Google matrix of business process management
aa r X i v : . [ c s . C Y ] S e p EPJ manuscript No. (will be inserted by the editor)
Google matrix of business process management
M. Abel and D.L. Shepelyansky Department of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Str 24, D-14476, Potsdam-Golm, Germany Laboratoire de Physique Th´eorique du CNRS, IRSAMC, Universit´e de Toulouse, UPS, F-31062 Toulouse, FranceReceived: September 14, 2010
Abstract.
Development of efficient business process models and determination of their characteristic prop-erties are subject of intense interdisciplinary research. Here, we consider a business process model as adirected graph. Its nodes correspond to the units identified by the modeler and the link direction indicatesthe causal dependencies between units. It is of primary interest to obtain the stationary flow on such adirected graph, which corresponds to the steady-state of a firm during the business process. Following theideas developed recently for the World Wide Web, we construct the Google matrix for our business processmodel and analyze its spectral properties. The importance of nodes is characterized by PageRank and re-cently proposed CheiRank and 2DRank, respectively. The results show that this two-dimensional rankinggives a significant information about the influence and communication properties of business model units.We argue that the Google matrix method, described here, provides a new efficient tool helping companiesto make their decisions on how to evolve in the exceedingly dynamic global market.
PACS.
Business process models are dynamical systems that de-scribe the interdependencies of functional units, or com-ponents, on a micro- or macroeconomic level. They depictthe way a company works and eventually makes moneywith the strategy it uses. The efficiency of a model isprimarily determined by the help a model can give forstrategic decisions, e.g. if a reorientation of products ormarketing is needed due to changes in the market or op-portunities because of technological developments (see e.g.[1,2] and Refs. therein).The building of a business model is a complicated task,because all important units in the company value pro-duction must be identified and properly linked at a cer-tain level of modeling. This involves a cancellation of non-important unit, which might be even harder. What mod-elers do further is a qualitative identification if a unit posi-tively or negatively stimulates a linked one (amplificationor damping, respectively). This yields a directed graph,where the units of the model are linked and the direc-tion reflects causality. The next step towards quantitativemodeling is the prescription of a functional dependenceof the units, which is basically a very heuristic proce-dure. Clearly, the functions have to be nonlinear, becausea growth to plus/minus infinity is not allowed, so typicalfunctions are of sigmoid-type, on the other hand minimalmodels are of predator-prey type, well known from biol- ogy. This reflects the modern point of view of a companyas a quasi-organic, dynamical system.In this work we introduce and analyze the Google Busi-ness Process Model (GBPM) of a real consulting company[3] whose major product is of intellectual nature. The de-tailed description of the original dynamical model can befound in [3] and thus we do not present it here. The modeldescribes a dynamical workflow propagation (see e.g. [4,5])which is simulated by certain dynamical equations.In our approach we trace parallels and similarities be-tween the directed graph of this model and the Googlematrix approach used for the ranking of the World WideWeb (WWW) [6,7,8]. Thus, we investigate only the modelgraph and do not enter the subject of dynamical simu-lations, because we want to reveal the underlying struc-ture of the stationary state of the model without usingthe quite heuristic functional dependencies which need tobe further supported by statistical analysis and measure-ment. This is not to say that the latter is a wrong ap-proach, however the determination of the stationary den-sity by the application of the PageRank algorithm for theGoogle matrix, which is a variant of Frobenius–Perronoperator [7], is a very powerful and well–established tech-nique which gives fundamental results on the network with-out solving the dynamical equations and using a vast studyof parameter variations.Indeed, the construction of the Google matrix for theWWW and the determination of the stationary probabil-ity distribution over WWW network via the PageRank
M.Abel and D.L.Shepelyansky: Google matrix of business management algorithm has been proposed by Brin and Page [6] andby now it became a powerful tool for classification of theWWW nodes (see e.g. [7,8,9,10]). The approach based onthe Google matrix construction for a directed network israther general and finds applications for various types ofnetworks including university WWW networks [11], Ulamnetworks of dynamical maps [12,13], brain neural networks[14], procedure call network of Linux kernel [15,16], hyper-link network of Wikipedia articles [17]. PageRank findsalso applications in blog analysis [18], citation network ofPhys. Rev. [19,20], and food flow network between speciesin ecosystems [21].In this work we extend this approach to the networkof business management. How is the model built? Basi-cally, one has identified major components of the com-pany, which are refined in their dynamics in respectivesubcomponents. By construction, the model is hierarchi-cal, but links between components can be set according tothe needs of the modeler. We only mention here the com-ponents and the nodes in the top component: managers,consultants,... ; subcomponents are: top, consultants, prod-ucts, proposals, customers, .... . The full list of nodes andlinks between them are given in Appendix. Dependingon the business process, one of the nodes is the mostimportant one, followed by others. This is the value ofour method: we identify without any bias the most impor-tant components in a model. This provides an extremelyhelpful information. If these components are not the oneswished by the shareholders or management, respectively,the model has to be changed and adapted. Since the com-putation is not very costly this gives a tool to simulatesmall changes, e.g. by linking different nodes, and study-ing their effect on the business process model. We considerthe GBPM as a first step in the application of the Googlematrix analysis to the business process management. Nextsteps should extend this approach and take into accountactual workflow between nodes inside a company[4,5] .Our network is small in comparison of typical applica-tions of Google Matrix, like the WWW [9,10], Linux ker-nel network [15,16] or Wikipedia network [17]. It consistsof 175 nodes only and is graphically displayed in Fig. 1.This size is comparable with the one of food network inecosystems [21]. Our purpose is an elementary study ofthe network properties using the spectral characteristicsof the Google matrix, PageRank and recently introducedCheiRank and 2DRank such that the order 10 is suffi-cient; the latter ranking algorithms are explained in detailbelow, Most big business models are proprietary (for un-derstandable reasons), and an application of the Googlematrix method is straightforward.Let us have a look on the network in terms of con-nectivity: the distribution of ingoing and outgoing links isshown in Fig. 2. Of course, with only one decade availableit is useless to try to identify exact scaling behaviour; nev-ertheless the global distribution is compatible with powerlaw scaling f ( d ) ∼ d − ν at ν ≈
3. The exponent ν = 3is not so far from the exponent ν = 2 . .
16 75 432119 12294 9291 9 12010113 103113 810111516 19140 141158153145 110 12141718132 143 133161156148 10920 212273 232932 24 2528 262733 30313435 3637 7038 4039 41 45 4342 4446474849 5051 52 53545556 59575860 7284 61626364 65666768 6974 777175 7812785 7679 8081 82838786 88 125 89 9093 104112 98 107106 9597 96100 99102105108 111114115116117 118121123124 126 129137138128130 131 134135136139 142144149 152 146147150 151166 154155157162 159160 163164165 167 170169 171168 174172173175
Fig. 1.
Google Business Process Model with links taken from[3]. The network is structured into several subgraphs reflectingthe functionality of the model. The names (or meaning) of thenodes and links between them are listed in the Appendix. generic scaling of business models in the future for net-works of larger size.
The Google matrix G underlies the determination of Page-Rank [6], which is a tool used by virtually every Internetuser when issuing an Internet search for some keywords.This approach gives a powerful and general way to analyzenetworks. For the construction of the Google matrix weuse the procedure described in [6,7]: G ij = αS ij + (1 − α ) /N , (1)where S ij is the normalized adjacency matrix of the graph.The elements of the adjacency matrix are zero (if thereis no link) or one (if there is a link). Due to the nor-malization the sum of all elements inside one column isequal to unity. Columns with zeros only are replaced by(1 /N, . . . , /N ), with N being the network size. Because .Abel and D.L.Shepelyansky: Google matrix of business management 3 degree110100 f r equen cy ingoing degreeoutgoing degreey=x^{-3} Fig. 2. (Color online) Distribution of ingoing (black points)and outgoing (blue points) links. An approximate global powerlaw scaling with the exponent 3 is shown by the straight redline. it is a full stochastic matrix of a Markov chain, the ma-trix S has N eigenvalues λ i , i = 1 , . . . , N which are gen-erally complex. In agreement with the Perron-Frobeniustheorem (see e.g. [7]) the largest eigenvalue is λ = 1.The damping parameter α denotes the possibility for arandom surfer on the graph to jump to any other node.Its effect is to bound away the eigenvalues with abso-lute value smaller than one: | λ i | ≤ α < i > α = 0 .
85, however this choice can be varied without es-sential impact on the results presented below. The righteigenvectors, ψ i , are defined by G ψ i = λ i ψ i , cf. [7,11].The PageRank vector is the one with λ = 1, and since G is a Frobenius-Perron operator, the corresponding righteigenvector, ψ = ( P (1) , . . . , P ( N )) T gives the stationaryprobability density P ( i ) that a random surfer is found atsite i with P i P ( i ) = 1. Once it is found, the nodes aresorted according to decreasing P ( i ), the node rank in thisindex, K ( i ) corresponds to its relevance.Other eigenvalues correspond to non-stationary, decay-ing modes. They are of transient nature and may play animportant role in non-stationary considerations, becausethey may live for a long time before dying out. This is,however, not the focus of this work. In a nutshell, the procedure uses the idea that a nodeis not only relevant if it is highly linked. One has also totake into account the relevance of the nodes pointing to it.Since this is an iterative procedure, the PageRank vectorcan be easily computed by the so–called power-iterationusing consecutive multiplication of initially random vectoron the Google matrix [7]. Of course, this vector is the mostimportant one, because it represents the stationary distri-bution on the graph. The relaxation process to the steady-state given by the PageRank is affected by the eigenmodeswith | λ | close to α . It is known that for the WWW there -0.5 0.0 0.5 1.0-0.4-0.20.00.20.4 λ -0.5 0.0 0.5 1.0-0.4-0.20.00.20.4 λ Fig. 3. (Color online) Distributions of eigenvalues λ of theGoogle matrix at α = 0 .
85 in the complex plane for matrix G (left panel) and matrix G ∗ with inverted link directions (rightpanel). are many eigenvalues which are close or even equal to α (see e.g. [7,11]). The spectrum of the Google matrix G ofthe GBPM is shown in the left panel of Fig. 3. The eigen-value next after λ = 1 is λ = 0 .
706 and other eigenvalueshave | λ | < .
52. There are only about 14% of eigenvalueswith | λ | > . N in analogy with the Linux kernelnetwork analyzed in [15,16]. The spectrum of the Googlematrix G ∗ , obtained from the network with the inverseddirection of links, is shown in the right panel of Fig. 3, itscharacteristics are similar to those of matrix G .The PageRank probability P ( i ) for our business modelis shown in Fig. 4 (top panel) as a function of rank K ( i ).Surprisingly, there is no dominant node, which means thatthis company is quite democratic - in terms of relevance.The first five nodes are: Identified Contact Loss (33), Iden-tified Contacts (32), Projects (5), Consultants (2), Deliv-ery Project Completion (87) . The numbers in brackets de-note the node indices, cf. the Appendix. Managers (nodeindex 1) do not appear before rank 18. This is quite sur-prising, since the management is expected to be at leastamong top ten positions. How can one understand thatbehaviour? The management plays typically the role of co-ordinating projects and keeping all together , which meansthat they decide which points are most important andhave many outgoing links related to orders given to oth-ers. However, the PageRank is proportional in average tothe number of ingoing links [7]. This implies the man-agement units are not most important according to thePageRank since they do not have a large number of ingo-ing links (not many units give order to managers). In theconsidered model of a consulting company the most rele-vant units are the customers, or contacts. Without them,no business is made, especially for consulting. The firsttwo ranks can be explained by this. The following ranksare
Projects and
Consultants . Of course, without goodprojects and correspondingly good workers the firm willdie, so this is of vital relevance. Rank 5 again involvesprojects, this time their delivery. This means that in thismodel the way the projects are completed is given a highimportance. This might not be necessarily true in all cases,however for the model of the firm under consideration itis. We recognize that in this view the result makes perfect
M.Abel and D.L.Shepelyansky: Google matrix of business management sense: customers, products and consultants are the mostrelevant units in the model of a consulting firm. Such afirm can only survive when its consultants are top leveland its products are alike - and if there are customers.The management is responsible only to get the firm run-ning well. This result may be surprising, but reveals thepower of the method. This means as well that the mostattention for refinement of the model should be put on thetop nodes given above. Nevertheless, one expects that themanagement plays somehow a very influential role.It is interesting to note that a similar situation takesplace for the procedure call network of the Linux kernelas it was shown in [15]. Indeed, for this network the Page-Rank gives at the top procedures which are often pointedon but which are not so much important for the codefunctionality. Thus it was proposed [15] to characterizethe network also by the PageRank of the Google matrixobtained from the network with inversed link directions.The rank P ∗ ( i ) of this inversed matrix G ∗ , named as theCheiRank [17], places on first positions rather influentialcode procedures. Hence, it is natural to use the CheiRankalso for our model of business process management.And indeed, using the CheiRank, introduced in [15] weobtain an adequate result. It corresponds to the station-ary distribution, P ∗ ( i ), of the inverted flow, or the infor-mation returned from the nodes to their precedent ones.Thus, it describes the influence or communication rankingof the nodes. Again, the eigenvector with the eigenvalue1 is computed and sorted according to the magnitudeof the entries. This yields a new rank, K ∗ ( i ), the men-tioned CheiRank. The result of the computation of P ∗ ( i )vs. K ∗ ( i ) is displayed in Fig. 4. (bottom panel). Here, wecan also give a tentative scaling P ∗ ( i ) ∼ K / ( ν − whichmust be compared and verified, respectively, with otherbusiness models of larger size. While the distribution of P ( i ) ∼ K / ( ν − is proportional to the distribution of in-going links, the distribution of P ∗ ( i ) is proportional to thedistribution of outgoing links (see e.g. [7,8,11,15]). Due toa small size of our network we do not try to use differ-ent values of ν for ingoing and outgoing links and for P and P ∗ respectively. According to the CheiRank the topnodes are: Principals (1), Projects (5), Consultants (2),Customers (6), Contacts (7) . The management now hasclearly first position in the ranking which is fully logical,since any management decision influences the whole com-pany, while the management is not necessarily the most important component, as explained above.Following [15] we also use the joint distribution ofnodes in the plane of probabilities ( P ( i ) , P ∗ ( i )) of Page-Rank and CheiRank shown in Fig. 5. That way, we seeboth ranks at once and can decide which emphasis to put,defining importance in a new way. In this sense, the mostimportant nodes are indicated in Fig. 5. The distribu-tion of all nodes in the plane of PageRank and CheiRank( K, K ∗ ) is shown in Fig. 6. In the plane ( K, K ∗ ) the mostimportant nodes are those with the smallest values of K and K ∗ . The zoom of this region of the plane is shown inFig. 7. K P ⋅ − ⋅ − K − K* P * ⋅ − ⋅ − ⋅ − K − Fig. 4.
Top panel: probability of PageRank vector P ( i ) as afunction of PagRank K ( i ) in log-log scale. Bottom panel: prob-ability of CheiRank vector P ∗ ( i ) in log-log scale. The straightlines show the approximate power law dependence with theslope 1 / ( ν −
1) = 1 /
2, corresponding the the average slope ν = 3 shown in Fig. 2. Of course, nodes might be both relevant (well-known)and influential (communicative). This can be character-ized by the correlator κ between PageRank and CheiRankwhich is defined as κ = N X i P ( i ) P ∗ ( i ) − . (2)For the WWW university networks [15] and Wikipedianetwork [17] it was found that the correlator is ratherlarge with κ ≈ κ ≈ − . ≪
1. For the GBPMwe have κ = 0 .
164 showing that there is practically nocorrelations between nodes with large number of outgo-ing and ingoing links. Thus the GBPM network has moresimilarities with the Linux kernel network in contrast tothe WWW and Wikipedia networks which are character-ized by high correlations between nodes which are highlyknown (high PageRank) and highly communicative (highCheiRank). .Abel and D.L.Shepelyansky: Google matrix of business management 5 . . . . . . P P * Principals ProjectsConsultantsCustomers Id. contacts loss
Fig. 5. (Color online) Distribution of nodes in the planeof probabilities of PageRank P ( i ) and CheiRank P ∗ ( i ). Themarked nodes illustrate the first four nodes in Chei rank (Prin-cipals, Projects, Consultants, Customers), and the top node inPageRank (Identified Contacts Loss) K K * Fig. 6. (Color online) Distribution of nodes in the plane ofPageRank K and CheiRank K ∗ , size of circles and their color isproportional to their listing node index with large radius (redcolor) for small index and small radius (blue-ros´e) for largeindex. With the appearance of CheiRank all nodes are nowdistributed in a two-dimensional plane (see Figs. 5,6,7).How can one combine both rankings in a way to find nodeswhich are both very relevant and influential? There aremany ways to find such a single-valued one-dimensionalranking which combines K and K ∗ : one can think of thedistance ( K + K ∗ ), or the absolute value, or some othercombination of K and K ∗ . Since P ( K ) and P ∗ ( K ∗ ) aremonotonic functions the plane ( K, K ∗ ) is mapped into( P, P ∗ ) plane in a unique way.A convenient way to order all nodes of the two-dimen-sional plane on a one-dimensional line was proposed in [17] K K * Fig. 7.
Zoom of the distribution of nodes in the plane of Page-Rank K and CheiRank K ∗ in the region of small K, K ∗ values.Numbers near circles give the listing node index, grayness isproportional to 2DRank K with black for minimum and lightgray for maximum K (see Appendix). K K * ( 2 , 2 ) ( 3 , 119 ) ( 4 , 1 ) ( 5 , 48 )( 6 , 73 )( 7 , 54 ) ( 8 , 3 )( 9 , 21 )( 10 , 28 )( 1 , 5 ) Fig. 8.
Illustration of the 2DRank algorithm to find rank K which combines PageRank K and CheiRank K ∗ . Specific nodesare drawn in the ( K, K ∗ ) plane when crawling through thesquares, indicated by the grey lines, from small to large ( K, K ∗ )the nodes are labeled by K ; numbers in brackets ( K i ) , i )give the value of found 2DRank K and the values of listingnode index i . One recognizes that at most 2 nodes can be foundon a square edge, and some edges might be empty. for Wikipedia articles being named 2DRank K . This rankis described by the algorithm presented below; it is dubbed2DRank K , since it combines the two ranks discussedabove. Remember that a ranking is basically a list of pairs(rank and nodes index), in our case K , i , or simply K ( i ).By K , we also use this ordering of nodes by the following,quite intuitive criterion: we look progressively if a point M.Abel and D.L.Shepelyansky: Google matrix of business management ( K, K ∗ ) lies on the square j × j , where j is a runningindex starting at 1. Since the ordering is unique, thereare only two possibilities for this to occur: either K = j or K ∗ = j . It may happen, that neither K nor K ∗ lieson the square, then one increases j by one and comparesagain with ( K, K ∗ ). The initial K list is empty. E.g. ifthere is no point with K = 1 and K ∗ = 1, then the firstsquare 1 × × j = 1, then welook if K = j , if yes, i ( K, K ∗ ) is determined and added tothe list K ( i ) whose own running index is increased; thenwe apply this procedure to K ∗ : if K ∗ = j , the node index i ( K, K ∗ ) is determined and added to the list K ( i ). Sincethere are no more points to check, we step from j to j + 1.The algorithm is finished if all nodes i have been visited.We can deliberately choose if we first look for K or K ∗ (we have chosen first K ). The procedure is illustrated forthe first ten nodes in K2 ranking in Fig. 8.According to this 2DRank algorithm we find for thefirst five nodes in 2DRank K : Projects (5), Consultants(2), Hire Rate (119), Principals (1), Required DeliveryProposal Effort (48) . The principals are still not the mostrelevant node, but obviously this ranking gives a quitebalanced characterization of the business process manage-ment under consideration.Top 30 nodes ordered according to PageRank, CheiRankand 2DRank are given in Appendix. Ranking of all nodesis available at the website [22].
We have presented a powerful method which quantita-tively describes the business process management in termsof the Google matrix, its eigenvectors and eigenvalues. Theapplication of the method yields the stationary distribu-tion on the directed graph which describes the businessprocess of a concrete company in the frame of our GBPM.Our results show that the importance and influence of theunits of business process are well characterized by two-dimensional ranking in the plane defined by PageRank andCheiRank. These ranks show that certain units (e.g.
Con-tacts ) perform important tasks being highlighted by Page-Rank, while other units (e.g.
Principals ) realize influentialcommunication processes highlighted by CheiRank. Thusthe two-dimensional ranking described here establishes abroad and detailed characterization of main operationalunits of business process management. In contrast to theWWW university networks and Wikipedia network, thenetwork of GBPM has rather small correlation betweentop units of PageRank and CheiRank that stresses a clearseparation between communication and realization tasksof business process. In this respect the GBPM network ismore similar to the procedure call network of Linux kernelwhich also has small correlation between these two ranks.Of course, the approach developed here is in its ini-tial stage and more advanced business process modelingwill need weighted graphs with subgraphs for the flows ofwork, information, money, products, etc. These generaliza-tions are straightforward and can be constructed at next more advanced stage. A study of changes in the modelis quick and straightforward, such that systematic studiesof future activities of a company are now feasible with-out sometimes very heuristic equations which can be usedat a final modeling stage. But now one is relieved fromthe task to determine fine–tune parameters and equationseach time a model is changed. We expect these results tohave significant impact in econometry for the evaluationof small, middle-size and large-scale models of businessprocess management. The application to macro-economyis straightforward, and global flows might be characterizedby the GBPM procedure.
Acknowledgements
We acknowledge fruitful discussion with O. Grasl whokindly provided his model [3] to us and explained the ba-sics of business process modeling.
References
1. J.D.W.Morecroft and J.D.Sterman (Eds.)
Modeling forLearning Organizations , Productivity Press, N.Y. (1994).2. M. Weske,
Business Process Management: Comcepts, Lan-guages, Architectures , Springer, Berlin (2007).3. O. Grasl,
Business model analysis: a multi-method ap-proach , System Dynamics Society, New York (2008).4. F. Lyemann and D. Roller,
Production Workflow , PrenticeHall, Inc., Upper Saddle River, NJ (2000).5. F. Lyemann, D. Roller, and M.-T.Schmidt, IBM SystemsJournal , 198 (2002).6. S. Brin and L. Page, Computer Networks and ISDN Sys-tems , 107 (1998).7. A. M. Langville and C. D. Meyer, Google’s PageRank andbeyond: the science of search engine rankings , PrincetonUniversity Press (Princeton, 2006).8. K. Avrachenkov, D. Donato and N. Litvak (Eds.),
Al-gorithms and Models for the Web-Graph: 6th Interna-tional Workshop, WAW 2009 Barcelona, Proceedings ,Springer-Verlag, Berlin, Lecture Notes Computer Sci. , Springer, Berlin (2009).9. D. Donato, L. Laura, S. Leonardi and S. Millozzi, Eur.Phys. J. B , 239 (2004).10. G. Pandurangan, P. Raghavan and E. Upfal, InternetMath. , 1 (2005).11. O. Giraud, B. Georgeot and D.L.Shepelyansky, Phys.Rev. E , 026107 (2009); B. Georgeot, O. Giraud andD.L.Shepelyansky, Phys. Rev. E , 056109 (2010).12. D.L.Shepelyansky and O.V.Zhirov, Phys. Rev. E ,036213 (2010).13. L.Ermann and D.L.Shepelyansky, Phys. Rev E , 036221(2010).14. D.L.Shepelyansky and O.V.Zhirov, Phys. Lett. A ,3206 (2010).15. A.D.Chepelianskii, arXiv:1003.5455v1 [cs.SE] (2010).16. L.Ermann, A.D.Chepelianskii and D.L.Shepelyansky,arXiv:1005.1395[cs.CE] (2010).17. A.O.Zhirov, O.V.Zhirov and D.L.Shepelyansky,arXiv:1006.4270[cs.IR] (2010)..Abel and D.L.Shepelyansky: Google matrix of business management 718. N. Perra and S. Fortunato, Phys. Rev E , 036107 (2008).19. S. Redner, Phys. Today , 49 (2005).20. F. Radicchi, S. Fortunato, B. Markines A. Vespignani,Phys. Rev. E , 056103 (2009).21. S. Allesina and M. Pascual, Ecology Lett. , 652 (2009);PLOS AppendixList of Nodes (node number is followed by its name andcomma):1 Principals, 2 Consultants, 3 Value, 4 Products, 5Projects, 6 Customers, 7 Contacts, 8 Heads Of Branch, 9Total Principals, 10 Maximum Principal Proposal Effort,11 Maximum Principal Hiring Effort, 12 Average Prin-cipal Work Effort, 13 Maximum Principal Work Effort,14 Maximum Project Time Share, 15 Maximum ContactMaintenance Effort, 16 Maximum Product Effort, 17 Con-tact Maintenance Effort, 18 Maximum Contact Mainte-nace Time Share, 19 Maximum Principal Project Effort,20 Contacting Effort, 21 Qualified Contacts, 22 RequiredContact Maintenance Effort, 23 Qualified Contact Main-tenance Effort, 24 Qualified Contact Lifetime, 25 Maxi-mum Qualified Contacts, 26 Minimum Qualification Du-ration, 27 Qualification Fraction, 28 Contact QualificationRate, 29 Qualified Contact Loss, 30 Maximum Qualifica-tion Rate, 31 Contact Identification, 32 Identified Con-tacts, 33 Identified Contact Loss, 34 New Customer Con-tact Potential, 35 Identificaton Duration, 36 IdentifiedContact Lifetime, 37 Identification Fraction, 38 DeliveryProposal Effort, 39 New Delivery Proposal Effort, 40 De-livery Proposal Writing Effort, 41 Principal Delivery Pro-posal Effort, 42 Delivery Proposal Effort Share, 43 De-livery Proposal Closing Rate, 44 Delivery Proposal Writ-ing Rate, 45 Minimum Duration Per Delivery Proposal,46 Delivery Project Effort, 47 Effort Per Delivery Pro-posal, 48 Required Delivery Proposal Effort, 49 DeliveryLead Success Rate, 50 Delivery Proposal Effort Fraction,51 First Time Delivery Lead Success, 52 Repeat DeliveryLead Success, 53 Repeat Delivery Lead Fraction, 54 Re-peat Delivery Lead Generation, 55 Repeat Delivery Leads,56 Repeat Delivery Lead Success, 57 Repeat Delivery Pro-posals, 58 Repeat Delivery Proposal Success, 59 RepeatDelivery Lead Loss, 60 Repeat Delivery Proposal Loss,61 Delivery Project Effort, 62 Customer Delivery LeadGeneration Duration, 63 Delivery Lead Closing Duration,64 Delivery Proposal Closing Rate, 65 Lead GenerationPressure, 66 Effect Of Delivery Project Per Principal, 67Repeat Delivery Lead Success Fraction, 68 Repeat De-livery Proposal Success Fraction, 69 First Time DeliveryLead Generation Duration, 70 First Time Delivery Leads,71 First Time Delivery Proposals, 72 Delivery ProjectsWon, 73 First Time Delivery Lead Generation, 74 FirstTime Delivery Lead Success, 75 First Time Delivery Pro-posal Success, 76 First Time Delivery Lead Fraction, 77First Time Delivery LeadLoss, 78 First Time Delivery Pro-posal Loss, 79 Delivery Proposal Closing Rate, 80 Deliv- ery Lead Closing Duration, 81 First Time Delivery Pro-posal Success Fraction, 82 First Time Delivery Lead Suc-cess Fraction, 83 Average Time To Delivery Project Start,84 Delivery Project Start, 85 Active Delivery Projects,86 Delivery Project Effort, 87 Delivery Project Comple-tion, 88 Delivery Project Completion Rate, 89 PrincipalProposal Effort, 90 Active Delivery Projects, 91 DeliveryProject Per Principal, 92 Total Consulting Staff, 93 De-livery Projects Staff Needed, 94 Consultants Needed, 95Active Consulting Projects, 96 Active Solution Projects,97 Consulting Projects Staff Needed, 98 Project WorkRate Needed, 99 Consulting Project Leverage, 100 So-lution Projects Staff Needed, 101 Maximum ConsultantWork Effort, 102 Solution Project Leverage, 103 Utiliza-tion Percentage, 104 Total Project Staff Needed, 105 So-lution Projects Staff Needed, 106 Solution Project De-livery Rate, 107 Delivery Project Completion Rate, 108Average Work Rate, 109 Actual Project Delivery Rate,110 Principal Project Effort, 111 Delivery Projects StaffNeeded, 112 Consulting Project Delivery Rate, 113 Max-imum Work Rate, 114 Hiring Effort Per Hire, 115 Hir-ing Effort, 116 Consultant Target, 117 Annual Consul-tant Growth Target Percentage, 118 Fluctuation Rate, 119Hire Rate, 120 Fluctuation, 121 Maximum Leverage, 122Leverage 123 Average Hiring Duration, 124 Total Cus-tomers, 125 New Customers, 126 Mature Customers, 127Customer Acquisition, 128 Customer Maturing, 129 Cus-tomer Attrition, 130 Customer Project Conversion, 131Maturing Duration, 132 New Customer Loss, 133 Ma-ture Customer Loss, 134 Customer Lifetime, 135 Cus-tomer ErosionTime, 136 Required New Customer Main-tenance Effort, 137 Required Mature Customer Mainte-nance Effort, 138 New Customer Contact MaintenanceEffort Share, 139 New Customer Maintenance Effort PerCustomer, 140 New Customer Contact Maintenance Ef-fort, 141 Mature Customer Contact Maintenance Effort,142 Mature Customer Maintenance Effort Per Customer,143 Customer Maintenance Effort, 144 Marketable Prod-uct, 145 Product Marketing Effort, 146 Product Market-ing Effort Percentage, 147 Required Product MarketingEffort, 148 Product Marketing Rate, 149 Marketing Re-ject, 150 Development Reject Duration, 151 DevelopmentReject Fraction, 152 Standardised Product, 153 ProductStandardisation Effort, 154 Product Standardisation Ef-fort Percentage, 155 Required Product StandardisationEffort, 156 Product Standardization Rate, 157 InnovationProduct, 158 Poduct Innovation Effort, 159 Product In-novation Effort Percentage, 160 Required Product Innova-tion Effort, 161 Product Innovation Rate, 162 InnovationReject, 163 Innovation Reject Fraction, 164 InnovationReject Duration, 165 Product Lifetime, 166 Product Ob-solescence Rate, 167 Time To Standardisation, 168 Lever-age Adjustment Time, 169 Leverage Loss, 170 LeverageWin, 171 Project Leverage, 172 Time To StandardizationExcellence, 173 Maximum Project Leverage, 174 ProjectLeverage Percentage, 175 Minimum Project Leverage.
M.Abel and D.L.Shepelyansky: Google matrix of business management
List of Links (node number marked by dot is followed bynumbers of nodes on which it points to, last node numberor blanc if empty is marked by comma):1. 2 3 4 5 6 7 9 91 92 94 119 122, 2. 1 3 5 92 101 119120 122, 3. 5, 4. 5 3, 5. 1 2 3 6, 6. 5 7 1, 7. 5 1, 8. 9, 9.13, 10. 11, 11. 19 15 16 119, 12. 13, 13. 10 103, 14. 19, 15.140 141, 16. 145 153 158, 17. 16, 18. 15, 19. 110 113, 20. ,21. 22 73, 22. 23 29, 23. 29, 24. 29, 25. 28, 26. 28, 27. 28,28. 21, 29. 32, 30. 28, 31. 32, 32. 33, 33. , 34. 31, 35. 31,36. 33, 37. 31, 38. 40, 39. 38, 40. , 41. 40 45, 42. 41, 43., 44. 43, 45. 43, 46. 47, 47. 48, 48. 39, 49. 48, 50. 47, 51.49, 52. 49, 53. 54, 54. 55, 55. 56 59, 56. 57, 57. 58 60, 58.72, 59. , 60. , 61. 62, 62. 54, 63. 56 59, 64. 58 60, 65. 5473, 66. 54 73, 67. 56 59, 68. 58 60, 69. 73, 70. 74 77, 71.75 78, 72. 84, 73. 70, 74. 71, 75. 72 127, 76. 73, 77. , 78. ,79. 75 78, 80. 74 77, 81. 75 78, 82. 74 77, 83. 84, 84. 85,85. 87, 86. 87, 87. , 88. 87, 89. 41, 90. 91 93, 91. , 92. ,93. 98 104 112, 94. , 95. 97, 96. 100, 97. 104 98, 98. 109,99. 97, 100. 98 104, 101. 103 113, 102. 105, 103. , 104. 106107 112, 105. 98 104 106 107, 106. 109, 107. , 108. 98 101,109. 103 110 112, 110. , 111. 107, 112. , 113. 109 110, 114.115 119, 115. , 116. 119, 117. 116, 118. 120, 119. 2, 120. ,121. 119, 122. , 123. 119, 124. , 125. 31 124, 126. 54 124129 133 137, 127. 125, 128. 126, 129. , 130. 127, 131. 128,132. , 133. , 134. 129, 135. 132 133, 136. 132 138 140, 137.133 138 141, 138. 140 141, 139. 136, 140. 132 143, 141. 133143, 142. 137, 143. , 144. 149 156, 145. 148, 146. 145, 147.148, 148. 144, 149. , 150. 149, 151. 149 156, 152. 166, 153.156, 154. 153, 155. 156, 156. 152, 157. 148 162, 158. 161,159. 158, 160. 161, 161. 157, 162. , 163. 148 162, 164. 162,165. 166, 166. , 167. 153 169 170, 168. 169 170, 169. , 170.171, 171. 62 93 169 170 174, 172. 169 170, 173. 170 174,174. , 175. 169 174,
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