An Application of Prim s Algorithm in Defining a SoS Operational Boundary

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1 Paper #37 n pplication of Prim s lgorithm in efining a SoS Operational oundary lex orod harles V. Schaefer, Jr. School of ngineering Stevens nstitute of Technology astle Point on udson oboken, N.J , US Tel agorodis@stevens.edu r. rian Sauser harles V. Schaefer, Jr. School of ngineering Stevens nstitute of Technology astle Point on udson oboken, N.J , US Tel bsauser@stevens.edu bstract This paper presents an application of Prim s lgorithm to define the operational boundaries of the System of Systems structure. The proposed application finds the minimum and maximum spanning trees of a connected weighted System of Systems based on a new method. The integral part of the new method is a model to define combined weights of a pair of nodes and a link between them. The minimum and maximum spanning trees can represent the upper or the lower operational boundaries of a connected weighted System of Systems, depending on categories and measures for each category. n this paper the proposed method is demonstrated using a nine-node cluster of a weighted System of Systems. ntroduction With the current research and practice of System of Systems (SoS), which has been SR 2007, Stevens nstitute of Technology, SN PRONS SR 2007, March 4-6, oboken, NJ, US proven to be challenging (ar-yam 2004; elaurentis 2005), a fundamental difficulty has been the identification, management, and optimization of the operational boundaries of these systems. The scope of this paper is to present a model for defining these boundaries with the potential to manage and optimize the problem space. efore we describe our proposed model, we would like to state some fundamental premises of our investigation. irst, we propose that principles of complex systems can be used to define a foundation for which to develop such a model. or example, ar Yam in his research based on a literature review of the state of knowledge on SoS concluded that complex systems are one of the characteristics of SoS (ar-yam 2004). One of these principles is that of nonlinearity (zerwinski 998). The Nobel Prize winner Murray ell-mann stated, When dealing with any nonlinear system, especially a complex one, it is not sufficient to think of the system in terms of parts or aspects identified in advance, then to analyze those parts or aspects separately, and finally to

2 combine those analyses in an attempt to describe the entire system. Such an approach is not, by itself, a successful way to understand the behavior of the system. n this sense there is truth in the old adage that the whole is more than the sum of its parts. t is of crucial importance that we learn to supplement those essential specialized studies with what call a crude look at the whole (ell-mann 2000). Therefore, a second premise is that the summation of the parts is not a representation of the whole. third premise is that SoS, as complex systems, can be represented as networks (orogovtsev and Mendes 2003; Newman, - L.arabási, and Watts 2006; Pastor-Satorras and Vespignani 2004; Strogatz 200). Shenhar was one of the first who described the SoS as a network of systems functioning together to achieve a common purpose (Shenhar 994). Later, others such as Maier identified other known network-centric systems as SoSs (i.e. The nternet, the global communications network) (Maier 996). This premise will be applied to support the use of network graph theory to define the operational boundaries of the SoSs. ased on these premises, this paper presents an adaptation and application of Prim s algorithm to define the operational boundaries of a SoS. The proposed method finds the minimum and maximum spanning trees of a connected, weighted SoS. The integral part of the method is a model to define combined weight of a pair of nodes and a link between them. The minimum and maximum spanning trees can represent the upper or the lower operational boundary depending on categories and measures for each category. We will use a documented SoS characterization as an example of categories that can be used in the model, but these categorizations can be organization specific. SoS haracterization n accepted universal definition of SoS is elusive. f for no other reason, definition PRONS SR 2007, March 4-6, oboken, NJ, US implies a specific purpose, and the purposes vary. The definition of a system being already abstract itself only makes matters worse, leading to an abstraction squared. oth systems and SoS conform to the accepted definition of system in that each consists of parts, relationships and a whole that is greater than the sum of the parts, and therefore in that sense they are the same. ut these terms differ in a fundamental sense, one that impacts their structure, behavior and realization, and the distinction comes from the manner in which parts and relationships are gathered together and therefore in the nature of the emergent whole. This difference can further be described in that a SoS s parts, acting as autonomous systems, forming their own connection and variety in their diversity, lead to enhanced functionality, that fulfills capability demands that sets a SoS apart. Therefore, to define a SoS within our proposed model we will use a characterization and not a definition. y doing so, we might provide a set of continua for a systems typology showing the quanta that perhaps define stages in emergence between the two and along their continuum. We have chosen the characteristics of oardman and Sauser (oardman and Sauser 2006), because these characteristics are traceable back to a centre of gravity in the arguments as to what constitutes a SoS, as evidenced in the literature. See ppendix for a summary of these characteristics. SoS Operational oundary Model To explain our model, throughout the paper we will use a simple illustrative example of a SoS cluster with nine nodes (i.e. systems). The purpose of this illustration is to use an example which in its simplest form is a network of systems. igure shows our chosen example with each component system of the cluster represented as a node with a letter assignment. The nodes are then interconnected with links to create a network. 2

3 Links that are not shown are assumed to not exist. Node s weight (c i ) relates to a system connected via link (a i ) to another system. Link s weight (a i ) represents the maturity of system integration between pair of nodes b i and c i. λ i and β i are parameters of the system depending on the category and measures for each category. w i is the combined weight of the i- th pair of nodes and link between them. i is an identifier of a pair of nodes and the link between them. or a none existing link w i = 0 igure. NY ab SoS luster Weight ssignments The first step in defining the model and an integral part of the method is to identify the combined weight of a pair of nodes and a link between them. or our example system, we will use the following nomenclature to describe the example system: Minimum Weight Spanning Tree The next step is to assign weights based on a predefined categorization for determining the minimum weight spanning tree. s stated earlier, we would be using the oardman and Sauser characterization of SoS to define these categories. xtracting a Minimum Spanning Tree (MST) is a classic problem. Significant research has been conducted to compare performance of various MST algorithms for different applications in various settings (redman and Tarjan 987; Kruskal 956; Moret and Shapiro 994; Noshita 985; Prim 957; ijkstra 959). System System b Node Link Weight Node i c Weight a i i Weight We use Prim s algorithm to calculate the ombined Weight w i igure 2. Weight ssignments igure 2 depicts two weighted nodes and a weighted link between them with a combined weight w i. The following formula represents the combined weight of a pair of nodes and a link between them: w i = a i + λ i b i +β i c i () Node s weight (b i ) relates to a particular system in a SoS. minimum cost spanning tree for a connected weighted graph (Prim 957). n the proposed algorithm, the sum of combined weights w i of the minimum spanning tree will represent this tree total value: W MinSpanTree = w i (2) i MinSpanTree Where each w i is defined by formula (). The value of WMinSpanTree that was calculated in formula (2) will be used as a lower operational boundary of a SoS based on specific category. 3 PRONS SR 2007, March 4-6, oboken, NJ, US

4 Maximum Weight Spanning Tree n this section we will find the maximum weight spanning tree based on a minimum spanning tree algorithm (ven 979). The value of the maximum weight spanning tree will be calculated as follows: W MaxSpanTree = i i MaxSpanTree The value of WMaxSpanTree calculated by formula (3) will be used as an upper operational boundary of a SoS based on specific category. pplication of the Method Let s assume that connectivity category of a SoS depicted in igure is represented by the following graph. w (3) or the purposes of our example we are using the onnectivity category and assuming that the combined weights have been calculated and assigned on a scale of 0 to, as illustrated in igure 3. The methodology for determining the weights will be organizational and characteristic specific. or the purposes of this paper, the weights are randomly selected to illustrate the model. inding Lower Operational oundary s previously suggested, we will find the value of the minimum spanning tree in order to define the lower operational boundary of the SoS depicted in igure 3. igure 3. Weighted SoS (onnectivity ategory) The extracted minimum cost spanning tree is represented in igure 4. The numbers allotted between vertices indicate the assigned combined weights. Vertices are components of the System of Systems represented by letters to. igure 4. Maximum Weight Spanning Tree The following is a detailed description of how the minimum cost spanning tree is extracted. Select an arbitrary node. Select a node neighboring node with the lowest weight value. is away, is and is ; as a result we select. Select the neighboring node with the lowest weight value away from either or. Node is away from, is away from. is away from. is away from but only from. Select. Now, are part of the tree. Select the neighboring node with the lowest weight value away from, or. is away from and also away from. is away from, and is from. s a result,,,, or can be selected. We select. s a result, are part of the tree. Select because, and all have weight value of, so again, any one of them can be selected. and are part of the tree. 4 PRONS SR 2007, March 4-6, oboken, NJ, US

5 2 3 4 Select, since is the lowest weight value away from. and are now part of the tree. Select, because is away from. Now,, and are part of the tree. Select, because is away from. Now,, and are part of the tree. inally, is selected because is away from., and are part of the minimum cost spanning tree. igure 5 depicts the presented minimum weight spanning progression. igure 5. Minimum Weight Spanning Tree Progression igure 6. Maximum Weight Spanning Tree The maximum weight spanning tree represents either the strongest or the weakest path depending on the category and category measures. The weakest link can be of particular significance in weighted networks (ranovetter 973). or example: n the connectivity category with capabilities measure, the larger the weight, the stronger the path. On the other hand, with risk measure, the larger the weight (higher the risk), the weaker the path. Optimization Space We agree with isner et al. that the optimization of each system does not guarantee the optimization of the overall system of systems (isner, Marciniak, and McMillan 99) owever, a SoS can be optimized by selecting the optimum tree between components depending on categories and measures for these categories. Therefore, the area between operational boundaries is an optimization space of a SoS, as illustrated in igure 7 based on calculations shown in igure 8. inding Upper Operational oundary The maximum weight spanning tree is used to define upper operational boundaries of a SoS. igure 6 depicts the maximum spanning tree found for the given SoS graph. 5 PRONS SR 2007, March 4-6, oboken, NJ, US

6 Total Weight of the Spanning Tree Number of Nodes in a Network (omponent Systems in a SoS) igure 7. Optimization Space of SoS (onnectivity ategory) # of Nodes Optimization in a Network W MinSpanTree W MaxSpanTree Space Optimization Space igure 8. Optimization Space alculation igure 8 also shows, as expected, that as the number of nodes on a network decreases the optimization space is also decreasing. The Minimum Operational oundary and the Maximum Operational oundary of the Optimization Space as shown in igure 7 are correspondingly the values of the weights of the minimum and the maximum spanning trees. These weights are defined in formula (2) and (3). The SoS optimization space and its minimum and maximum boundaries depend on the SoS components categories and measures. ategories and Measures or this paper we used defined categories for characterization of a SoS as defined in PRONS SR 2007, March 4-6, oboken, NJ, US.2 raph ppendix. While we did not define how these categories may be quantified, each category there can have a single or multiple measures, for example: vailability unctionality ntegration Supportability ompatibility The characterization of a system boundary need not ask was it good or bad, but was it the right boundary to the situation, the task and the environment. We presented a model for characterizing a SoS operation boundary, but a characterization of classification system should provide a framework through which to define and establish guidelines with attributes that ensure meaningful differentiation are recognized and addressed (Sauser, 2006). Thus any model such as the one we have presented should be organization-specific (no project or organization is the same), maintain simplicity for a universal use within an organization, and provide two fundamental functions, definition and arrangement. efinition is the determination of classes of entities that share characteristic attributes; and arrangement involves a systematic ordering of classes that expresses conceptual relationships within the overall structure (Jacob, 99). onclusion Maturity apability Security Risk tc. n this paper we introduced a new method to define SoS operational boundaries based on a characterization of a SoS and to determine the optimum connectivity path between SoS components. We proposed a model to define the combined weight of a pair of nodes and a link between them. The research suggests that by extracting the minimum and maximum spanning trees from the connected weighted graph we can 6

7 determine the operational boundaries of the SoS structure. The new method and accompanying model will be used as a framework for future research on the topic. urthermore, a practical tool to define operational boundaries of a SoS based on the proposed framework is a part of current and future research by the authors of this paper. uture Research s part of future research the proposed method will be applied to the rest of the oardman-sauser (oardman and Sauser 2006) distinguishing characteristics of SoSs, as defined in ppendix, and potential categories measures. This can serve as a basis to create a omplexity Model for a SoS, and to define its operational boundaries. ppendix presents the first draft of a omplexity Model of a SoS showing how the characteristics will translate into a defined operational boundary. The diagram in ppendix represents the possible lower and upper boundaries for each of different categories, both of a SoS and it s polar opposite that could be represented as a ninenode graph shown on igure 3. References ar-yam, Y The haracteristics and merging ehaviors System of Systems. Paper read at NS: omplex Physical, iological and Social Systems Project. oardman, J., and. Sauser System of Systems: The Meaning of of. Paper read at nternational System of Systems onference, pril 24-26, at Los ngeles,. zerwinski, T oping with the bounds: speculations on nonlinearity in militaryaffairs, RP. elaurentis, Understanding Transportation as a System of Systems esign Problem. Paper read at 43rd erospace Sciences Meeting, January 0-3, at Reno, NV. ijkstra, Note on Two Problems in onnexion with raphs. Numerische Mathematik : orogovtsev, S., and J. Mendes volution of Networks: rom iological Nets to the nternet and WWW, Oxford Press. isner,., J. Marciniak, and R. McMillan. 99. omputer-ided System of Systems (S2) ngineering. Paper read at nternational onference on Systems, Man, ybernetics, October 3-6, at harlottesville, V. ven, S raph lgorithms, omputer Science Press, Rockville, M. redman, M., and R. Tarjan ubonacci heaps and their use in improved network optimization algorithms. Journal of M ell-mann, M n nlarged oncept of Sustainability. Talk at meeting of S usiness Network. ranovetter, M The Strength of Weak Ties. merican Journal of Sociology Jacob,. 99. lassification and categorization: rawing the line, Proceedings of the 2nd SS S/R lassification Research Workshop: dvances in lassification Research Kruskal, J On the shortest spanning tree of a graph and the travelling salesman problem. Proceedings of the merican Mathematics Society Maier, M rchitecting principles of systems-of-systems. Paper read at 6th nnual nternational Symposium of the 7 PRONS SR 2007, March 4-6, oboken, NJ, US

8 nternational ounsil on Systems ngineering, at oston, M. Moret,., and. Shapiro n empirical assessment of algorithms for constructing a minimum spanning tree. MS Series in iscrete Mathematics and Theoretical omputer Science Newman, M., -L.arabási, and. Watts The Structure and ynamics of Networks, Princeton University Press. Noshita, K theorem on the expected complexity of ijkstra s shortest path algorithm. Journal of lgorithms Pastor-Satorras, R., and. Vespignani volution and Structure of the nternet: Statistical Physics pproach, ambridge University Press. Prim, R Shortest connection networks and some generalizations. ell System Technical Journal Sauser,., and J. oardman rom Prescience to mergence: Taking old of System of Systems Management. Paper read at 27th nnual National onference of merican Society for ngineering Management, October 25-28, at untsville, L. Sauser, Toward Mission ssurance: ramework for Systems ngineering Management. Systems ngineering 9(3) Shenhar, New Systems ngineering Taxonomy. Paper read at 4th nternational Symposium of the National ouncil on System ngineering. Strogatz, S xploring complex networks. Nature iography lex orod received a.s. in nformation Systems and an M.S. in Telecommunications degrees from Pace University. Prior to the graduate studies he held a Research nalyst position at Salomon Smith arney. urrently he is a Research ssistant as well as a Ph. student of Systems ngineering and ngineering Management (SM) at Stevens nstitute of Technology. rian Sauser holds a.s. from Texas &M University in griculture evelopment with an emphasis in orticulture Technology, a M.S. from Rutgers University in ioresource ngineering, and a Ph.. from Stevens nstitute of Technology in Technology Management. e has worked in government, industry, and academia for more than 2 years as both a researcher/engineer and director of programs. e is currently an ssistant Professor at Stevens nstitute of Technology in the Systems ngineering and ngineering Management (SM) epartment and irector of the SM Systems ngineering Management Program. 8 PRONS SR 2007, March 4-6, oboken, NJ, US

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10 ppendix. oundation and escriptions of the SoS haracteristics 0 PRONS SR 2007, March 4-6, oboken, NJ, US