a node-centric analysis of metagraphs and its applications to workflow models

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A node-centric analysis of metagraphsand its applications to workflow modelsVincent Yen aa Department of Information Systems and OperationsManagement , Raji Soin College of Business, Wright StateUniversity , Dayton, OH 45435, USAPublished online: 24 May 2007.

To cite this article: Vincent Yen (2007) A node-centric analysis of metagraphs and itsapplications to workflow models, Enterprise Information Systems, 1:2, 139-159, DOI:10.1080/17517570701249320

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Enterprise Information Systems,Vol. 1, No. 2, May 2007, 139–159

A node-centric analysis of metagraphs and

its applications to workflow models

VINCENT YEN*

Department of Information Systems and Operations Management,

Raji Soin College of Business, Wright State University, Dayton, OH 45435, USA

(Received November 2006)

Business processes and its related workflow systems have received greater interestin practice and research in the last decade. Many analytical methodologiesfor analysis and design of workflow systems emerged. A recent formal approachto study workflows using a graph-theoretic method called ‘metagraphs’ hasdemonstrated effectiveness for analysing connectivity and interactions ofinformation and resources between workflow components. However, pastworks in analysis of metagraph are element-based. Since nodes in metagraphsrepresent either the input or output of an activity it is natural to processinformation contained in a node taken as a unit. This paper takes a node-centricview on metagraphs that is a major departure from the element-based approachtoday. The change in focus requires provisioning an analysis framework under thenode-centric views. New basic constructs including, but not limited to, conceptssuch as: ‘surplus sets’, ‘deficit sets’, ‘state of a path’, and ‘node-centric viewof adjacency matrices’ are introduced. The approach produces computationalfeasible systems for elements that are over supplied and/or under supplied froma source node to a target node of any path of the metagraph. Such informationcould be valuable for designing workflow systems. Also, the node-centricapproach is shown to be an extension of the basic constructs of element-viewmetagraphs and is a complementary method for validating informationrequirements of workflow modelling. Illustrative examples are given.

Keywords: Metagraphs; Workflow; Workflow analysis; Workflow modelling;Business process analysis; Business process modelling; Adjacency matrix

1. Introduction

Business processes and its related workflow systems have gained greater interestsince the early 1990s (Xu 2007). Studies in this area have produced manyanalytical and formal models for the planning, design, evaluation, validation andsimulation (Zang and Fan 2007). These models also feature greater agility forbusiness process re-designs and management—an important attribute in gaining

*Email: [email protected]

Enterprise Information Systems

ISSN 1751–7575 print/ISSN 1751–7583 online � 2007 Taylor & Francis

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DOI: 10.1080/17517570701249320

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a competitive advantage. Business processes redesign is central to ‘re-engineering’.

Daveport (1995) calls the 1990s ‘the decade of re-engineering’. However, business

processes are normally quite complex because it involves data, documents,

personnel, machines, business rules, control and sequencing issues, and

co-ordination among functional units. An example is an order fulfilment process

that typically involves such activities as: receiving orders from customers,

checking product availability, checking customers’ credit, preparing invoices and

billing, and preparing packaging and delivery. These activities require resources,

e.g. information, people and/or machines. In this paper the term ‘business

process’ and ‘workflow’ will be used interchangeably. The Workflow

Management Coalition (WfMC) (www.wfmc.org), a workflow standard setting

organization, defines workflow as:

The automation of a business process, in whole or part, during which

documents, information or tasks are passed from one participant to another

for action, according to a set of procedural rules.

This definition highlights several important characteristics of a workflow:

1. The work involves documents or information.2. It is processed by resources such as persons and/or computers.3. The sequence of the processes is constrained by procedural rules.4. Being a business process, it must achieve pre-set business goals.

The core of the paper addresses component 1 with some considerations for

2 and 3.Because of practical significance, the need for software tools, mathematical

models, and methodologies for the analysis and design of complex business

processes and workflow systems always exists. In the middle of the last decade

there were more than 250 workflow software systems either available or

underdevelopment (ven der Aalst 1996). Some well known software tools are

Lotus Notes, MQSeries Workflow (www.ibm.com/mqseries), Microsoft Exchange

(www.microsoft.com/exchange/default.mspx), and Rational Rose (www-

306.ibm.com/software/rational). A recent trend in the development and construc-

tion of business processes applications is by means of Web services—an

information technology despite the fact that some aspects are still far from

completion, e.g. security, transaction. Nevertheless, enterprises have been using

Web services successfully within company boundaries for software integration

and reuse. New initiatives for developing business applications that utilize Web

services have been in the works for several years. For example, the Business

Process Management Initiative (BPMI, www.bpmi.org) has developed a Business

Process Modelling Language (BPML) and a standard Business Process Modelling

Notation (BPMN). BPMN contains a small set of readily understandable

graphical notations for designing of business processes and is convertible to

BPML and BPEL4WS—the Business Process Execution Language for Web

Services.

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In designing or analysing workflow systems, it is important to understand the

underlying information requirements and connectivity issues involved. Many

analytical workflow models (many with comprehensive graphical representations)

have appeared in the literature. Formal approaches to analyse workflows have

been proposed by Murata (1989), van der Aalst (1995), and Curtis et al. (1992),

using Petri nets, coloured Petri nets, and state charts techniques, respectively.

A collection of business process models primarily based on Petri nets is in van

der Aalst et al. (2000). Petri nets are used primarily for the analysis of timing

and conflict resolution issues (Basu and Blanning, 2000) and the state charts

approach could easily be bogged down by its complexity. Basu and Blanning

(1994) proposed a new approach called ‘metagraphs’ that utilize a graph-

theoretic construct. Individual element in the nodes of the metagraph is the

central point for analyses, thus the analyses are said to be element-centric.

Metagraphs use element-centric analysis show effective results in connectivity and

interactions issues of information and resources between workflow components.

Since nodes in metagraphs represent either the input or output of an activity it is

natural to process information contained in a node taken as a unit. This study

will take nodes-centric view in analyzing metagraphs that is the major departure

from the element based approach today.The main problems addressed in this paper are now described. Starting from any

source node to a target node of the metagraph, assuming the information elements in

the source node are available, what are the missing information elements and/or the

surplus information elements on each path (a sequence of activities/edges) from the

source node to the target node? How could such information elements be effectively

calculated for all possible paths on the metagraph? What are the relationships

and differences between the two approaches: the element-centric views and the

node-centric views?The organization of the paper is as follows. Sections 2 and 3 provide an example

and an overview of a workflow model and also review the concept of element-centric

metagraphs as developed in (Basu and Blanning 1994). Section 4 develops the main

theme of the paper: the node-centric view of metagraphs. It begins by defining the

concept of ‘paths’ and ‘chains’. Next, it defines new concepts such as surplus sets and

deficit sets, input states and output states, edge functions, and the node-state. Under

the node-centric view node A (the source node) is connected to node B (the target

node) if node A intersects with the invertex of an edge, say e, and node B intersects

with the outvertex of the same edge e. Since node A and B may not be the invertex

and outvertex of the edge e these new concepts offer useful/extended information as

the source node moves to the target node along a path of the metagraph. Section 4.3

shows how the basic notions of co-inputs/co-outputs under the element-centric view

of metagraphs are related to surplus/deficit sets under the node-centric view of

metagraphs, in fact, the latter is an extension of the former. Section 4.4 defines

the node-centric adjacency matrix and its algebraic properties. Section 5 shows

computational details for the two running examples, e.g. adjacency matrices and the

closure matrices. The last section provides a discussion, a brief summery and future

direction for research.

A node-centric analysis of metagraphs and its applications to workflow models 141

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2. A model of workflow analysis

As a way to illustrate a workflow system and its accompanying information

requirements, consider an example of an application process of automobile

insurance. First an applicant completes an application form that contains four parts:

Part (i), the basic information (BI), concerns the location the car is normally

parked and the applicant’s home property profile.Part (ii), the driver’s information (DI), is about the driver’s demographic

information, driving violations and accident history.Part (iii), the car data (CD), is information about the car itself.Part (iv), the type of insurance (TI), is about the type of car insurance requested

(i.e. comprehensive, medical requirements).

Parts (i) and (ii) are used to evaluate the applicant’s risk index (RI). The risk index

and data from parts (iii) and (iv) determine the premium of the requested insurance

(PM). All four parts are used to determine the discount rate (DR) of the premium.

And finally the discount rate and the premium are used to calculate the adjusted

premium (AP) and issue the insurance policy (IP). The example can be shown

graphically in figure 1.In figure 1 there are four tasks or activities {e1, e2, e3, e4} defined by:

e1 A risk analyst calculates the risk index (RI) based on the applicant’s basic

information (BI) and driver’s information (DI).e2 A premium staff calculates the premium (PM) using the risk index (RI),

car data (CD), and the type of insurance (TI) requested.e3 A staff member determines the discount rate (DR) for the applicant

based on the basic information (BI), driver’s information (DI), car data

(CD), and the type of insurance (TI) requested.e4 The manager of the branch determines the adapted premium (AP) and

issues the insurance policy (IP) using the premium (PM) and discount

rate (DR) data.

Each node in figure 1 contains either a set of input information or output

information of some task. Node 1 has sufficient information to launch activities

e1 and e3 thus they can produce outputs of node 4 and node 6. Given node 1 if

activity e1 is performed then the unused ‘surplus elements’ of node 1, i.e. {CD,TI},

BIDI

RI

DR

PM

AP, IP

1 2

3

4

5

6

7

8

e1

e1

e3 e4

CDTI

Figure 1. An auto insurance application workflow.

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relative to this activity should be of interest for tracking. Notice that activity e1produces node 4 that is contained in node 3, i.e. node 4 is a subset of node 3. Hencenode 3 is considered ‘reachable’ (or connected) from node 2 and this is the only wayto reach node 3 from any other node of figure 1. However, activity 1 can only reachnode 3 partially because it produces one (i.e. RI) of the three pieces of informationrequired by node 3. So there are deficits or unrealized information elements by e1relative to node 3. If a partially reachable node is defined as unreachable thenpotentially useful information as to why it is not reachable gets lost. Therefore in thispaper, partially reachable nodes are considered reachable. A similar situationhappens at node 3 when considering whether node 3 could launch activity 3 becausenode 3 does not possess enough input elements required by activity e3. That is node 3partially fulfils the input requirement of e3. Again e3 should be included as an activityavailable at node 3 along with the deficit elements because this information may bevaluable for designing/implementing the workflow system involving e3. Also, notethat activity e2 cannot begin unless activity e1 is completed; similarly e4 cannot beginunless e2 and e3 have been performed.

In practice, information contained in any node of a workflow system are oftenprocessed together as a set by an activity; for example, can I get a premium quote(PM) if I supply {CD,TI}? The answer is no because the given information (i.e. CD,TI) is insufficient to generate the output for ‘premium quote’ at node 3. The simpleexample demonstrates the view that nodes on a workflow system could be as basic aselements in the metagraphs studied by Basu and Blanning (2000). What follows is alist of entities and assumptions that provide the analysis and modelling frameworkunder the node-centric view.

1. An information element is a data field.2. A document or report is a collection of information elements.3. A node may contain information elements, documents, assumptions and

resources.4. A task or activity is viewed as a function which takes the information at input

node and produces the information at the output node. The detail procedureused for producing the output at each activity is not concerned.

5. A node is reachable by a task or activity if the output of the activity has a non-empty intersection with that node. The set of all available activities of a nodeare those activities whose invertex has a non-empty intersection with the node.

6. Sequence dependency is a precedence relationship between tasks/activities,for example, task B is dependent on task A if task B cannot begin unlesstask A is completed.

7. An assumption also called a condition is a proposition associated with a task,such that the assumption must be true for the task to be executed.

8. A resource is an entity associated with one or more tasks, and the resourcemust be available if the tasks are to be executed. The resources may be people,machines, or computer software.

Some items assumed in the above list are similar to those in Basu and Blanning(2000). Their approach to workflow analysis using metagraphs has been provenvaluable in answering some important questions regarding (1) information elements,(2) tasks, (3) resources, and (4) interactions among components. A brief review ofmetagraphs follows.


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3. Metagraphs

Because a metagraph is assumed in both approaches of the analysis it should beuseful to provide a brief review of metagraphs first. This is given by way of amodified example from Basu and Blanning (1994) as shown in figure 2.

In figure 2, x1, . . . , x7 are elements and they can be interpreted as variables, dataattributes, propositions, documents, etc.; e1, e2, . . . , e4 are edges and they can betasks, rules, decision models, etc. Each edge has an input node and an output node,the edge takes the elements of the input node generates the elements of the outputnode. In this paper when the edges are models each of them is treated as‘black boxes’. This view allows analysts to focus on the interrelationships betweendifferent modules and facilitate understanding and use of these relationships.Also, note that a metagraph is always a directed graph or digraph.

Using metagraphs for any workflow analysis must first begin with a model of theworkflow; then the calculation of the adjacency matrices of the metagraph. Eachentry (a pair of information elements; xi and xj) of the adjacency matrix is a tripleconsisting of the co-input (all information elements in the input node except the xi),the co-output (all information elements in the output node except the xj), andthe edge between two nodes joining the elements xi and xj in the metagraph.The metagraph and the adjacency matrix used in that context are referred to aselement-centric because elements are the centre of the analysis. Although elementsare indispensable the set of elements of each node is often treated as a whole inpractice therefore node-based studies are needed. The analysis of metagraphs in thispaper employs the node-centric views of the graph; consequently, the approach iscalled node-centric analysis of metagraphs.

As in element-centric analysis, the node-centric approach also relies on ‘paths’and ‘adjacency matrices’; however, their definitions must be re-defined in the contextof nodes.

4. Properties of node-centric view of metagraphs

Since the analysis of metagraphs will be node-based it is useful to redefinemetagraphs so that nodes can be more directly named and analysed. Given ametagraph, the modified definition of metagraphs is same except that a new setconsisting of all vertices of the metagraph is added and each node is labelled with anumerical identification.

x1

x2

x3

x4

x5

x6

x7

1 2

3 4

5

6

e1

e2

e3

e4

Figure 2. A metagraph.

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Definition 1: Given a finite generating set X¼ {xi, i¼ 1, . . . , I }, a metagraph is an

ordered triple S¼ {X, N, E}, where E is a set of edges E¼ {ek, k¼ 1, . . . ,K}, each of

which is an ordered pair ek¼hVk,Wki and Vk 6¼ ;, Wk 6¼ ;; and N is the set of all

vertices of Vk and Wk relabelled by natural numbers, or N¼ {Nj |Nj2[Kk¼1fVk,Wkg,

j¼ 1, . . . , J}. Each Vk is the invertex of the edge ek, and Wk is its outvertex.

Under this definition a node in the metagraph may be either referred to by Nj for

some j2 {1, . . . , J }, Vk, or Wk for k2 {1, . . . ,K } depending upon the context. It is

possible that a node may contain other nodes, in which case we refer to the contained

nodes as sub-nodes. In figure 2, the metagraph has six nodes. Using the modified

definition of metagraphs, a node number is assigned to each node; it has no

sequential meaning in the metagraph. Notice in figure 2, node 5 is a sub-node of

node 4. The concept of a path from one node to another node in the context of node-

centric view has two forms. One form of a path is defined similar to the definition

of the path in the element-centric view; another form of a path is an extension of the

former. Definitions for these two types of paths are given below.

Definition 2: Given a finite generating set X¼ {xi, i¼ 1, . . . , I }, a metagraph

S¼ {X,N,E }, and two nodes A, B2N, a simple path from node A to node B,

denoted by path(A,B) is a sequence of edges {ek, k¼ 1, . . . , p} with A¼V1,

Wk\Vkþ1 6¼ ; for k¼ 1, . . . , p� 1, and Wp\B 6¼ ;. A generalized path between

two nodes A and B is a sequence of edges {ek, k¼ 1, . . . , p} that satisfy either (1)

A\V1 6¼ ;, and A 6¼V1; or Wp\B 6¼ ; or Wp 6¼B; or (2) if there exists one Nkþ12N,

such that Wk\Nkþ1 6¼ ;, Nkþ1\Vkþ1 6¼ ;, and Wk\Vkþ1¼;, k¼ 1, . . . , p� 1. The

path(A,B) may be written explicitly as e1, e2, . . . , ep(A,B) when a path is known.

Node A is called the source node and node B is the target node of the path. Node B is

connected to (linked to or reachable from) node A if a path (simple or generalized)

exists from node A to node B. The number of edges in the path is called the length of

the path.

Obviously a simple path is a special case of the generalized path. Figure 3 shows

an example of both types of paths. A path used from hereon means either a

generalized path or a simple path. The simple path in element-centric analysis is

different from the above definition because the source and the target of the path are

elements, not nodes. Under the definition of a generalized path, it is easy to see that

‘connectivity’ between two nodes is transitive, i.e. if node A is connected to node B

and node B is connected to node C then node A is connected to node C. Another

point is that the path from node A to node B is not unique. For example, in figure 2

there are two paths from node 1 to node 6, namely {e1, e2, e4} and {e1, e3, e4}.

Therefore path(A,B) is a set function containing all paths from A to B. On a given

path there might be many sequences of connected nodes between two nodes.

The sequence of connected nodes is called a chain as defined below.

e1 e2

N2

W1 V2

e1 e2N2

W1V2

(a) (b)A B

A = V1V1

B = W2 W2

Figure 3. Two forms of a path. (a) {e1, e2} is a simple path. (b) {e1, e2} is a generalized path.


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Definition 3: Given a finite generating set X¼ {xi, i¼ 1, . . . , I } and a metagraphS¼ {X,N,E}, the set of all chains on a path p¼ {ek, k¼ 1, . . . , p} from node A tonode B, denoted by c(p(A,B)) or c(e1, e2, . . . , ep(A,B)) is the set of ordered sequenceof nodes, i.e. c(p(A,B))¼ {A¼V1,W1,V2,W2, . . . ,Vp,Wp¼B} if p(A,B) is a simplepath, and c( p(A,B)) is a set containing ordered sequences of the form:{A,V1,W1, . . . ,Wk�1,Nk,Vk, . . . ,Vp,Wp,B} where {Nk2N |Nk\Wk�1 6¼ ;,Nk\Vk 6¼ ;, for some k¼ 2, . . . , p� 1} if it is a generalized path; (Vk,Wk) is aninvertex and outvertex pair of ek.

Note that without ambiguity the path: e1, e2, . . . , ep(A,B) also represents the setof chains on the path e1, e2, . . . , ep from A to B thus c(e1, e2, . . . , ep(A,B))¼ e1,e2, . . . , ep(A,B). Generally, a chain on a path is a set of nodes on the path. If it is ageneralized path then the sequence may also include some nodes Nk that intersectswith Wk�1 and Vk for some k¼ 2, . . . , p� 1. For example a chain of c(e1, e2,e4(N1,N6)) of figure 2 is {N1,N2,N3,N4,N5,N6}.

4.1 Node-state, chain-state, edge functions and path functions

In the context of element-centric analysis of metagraphs the concept of the co-input/co-output of each element x2X is quite useful. These are sets that show elements ofX other than x that participated in the input and output of an activity. They can begeneralized to the co-input/co-output of a path. Similarly, in the node-centricanalysis of metagraphs, it is important to know the state of a node relative to theinvertex and/or the outvertex of some edge that the node intersects. To illustrate theconcept of the ‘state’, let A and B be two nodes and V and W are the invertex andoutvertex of their edge. The state of a node, or simply the node-state, depends onwhether the node evaluates with an invertext or an outvertex. Assuming A intersectsV and B intersects W, the state of node A, referred to as input state of A because itintersects with the invertex V, is described by a pair of two sets: h{the surplus set of Arelative to the invertex V }, {the deficit set of A relative to the invertex V }i. The stateof node B, referred to as the output state of B because it intersects with theoutvertex W, is defined similarly by: h{the surplus set of the outvertex W relative toB}, {the deficit set ofW relative to B}i. The surplus and deficit sets are set differencesbetween the node and the vertex. To illustrate, let A be the node and V be theinvertex (in general, it does not need to be an invertex) of some edge then thereare two cases between the node A and the invertex V.

Case (i): A\V 6¼ ;. Then there are three possibilities: (a) V�A, (b) A�V, and(c) A 6�V and V 6�A.

Case (ii): A\V¼;. Figure 4 shows three possibilities of case (i).

Under the case (i)(a), the set A is equal to or larger than the invertex V, thesurplus set of the node A relative to V is the difference set A\V; for the case (i)(b)the deficit set of the node A relative to V is the difference set V \A; for the case (i)(c)the surplus set (A\V ) and deficit set (V\A) coexist. The example does not use anyproperty of the invertex V therefore can be replaced by any node. For case (ii) it isobvious that the surplus/deficit set of A relative to the set V is empty. Both thesurplus/deficit set of a node relative to a vertex serve as a measure of ‘closeness’

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between the node and the vertex. Formally, the input/output state of a node(a source node or a target node) is defined below.

Definition 4: LetA, B,V,W2N, and e2E an edge with invertexV and outvertexW,and A\V 6¼ ;, B\W 6¼ ;. The input state of node A on the edge e or denoted byin (A | e)¼hA\V, V\Ai is the pair of the surplus set and the deficit set of the node Arelative to the invertex V. Similarly, the output state of node B on the edge e or simplydenoted by out(B | e)¼hW\B, B\Wi is the pair of surplus set and deficit set of node Brelative to the outvertex W. The state of the chain from A to B on edge e is definedby e(A,B)¼h(A\V )[ (W\B), (V \A)[ (B |W )i whose first component is a surplus setand the second is a deficit set. Let sur[e(A,B)]¼(A\V)[ (W\B), and dft[e(A,B)]¼ (V\A)[ (B |W) then e(A,B) is represented either by hsur[e(A,B)], dft[s(A,B)]i,or a triple hsur[e(A,B)], dft[s(A,B)], ei.

Definition 4 applies to nodes A and B that may not belong to the node set N ofthe metagraph. This feature is useful because it enables the calculation of the stateof e(A,B) for any set A and B.

The reason in defining the output state of node B by hW\B,B\W i is that it is ameasure of how ‘close’ the output W meets node B. If there are elements common inA and B and they both in the surplus set and the deficit set of e(A,B) then theyshould be deleted from the surplus set and the deficit set. More precisely ifC¼ (A\V )\ (B |W ) 6¼ ; then e(A,B)¼h[(A\V )[ (W \B)]\C, [(V \A)[ (B |W )]\C i.

In definition 4, edge e is being treated as a function over all pairs of nodes in N.Formally, assume V and W are the invertex and outvertex of e then the value of thesimple edge function e on (A,B) is defined by (i) e(A,B)¼; if A\V¼; orB\W¼;, and (ii) if A\V 6¼ ;, and B\W 6¼ ; then

e : ðA,BÞ 2 N�N ! eðA,BÞ ¼ AnV [ ðW nBÞ, ðV nAÞ [ ðBjW Þ� �

:

Sets A and B are not limited to members of N. This function may be extended toa path of length more than one—called the path function.

Definition 5: Let e1, . . . , em�1(A,B)) be the set of chains on a pathp¼ {e1, . . . , em�1}) of length (m� 1), m� 2, and (Vi,Wi) the invertex and outvertexpair of ei, i¼ 1, . . . ,m� 1. The state (a vector of two sets) of a given path p isuniquely defined by: p(A,B)¼h(A\V1)[ (W1\V2)[ � � � [ (Wm�2\Vm�1)[ (Wm�1\B),(V1\A)[ (V2\W1)[ � � � [(Vm�1\Wm�2)[ (B\Wm�1)i whether the path is simple orgeneralized. The path p evaluated in this sense is called the path function.

A V A

V

(a) (b)

V A

(c)

Figure 4. Surplus and deficit sets of the state of node A. (a) A\V¼ surplus of A.(b) V \A¼ deficit of A. (c) V \A¼A has surplus and deficit.


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The path function is only dependent on the source node, the target node, and theinvertex and outvertex pairs of each edge of the path; it is independent of choicesof the chain’s intermediate nodes. Thus, given the path p¼ {e1, . . . , em�1} fromnode N1, to node Nm, e1, . . . , em�1(N1, . . . ,Nm) can be simplified to p(N1,Nm).Again, like simple edge functions the source and target nodes of the chain do notneed to be the nodes of the metagraph. Edge functions and path functions arestate-valued functions.

Consider the chain ({N1,V1,W1,V2,W2,N3} on path p¼ {e1, e2}) in figure 3(a),it can be viewed as a linked chain of e1(N1,W1) and e2 (V2,N3). They are linkedbecause the end node of chain 1 and the starting node of chain 2 intersect; similarexample is in figure 3(b). The operation of linking two chains is denoted by theoperator ‘^’ as follows.

Definition 6: Let S¼ {X,N,E} be a metagraph; p1 (A,B)¼ ei, eiþ1, . . . , em�1 (A,B)and p2 (B,C)¼ em, emþ1, . . . , ej�1 (B,C) are two paths and (Vk,Wk) is an invertexand outvertex pair for k¼ i, iþ 1, . . . , m, mþ 1, . . . , j� 1. Then the product of twopath functions is denoted by p1(A,B)^ p2(B,C)¼ p1p2 (A, . . . ,B, . . . ,C) wherep1p2¼ {ei, eiþ1, . . . , em�1, em, emþ1, . . . , ej�1}. The product of two path functions isdefined to be the path function of the concatenated path, i.e.

p1ðA,BÞ ^ p2ðB,CÞ ¼ p1p2ðA, . . . ,B, . . . ,CÞ ¼ p1p2ðA,CÞ:

For example, consider two path functions of length one: e1(N1, N3) ande3(N3, N4) of figure 2. The product of these two path functions is evaluatedby the concatenated path. By definition 5, the state of e1e3 (N1,N2,N4)¼{N1\V1}¼;[ {W1\V3}¼;[ {W3\N4}¼;, {V1\N1}¼;[ {V3\W1}¼ ;[ {N4\W3¼

{x5}}¼h;, {x5}i, i.e. the chain has a deficit element. By definition 6, e1(N1,N3) ^e3(N3,N4)¼ e1e3 (N1,N3,N4)¼h;, {x5}i.

Proposition 1: A chain is simple if and only if its surplus state and the deficit state areempty sets.

Proof: This is a direct result of the definition of simple chains. œ

4.2 Relationships between element-centric and node-centric approaches

Certain important basic properties of node-centric and element-centric analysesof metagraphs are related. In fact, the concepts discussed in this paper areextensions of those in element-centric analysis. A fundamental concept inelement-centric views is co-input/co-output of an element x. In figure 2, node 1(V1) and node 2 (W1) are invertex and outvertex of edge e1. The co-input of x1is the set (V1\{x1}) and the co-output of x3 is the set (W1\{x3}). Treating edgesas functions, the co-input of x1 and co-output of x2 on edge e1 can be expressedby e1(x1, x2)¼hV1\{x1},W1\{x3}i. If each element x1 and x2 is considered as anode then from the node-centric view e1({x1}, {x2})¼hW1\{x3},V1\{x1}i. It turnsout that co-input of x1 equals the deficit of the node {x1} and co-output ofx3 is the surplus of the node {x3}. The vector e1(x1, x2) can be derived frome1({x1}, {x2}) by reverse the order of the latter, and vice versa. Another funda-mental concept is the co-input and co-output of a ‘path’ of length L (L41).

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Assuming the path has a sequence of edges he1, e2, . . . , eLi, x2V1, x0 2WL, andWl\V1þ1 6¼ ; for l¼ 1, 2, . . . ,L� 1, then the co-input and the co-output of thepath between elements x and x0 (in element-centric view) is:

ðV1nfxgÞ [[L

l¼2ðVlnWl�1Þ,

[L

lWlnfx

0g

D E:

Again treating elements x and x0 as singleton nodes the surplus and deficit sets forthe same path under the node-centric views are contained in the state vector of thepath following definition 6:

ðfxgnV1Þ [ ðW1nV2Þ [ � � � [ ðWL�1nVLÞ [ ðWLnfx0gÞ,

�

ðV1nfxgÞ [ ðV2nW1Þ [ � � � [ ðVLnWL�1Þ [ ðfx0gnWLÞ:�

Since ({x}\V1)¼; and ({x0}\WL)¼; last vector can be simplified to:

ðW1nV2Þ [ � � � [ ðWL�1nVLÞ [ ðWLnfx0gÞ,

�ðV1nfxgÞ [ ðV2nW1Þ [ � � � [ðVLnWL�1Þ:

�

By comparison, the co-input of the simple path equals to the deficit component, butthe co-output is not the same as the surplus component. However, if the set(W1[ � � � [WL�1) is added to the surplus component, i.e. (W1[ � � � [WL�1)[{(W1\V2)[ � � � [ (WL�1\VL)[ (WL\{x

0})}¼SL

l Wlnfx0g it produces the co-output.

The relationships between the surplus/deficit sets and co-input/co-outputdiscussed above can be applied to convert element-centric adjacency matrices(discussed later) adjacency matrices to node-centric adjacency matrices andvice versa.

In the element-centric view a simple path h(x, x) from an element x to itself iscalled a cycle, and a metagraph containing one or more cycles is cyclic. A metagraphwithout cycles is called acyclic. The notion of cycles may also be defined undernode-centric views. A node-centric metagraph is cyclic if there is a path (simple orgeneralized) from a node to itself.

Proposition 2: If a metagraph is cyclic under the element-centric view then the samemetagraph is also cyclic under the node-centric view. The reverse is not true in general.

Proof: Let h(x, x), x2X be a cycle under the element-centric view of the metagraphS¼ {X,E}. Then there exists a sequence of edges h(x, x)¼ {el, l¼ 1, . . . ,L} such thatel¼hVl,Wli 2E, and x2Vl, x2WL and Vl\Wl 6¼ ; for l¼ 1, . . . ,L, and WL¼Vl.In the node-centric view of metagraph S 0 ¼ {X,N,E }, the existence of the pathh(x, x) means that node Vl can reach the node WL¼Vl. Thus, S0 is cyclic bydefinition. œ

The reverse of the proposition is shown by an example. Figure 5 is a cyclic underthe node-centric view because there exists two cyclic chains: (1) e1, e2(N1,N1) and (2)e1, e2(V1,V1) on the common path {e1, e2} that is not a ‘simple path’. Since {e1, e2}is not a simple path any element x2N1\Vl in figure 5 does not form a cycle fromthe element-centric view and thus proves the reverse proposition.

4.3 Node-centric views of adjacency matrices

The adjacency matrix under the node-centric views (as opposed to the element-centric views) is a useful construct in practical applications. An adjacency matrix


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under the node-centric view can be defined by computing state-valued path functions

of length one between any two nodes.

Definition 7: Given a finite generating set X¼ {xi, i¼ 1, . . . , I }, and a metagraph

S¼ {X,N,E}, where N is a set of nodes N¼ {Nj |Nj�X, j¼ 1, . . . , J}, and E is a set

of edges E¼ {ek, k¼ 1, . . . ,K}, the state-valued node-centric adjacency matrix, A, of

S is a J� J matrix with components aij, i, j2 {1, . . . , J} defined as follows:

aij ¼[K

k¼1fekðNi,NjÞg

is the set of paths from Ni, to Nj of length one, and ek(Ni,Nj)¼h(Ni\Vk)[ (Wi\Nj),

(Vk\Ni)[ (Nj\Wk), eki.

An example from figure 2, the value of the matrix at (N1,N3) is e1 (N1,N3) that

evaluates to e1(N1,N3)¼h{x4}, {x2}, e1i. The state-valued node-centric adjacency

matrix A of the metagraph of figure 2 is shown in table 1.Node-centric adjacency matrices defined over the same generating set and the

node set can be added and multiplied. The addition of two node-centric adjacency

matrices can be carried out by performing union of corresponding elements of the

two matrices.

Definition 8: Given a generating set X, two metagraphs S¼hX,N,E i and

S0 ¼ hX,N,E 0i and two node-centric adjacency matrices A with components {aij}

and B with components {bij}, then AþB is the adjacency matrix of the metagraph

hX,N,E[E 0i, with components (AþB)ij¼ aij[ bij.

N1

V2

V1

N2

N3W1

x

ye2

e1

e3

Figure 5. A cyclic metagraph from the node-centric view.

Table 1. State-valued node-centric adjacency matrix A.

Node N1 N2 N3 N4 N5 N6

N1 ; h;, ;, e1i h{x4}, {x2}, e1i ; ; ;

N2 ; ; ; h{x4}, {x2}, e2ih;, {x5}, e3i

h{x4, x5}, {x2}, e2ih;, ;, e3i

;

N3 ; ; ; h;, ;, e2i, h{x2},{x4, x5}, e3i

h{x5},;, e2i,h{x2}, {x4}, e3i

;

N4 ; ; ; ; ; h{x5},;, e4iN5 ; ; ; ; ; h;, ;, e4iN6 ; ; ; ; ; ;

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Since the addition is defined by the set union it is easy to see that addition of

matrices is idempotent AþA¼A, and satisfies the associative (Aþ (BþC))¼

((AþB)þC) and commutative AþB¼BþA properties.The product of two node-centric adjacency matrices can be defined by using

definitions 5 and 6. Specifically, the product is the union of the corresponding

components from all such vectors over J (the number of nodes in the node set),

L (the number of edges in the set E ), and M (the number of edges in the set E 0),

as follows.

Definition 9: Assume a generating set X, two metagraphs S¼hX,N,E i and

S0 ¼ hX,N,E 0i and two state-valued node-centric adjacency matrices A and B of

S and S0. Let (aik)l¼ el(Ni,Nk) and (bkj)m¼ em(Nk,Nj) be the edge functions of length

one in aik and bkj such that:

aik ¼ fðaikÞl, l ¼ 1, . . . ,Lg,

bkj ¼ fðbkjÞm, m ¼ 1, . . . ,Mg:

The product of the state-valued node-centric adjacency matrices A and B is a

state-valued matrix C¼A�B, with components

Cij ¼[J

k¼1

[L

l¼1

[Mm¼1

ððaikÞl ^ ðbkjÞmÞ:

The ‘^’ is the product operator defined by definition 6 (Appendix A).

The effect of the operator ‘^’ is to form a new path of length two from node Ni to

node Nj. The element Cij is a set of such products union over l2L, m2M, and k2K.For example, it is easy to compute c14 of the square matrix C from node 1 to

node 4 of figure 2. There are two paths. By definition 6, the state of the path

is independent of intermediate nodes of on the path therefore only two state vectors

of c14:

1. e1e2(N1,N4)¼h{x4}, {x2}, he1, e2ii.2. e1e3(N1,N4)¼h;, {x5}, he1, e3ii.

Obviously, the computation of the product of two node-centric adjacency

matrices could be quite complicated and time consuming as the number of nodes

increases; however, due to the algebraic nature of the matrix operations this work

can be automated.Following the product definition, the square of the adjacency matrix A (i.e. A2) is

calculated and given in table 2. Note that table 2 is not a complete matrix

because empty rows and columns are deleted from the product matrix for

saving spaces. The third power of the adjacency matrix A (i.e. A3) is calculated

in table 3.If the generating sets are not the same, and/or the node sets are not the same, we

may augment the generating set and the node set so that both metagraphs share the

same generating set and the node set. The node-centric algebraic operations of

additions and multiplications are then applicable to the matrices of the augmented

metagraphs.


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The adjacency matrix multiplication just defined possesses the same properties asthe multiplication of element-centric adjacency matrices. Namely, matrix multi-plication is neither idempotent nor commutative, but is associative and distributiveunder addition.

Proposition 3: Assume X is a generating set and three metagraphs S¼hX,N,E i,S 0 ¼ hX,N,E0i, and S00 ¼ hX,N,E00i, where N¼ {Nj |Nj "[

Kk¼1fVk,Wkg, j¼ 1, . . . , J}

each with a state-valued node-centric adjacency matrices A, B, and C respectively.Then:

1. A� (B�C)¼ (A�B)�C.2. (AþB)�C¼ (A�C)þ (B�C).

(The proof is given in Appendix B.)The closure of a node-centric adjacency matrix A is defined as the infinite sum

A�¼AþA2

þA3þ � � � , and this limit exists if we adopt the definition of the Cat( )

function and the Trc( ) function (Basu and Blanning 1994) as included in theappendix. The result of A3 is given in table 3.

Each entry of the matrix A� contains state vectors associated with the complete

set of paths between any two nodes.

Proposition 4: Assume X is a generating set X and S¼hX,N,Ei is a metagraph,where E is a set of edges E¼ {ek, k¼ 1, . . . ,K }, and N¼ {Nj |Nj " [K

k¼1fVk,Wkg,j¼ 1, . . . , J}. Then anij ¼; iff there is no path of length n from node Ni to Nj. Otherwisefor each triple in anij there is a path from node Ni to Nj such that:

1. The first member of anij is a surplus set of the path.2. The second member of anij is a deficit set of the path.3. The third member of anij is the path.

And all such paths have state vectors represented by a triple in anij.

Corollary: Under the conditions of Proposition 2:

1. A�¼AþA2

þ �� þAK, where K is the number of edges in S.2. anij ¼; for all i 6¼ j and n4K.

Table 2. The square of the adjacency matrix A.

Node N4 N5 N6

N1 h{x4}, {x2}, he1, e2ii,h;, {x5}, he1, e3ii

h{x4, x5}, {x2},he1, e2ii, h;, ;, he1, e3ii

;

N2 ; ; h{x4, x5}, {x2}, he2, e4ii,h;, ;, he3, e4ii

N3 ; ; h{x5}, ;, he2, e4ii,h{x2}, {x4}, he3, e4ii

Table 3. A3.

Node N6

N1 h{x4, x5}, {x2}, he1, e2, e4ii, h;, ;, he1, e3, e4ii

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3. anij ¼ amij for all m, n4K.4. a�ij ¼; iff Ni is not a member of a cycle in S.

The closure, A�, of the adjacency matrix of figure 2 appears in table 4. Since themaximum length of any path of the metagraph in figure 2 is 3 thereforeA�

¼AþA2þA3.

5. Applications

In this section, the workflow example from car insurance given in section 2 is used toillustrate the node-centric adjacency matrix, and its closure matrix. Table 5 is thenode-centric adjacency matrix B derived from figure 1. Table 6 is the square of theadjacency matrix.

Several values of the square matrix could be simplified if the informationelements of the source node intersect with the target node. For example,N1\N3¼ {CD,TI}, the state value at (N1,N3) equals to h;, ;, e1i after deleting{CD,TI} from the surplus set and the deficit set of the triple. The simplified versionof the adjacency matrix squared is given in Appendix C.

The third power of the adjacency matrix has only two non-empty entries asshown in table 7 and the closure of the adjacency matrix B is give in table 8.

6. Conclusion

In its original form, metagraphs are primarily formulated and analysed in terms of‘information elements’ contained in the nodes. However, analysis of metagraphsbased on the ‘nodes’ view provides information complementary to ‘element’ views.For example, in the workflow system the input at one node often requires multipleinformation elements, assumptions and resources taken together for processing bythe activity associated with the node. The output of the activity is a set of requiredbusiness information elements. In the real world a person who is responsible for anactivity normally uses a group of information received and produces a set of requiredbusiness data to the subsequent activities. The node-centric view discussed in the

Table 4. The closure of the node-centric adjacency matrix A.


N1 ; h;, ;, e1i h{x4},{x2}, e1i

h{x4}, {x2},he1, e2ii,

h;,{x5}, he1, e3ii

h{x4, x5}, {x2},he1, e2ii, h;, ;, he1, e3ii

h{x4, x5}, {x2},he1, e2, e4ii,

h;, ;, he1, e3, e4iiN2 ; ; ; h{x4}, {x2}, e2i

h;, {x5}, e3ih{x4, x5}, {x2}, e2i

h;, ;, e3ih{x4, x5}, {x2},

he2, e4ii,h;, ;, he3, e4ii

N3 ; ; ; h;, ;, e2i, h{x2},{x4, x5}, e3i

h{x5},;, e2i,h{x2}, {x4}, e3i

h{x5}, ;, he2, e4ii,h{x2},

{x4}, he3, e4iiN4 ; ; ; ; ; h{x5}, ;, e4iN5 ; ; ; ; ; h;, ;, e4iN6 ; ; ; ; ; ;


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paper provides a framework to answer many questions such as: Would

the information given at a source node be sufficient to produce the information at

the target node? If it is not sufficient at the source node what are the elements

needed? If it is sufficient at the source node what are the surplus elements that are not

needed? The closure of the node-centric matrix contains information on where non-

empty surplus and/or deficit set occur, and also which path(s) that minimizes the

deficit set from a source node to a target node.

Table 5. The adjacency matrix B for the car insurance workflow.


N1 h{CD,TI},{CD,TI}, e1i

h{CD,TI},;, e1i

h{BI,DI},{RI}, e2i

h ;, ;, e3i h;, {PM}, e3i,h{BI,DI},

{RI,DR}, e2i

;

N2 h;, {CD,TI}, e1i

h;, ;, e1i ; h;, {CD,TI}, e3i h{;, {CD,TI, PM}i, e3i

;

N3 ; ; h;, ;, e2i h{RI}, {BI,DI},e34

h;, {DR}, e24,h{RI},

{BI,DI, PM}, e3i

;

N4 ; ; h ;, {CD,TI}, e2i

; h;, {CD,TI,DR}, e2i

;

N5 ; ; ; ; ; h;, {DR}, e4iN6 ; ; ; ; ; h;, {PM}, e4iN7 ; ; ; ; ; h;, ;, e4iN8 ; ; ; ; ; ;

Table 6. The square of the adjacency matrix B.

Node N5 N6 N7 N8

N1 h{CD,TI},{CD,TI},he1, e2ii

h{RI, CD,TI},{CD,TI, BI,DI},

he1, e3ii

h{CD,TI}, {DR,CD,TI},he1, e2ii; h{CD,TI,RI},

{BI,DI, CD,TI, PM}, he1, e3ii

h{BI,DI}, {RI,DR},he2, e4ii;

h;, {PM}, he3, e4iiN2 h;, {CD,TI},

he1, e2iih{RI}, {BI,DI,CD,TI}, he1, e3ii

h;, {CD,TI,DR},he1, e2i; h{RI}, {BI,DI,CD,TI, PM}, he1, e3ii

h;, {CD,TI, PM},he3, e4ii

N3 h;, {DR}, he2, e4ii;h{RI},

{BI,DI, PM},he3, e4ii

N4 h;, {CD,TI,DR}i,he2, e4ii

Table 7. The cubic power of the adjacency matrix B.

Node N8

N1 h{CD,TI}, {CD,TI,DR},he1, e2, e4 ii; hCD,TI,RI}, {BI,DI, CD,TI, PM}, he1, e3, e4i

N2 h;, {CD,TI,DR}i,he1, e2, e4ii; h{RI}, {CD, TI, BI,DI, PM}i, he1, e3, e4ii

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Table

8.

Theclosure

oftheadjacency

matrix

B.

Node

N3

N4

N5

N6

N7

N8

N1

h{CD,T

I},

{CD,T

I},e

1i

h{CD,T

I},;,e

1i

h{BI,DI},{RI},e

2i;

h{CD,T

I},

{CD,T

I},he 1,e

2ii

h;,;,e

3i;h{RI,CD,T

I},

{CD,T

I,BI,DI},

he1,e

3ii

h;,{PM},e 3i;

h{BI,DI},{RI,DR},e 2i;

hh{C

D,T

I},

{DR,C

D,T

I},

he1,e

2ii;h{RI,CD,T

I},

{BI,DI,CD,T

I,PM},

he1,e 3ii

h{BI,DI},{RI,DR},he

2,e

4ii;

h;,{PM},he

3,e

4ii;h{CD,T

I},

{CD,T

I,DR},he

1,e

2,e

4ii;

h{RI,CD,T

I},{CD,T

I,PM},

he1,e

3,e

4ii

N2

h;,{CD,T

I},e

1i

h;,;,e

1i

h;,{CD,T

I},he 1,e

2ii

h;,{CD,T

I},e

3i;h{RI},

{BI,DI,CD,T

I},

he1,e

3ii

h{;,{CD,T

I,PM},e 3i;

h;,{CD,T

I,DR},

he1,e

2ii;h{RI},{BI,DI,

CD,T

I,PM},he

1,e

3ii

h;,{CD,T

I,PM},he

3,e

4i;

h{CD,T

I},{CD,T

I,DR},

he1,e

2,e

4ii;hh{R

I},{CD,T

I,BI,DI,PM}4

,he 1,e

3,e

4ii

N3

;;

h;,;,e

2i

h{RI},{BI,DI},e

3i

h;,{DR},e 2i;h{RI},{BI,

DI,PM},e 3i

h;,{DR},he

2,e

4ii;h{RI},{BI,

DI,PM},he

3,e

4ii

N4

;;

h;,{CD,T

I},e

2i

;h;,{CD,T

I,DR},e 2i

h;,{CD,T

I,DR}i,he 2,e

4ii

N5

;;

;;

;h;,{DR},e 4i

N6

;;

;;

;h;,{PM},e 4i

N7

;;

;;

;h;,;,e

4i

N8

;;

;;

;;


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Section 4.3 shows that the very basic notions of co-input and co-output of anedge and the co-input and co-output of a simple path under the element-centricview are related with state-valued edge functions and path functions of thenode-centric view. Consequently adjacency matrices developed under either view ofmetagraphs have fixed relationships. The framework defined in this paperhas proven to be an extension of the framework from the element-centric view ofmetagraphs.

The concept of ‘connectivity’ plays an important role in both approaches to theanalysis of metagraphs. Many results of connectivity obtained from the node-centricanalysis are parallel to those under element-centric analysis. However, there aremajor difference in computing paths and adjacency matrices. Each path underthe node-centric view will have the surplus set and the deficit set calculated thatindicate the adequacy of the information elements, assumptions and resources onthe path.

The paper has constructed a basic framework for node-based analysis ofmetagraphs. Topics for further research in the context of the node-centric views arerich and can be theoretical and/or applied. A practical project which iscomputationally desirable and feasible is to develop software that computes paths,adjacency matrices, and the closure of adjacency matrix under the node-centricapproach. The core of the software development is on the computation andinterpretations of adjacency matrices.

Further research with node-centric analysis of metagraphs can include theanalysis of resources and assumptions within a node as sub-nodes and find ways toresolve potential inadequacies, conflicts, and wastes. Other areas of research mayapply the node-centric analysis to decision support systems and knowledgemanagement systems, as they have been shown to be effective with element-centricanalysis.

Appendix A: The wedge operator for the multiplication of node-centric

adjacency matrices

Following Basu and Blanning (1994), the catenation function, Cat( ), applies to apair of ordered sets and returns a new combined ordered set as in:

Catð a, bh i, c, d, eh iÞ ¼ a, b, c, d, eh i:

The truncation function, Trc( ) takes an ordered set and returns an ordered set withthe truncation beginning at the first repeating element in the ordered set, e.g.

Trcð a, b, f, g, a, t� �

Þ ¼ a, b, f, g� �

:

Definition 10: Assume a generating set X, two metagraphs S¼hX,N,Ei andS0 ¼ hX,N,E0i,N¼ {Nj |Nj2[K

k¼1fVk,Wkg, j¼ 1, . . . , J} and the chain-valued node-centric adjacency matrices A and B. Let (aik)l and (bkj)m be the ordered triples withrespect to edge l (el) and edge m (em) in aik and bkj such that:

aik ¼ fðaikÞl, l ¼ 1, . . . ,Lg,

bkj ¼ fðbkjÞm,m ¼ 1, . . . ,Mg:

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The product of the state-valued node-centric adjacency matrices A and B is a state-

valued matrix C¼A�B, with components

Cij ¼[J

k¼1

[L

l¼1

[Mm¼1

ððaikÞl ^ ðbkjÞmÞ:

The ‘^’ is the product operator defined by definition 7, it is either a set of chains or a

null set, as follows:

1. If (aik)l¼; or (bkj)m¼; for some l and m then (aik)l^ (bkj)m¼;.2. If (aik)l 6¼ ; and (bkj)m 6¼ ; for some l and m then (aik)l ^ (bkj)m forms a path

hel, emi of length 2 from node i to node k, and node k to node j. Since

ðaikÞl ¼ elðNi,NkÞ, el 2 E, and ðbkjÞm ¼ emðNk,NjÞ�, em 2 E 0:

ðaikÞl ^ ðbkjÞm ¼ elðNi,NkÞ ^ emðNk,NjÞ ¼ el, emðNi,Nk,NjÞ

The product yields to a stated-valued vector by definition 6, i.e.

ðaikÞl ^ ðbkjÞm ¼ ðNinVlÞ [ ðWlnVmÞ [ ðWmnNjÞ, ðVlnNiÞ [ ðVmnWlÞ [ ðNjnWmÞ, ei, ej� ��

:

3. Cij is the set of all triples obtained from (1) and (2) above by varying l and m.

The effect of the operator ‘^’ is to combine the surplus states, deficit states, and

edges from node Ni to node Nk and from node Nk to node Nj. Cij is a set that is the

union of the above products over l2L, m2M, and k2K.

Appendix B: Proof of proposition 3

Proposition 3: Assume X is a generating set and three metagraphs

S¼hX,N,Ei,S0 ¼ hX,N,E 0i, and S00 ¼ hX,N,E 00i, where N¼ {Nj |Nj "[Kk¼1fVk,Wkg, j¼ 1, . . . , J} each with a chain-valued node-centric adjacency matrices

A, B, and C respectively. Then:

1. A� (B�C)¼ (A�B)�C,2. (AþB)�C¼ (A�C)þ (B�C).

Proof of part 1: Let the chain-valued node-centric adjacency matrices A and B as

defined in Appendix A. Thus, (aik)l and (bkj)m are the ordered triples with respect

to edge l (el2E ) and edge m (em2E0). Let an element of C be denoted by

cjq¼ {(cjq)r | r¼ 1, . . . ,R} where (cjq)r¼ er(Nj,Nq), er2E00. Since an element of (A�B)

is a set of paths of the form: (aik)l^ (bkj)m¼ el em(Ni,Nk,Nj) when it multiplies

with C, the elements of C are sets of the products: [(aik)l^ (bkj)m]^

[(cjq)r]¼ el em(Ni,Nk,Nj)^ er(Nj,Nq)¼ el em er(Ni,Nk,Nj,Nq)¼ el em er(Ni,Nq). Elements

for the product A� (B�C) can be explained in the same manner. That is,

[(aik)l^ [(bkj)m^ (cjq)r]¼ el(Ni,Nk)^ em er(Nk,Nj,Nq)¼ el em er(Ni,Nk,Nj,Nq)¼ el em er(Ni,Nq). This completes the proof for part 1.


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Proof of part 2: Continuing the notations used in the proof above for matrices A, B

and C, the element (AþB)ij¼ (aij)l[ (bij)m is the (i, j ) element of the (AþB) by

definition 9. By Appendix A, the (i, j ) element in the product (AþB)�C is a set

containing [(aij)l[ (bij)m]^ (cjq)r¼ [(aij)l^ (cjq)r][ [(bij)m^ (cjq)r], which is the (i, j)

element in (A�C)þ (B�C), and it equals to el er(Ni,Nj,Nq)[ em er(Ni,Nj,Nq).

This completes the proof for part 2. œ

Each chain-valued element in the matrix can be easily converted to state-valued by

definition 6.

Appendix C: The square of the adjacency matrix B simplified—table 9

Acknowledgement

The research was supported by a professional development grant from Wright State

University, 2005–2006.

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Table 9. The square of the adjacency matrix B of table 6 after removing source nodeelements from the deficit set.

Node N5 N6 N7 N8

N1 h;, ;, he1, e2ii h{RI},;, he1, e3ii

h;, {DR}, he1, e2ii; h{RI},{PM}, he1, e3ii

h{BI,DI}, {RI,DR},he2, e4ii;

h;, {PM}, he3, e4iiN2 h;, {CD,TI},

he1, e2iih{RI},

{CD,TI},he1, e3ii

h;, {CD,TI,DR},he1, e2ii; h{RI}, {CD,TI, PM},

he1, e3ii

h;, {CD, TI, PM},he3, e4ii

N3 h;, {DR}, he2, e4ii; h{RI},{BI,DI, PM}, he3, e4ii

N4 h;, {CD, TI,DR}i, he2, e4i

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a node-centric analysis of metagraphs and its applications to workflow models

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