software requirement specification ( srs ) document forms the basis of problem analysis between client and developer .srs needs to be very specific , while serving as a basis , to proceed towards implementation of desired software .it is very often observed that srs is expressed in any natural language as comprehensible by the client .but it may be ambiguous , possibly inconsistent , and probably unmanageably large from the software analyst s point of view .  identifying major functionalities from the oo analysis point of view plays an important role in project success .the use of formal languages like unified modeling language ( uml ) have been applied to avoid the inherent problems of natural language such as incompleteness and ambiguity .earlier analysis was used to help for an explanatory model called as build and fix programming style .but this style was observed to be very informal and there are no set of rules as to which one is superior .every programmer formulates his own software development technique solely guided by his expertize and in his own language and style .  in recent years, the object - oriented software development style is a preferred style over conventional style by developers as the present day software development languages are object oriented in nature .hence , oo analysis of software helps to find out the candidate for class , function , and the attributes associated with those classes .natural language processing ( nlp ) combines the effect of computer science and linguistics branch which are concerned with the interaction between the computer and human languages .natural language generation systems mostly extracts right information from statements which are in human readable form .the aim of the work is to present a review on existing literature of application of nlp in object oriented analysis ( ooa ) based on literature available such as : abbott , saekai and enamoto , nanduri and rugaber , juristo and moreno , d. popescu et al . , ibrahim and ahmed , harmain and gaizauskas , overmyer and rambow , mich . among these literaturesfew authors suggested automated tools , other use manual approach and few combine both tool and manual approach to obtain the different elements of oo analysis .  in order to examine each proposal, the following dimensions may be considered  * * steps to modify the srs into required form : * as discussed above the srs is written in a form i.e. , convenient to the user . during this step the srsis modified to a format by which the process of finding the keywords became easier for analysts . ** finding out the candidate for class and object from modified srs : * after transforming the srs to the required format for analysis , then the candidates for the class name and its details are traced out .the process of identification of the class name and its detail can be a manual or a automated process . in manual process, the domain expert analyzes text to bring out `` intermediate output '' then automated process considers the intermediate output to generate the desired output .the objective of this work is two - fold :  1 .this analysis provides information on available techniques for use of nlp , further to be considered under oo analysis .2 .  it provides an overview of the current state for the use of nlp in oo analysis , focusing on the strengths and weaknesses of existing proposal .thus researchers can have a broad knowledge into the work that already being done and that still can be carried out in this field .the paper is organized as follows : in section 2 the existing proposals of the use of nlp for ooa analysis are presented .section 3 presents an analysis on all proposals .finally , section 4 discusses on some concluding remarks and highlights the future trend .jones in his paper presented a review paper on nlp based on the historical study .the paper reviews study of nlp from late 1940 s to present by distinguishing four phases in the history of nlp . the impact of use of machine translation , artificial intelligence influence , logico - grammatical style adaption and massive language data attack act as the basis of the phase division .  1.  the early phase of study on nlp was during * late 1940s to late 1960s * and in this period focus was mainly in machine translation ( mt ) .noticeable amount of work done in ussr , usa , europe and japan during this period .thus the languages considered for research in this period were mostly russian and english .language syntax was mainly the area of research in this period as syntactic processing was manifestly necessary , and partly carried out through implicit or explicit endorsement of the idea of syntax - driven processing .though during this period use of computers for literary and linguistic study has began , but it has never been linked with nlp .next phase of study was undertaken from * late 1960s to late 1970s * and the work mainly focused on use of artificial intelligence ( ai ) in nlp , with much more priority on word knowledge and on its implementation in the construction and manipulation of meaning representations .ai was mainly considered during this period for construction and addressing of knowledge base or data . in late 1960s , the prevalent theory of linguistic is transformational grammar , which provides the semantic information about nlp .this phase was mainly concerned to the period of * late 1970s to late 1980s * and characterized as grammatico - logical phase .the requirement of development of grammatical theory and movement towards incorporation of logic in knowledge representation and reasoning triggered during 70s . during this phase , deliberate attempts made to transform the commercially available dictionary to machine - readable form which further helps in text corpora validation and customizing lexical data .the forth and the final phase can be attributed as the study carried out from * late 1980 s onwards*. during this phase , the main area of research is statistical language data processing .the identification of linguistic occurrence and patterns in the corpus for both syntactic and semantic analysis , drew interest in this period . the present attention on lexicon , retrieving statistical information , and restore interest in mt .table [ time ] compares the nlp research based on time period   in course of this section , the study made by various authors are analyzed on the basic of how they transform the srs and how the candidate for the class name and its details are found out from transformed srs .the study made by * abbott * proposes a method to derive the elements of object oriented analysis i.e. , data type , variables , operators , attributes and candidate for class name form english statements .this paper shows the process of analysis of the english statement of srs and helps to generate elements of oo analysis .the approach used in this paper is divided into three different sections .these are as follows  a.  * development of informal strategy for the problem : * the informal strategy should suggest the problem solution on the conceptual level .this step express the solution of the problem in terms of problem domain .b.  * formalize the strategy : * the second step is formalizing the solution by finding out its data types , objects , operators , and control constructs .the steps of formalization are : 1 ._ identify the data types : _ the data types are suggested by the common nouns .the name of a class of beings or things are known as common noun .identify the objects of those types : _ objects are suggested by proper nouns or direct references .the name of specific things or beings is known as proper noun .a specific , previously identified being or thing without necessarily referring to it by name is known as direct reference .identify the operators for the objects : _ operators are suggested by verb , attribute , predicate or descriptive expression .attribute is a property , association , characteristics or component of something .predicate designates a property or relation that can be consider true or false i.e. , to hold or not to hold .a descriptive expression is a characterization for which there may be some particular object .the _ control structures _ are directly provided by the english language .segregate the solution into two parts , a package and subprogram : * the package will contain the formalization of the problem domain , that is , the data types and their operators. then subprogram(s ) will contain the specific steps ( expressed in terms of the data types and operators defined in the package ) for solving the particular problem .during the course of the paper , different types of nouns are analyzed i.e. , the difference of common noun with proper noun , direct references and mass nouns are provided .classes of objects are referred by common noun but specific and individual objects are referred by proper noun .mass nouns are names of qualities , substances , and activity that do not have an a priori organization into individual units or instances .  in this paper abbotthad taken an example of _ `` calculating the days between two given dates '' _ to explain his technique .as per the analysis referred in the paper , the process is divided into three different steps :  * in step 1 , an informal strategy of the problem analysis is provided . in this stepthe process of getting the detailed solution is being analyzed . * in step 2 , the data types , objects , operators , and control structures are found for the specific problem form the informal specification . * in step 3 , the final solution is being prepared .the package for the problem is assigned and the subprogram details are provided in this step .though this paper is comparatively easy in solving the problem of finding the candidate for class name and its details but the process is manual .a software engineer having a good knowledge about the domain requires to provide the step - wise solution of the problem .the paper by * saeki et al . * discusses the process of derivation of formal specification from an informal specification written in natural language .the informal specification contains important information leading to their formal specification or the prototype program .then the similarity between the structure of the words and the structure of software component is analyzed . in this paper ,the * `` lift control system '' * example has been explained as an informal specification to explain the process .the process consist of three major steps as follows :  a.  * design activity : * the purpose of this step is to construct a module design document from informal specification .the modular design document presents the modular structure of a formal specification , i.e. , external design of class modules which contain class names , method names and message protocols .this design activity consists of several sub - activities .each of them produce an intermediate product using the informal specification .the intermediate products obtained are as follows : * * noun table * : this table contains information about extracted nouns .according to the author , noun can be classified into different group , i.e. , _ class noun _ identifies object or set of objects , _ value noun _ identifies the values or set of values , _ attribute noun _ identifies the attribute of the objects and _ action noun _ identifies the actions to be carried out . * * verb table : * this table contains information about extracted verbs , i.e. , verb names , their categories , their subjects and objectives .according to the author , verbs can also be classified into different groups ._ relation verb _ specifies the relation between objects or between objects and their attributes . _state verb _ specifies the internal state of the object or the attribute values of the object ._ action verb _ specifies the action to the objects and _ action relational verb _ specifies the relation between the actions . * * action table * : this table presents the extracted actions , their agents , target objects and input output parameter associated with them . for each action verb, there is always an agent and its target object .the action verb changes the state of the target object . in order to extract a target object from a sentence , verb patterns that appear in various kinds of natural language specificationare examined .* * action relation table : * the informal specification , its verb table and its action verb are needed to identify the relationship between the actions . for every action to be performed ,the sender , receiver and the message transmission between them need to be identified . in this paper , the authors have used a rule called `` action relation rule '' to generate a few candidate for a sender - receiver pair . ** module design document : * a module design document is constructed using above mentioned tables .the noun table helps to identify objects and their attributes .the verb table is used to extract relationship among objects and kind of attributes , each object possess .the module design document can have both graphical and textual representation .both of them are based on syntax of formal specification language tell and object oriented language smalltalk80 .b.  * elaborate - design activity cycle : * the task of this activity is to refine and rewrite the informal specification as per module design document .the output of this activity is a natural language description which is accurate , detailed , structured yet informal .whenever the elaborate - design cycle is carried out , a pair of informal specification and its module design document is obtained .the elaborate activity consists of sub - activities such as paste , refine and an intermediate product a paste document is generated . *_ paste : _ during this phase , the sentences of informal specification may be paraphrased to accurate sentence and then replace the original one . in the updated expression , the nouns and verbs are extracted as classes , and attributes or method to be used respectively . * _ refine : _ the informal specification of each pasted module are constructed during refine activity . during this phase , the internal behavior and property of each class module is rewritten again , which are used for construction of each class . * _ design activity for elaborate informal specification : _ before this activity , the module design document for each class is constructed from elaborate informal specification . during this activity ,each class module and method module are composed into small sub module to realize the internal behavior and property .+ the cycle continues until a formal specification is obtained from the informal specification .the requirements need to be refined and made simpler and smaller during this cycle .c.  * software process based on natural language : * during this process the design and elaborate process are embedded . before this phase , the informal specification is already converted to formal specification , the rest steps are as follows * _analyze activity : _ this step acquires a problem description by means of interaction between customer and developer .an informal specification is obtained as an output of this step . * _ evaluate activity : _ during this activity , the obtained formal specification is executed and verified .the diagnostic document keeps a record of the execution and verification . * _ evolve activity : _ during this activity , the developers check the diagnostics document and find whether something is to be modified or not .if so , they create a new version of informal specification , this process is called as `` evolve activity '' . from this new document , new module design document , new formal specification , new diagnosis document are produced . *_ instantiate activity : _ when the developers judge that the formal specification is accepted to the customer s need , they finally produce the concrete program code .this activity is called as instantiate activity .a formal specification is considered to be a generalization or abstraction of program code .  in this paper , the author has presented a software design process based on natural language and obtained formal specification from informal one through the design and elaborate activity .the technique used in this paper is verb - oriented which has an impact on the dynamic nature of informal specification .along with this , nouns also needed to extract the hierarchy of class .this paper by * d. popescu et .al . * proposed an approach to help the writer or reviewer in identifying the ambiguities in nl srs .a tool named `` dowser '' is proposed by them which creates oo digram from nl srs .their approach consists of three steps  1 .the nl srs is parsed according to a constraining grammar .2 .  from the obtained relationship after parsing, the tool creates the elements of object - oriented analysis model of the specified system .the diagram of the model is generated , which is reviewed by human reviewer to detect any inconsistency or ambiguity .the authors in this paper prefer the use of model over analysis of the whole nl srs due to following reason  * all software companies irrespective of their domain use nl srs for description of software system .* an object oriented analysis module ( ooam ) shows the concepts and the relationship among them . * the ooam model for each sentenceis identified while oo design selects parts of the sentences , such as class from subject , attributes from adjective and methods from verb .during the course of the paper , the authors had used few concepts such as  * * constraining grammar : * it is different from the internal grammar that parser uses .the constraining grammar attempts to reduce the number of way a statement can be represented and also try to make it uniguous ( not ambiguous ) . on the other hand ,the nl parser grammar checks the legitimacy of the sentence only , without checking whether it is ambiguous or not .+ this paper used the constraining grammar proposed by juristo et.al . , in their paper .it attempts to generate an unambiguous mapping form this grammar to ooam .the constraining grammar influences the structure of nl srs as it uses simple sentence consisting of subject , object and verbs . * * natural language parsing :* as ooam is generated using syntactic information automatically , parsing of nl srs is needed to obtain the required information .the parser uses the link - grammar that consists of set of words .these words act as terminal symbols and have different liner requirements .* * transformation rules : * these rule helps to transform the obtained syntactic information to targeted ooam .dowser have thirteen different transformation rules , the most generally used rules are : * * if the sentence contains both subject and object link after parsing , then two different classes are created with association named as verb . ** to find aggregation , if the parsed sentence consists of both subject and object link and the verb is one of `` have , posses , contain or include '' then the object is aggregated to subject .* * `` if or when '' always represent the start of an event .if an event is detected by dowser and the main clause only consists of subject link , then class is created with the noun present in subject link and verb acts as a method to it . * * if genitivity detected by dowser , two different classes are created with one linking aggregation . in order to fix whether both became aggregated class or one became attribute of other , both syntactic and semantic information of the sentence are needed .* * though active clauses preferred in nl srs still passive clause helps to describe relations and states .the passive verb and its connecting word describe the association .+ dowser applied two post processing rules after transforming nl srs .these are a.  it converts all classes that aggregated to another class as attribute of other class .b.  it removes class `` system '' from ooam . * * domain - specific terms(dst ) : * special domain data dictionary is needed by the parser to interpret the dst .but for each domain such dictionary does not exist . hence , to improve the dst recall , the link grammar has a guessing mode that uses the syntactic role of unknown terms .* * diagramming ooam : * the textual ooam is created using previous steps .the tool _ umlgraph _ transforms the textual information into _ a dot file _ while then transforms into graphic format using _ graphviz _ tool . * * interpretation of ooam : * in this step , a human analyst checks for ambiguity in generated diagram .the defects that can be found out in ooam are as follows * * an association can be ambiguous ; so , the analyst checks whether different classes transmit message to same class or not . ** the classes should represent only one concept .as dowser does not allow generalization principle , the concepts such as _ cash payment _ and _ on - line payment _ are not combined to form payment as a whole ; but can be identified as two different classes . * * if the classes have attributes that are not of primitive type , proper definition added to it so , it can be represented in own class . * * the class must be associated with other class otherwise it becomes unspecified .this paper only supports the static behavior/ relationship of ooam present in nl srs .it does not manage the modeling behavior .this paper of * ibrahim and ahmad * proposes method to facilitate requirement analysis process and extraction of class diagram from requirements using nlp and domain ontology .a tool named `` requirements analysis and class diagram extraction ( race ) '' is being proposed by the authors that analyzes the textual requirements , finding out the relationships and finally extracts the class diagram .the race system consists of different internal and external components or sub - systems .these can be described as follows :  a.  opennlp parser : the opennlp parser used in this paper for lexical and syntactic parsing .the parser takes english text as input and provides corresponding pos tag for each word as output .b.  race stemming algorithm : stemming is a process of removing affixes and suffixes from a word and generating the base word .the generated base word reduces the redundancy and increases efficiency . c.  wordnet: it is used to validate the semantic of the sentences that generated after syntactic analysis .it also helps to display hyponyms for a selected noun , which helps to know the `` a kind of '' relationship .d.  concept extraction engine : this module is used to extract concepts according to the requirement document .the algorithm for this module is as follows : 1 .requirement document is taken as input .2 .  stop words are identified and stored as * stop - words_found * list 3 .calculate the frequency of each words in the document except the stop words .4 .  use race stemming algorithm to stem each words and store them in a list .use opennlp to parse whole document. 6 .  from the parsed outputextract the words with pos proper nouns ( nn ) , noun phrases ( np ) , verb ( vb ) and store them in concept - list .7 .  for each concept in concept - list ,if any other concept is synonym with present one , then it can be conveyed that both are semantically related .8 .  for each concept in concept - list ,if any other concept is item requirement document is taken as input .stop words are identified and stored as * stop - words_found * list 10 . calculate the frequency of each words in the document , except the stop words .use race stemming algorithm to stem each word and store them in a list .use opennlp to parse whole document . 13 . from the parsed outputextract the words with pos proper nouns(nn ) , noun phrases ( np ) , verb ( vb ) and store them in concept - list .14 . for each concept in concept - list , if any other concept is synonym with present one then it can be conveyed that both are semantically related .15 . for each concept in concept - list , if any other concept is hyponyms with present one i.e. , lexically same then it can be conveyed that former is a kind of later and saved in generalization - list . with present one i.e., lexically same then it can be conveyed that former is a kind of later and saved in generalization - list .e.  domain ontology : it is used to improve the performance of concept identification . in race system library systemontology is being used .xml is being used to build the ontology .f.  class extraction engine : the input to this module is the output of `` concept extraction engine '' . during this step , some heuristic rules are used by the authors to extract the class diagram .the rules are as follows * class identification rules : the rules used for extraction of classes are : * * if the occurrence of the concept is only one or frequency is 2% , then the concept is ignored as class .* * if the concept is related to design elements , location name or person name , then ignore as class . * * if the concept found in high level of hypernyms tree or an attribute then ignore it as a class . * * if the concept is a noun phase and the second part is an attribute then consider the first part for class name . * attribute identification rules : the rules for attribute identification is as follows * * if the concept is a noun phase including underscore between two nouns , then the first noun is a candidate for class name and the second part is attribute of that class .* * if the concept has only one value then it is an attribute .* relationship identification rules : the rules for relationship identification is as follows : * * using step 8 of concept extraction engine , the element of generalization - list transferred as generalization ( is - a ) relationship . * * if there exists a sentence having ( ct1-vb - ct2 ) where ct1 and ct2 are classes , then vb is an association rule . * * if the sentence is of the form ct1+r1+ct2+``and''+ct3 where ct1 , ct2 and ct3 are classes and r1 is the relationship , then there exists relationship between ( ct1,ct2 ) and ( ct1,ct3 ) .g.  race concept management : user interaction is important in race system .the ui helps to perform tasks such as creating and printing requirement and acts as an interface to add , modify , view and organize relationships .the race system is implemented using c # , ms access is being used for database operation , and to open textual requirements word document , text file , rich text file , and html file are being used .this paper of * overmyer and rambow * proposes a tool called linguistic assistant for domain analysis ( lida ) that provides linguistic assistance for model development process .this tool helps to obtain the oo model for a domain using uml . in order to perform this task , large volume of text from `` legacy system '' is collected .the lida tool considers the following features  * domain independent linguistic processing used to group different form of base words using pos and to find multi - word terms . *the final output is in the form of full text , word - list and uml model in parallel .so the user can compare all of them . * key word in context ` kwic ' view displays the words or group of words in sentences .* hypertext description model used to help in documentation of the model . *completed model can be exported any case tool or any model can be imported from any case tool to lida .lida consists of following components :  a.  _ text analysis environment : _ this component is the main component of lida as it provides the central functionality .the main functionalities this component performs are : * it takes the text input in rtf and ascii format . *then it assigns pos tag to each word . for postagging , it uses mxpost , a software tool developed at university of pennsylvania , usa . * base word is obtained form each word and their frequency is calculated .* multi - word phases are checked for a given base word . *users are allowed to mark the words or phases as candidate model and highlights these words in the text . *retrieve textual context of marked words .+ _ mode editing environment : _ this model offers the functionality requirement to generate a model from the proposed model marked in lida s text analyzing environment .the functional features of this components are : * * display * list of candidate model element marked and add them to model editing environment .transfer of information between text analysis environment and mode editing environment helps developer to analyze the problem in details . * * suggest * operations for combining elements to class model . * * add * textual context helpful for processing model builder . * * generate * textual description of model for documentation and validation of model with domain expert .b.  _ lida text description : _ lida uses model - explainer an integrated tool , which generates the hypertext description document for object model .this document is generated from customized text which includes the class information like superclass , subclass , attributes , operation and association with other class .these descriptions help to obtain additional information about the final result .the following table [ review ] provides a comparative analysis of the approaches to obtain the elements of oo analysis from srs .in present day scenario , the use of object oriented system is widely applied for software development . the customer mention all it s requirements in a document called software requirement specification .this srs document is written in nl which is understandable by the customer side , but it is sometimes incomplete and ambiguous .the development team need to go through these document and generate uml diagram and analyze on basic of oo analysis .the uml being very often used for ooa tries to fix the class diagram where class is also basic element of oo system .during the course of the paper , it can be found out that different approaches are adapted to generate the class diagram and its corresponding details .these approaches can be mentioned as follows :  * the software requirement document is considered as an input for the analysis . * as it is written in nl , it contains some ambiguities or unwanted information .so , in order to remove that different steps are carried out in all papers . *each words in the text is tagged with a pos. then the words are combined together depending upon their pos . *the noun and verb tagged words are mainly used for class name and their operations respectively .so these words are then stemmed to obtain the root word and their suffixes . * after obtaining the root noun words , the higher frequency nouns are considered and they are the most eligible onces for fixing class name .* for operations in class , the verbs present in the sentence are the best candidate . * the adjectives present in text act as an attribute for the class for that noun it tries to modify . * in order to find the relationship between classes , the relation between the subject and object of a sentence is found out . * for other rules likemultiplicity determines are used that specify the relationship like one - one , one - many , many - one , many - many .there are different tools that have been developed to analyze the text ; but as there is no exhaustive dictionary which helps to provide pos for each words .although few tools generate the class diagram but different authors suggest that a manual intervention is needed to improve the final result . until and unless there is specific rules for writing the srs document , the ambiguities continue to be present in it and that cause issue in compiling the srs .though many approaches have been proposed and also are used to obtain the elements of oo analysis still there is scope for research in this area . to automated understanding the srs written in informalnl is also an issue in research .m. saeki , h. horai , and h. enomoto , `` software development process from natural language specification , '' in _ proceedings of the 11th international conference on software engineering _ ,icse 89.1em plus 0.5em minus 0.4emnew york , ny , usa : acm , 1989 , pp .s. nanduri and s. rugaber , `` requirements validation via automated natural language parsing , '' in _ system sciences , 1995 .proceedings of the twenty - eighth hawaii international conference on _ , vol .3.1em plus 0.5em minus 0.4emieee , 1995 , pp .362368 .d. popescu , s. rugaber , n. medvidovic , and d. m. berry , `` reducing ambiguities in requirements specifications via automatically created object - oriented models , '' in _ innovations for requirement analysis . from stakeholders needs to formal designs_.1em plus 0.5em minus 0.4emspringer , 2008 , pp .103124 .m. ibrahim and r. ahmad , `` class diagram extraction from textual requirements using natural language processing ( nlp ) techniques , '' in _ computer research and development , 2010 second international conference on_.1em plus 0.5em minus 0.4emieee , 2010 , pp .200204 .h. m. harmain and r. gaizauskas , `` cm - builder : an automated nl - based case tool , '' in _ automated software engineering , 2000 .proceedings ase 2000 .the fifteenth ieee international conference on_.1em plus 0.5em minus 0.4emieee , 2000 , pp .s. p. overmyer , b. lavoie , and o. rambow , `` conceptual modeling through linguistic analysis using lida , '' in _ proceedings of the 23rd international conference on software engineering_.1em plus 0.5em minus 0.4emieee computer society , 2001 , pp .401410 .l. mich and r. garigliano , `` nl - oops : a requirements analysis tool based on natural language processing , '' in _ proceedings of third international conference on data mining methods and databases for engineering , bologna , italy _ , 2002 .r. gaizauskas , k. humphreys , h. cunningham , and y. wilks , `` university of sheffield : description of the lasie system as used for muc-6 , '' in _ proceedings of the 6th conference on message understanding_.1em plus 0.5em minus 0.4emassociation for computational linguistics , 1995 , pp .

Abstract: the software development life cycle ( sdlc ) starts with eliciting requirements of the customers in the form of software requirement specification ( srs ) . srs document needed for software development is mostly written in natural language(nl ) convenient for the client . from the srs document only , the class name , its attributes and the functions incorporated in the body of the class are traced based on pre - knowledge of analyst .   the paper intends to present a review on object oriented ( oo ) analysis using natural language processing ( nlp ) techniques . this analysis can be manual where domain expert helps to generate the required diagram or automated system , where the system generates the required digram , from the input in the form of srs .   * keywords :* software development life cycle ( sdlc ) , software requirement specification ( srs ) , natural language processing ( nlp ) , natural language ( nl ) , parts of speech ( pos ) , object oriented ( oo ) .


promises new solutions for many applications in the biomedical and industrial fields .this new paradigm utilizes various methods to communicate information between nano - scale machines .a promising communication method at this scale , is _molecular communication_. different molecular communication schemes have been proposed that can be categorized based on their effective range of communication , namely short , medium or large .  in this paper , we focus on a diffusion based communication system for short and medium range nanonetworks . in this system, molecule concentration represents information .the transmitter emits molecules whose type and intensity depend on the current input symbol .these molecules propagate through the environment , with a portion hitting the receiver s surface .receptors located on receiver s surface form chemical bonds with the incoming molecules , initiating a process that eventually results in decoding of the transmitted information .different channel models for molecular communication have been developed and their related channel capacities have been evaluated in - .authors in have proposed two modulation methods called concentration shift keying ( csk ) and molecular shift keying ( mosk ) . in these schemes ,information is encoded in molecule diffusion rate and molecule type , respectively .both csk and mosk suffer from interference caused by molecules from previous transmissions , hitting the receiver after a long delay .to control this interference in csk ( where only one molecule type is used ) , the decoding of the current symbol is adapted to the last decoded symbol .although this helps to manage the interference , it results in an error leakage to the current symbol if the last symbol is decoded incorrectly . using multiple molecule types , mosk is more resistant to interference than csk but it requires complex molecular mechanisms at both the transmitter and the receiver for message synthesis and decoding .  in this paperwe propose a new modulation method based on the idea of using distinct types of molecules for consecutive slots at the transmitter , thus effectively suppressing the interference .although simple in its formulation , the method outperforms the existing ones . to evaluate the schemes , we propose a new system model for the molecule propagation process and the channel based on the poisson distribution .this model is more realistic and amenable to analysis than the one proposed in based on the binomial distribution .we evaluate and formulate the probability of error for our proposed scheme as well as the previous ones and compare the results . specially , we take into account the dependency of decoding of the current symbol to the last transmitted and decoded symbols to compute these metrics .numerical results indicate our scheme has a lower probability of error compared to csk and mosk . to understand the efficiency of our proposed scheme , we use information theoretic tools to derive a fundamental limit on the minimum probability of error of an arbitrary scheme .a comparison indicates that although our scheme outperforms the existing ones , its error exponent ( in terms of transmission power ) is larger than our fundamental lower bound , suggesting room for further improvement .the rest of the paper is organized as follows .proposed modulation scheme and system model are described in section ii . in section iii performance of different modulation techniques are discussed .also in this section a more general molecular communication system is introduced for which a lower bound on probability of error is computed .lastly numerical results are presented in section iv .the concentration shift keying ( csk ) scheme is inspired by the amplitude shift keying ( ask ) used in the classical communication . in this scheme ,symbols are encoded in the number of molecules that the transmitter diffuses ; this is called _ diffusion rate _ or _transmission power_. for example , to represent bits , different molecule diffusion rates are utilized . at the receiver ,the concentration ( number of the received molecules ) is compared with decision thresholds in order to decode the transmitted symbol .the molecules from the previous symbol in the last time slot may hit the receiver in the current time slot .thus , to reduce the effect of the interference in this method , decision thresholds for the current symbol need to be adapted to the last decoded symbol .molecular shift keying ( mosk ) scheme resembles the orthogonal modulation in the classical communications . to transmit bits per time slot , different types of molecule are utilized .the transmitter releases a specified number of molecules of the type corresponding to the current input symbol . to decode the transmitted signal at a certain time slot, the receiver looks for a unique molecule type whose concentration exceeds a certain threshold .an error occurs if the concentration of none of the molecule types or more than one molecule type exceeds .although interference due to the previous transmission in mosk is less than that in the csk , but it still exists . also using different molecule types complicates the hardware structure at both the transmitter and the receiver as the number of molecule types increases exponentially with .the performance of both the mosk and csk schemes is degraded by the interference caused by molecules of previous transmissions .in addition csk suffers from propagation of errors since the thresholds for each time slot is adapted to the previously decoded symbols .therefore , if the previous symbol is falsely decoded , its error leaks to the current symbol .our scheme borrows ideas from csk and mosk . having molecules of types and , the transmitter uses type in odd time slots , and type in even time slots .the use of two molecule types resembles mosk but the difference is that the molecule type is not used for signalling . to convey information in each time slot ,different diffusion rates are used ( similar to csk ) .thus , to match each symbol to bits , different propagation rates are utilized in each time slot . as the molecule types are different in two subsequent time slots , ( i ) the previous symbol interference has almost disappeared ( ii ) the decision threshold of the current symbol is independent of the last decoded symbol . since the data is not encoded in the molecule types ( but in the concentrations ), the number of molecule types ( and as a result the complexity ) , does not increase with .we refer to this scheme as molecular - concentration shift keying ( mocsk ) .in this section we develop a model for a general molecular transmission .later we use this model to compare probabilities of error of mosk , csk and mocsk .it is assumed that the symbols are transmitted in equal time slots called symbol duration , .generally the signal is decoded based on the concentration of molecule types received during a time slot .we assume that in a time slot three sources contribute to the received molecules of a specific type : ( i ) molecules due to transmission in the current time slot ( ii ) the residue molecules due to transmission in the last time slot ( the residue molecules from two or more earlier time slots are ignored ) ( iii ) the molecules from other sources as the noise of environment .more precisely , each molecule either misses hitting the receiver , or hits the receiver in the current or next time slot .let the hitting probabilities in the current and next time slot be and , respectively ( ) .we assume intra - molecule collisions have little effect on the molecule s movement .  for each molecule type, the transmitter has a storage .every storage has an outlet whose size controls how many molecules can exit the storage to be diffused in the environment .the distribution of the number of molecules that exit from the storage in a time slot is poisson .this follows from the fact that the number of molecules in the storage is large and probability of each passing through the outlet is small .we use to denote the number of molecules exiting the storage ( and diffused in the environment ) .the parameter is called the diffusion rate and is determined by the size of the opening of the outlet .parameter itself is a random variable that is decided by the transmitter based on the message it wishes to transmit .assume that the transmitter has chosen diffusion rates of and of a molecule type like in two consecutive time slots in a modulation scheme . also , consider the environmet noise independent of the source signal as a poisson noise with parameter .considering hitting probabilities and , and using the thinning property of poisson and the fact that sum of independent poisson rv s is itself a poission , the number of type molecules received within the current time slot ( denoted by ) follows a poisson distribution with parameter where .this poisson model has advantages over the binomial model considered in : ( i ) the poisson model is more realistic and practical in comparison with the binomial model .for instance , in binomial model the transmitter needs to count the number of molecules that it diffuses in comparison with the poisson model in which the size of the storage s outlet is controlled ( without counting the exact number of molecule passing through the outlet ) .( ii ) the poisson model is easier to work with analytically because of the nice properties of poisson ( e.g. thinning property , etc . ) .   in the csk method , only one type of molecule for instance is used in all time slots .so , the decoding is made by comparing the number of received molecules , i.e. , with the thresholds determined based on the last decoded symbol .  in mosk , as information is encoded in molecule type , if two subsequent symbols are the same , only one molecule type for instance is received , otherwise two molecule types for instance from the current symbol and type from the previous symbol are received . in the former case ,the symbol is decoded correctly if the total sum of received molecules , i.e. , exceeds the threshold . in the latter, decision is made based on , and the threshold .  in the proposed mocsk scheme , as stated above , 2 different types of molecules in 2 subsequent time slots are used .as the receiver knows the type of the current symbol , it needs only to compare the number of the molecules of this type with the thresholds to decode the current symbol .we now compute the average probability of error for csk , mosk and mocsk .  as in , we only consider the class of simple symbol - by - symbol decoders that are practically appealing .comparisons are made consistently for decoders in this class with the same transmission powers .for binary csk ( bcsk ) using molecules of type , if the current input symbol is and the last transmitted and decoded symbols are and respectively , the probability of successful decoding is computed as is the decoded current symbol and is the decision threshold corresponding to the value of .note that in this equation , is not necessarily equal to the previous transmitted symbol , i.e. . in fact , the decision threshold for the current symbol is determined based on the previous decoded symbol regardless of being decoded correctly or not.  for binary mosk ( bmosk ) , assume molecules type and are sent for 0 and 1 , respectively .for the current transmitted symbol of , probability of successful decoding for and are computed as follows : + for the proposed binary mocsk ( bmocsk ) , if the molecules of type is used in the current slot , the probability of successful decoding can be computed as follows : conditional probabilities for quaternary and higher modulation level cases can be calculated similarly .let be the error event . for bcsk , the average probability of error is equal to . as the decoding of the current symbol is dependent onboth the last transmitted and decoded symbols in bcsk , we have: , is the probability that the previous symbol is falsely decoded to and can be substituted by .so when , will be equal to . therefore , and can be computed recursively from ( [ eq : a ] ) which is a system of linear equations . for qcsk ( and higher ) levels ,the probability of error can be computed similarly . for the mosk ,the decoding of the current symbol is independent of the previous decoded symbol , i.e. .therefore , is obtained as   for the proposed mocsk , since the decoding of the current symbol is independent of both the transmitted and previous decoded symbols , is computed as follows    in order to determine how well the modulation schemes considered above are , we examine a general molecular communication system with two molecule types and and a memory of only one symbol at the decoder ( since we are considering the class of simple and practical symbol - by - symbol decoders only ) .let be the input symbol ( assumed to be i.i.d . anduniform rv s over a set ) .the encoder uses to determine the diffusion rates and of molecules of type and respectively at the time slot .thus the actual number of molecules that are diffused would follow independent poisson distributions with parameters , . using the thinning property of poisson , the distribution of the numbers of molecules of type , received at the receiver in the time slot is , where , and are defined in section ii - c .the current input symbol is decoded as based on the and ( the previous decoded symbol ) .the average probability of error is : the following lemma , a lower bound on is derived . + * lemma 1 .* is bounded from above as follows :   the following chain of inequalities proves ( [ eq : lemma]).  inequality ( a ) holds because is function of ; ( b ) holds as does not provide further information about conditioned on . equality ( c ) holds because is independent of previous message bits .( d ) holds because of the markov chain .  using the well - known trick of taking a random index uniformly distributed on and independent of previously defined rv s, one can show , and the statement of lemma 1 for random index instead of fix index .considering constraints and \leq r_{av} ] we have : if  tends to when tends to infinity and is the solution of . denoting the rhs of the upper bound inequality ( [ eq : d ] ) or ( [ eq : e ] ) by , and using the upper bound in ( [ eq : lemma ] ) we have , the fano s inequality , and the convexity of entropy function , we obtain :  is the size of the alphabet of variable .this inequality gives a lower bound on the average probability of error .our analysis also extends to types of molecules in a straightforward way ; it is omitted due to lack of space .for numerical evaluations , we have assumed the distance between transmitter and receiver and time slot duration are and sec . respectively and hence the probabilities of hitting as and ( unless explicitly stated otherwise as in the description of two of the curves in fig 1 ) .also , we have considered .1(top ) shows the plots of the probability of error versus the average transmission power ( diffusion rate ) per bit for binary and quaternary csk , mosk and proposed mocsk modulations .this figure shows that bmocsk substantially outperforms the bmosk , with two types of molecules being employed in the two schemes .for the quaternary level , as expected , mosk with more complicated transmitter and receivers employing four molecule types , yields a better performance compared to the proposed quaternary mocsk ( qmocsk ) that uses only two molecules types .the same results hold for the radio communication as well , where m - ary orthogonal modulation substantially outperforms m - ary amplitude modulation for , at the cost of higher complexity .also , fig .1(top ) shows that mocsk outperforms csk scheme in binary and higher levels .further this improvement becomes more significant as we increase the hitting probability . to illustrate this ,probability of error of csk has been plotted for in binary and quaternary levels in the fig .1(top ) . increasing results in more interference from the previous symbol , thus degrading the csk performance . generally speaking , as the decoding of the current symbol in the mocsk is independent of both the previous transmitted and decoded symbols , this scheme outperforms the csk and mosk schemes in a fair comparison .1(bottom ) compares the probability of error of the proposed bmocsk and the lower bound derived in ( [ eq : lb ] ) versus the maximum transmission power . in the bmocsk , and2 molecule types are used . for a fair comparison ,the lower bound has been computed for , and .it is observed that the gap between the lower bound and the proposed scheme increases rapidly . to see how the lower bound changes with increasing , the lower bound has been also plotted for and .

Abstract: in this letter , we propose a new molecular modulation scheme for nanonetworks . to evaluate the scheme we introduce a more realistic system model for molecule dissemination and propagation processes based on the poisson distribution . we derive the probability of error of our proposed scheme as well as the previously introduced schemes , including concentration and molecular shift keying modulations by taking into account the error propagation effect of previously decoded symbols . since in our scheme the decoding of the current symbol does not depend on the previously transmitted and decoded symbols , we do not encounter error propagation ; and so as our numerical results indicate , the proposed scheme outperforms the previously introduced schemes . we then introduce a general molecular communication system and use information theoretic tools to derive fundamental limits on its probability of error .   nanonetworking , molecular communication , modulation , poisson distribution .


a majority of infectious diseases , ranging from seasonal influenza to ebola outbreaks and sexually transmitted infections , can be viewed to occur on contact networks of humans and animals , which are composed of individuals and dyadic links between them .epidemic processes are one of the most widespread applications of network analysis .structure of contact networks has been shown to affect , for example , the likelihood and speed of an infection penetrating into a significant part of a population , effectiveness of intervention strategies , and identification of super - spreading individuals .accumulating data evince that contact networks underlying epidemic and other processes are often highly dynamic , constituting temporal networks .for example , links may be only occasionally used for actual physical contacts .individuals may be socially active in some restricted periods of time .temporality of networks may alter effects of networks on epidemic processes .this is a practical enquiry because various instances of epidemics in human and animal populations , and also viral spreading of information in human society , seem to occur on temporally varying networks .however , our understanding of epidemic processes in temporal networks is still limited . theory based on the branching processenables us to understand long - tail behaviour of the number of newly infected individuals .other theoretical approaches include analysis of epidemic spreading on theoretically tractable generative models of temporal networks .however , these studies assume network models such that they may miss effects of the properties of temporal contact networks that are present in empirical data but not modelled . on empirical temporal networks, the most popular approach has probably been to run numerical simulations of epidemic processes on the empirical networks and their variants ( e.g. , ) .valdano and colleagues introduced a temporal - network variant of the individual - based approximation to understand the susceptible - infected - susceptible ( sis ) model of epidemic spreading  ( also see for a similar approach to a different disease model ) . in this approach , the probability that each node is infected is tracked over time using the matrix algebra .they showed how to calculate the epidemic threshold ( i.e. , strength of infection above which infection can remain prevalent in the population ) and the prevalence ( i.e. , fraction of infected nodes in the stationary state ) in terms of the spectral radius of a relevant matrix .their theory is applicable to arbitrary temporal network data .  despite these and other developments ,we are yet left with fundamental questions regarding mechanisms of infections in given temporal networks .why is the epidemic threshold large in one temporal network and small in another ?how do two time scales , one of network dynamics and the other of the epidemic process , interact ? in the present study , we use the individual - based approximation to reveal factors controlling the epidemic threshold for the sis model on arbitrary temporal networks .we show theoretical and numerical evidence that the epidemic threshold decreases ( i.e. , infection is more likely to occur ) as the network becomes more temporal in the sense that the network changes more slowly ( but not quiescent ) relative to the time scale of the epidemic process .we use the continuous - time sis model on networks switching with regular intervals to reach this conclusion .this result is consistent with those derived for particular temporal network models .impacts of the temporality of networks on the epidemic threshold vary across networks .we find that non - commutativity of the adjacency matrices at different times , as quantified by the lie bracket of the adjacency matrices , is a key indicator that influences the epidemic threshold .nodes and snapshots , and the corresponding aggregate network .the link weight in the aggregate adjacency matrix , , is equal to the sum of the link weight over the four snapshots divided by four.,scaledwidth=70.0% ]we consider the continuous - time sis model on undirected temporal networks having nodes , as schematically shown in fig .[ fig : schem ] .each node assumes either the susceptible or infected state .an infected node infects each of its susceptible neighbours at rate .an infected node transits to return to the susceptible state at rate .to model exogenous dynamics of networks , we consider an infinite sequence of adjacency matrices and sequentially apply each of them for time . in other words , is applied between time and .we refer to each network applied for time as the snapshot network , or snapshot in short .this switching network modelling of temporal networks is common in studying synchronisation processes .it is also in accordance with observation of temporal network data at regular time intervals .here we regard as a free parameter .it controls the relative time scale of the epidemic and network dynamics ; multiplying by a constant is equivalent to not changing and instead changing and to and , respectively .in addition , changing ( , , ) to ( , , ) does not change the dynamics .therefore , we set without loss of generality .the model simplifies over real epidemic processes .heterogeneities are considered only through the quenched disorder of the network .a more refined model would account for heterogeneity also in the parameters , drawing and separately for each node and for each snapshot from probability distributions .we suppress these refinements for the sake of tractability of the model ..properties of empirical temporal networks . the number of nodes ( ) , that of links ( ) , that of events ( ) , and that of snapshots ( ) are shown . the largest connected component of the aggregate undirected network is used for each data set . the length of the aggregating time window used for figs .[ fig : empirical ] and [ fig : plot_cdependency ] is also shown . in the reality mining data set ,we have ignored the first 27 weeks , because the time stamps for these entries are false  . [tab : data ] [ cols="<,>,>,>,>,^",options="header " , ]   for details on the data sets .we imposed periodic boundary conditions .the theoretically obtained is shown by the arrows .we calculated the theoretical value by a bisection method on the leading eigenvalue of .we expanded up to the term of the order of and calculated its leading eigenvalue using the power method .[ fig : empirical ] ]  we denote the probability that node ( ) is in the infected state at time by . under the assumption that the states of different nodes are independent of each other ,the individual - based approximation to the sis dynamics linearised around the disease - free configuration ( i.e. , for all ) is given by where , denotes the transposition , and is the identity matrix . when the network is static , eq .is simplified to , where is the adjacency matrix .the epidemic threshold , denoted by , is defined as the value of above which infection can persist in the network . according to the individual - based approximation , for a static networkis given by the value of at which the leading eigenvalue of is equal to zero .the leading eigenvalue of is given by , where is the leading eigenvalue of .therefore , we obtain  .  when the network varies in time , eq.  yields where \cdots \exp\left[(\beta a^{(0)}-i ) \tau\right ] .\label{eq : t(s;tau),general}\ ] ] the leading eigenvalue of , which depends on sequence , is denoted by .the epidemic threshold , , satisfies .valdano and colleagues were the first to derive this result by analysing the sis model in discrete time .for two temporal networks , we simulated the sis model dynamics using the quasistationary state ( qs ) method tailored to the case of temporal networks ( appendix [ sec : qs method ] ) .we aggregated the original data over several time windows to define snapshot networks ; the size of the aggregating window is shown in table [ tab : data ] . the average prevalence of infection for values of and is shown in fig .[ fig : empirical ] . similarly to ref . , we assumed a periodic boundary condition such that the first snapshot ensues after the last snapshot .the theoretical estimates of are shown by the arrows in fig .[ fig : empirical](a ) and ( b ) , which are fairly accurate .  figure [ fig : empirical ] suggests that the epidemic threshold , denoted by , decreases as increases in both networks . to formulate this point, we compare with the epidemic threshold for the sis dynamics occurring on the aggregate , static network , denoted by .we start with defining the aggregate network as the adjacency matrix applied for . with this normalisation, the temporal and aggregate networks have the same time average of the weight of each link ( fig .[ fig : schem ] ) . the epidemic threshold for the aggregate networkis given by , where is the largest eigenvalue of .any real - valued , continuous spectral function that acts on the spectrum of its matrix argument and only attains finite values satisfies for arbitrary symmetric matrices and  .although a generalisation of this inequality to the case of more than two matrices is false in general , we conjecture that it remains true if the involved matrices have only non - negative entries and is the spectral radius .we have the following evidence .first , to the best of our knowledge , all counterexamples involve matrices with negative entries  .second , the theoretical results for two model temporal networks presented in sec . [ sec : modelnetwork ] are consistent with this inequality .third , all numerical calculations performed in this paper on real data sets and synthetic networks are consistent with this inequality . by admitting this generalised inequality , and applying it to matrices ( ), we obtain \cdots \exp\left[(\beta a^{(0)}-i)\tau\right ] \right ) \nonumber\\ & = \exp\left(-\ell\tau\right ) \phi \left ( \exp\left[\beta \tau a^{(\ell-1 ) } \right ] \cdots  \exp\left[\beta \tau a^{(0 ) } \right ] \right ) \nonumber\\ & \ge \exp\left(-\ell\tau\right ) \phi \left ( \exp\left[\beta \tau \sum_{\ell^{\prime}=0}^{\ell-1 } a^{(\ell^{\prime } ) } \right ] \right ) \nonumber\\ & =  \phi \left ( \exp\left[\left(\beta a^*- i \right ) \ell \tau \right]\right ) .\label{eq : mu_max_general}\end{aligned}\ ] ] equation  implies that at .therefore , .several remarks are in order .first , eq .  implicitly assumes the periodic boundary condition for the temporal networks .however , because is arbitrary , eq .  holds true for arbitrary sequences of networks .second , in the limit , eq .  is satisfied with equality such that . third , if each snapshot is composed of a single link , numerical results suggest that the epidemic threshold increases as the temporality of the network increases ( appendix [ sec:1 edge ] ) , which is opposite to the current result .in this situation , the probability that the infection is extinguished is not negligible even for a large infection rate .a theory that accounts for stochasticity , which is different from the deterministic individual - based approximation , correctly predicts the direction of the change in the epidemic threshold as increases ( appendix [ sec:1 edge ] ) . it should be noted that a network mainly composed of isolated single links is realistic for sexually transmitted infections through monogamous relationships  situation is out of the scope of the following analysis .our theory requires that each snapshot has a relatively large connected component such that the sis dynamics on the snapshot are not significantly influenced by disconnected single links even if they are present .the stochasticity and the absorbing configuration ( all nodes susceptible ) become relevant not only for disconnected single links ( appendix [ sec:1 edge ] ) but also snapshots of increasing size as . if exceeds the typical time of reaching the absorbing configuration on a snapshot , the epidemic ceases due to stochasticity .to investigate the distance between the epidemic threshold for aggregate and temporal networks and its dependence on parameters , we start by calculating the epidemic threshold for two temporal network models . in the first model , snapshots consist of a disjoint union of cliques , each with nodes , and isolated nodes. a clique may be a suitable model for conversation events in a small group  .the clique size remains the same across different cliques and snapshots .the number of nodes stays constant across snapshots , but the number of cliques may depend on a snapshot .we assume that snapshots are randomly and independently drawn from a set of possible snapshots with equal probability , which we call the random sampling with replacement . whether the expected prevalence is positive or not can be determined by , which is equal to the maximum lyapunov exponent associated with the switching linear dynamics induced by operator ( eq . ) as  .the epidemic threshold corresponds to .it is algorithmically undecidable to determine whether or not  , hindering us from deriving the exact value of .therefore , we evaluate the expected state vector ,  x ] , coincide in this case . to quantify the difference between and matrices are non - commuting , we use zassenhaus formula  given by = \exp\left[sm_1\right ] \exp\left[sm_2\right ] \prod_{n \ge 2 } \exp\left[s^n c_n(m_1,m_2)\right ] ,  \label{eq : zassenhaus}\ ] ] where and are matrices , is a real number , and indicates a matrix product in ascending order of the indices . matrices ( ) are given by linear combinations of nested commutator brackets , where the commutator bracket of and is defined by \equiv m_1m_2 - m_2m_1.\ ] ] for instance , the first two matrices are given by \ ] ] and + [ m_1,[m_1,m_2 ] ] \right).\ ] ] by iteratively applying eq .  with , we obtain & = t(\tau ) \prod_{\ell^\prime=1}^{\ell - 1 } \prod_{n \ge 2 } \exp\left [ \left(\tau\beta\right)^n c_n\left(a^{(\ell^\prime ) } , \sum_{\ell^{\prime\prime}=0}^{\ell^\prime-1 } a^{(\ell^{\prime\prime } ) } \right ) \right],\end{aligned}\ ] ] which relates the time evolution operator for epidemic dynamics on the aggregate network on the left - hand side to that of the temporal network , , on the right - hand side .in particular , equals \left\ { i - ( \tau \beta)^2 \sum_{\ell^{\prime } = 1}^{\ell-1 } c_2\left(a^{(\ell^\prime ) } , \sum_{\ell^{\prime\prime } = 0}^{\ell^\prime-1 } a^{(\ell^{\prime\prime } ) } \right ) - ( \tau \beta)^3 \sum_{\ell^{\prime } = 1}^{\ell-1 } c_3\left(a^{(\ell^{\prime})},\sum_{\ell^{\prime\prime } = 0}^{\ell^\prime-1 } a^{(\ell^{\prime\prime } ) } \right ) +  \cdots \right\ }  \right ) \nonumber\\ & = \phi\left (  i + \left(\beta a^ * - i \right ) \ell \tau + \left\ { \left(\beta a^ * - i\right)^2 + \frac{\beta^2}{\ell^2 } \sum_{\ell^{\prime } = 1}^{\ell-1 }  \sum_{\ell^{\prime\prime } = 0}^{\ell^\prime-1 }  \left[a^{(\ell^\prime)},a^{(\ell^{\prime\prime } ) } \right ]  \right\ } \frac { ( \ell \tau)^2}{2 } + \cdots \right ) .\label{eq : mu_max , expanded}\end{aligned}\ ] ] equation  suggests that nested commutator brackets between adjacency matrices of snapshots control the difference between and ) ] , we obtain = \lambda .\label{eq : lyapunov , entrywise}\ ] ] the condition of the theorem is satisfied if the aggregate network is connected .we denote the ensemble average over the set of snapshots by . by using jensen s inequality, we obtain \right ] \le \lim_{\ell \to \infty } \frac{1}{\ell \tau } \ln \mathbf{e}\left [ t(\tau)_{ij } \right ] = \lim_{\ell \to \infty } \frac{1}{\ell \tau } \ln \left[\left(\hat{t}^{\ell}\right)_{ij } \right ] .\label{eq : ensemble_average , hat{t}}\ ] ] if = 0 ] , we combine eqs .  and to obtain \ge e[\lambda ] = \lambda .\label{eq : loghatt / tau > lambda}\ ] ]  because the largest eigenvalue of is equal to and is symmetric in the case of undirected snapshots , we obtain we distinguish two cases .if there is an such that \neq 0 ] for all , and eq .  with true . in this case, we let to have in the and elements and zero elsewhere in eq .  to obtain /2 ] . by combining these equations ,we obtain \ge \lim_{\ell \to \infty } \frac{1}{\ell\tau } \ln \left [ ( \hat{t}^{\ell})_{ij } \right ] \ge \lambda.\ ] ]we denote by the adjacency matrix of the network in which nodes form a clique and nodes are isolated . restricted to the clique part ( i.e. , principal minor of size ) , the eigenvalues of are equal to and , where has multiplicity .we define a matrix by where the first column of is an eigenvector corresponding to , restricted to the clique part , and the remaining columns span the eigenspace of the eigenvalue . then , we obtain .\label{eq : exp(taum),calc , clique}\end{aligned}\ ] ] equation  remains true if a snapshot consists of multiple cliques .  by combining eqs .  and, we obtain ,\ ] ] where is the aggregate network , and is the diagonal matrix with degree of the aggregate network on the diagonal .we denote the eigenvector corresponding to by .the normalisation is given by . by using eq ., we obtain where we abbreviated the row of matrix by and denotes the scalar product of two vectors .summation of eq .  over yields where . by solving eq ., we obtain .\label{eq : beta_c , general , clique}\ ] ] it should be noted that depends on .when , is the eigenvector of corresponding to such that therefore , converges to in the limit . in the limit , converges to , representing the fact that just one snapshot is used indefinitely long , and the epidemic threshold in a single snapshot is that of a clique .finally , we approximate for general values in eq .  to obtain eq . .it should be noted that eq .is independent of in the limit and such that eq .is exact in these two limits .we denote by the adjacency matrix of the network in which a star of size is embedded . without loss of generality, we assume that node 1 is the hub , nodes 2 to are leaves connected only to the hub , and nodes to are isolated . the adjacency matrix restricted to the star part ( i.e. , principal minor of size ) has three distinct eigenvalues and , where the eigenvalue 0 has multiplicity .we define a matrix by where the first column is an eigenvector corresponding to eigenvalue , the second column is an eigenvector corresponding to eigenvalue , and the remaining columns span the eigenspace of the zero eigenvalue .then , we obtain    & = e^{-\tau } \begin{pmatrix } q_{\rm st } & 0\cr 0 & i \end{pmatrix }  \begin{pmatrix }   e^{\tau\sqrt{d_{\rm hub}}\beta } &             & \cr              & e^{-\tau\sqrt{d_{\rm hub}}\beta } & \cr              &             & i  \end{pmatrix }  \begin{pmatrix }  q^\top_{\rm st } & 0\cr  0 & i  \end{pmatrix } \nonumber\\  & = e^{-\tau}\left [ \frac{1}{\sqrt{d_{\rm hub } } } \sinh \left(\tau \sqrt{d_{\rm hub } } \beta \right ) a_{\rm st } + \frac{1}{d_{\rm hub } } \left[\cosh\left(\tau \sqrt{d_{\rm hub } } \beta\right ) - 1\right ] a_{\rm st}^2 + i \right ] .\label{eq : exp(taum),calc , star}\end{aligned}\ ] ] if more than one star is embedded in a snapshot , appropriately permutated versions of eq .  are added together . by applying eq .  to eq ., we obtain \frac{\sum a^2}{r } + i \right],\ ] ] where the summation of runs over all possible snapshots and is the number of the possible snapshots .when , we obtain such that eq .is consistent with eq . .as a special case , we consider the discrete - time version of the activity driven model  . in each snapshot , every node is assumed to be activated with probability independently of the other nodes .if node is a hub , the probability that it connects to a node in a snapshot is equal to .therefore , we obtain up to the order of  where we neglected the probability that both and are hubs and are connected to each other .if node is a hub , the probability that it selects both nodes and ( ) as leaves is equal to  ] and  a ] , we obtain regardless of . under the restriction that , matrices are non - negative .then , we can apply arguments similar to those in appendix [ sec : lyapunov proof ] to show that .  as a demonstration, we calculated the epidemic threshold for the discrete - time sis model for the ht09 data set ( table [ tab : data ] ) .the epidemic threshold for the aggregate network was equal to . with and , we obtained and , respectively , confirming that the epidemic threshold increases as increases in the discrete - time sis model .ls acknowledges the support provided through the engineering and physical sciences research council ( epsrc ) [ grant number ep / g03706x/1 ] . ls andnm acknowledge the support provided through jst , erato , kawarabayashi large graph project .kk acknowledges funding from the ramn y cajal program of mineco .vme acknowledges support from mineco ( spain ) [ project nomaq ( fis2014 - 60343-p ) ] .nm acknowledges the support provided through jst , crest .p. vanhems , a. barrat , c. cattuto , j .- f .pinton , n. khanafer , c. rgis , b. kim , b. comte , and n. voirin .estimating potential infection transmission routes in hospital wards using wearable proximity sensors ., 8:e73970 , 2013 .j. stehl , n. voirin , a. barrat , c. cattuto , l. isella , j .- f .pinton , m. quaggiotto , w. van den broeck , c. rgis , b. lina , and p. vanhems .high - resolution measurements of face - to - face contact patterns in a primary school . , 6:e23176 , 2011 .y. wang , d. chakrabarti , c. wang , and c. faloutsos .epidemic spreading in real networks : an eigenvalue viewpoint . in _ proceedings of the 22nd international symposium on reliable distributed systems _, srds15 , pages 2534 , los alamitos , usa , 2003 .ieee .       c. tantipathananandh , t. berger - wolf , and d. kempe .a framework for community identification in dynamic social networks . in _ proceedings of the 13th acm sigkdd international conference on knowledge discovery and data mining _ , kdd07 , pages 717726 , new york , usa , 2007 .

Abstract: data of physical contacts and face - to - face communications suggest temporally varying networks as the media on which infections take place among humans and animals . epidemic processes on temporal networks are complicated by complexity of both network structure and temporal dimensions . theoretical approaches are much needed for identifying key factors that affect dynamics of epidemics . in particular , what factors make some temporal networks stronger media of infection than other temporal networks is under debate . we develop a theory to understand the susceptible - infected - susceptible epidemic model on arbitrary temporal networks , where each contact is used for a finite duration . we show that temporality of networks lessens the epidemic threshold such that infections persist more easily in temporal networks than in their static counterparts . we further show that the lie commutator bracket of the adjacency matrices at different times is a key determinant of the epidemic threshold in temporal networks . the effect of temporality on the epidemic threshold , which depends on a data set , is approximately predicted by the magnitude of a commutator norm . + _ keywords _ : temporal networks , sis model , epidemic threshold + pacs numbers : : 64.60.aq , 89.75.hc