manifold .more precisely , random variables are associated to each node of a network and their correlations are considered as weighted links among the nodes .the nature of these variables characterizes the network ( for example , each node can contain energy , or information , or represent some internal parameter of a neuron in a neural network , or the concentration of a biomolecule in a complex network of biochemical reactions , and so on ) .the variables are assumed to be random either because of the difficulty of perfectly knowing their values or because of their intrinsic random dynamical properties .thus , as it is customary for probabilistic graphs models , a joint probability mass function is associated to the description of the network . at this point, we assume gaussian joint probability mass functions because of their tractability and since they are used extensively in many applications ranging from neural networks , to wireless communication , from proteins to electronic circuits , etc .finally , the geometric complexity measure of networks is obtained by resorting to the afore mentioned relation between networks and joint probability mass functions and introducing in the space of these mass functions a riemannian structure borrowed from information geometry .in addition to the statistical methods in network complexity we also consider topological methods by encoding a network into a _ clique graph _ , that has the complete subgraphs as simplexes and the nodes of the graph ( network ) as its nodes so that it is essentially the complete subgraph complex .the maximal simplexes are given by the collection of nodes that make up the cliques of .in particular , we are interested in the information about the topological space stored in the number and type of holes it contains .so , we exploit algebraic topology tools in order to describe a network by means of its homology groups .the ability of the geometric entropy to capture algebraic topological features of networks is investigated by a bottom - up approach .first , we consider networks with low number of nodes ( _ small size _ ) and given homology groups .then , we compute the dimension of the homology groups of large size networks . hence , we compare the geometric entropy against the dimensions of homology groups ( betti numbers ) revealing a clear detection of topological properties of the considered networks . in particular , when dealing with large size networks , we consider random graphs and scale free networks .according to the well known transition in the appearance of a giant component , a description of networks through their betti numbers shows a clear correlation with the growth of the size of the largest components .this perfectly matches the behaviour of the geometric entropy which in turn , when compared to betti numbers of the the networks , clearly appears to probe relevant topological aspects of the networks .the organization of the present paper is as follows . in section [ sec2 ]we review some methods of the algebraic topology useful to describe topological properties of networks encoded in simplicial complexes . in section [ sec3 ]we describe the geometric entropy stemming from both statistical methods and ig methods . in section [ sec4 ]we compute the homology groups of small size networks as well as of large size networks within the ensembles of random graphs and scale free networks .then we compare the geometric entropy computed on these networks against their betti numbers .concluding remarks are given in section [ sec5 ] .in order to make the present work self contained , we start by reviewing some methods of combinatorial algebraic topology by referring to ; these methods allow a topological characterization of networks .in particular , we focus on simplicial algebraic invariants ; among them , we select homologies since they are easier to compute than , for example , homotopy groups .let be a set of vertices ( or nodes ) .a _ simplex _ in with dimension is any its subset with cardinality equal to , and it is called a -simplex ; in particular the _ empty set _ is the only -simplex .face _ of a -simplex is the simplex whose vertices consist of any nonempty subset of the ; if is a -simplex , with , it is called a -face of .the subset needs not be a proper subset , so is regarded as a face of itself .  a _ simplicial complex _ consists of a set of vertices and a set of simplexes such that ( i ) any set consisting of exactly one vertex is a simplex ; ( ii ) any nonempty subset of a simplex is a simplex .it follows from condition ( i ) that -simplexes of correspond bijectively to vertices of .analogously , from condition ( ii ) it follows that any simplex is determined by its -faces .thus , we can identify as the set of its simplexes , and a vertex of as the -simplex corresponding to it .for example , let be the set of vertices , and consider the simplicial complex on with set of simplexes . intuitively , is a set of vertices and links ( edges ) among them . therefore , is called the _ standard polygon _ with edges .if , we add to it the -simplexes , for , we arrive at the simplicial complex called the _ standard polygonal disk_. less formally , in order to obtain we add to the standard polygon the triangles that we can construct among triplets of the set of vertices . as an example , we draw the difference between a polygon and a polygonal disk in figure [ polygon ] when .  = [ circle , draw , fill = green!50 , inner sep=0pt , minimum width=4pt ]  c  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ( : 2 ) ( + 144:2 ) ( : 2 ) ( + 216:2 ) ;  c  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  the _ dimension _ of a simplicial complex is . is said to be if it contains only a finite number of simplexes .in such a case ; however , if , needs not be finite .indeed , consider the simplicial complex with set of vertices , and as the set of simplexes the edges as varies in ; then . in this case , is not finite but .a simplicial complex , that we have abstractly defined upon a set of vertices , can be also assigned by means of a _ geometric realisation _ by assuming that its vertices are points in .for example , we get what is known as the natural realisation if we take and , where the are the standard basis vectors in .we note that although may be -dimensional , a realisation may not `` fit '' into .indeed , the standard polygon on vertices can not be embedded into even though its dimension amounts to .for this reason , being networks abstract discrete objects , we rely on the combinatorial approach avoiding any particular geometric realisation .when a simplicial complex is finite , it is possible to introduce the _ euler - poincar characteristic _ as the summation along the number of all its simplexes where is the number of -simplexes . in particular , we have that .let be a set of vertices , a _ step _ into a simplicial complex on is an element such that ; is the _ initial point _ and is the _ final point _ of the steptwo steps are _ consecutive _ if the final point of is the initial point of .path _ is a sequence of consecutive steps .two paths and can be multiplied if is a path , i.e. and are consecutive ; such a path is called the product .consider such that ; we say that the path is elementary contractible in the step . by referring to the fig.[polygon]-leftthe path can be contracted to the path , whereas there is no way to contract any paths in fig.[polygon]-right .in general , if a path has two consecutive steps , we can substitute them with the step ; in this way we obtain a new path , and we say that is obtained from by an _elementary contraction_. vice versa , we say that is obtained from by an _elementary expansion_.  two paths and in are called _ homotopic _ if it is possible to go from one to the other by means of a finite number of elementary expansions or contractions ; in this case we write .the latter is an equivalence relation that preserves the initial and the final points ; moreover the product  " passes to the quotient , meaning that if and , then .consider now ; a _ loop _ with base point is a path in which starts from and ends at .the set of the homotopy classes of loops at forms a group under multiplication , called the _fundamental group _ of at .given the usual set of vertices , an _ oriented _ -simplex of the simplicial complex on is a -tuple such that . for are no oriented -simplexes ; for every vertex there is a unique oriented -simplex .then we define the module of chains in as the free -module generated by oriented -simplexes .a -chain is basically a linear combination of -simplexes with coefficients in .then it straightforwardly follows that for .consider now the module homomorphisms for .their values are uniquely determined on the -simplexes .so , we can define where means the oriented -simplex obtained by omitting . actually , the homomorphism is a boundary operator in the sense that it acts on a by giving rise to its faces .it is not difficult to show that for all .let be an oriented -chain in , i.e. ; then , is called a -_cycle _ if ; is called a -_boundary _ if there exists such that .the set of the -cycles , and the set of the -boundaries are submodules of , and the relation holds true for . less formally , a cycle is a member of if it `` bounds '' something contained in the simplicial complex . for example , by referring to the polygonal disk of fig .[ polygon ] , we can see that the oriented chain is a boundary , while the chain is not .since for all , we can consider the quotient module ( called - module of homology ) .intuitively , the construction of homology assumes that we are removing the cycles that are boundaries of higher dimension from the set of all -cycles , so that the ones that remain carry information about -dimensional holes of the simplicial complex .if is finitely generated ( which is necessarily true if has finitely many simplexes ) from the structure theorem it follows the is isomorphic to the direct sum of a finite free -module and a finite number of finite cyclic groups , where divides .thus , the is defined as the number of basis elements of on .such a rank is also the betti number of , i.e. , hence the euler - poincar characteristic becomes .let us now compute the lower homology groups .consider first an empty simplicial complex . in this case, it has only one -simplex , the empty set ; thus , the complex chain is isomorphic to .in addition , for . for these reasons and for . for non empty the homomorphism is surjective and .concerning , notice that is defined by for any . from andthe relation it follows that , if is a -simplex in , then . for this reason when is a path in from to .let us now assume that is connected and .then , every -cycle can be written as and it follows that .in contrast , if is not connected and is the set of vertices one for any connected component , we have that is the -free module on the classes for .hence , the notion of connectivity in is reflected on , the dimension of which , that is the betti number , counts the number of connected components of a simplicial complex .again , consider a connected simplicial complex and ad vertex .an homomorphism from the fundamental group of homotopy to the first homology group can be obtained via the morphism where is the -edges closed polygon . in this way , the homomorphism is defined as follows {\pi_1}\right)= \sigma_*[p_n]_{\pi_1}:=[\sigma(p_n)]_{h_1},\ ] ] where {\pi_1} ] is the homology class of .it is not difficult to prove that is surjective and its kernel is given by the commutators =\alpha*\beta*\alpha^{-1}*\beta^{-1} ] as a system of local coordinates for .hence , assuming parametrizations which are we can turn into a differentiable manifold , that is called a _ statistical manifold _ .  given ,the fisher information matrix of at is the matrix  ] , for and .finally , we can associate to the bare network the statistical riemannian manifold , with   at this point , in order to take into account the possible connections among vertices of a network , we consider its adjacency matrix .then , let the map be defined by where denotes the set of symmetric matrices over with vanishing diagonal elements that can represent any simple undirected graph .the manifold associated to the network is the riemannian manifold given by and with components where is entry of inverse of the invertible matrix .hence , given a network with adjacency matrix and associated manifold , we put forward the following network complexity measure , where is the volume of the manifold obtained from the volume element unfortunately , in such a way is ill - defined since the parameter space is not compact and since diverges as approaches to zero for some .thus , as it is usual , we regularize it as follows where is any suitable regularizing function .this should be a kind of compactification of the parameter space and with the excision of those sets of values which make divergent .in this section we apply the methods so far described , i.e. we compute the homology groups of given networks and compare them against the geometric entropy values of the same networks .actually , we refer to the dimensions of those groups via the betti numbers and work out a relation between them and the geometric entropy .in such a way , we show a clear detection of this kind of network topological features by means of the entropy of the eq . .the aim of this work consists of checking to what extent the geometric entropy can be sensitive to some topological property of networks beyond its effectiveness beforehand checked as a network complexity measure . in order to get well - grounded insights about the relation between the betti numbers and the values of , we firstly consider low size graphs upon which analytical methods can be worked out .once that is set up , we investigate large size networks particularly focusing on two graph ensembles , namely the random graphs and the scale free networks .let us consider five nodes graphs for our preliminary investigation .the clique graph interpretation thereby ascribes the simplicial complex as a , i.e. a nodes fully connected network . in order to show how the entropy works, we refer to two chains within as in tab .[ tab1 ] .[ ht ]    in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & 1 & & +    in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ( + 144:2 ) node2 ( : 2 ) node0 ;  & & & & 2 & & +  the network on the left side of tab .[ tab1 ] is a chain with one -simplex and three -simplexes ; from relation it follows that its topological dimension is . whereas , the network on the right side of tab .[ tab1 ] is a chain with one -simplex and two -simplexes and from we have that its topological dimension is .according to graph theory , the adjacency matrices of these networks are respectively .consider now independent random variables sitting on the five completely disconnected nodes of the network , with gaussian joint probability distribution given by }{\sqrt{(2\pi)^5\det c_0(\theta)}},\ ] ] where and covariance matrix is given by   from eq .we obtain that in order to associate a parameter space to each network , the matrices and must be positive definite , as indicated by eq . .such a condition is fulfilled by imposing that each main minor of the matrix has positive determinant .thereby , we arrive at moreover , from eq .we endow and , defined in , with riemannian metrics , hence obtaining and as manifolds associated to the networks . here , and given by the next ingredients necessary to compute the geometric entropy values against the two networks are the volume elements computed through eq .however , as we can see from the analytical expressions for and , it turns out that both and are ill - defined .in fact , on the one side , the numerator of diverges as goes to infinity for some ; on the other side , the denominator of goes to zero as approaches the lower bound of for some .  in order to overcome this difficulty, we introduce a regularizing function .this issue has been widely tackled in the literature ( see for instance ref . ) . as far as our approach is concerned , the aim is to make meaningful the following integrals where and . supported by a general result ( see corollary 2 in ) we consider the following regularizing function , here , denotes the heaviside step function and is the trace operator , while is a _real _ number accounting for a finite range of values where integrals in are substantial .regularizing function in depends on structure of networks through adjacency matrix and cures all the possible divergences of integrals , due to the functional forms in .indeed , if goes to infinity for some , then the integrand is killed to zero by function . moreover ,if goes to zero then the singularity is removed by the exploiting the well - known relation .we are now ready to compute the values of the integrals in through the regularizing functions as in which are shown in tab .the difference of entropic values between the -dimensional chain and the -dimensional chain is a particular result of a more general one .in addition , it is worth remarking that integrals in do not depend on the particular label permutation of the network .hence , if two networks are isomorphic , then the values of as in coincide .  actually , two isomorphic networks , intended as clique graphs , have the same topological features ; thereby , they have the same homotopy and homology groups . as a consequence , whenever the betti numbers and are different , then the networks are not isomorphic . in tab .[ tab2 ] we show the geometric entropic values of as in computed for several networks . in particular , we point out that reflects the differences of their first homology groups . indeed , by following the values on in tab .[ tab2 ] , we can see that the larger the number of connected components of a network , the lower the value of the entropy . on the contrary , according to the values of in tab .[ tab2 ] , the larger the number of cycles the more complex the network .in addition , cycles mostly influence the complexity with respect to topological dimension as it follows by comparing the values of the entropy of the last two networks on the right side of tab .[ tab2 ] .[ ht ]   & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & & & & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & & & & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & & & & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & & & & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & & & & +   & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ( : 2 ) node0 ( + 216:2 ) node3 ;  & & & & & & & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & & & & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 288:2 ) node3 ( + 216:2 ) node4 ( + 360:2 ) node0 ( + 144:2 ) node2 ( : 2 ) node0 ( + 72:2 ) node1 ( + 288:2 ) node3 ( + 144:2 ) node2 ( + 216:2 ) node4 ( + 288:2 ) node3 ( : 2 ) node0 ;  & & & & & & & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 252:1 ) node3 ( + 252:2 ) node4 ( + 360:2 ) node0 ( + 72:2 ) node1 ( + 252:1 ) node3 ( + 144:2 ) node2 ( + 252:2 ) node4 ( + 252:1 ) node3 ( : 2 ) node0 ;  & & & & & & & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 45:0 ) node3 ( + 252:2 ) node4 ( + 360:2 ) node0 ( + 144:2 ) node2 ( + 252:2 ) node4 ( + 45:0 ) node3 ( : 2 ) node0 ;  & & & & & & & +  in tab .[ tab3 ] we can see that networks with the same betti numbers and do not have the same values of .this is in agreement with results in where it has been shown that networks with different degree distributions do not have the same complexity computed by .hence , characterizing networks complexity through their first homology groups is not enough . beyond these topological features , it is necessary to consider also other aspects as it is well known in the vast literature around . for this reason , the fact that takes different values for networks having the same betti numbers and is not a validity restriction of the geometric entropy .[ ht ]   & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & & & & +   & +  in 0 ( : 2 ) node0 ( + 72:2 ) node1 ( + 144:2 ) node2 ( + 216:2 ) node3 ( + 288:2 ) node4 ( + 360:2 ) node0 ;  & & & & & & & +   we now investigate the behaviour of the geometric entropy , given in , by comparing it with respect to the behaviour of the betti numbers computed for two ensembles of networks , namely the random graphs and the scale free networks .the numerical computation of has been performed on networks by exploiting the functional form of the components of the varied fisher rao metric .then , the volume regularization has been obtained at first by restricting the manifold support to an hypercube , and we worked out monte carlo estimates of the average by generating markov chains inside . in this case , the number of random configurations considered varies between and .finally , the regularization procedure of the volume is obtained by excluding those points where the value of exceeds ( the numerical overflow limit of the computers used ) .one of the basic models of random graphs is the uniform random graph .this is devised by choosing with uniform probability a graph from the set of all the graphs having vertices and edges , with a nonnegative integer .free networks can be obtained as special cases of random graphs with a given degree distribution showing thereof a power law degree distribution .these are described by two parameters , and , which define the size and the density of a network ; hence , given the number of nodes with degree , these models , denoted , assigns a uniform probability to all graphs with  .  in fig .[ rg ] we report the behaviour of of vs for a fixed value of , namely , together with the behaviour of the betti numbers and of  . by interpreting as clique graphs, we can see a perfect correlation between and the betti number .this is not surprising as reflects the number of connected components of and the appearance of a giant component is accounted for when . whereas the correlation of with the betti number of is subtler to interpret .indeed , rapidly increases its value as increases , contrary to which shows a saturation when the numbers of is large . actually , counts the cycles of and if on one side equals zero when reflecting a well known theoretical result , on the other side it is not enough for completely describing the topology of when is larger than .however , the geometric entropy properly correlates with the betti numbers and when and the only homology groups of are and .   and the betti numbers , of as functions of the number of randomly chosen links of weights equal to . ]the scale free network ensemble is recovered in the case of networks with a finite number of cycles . in order to highlight the connection of the geometric entropy of eq .with the topology of the power law random graph model intended as a clique graph we have proceeded as follows .we considered networks of nodes for which , without loss of generality , we set .for each value of , we selected different realisations of the networks , each realisation having the same value of .actually , because of the practical difficulty of getting different realisations of a scale free network with exactly the same value of at different values , we accepted a spread of values in the range .  in fig .[ sf ] we report the behaviour of the geometric entropy of the power law random graphs when , , and the exponent is in the range together with the behaviours of the betti numbers of the same networks  .the pattern of displays a clear correlation with the betti number . in addition , in the range of considered here , the only topological features of consist of the homology groups and .hence , a tight correlation between the geometric entropy and the topology of the power law random graphs is found to exist . indeed , for the euler poincar characteristic is supplied only by and as follows from fig .[ sf ] and eq . .   and betti numbers , of power - law networks as a function of the exponent ; values varied - according to the realization of the rg and , independently , of - approximately in the range . ]  finally , in both figures , fig .[ rg ] and fig .[ sf ] , the pattern of shows a sort of saturation of its values .the reason of this relies on the numerical methods for regularizing the volume computed through the volume form .interestingly , the apical values of and are different thus allowing to characterise different network ensembles via the geometric entropy defined in eq . .in this work we have pursued our investigation about the potential applications of a recently defined geometric entropy which has been shown to perform very well in quantifying networks complexity .the present investigation focused on the possible use of the mentioned geometric entropy to catch some topological property of networks . by considering networks as clique graphs , we proceeded in a bottom up analysis . to begin with ,small size networks have been considered and analytical computations have been applied . then , numerical computations has been used to tackle large size networks .the entropy of a network is obtained after having associated to it - on the basis of a probabilistic approach - a riemannian manifold , and then by computing the volume of this manifold . since in general thismanifold is not compact , we introduced an `` infra red '' and `` ultra violet '' regularising function to compactify it .this procedure is independent of networks topology in as much as it is defined up to graph isomorphisms .small size networks are ascribed to a simplicial complex .the analytical computation of for these networks displays a monotonic behaviour of with respect to the betti numbers and .however , the information about network complexity retained by the geometric entropy can not be reduced to the only knowledge of the topological properties described by and .this is due to the fact that and do not exhaustively account for the network connectivity .this explains why different values of are found for networks with same or ( see tab .[ tab3 ] ) . then , passing to a `` coarse grained '' description for large networks , the connection between and topology becomes clearer .we have considered two different network ensembles in tackling large size graphs , that is , the random graphs and the scale free networks .the entropy of random graphs perfectly correlates with their .this is not very surprising because counts the number of connected components of a graph .thus , due to the appearance of a giant component in random graphs , is expected to asymptotically reach the value , as well saturates for large values of .as far as is concerned , we found a proper correlation between the entropy of scale free networks and their number of cycles .indeed , scale free models account for networks with a finite numbers of cycles and the behaviour of properly agrees with the pattern displayed by the betti number of the considered power law random graphs .this suggests a strong correlation between and the topology of scale free networks .finally , from both figure [ rg ] and figure [ sf ] , we can see that the entropy saturates for some values of and , respectively .since these apical values are different , it seems that the geometric entropy is able to characterise some difference within the network ensemble .this paves the way to further and deeper studies also about this issue .we are indebted with m. piangerelli , m. quadrini , and v. cipriani of the computer science division of the university of camerino for computational help . d.f .also thanks e. andreotti for useful discussions .data for betti numbers were computed by using tools from : j. binchi , e. merelli , m. rucco , g. petri , f. vaccarino , _ jholes : a tool for understanding biological complex networks via clique weight rank persistent homology _ , electronic notes in theoretical computer science ( ( proceedings of the 5th international workshop on interactions between computer science and biology ( cs2bio14 ) ) ) * 306 * , 518 ( 2014 ) .

Abstract: a geometric entropy is defined as the riemannian volume of the parameter space of a statistical manifold associated with a given network . as such it can be a good candidate for measuring networks complexity . here we investigate its ability to single out topological features of networks proceeding in a bottom - up manner : first we consider small size networks by analytical methods and then large size networks by numerical techniques . two different classes of networks , the random graphs and the scale free networks , are investigated computing their betti numbers and then showing the capability of geometric entropy of detecting homologies .