Lattice investment projects support process model with corruption
LLattice investment projects support process modelwith corruption
O.A. Malafeyev ∗ and S.A. Nemnyugin † Saint-Petersburg State University, Russia
Abstract
Lattice investment projects support process model with corruptionis formulated and analyzed. The model is based on the Ising latticemodel of ferromagnetic but takes deal with the social phenomenon.Set of corruption agents is considered. It is supposed that agents areplaced in sites of the lattice. Agents take decision about participationin corruption activity at discrete moments of time. The decision maylead to profit or to loss. It depends on prehistory of the system. Profitand its dynamics are defined by stochastic Markov process. Stochasticnature of the process models influence of external and individual fac-tors on agents profits. The model is formulated algorithmically and isstudied by means of computer simulation. Numerical results are givenwhich demonstrate different asymptotic state of a corruption networkfor various conditions of simulation.
Keywords: corruption, lattice model, Monte-Carlo method
Mathematics Subject Classification (2010):
Stochastic methods are widely used in studies of complex systems which op-erate in complex environment under influence of many unpredictable factors[36],[40].Both dynamics and equilibrium states of social-economic systems arestudied by various methods and approaches including models similar to thosewhich are used to study physical phenomena [50]. Stochastic models andother kinds of models are also used [34],[19]. The problem of corruptionattracts significant attention of researchers because corruption phenomena ∗ [email protected] † [email protected] a r X i v : . [ q -f i n . M F ] J a n ay have significant social-economic consequences. Various approaches tothe problem are presented in [1–30,40–49].In the present article the model is proposed to study possible corruptionstates of a set of agents. These kind of phenomena is caused by socialrelations of agents and state of the economics and/or public administration.In Section 1 model is formulated. In Section 2 numerical results are given. In the model social relations are presented by links which connect nodesrepresenting agents of corruption activity. Length of all links is same. Wesuppose that both intensity of relations is equal for all agent-to-agent rela-tions and coordination number which characterizes number of social links ofan agent is fixed. Such conditions correspond to the Cartesian lattice. Forgiven topology of the lattice the coordination number defines dimensionalityof the lattice. It is supposed also that agents of corruption activity may bein one of q states.Social dynamic in the model is supposed to be Markovian. Time is dis-cretized and divided onto equal intervals. Every discrete moment randomlyor regularly chosen agent takes decision about change of his state.In simplest case q = 2. Let us suppose that one state corresponds tothe decision to participate in corruption activity and second one to nonparticipation in this activity. Transition to the first state corresponds toagent’s profit. Second kind of transition leads to his loss. Every decisionis taken with accounting of agent interactions with other participants ofcorruption activity. Effect of boundary is excluded by imposing periodicboundary conditions.Topology of two-dimensional model and social relations are presentedon Fig. 1. In the article three-dimensional lattice model is studied so Fig. 1may be considered as two-dimensional section of the models lattice alongone of its planes.Let c i — is the state of i -th agent and c = { c , c , ..., c M } — is the statevector of the social system. M — is number of agents. Dynamic of thesystem is defined by the function: W ( c ) = M (cid:88) i =1 φ i ( c ) , where functions φ i ( c ) describe interactions between agents. Dynamics isalso defined by the procedure of decision making about change of the stateof i -th agent. Algorithm of modelling of decision making is analogous toMetropolis algorithm, which is widely used for numerical study of latticesystems in physics [50]. 2igure 1: Topology of two-dimensional model and social relations betweenagents of corruption activity.As an example following definition of the function φ may be given: φ i ( c ) = − J (cid:88) j = { S,W,E,N,U,D } c i c j , where { S, W, E, N, U, D } — are nearest neighbours of the i -th agent (six forthe Cartesian three-dimensional lattice), J — is constant, which is one ofthe model’s parameters and describes intensity of social interactions betweenagents. In more complicated models set J = { J , J , ..., J MM } may be usedas well as other definition of the function φ : φ i ( c ) = − (cid:88) j = { S,W,E,N,U,D } J ij c i c j . This case corresponds to the spin glass model. Various kinds of lattices maybe also used. Here β — is the model’s parameter. Its inverse value β = T characterizes social-economic activity. Less values of T correspond to morestagnation of the social-economic activity.In according to the Metropolis algorithm initial state of the system maybe chosen randomly then at every step an agent ( k ) is taken randomly andhis state is replaced by the trial one: c k ← ( c k ) trial . Replacement of the previous state by the trial one leads to the change ofthe value of the function W :∆ W ( c ) = W ( c trial ) − W ( c ). If ∆ W ( c ) <
0, then trial state must be taken as the next state of theMarkov chain. If ∆ W ( c ) >
0, then pseudorandom number ξ is generated3hich belongs to the interval [0, 1] and if exp ( − β ∆ W ) ≥ ξ , then trial stateis taken as the next state of the Markov chain otherwise the state is notchanged.From the point of view of social dynamics this algorithm may be inter-preted as follows. At discrete time moment some agent feels temptation totake part in corruption activity. He tries to estimate possible profit andrisks of this decision. Decision is based both on external factors and ownlife experience which in part is defined by random facts. Metropolis acceptand rejection rule is one of possible models of social dynamics. Its reliabilityshould be verified by the statistical data.Despite of the initial state of the system stochastic evolution governed byMetropolis algorithm converges to the unique final distribution. Modellingof stochastic dynamics of a system let us estimate influence of characteris-tics of social interactions and state of economic or public administration ontransition processes and setting of the stationary state of corruption activity.Total profit of agents from taking part in corruption activity may bedefined as follows: U = (cid:88) j : c j > c j , It characterizes loss of economic due to corruption activity. Average sizeof clusters of similar values of state variables are characteristics of scale ofcorruption.
Numerical simulations have been performed for the simple model of cor-ruption activity on cartesian three-dimensional lattice with size 60x60x60and two possible local states with periodic boundary conditions. Numberof Markov steps was taken to be 10 . This value guarantees for the latticeof a given size that transition to the final stationary distribution is nearlycompleted. Examples of configurations for low and high economic activitycases are given on Fig. 2 and Fig. 3) respectively. Figures demonstrate two-dimensional sections of the three-dimensional lattice. It may be seen thatin the case of low social activity dense clusters of various sizes are appearedand corruption has tendency to be widespread.A cluster is defined as the set of connected agents which made decisionto take part in the corruption activity. As a rule it doesn’t contains agentsof other kind.High social-economic activity leads to subdivision of clusters into smallerparts interleaving with groups of agents who don’t participate in the corrup-tion activity. Moreover second case show faster dynamics with appearingand disappearing of small clusters. 4igure 2: Examples of configurations for the lattice model and the case oflow activity (T = 0.5)Figure 3: Examples of configurations for the lattice model and the case ofhigh activity (a) T= 4.0; (b) T = 5.0 In the article simple investment projects support process model with corrup-tion is proposed. It is formulated as three-dimensional lattice model withnearest neighbours interactions and periodic boundary conditions. Due tothis model agents of corruption activity take decisions about participationin corruption activity at discrete moments of time. Qualitative results ofsimulation seems to be reasonable but should be verified by empirical data.The study was performed using computational resources provided bythe Resource Physics Educational Centre of the Research park of Saint-Petersburg State University.
The work is partly supported by work RFBR No. 18-01-00796.5 eferences [1] Malafeev O. A., Zubova A. F. Mathematical and computer mod-eling of socio-economic systems at the level of multi-agent interaction(Introduction to the problems of equilibrium, stability and reliability).SPb.:Publishing SPbSU, 2006. P. 1006.[2] Malafeev O. A., Sosnina V. V. Management model process ofcooperative three-agent interaction. Problems of mechanics and control:nonlinear dynamical systems, 2007. №
39, p. 131-144.[3] Grigorieva K. V., Malafeev O. A. The dynamic process of cooperativeinteraction in the multi-criteria (multi-agent) postman task. The Bulletinof Civil Engineers, 2011. №
1, p. 150-156.[4] Malafeev O. A. Managed conflict systems. Saint-Petersburg, 2000.P. 280.[5] Malafeev O. A., Kolokoltsov V. N. Understanding game theory.New Jersey, 2010. P. 286.[6] Malafeev O. A., Zenovich O. S., Sevek V. K. Multi-agentinteraction in the dynamic problem of managing construction projects.Economic Revival of Russia, 2012. №
1, p. 124-131.[7] Drozdova I. V., Malafeev O. A., Parshina L. G. Efficiency ofoptions for the reconstruction of urban housing. Economic Revival ofRussia, 2008. №
3, p. 63-67.[8] Malafeev O. A., Pachar O. V.Dynamic, non-stationary task ofinvesting projects in a competitive environment. Problems of mechanicsand control: Nonlinear dynamical systems, 2009. №
41, p. 103-108.[9] Gordeev D. A., Malafeev O. A., Titova N. D. Probabilisticand deterministic model of the influence factors on the activities of theorganization to innovate. Economic Revival of Russia, 2011. №
1, p. 73-82.[10] Grigorieva K. V., Ivanov A. S., Malafeev O. A. Static coalitionmodel of investment of innovative projects. Economic Revival of Russia,2011. №
4, p. 90-98.[11] Malafeev O. A., Chernych K. S. Mathematical modeling of thecompany’s development. Economic Revival of Russia, 2004. № , p. 60-66.[12] Gordeev D. A., Malafeev O. A., Titova N. D. Stochastic model ofdecision-making about bringing to market an innovative product. Heraldof civil engineers, 2011. №
2, p. 161-166.613] Kolokoltsov V. N., Malafeev O. A. Mathematical modeling ofsystems of competition and cooperation (game theory for all), textbook.Saint-Petersburg, 2012. P. 624.[14] Gricai K. N., Malafeev O. A. The problem of competitive man-agement in the model of multi-agent interaction of the auction type.Problems of mechanics and control: nonlinear dynamical systems, 2007. №
39, p. 36-45.[15] Akulenkova I. V., Drozdov G. D., Malafeev O. A. Prob-lems of reconstruction of housing and communal services of a megacity,monograph. Ministry of Education and Science of the Russian Federa-tion, Federal Agency for Education, St. Petersburg State University ofService and Economics, Saint-Petersburg, 2007. P. 187.[16] Parfenov A. P., Malafeev O. A. Equilibrium and compromise con-trol in network models of multi-agent interaction. Problems of mechanicsand control: nonlinear dynamical systems, 2007. № №
10 (69), p. 14-17.[20] Malafeev O. A., Gritsai K. N., Competitive management inauction models. Problems of mechanics and control: nonlinear dynamicalsystems, 2004. №
36, p. 74-82.[21] Ershova T. A., Malafeev O. A. Conflict management in the modelof entering the market. Problems of mechanics and control: nonlineardynamical systems, 2004. №
36, p. 19-27.[22] Grigorieva K. V., Malafeev O. A. Methods for solving the dynamicmulticriteria mailman problem. Herald of civil engineers, 2011. № №
4, p. 41-46.[26] Malafeev O. A., Muraviev A. I. Mathematical models of conflictsituations and their resolution. Volume 1. General theory and all sortsof information, Saint-Petersburg, 2000. P. 283.[27] Malafeev O. A., Drozdov G. D. Modeling processes in the systemof urban construction management. Volume 1, Saint-Petersburg, 2001.P. 401.[28] Malafeev O. A., Koroleva O. A. The model of corruption incontracting. Proceedings Edited by Smirnov N. V., Tamasyan G.Sh. In the collection: Management processes and persistence. Proceed-ings of the XXXIX International Scientific Conference of Post-GraduateStudents and Students, 2008. P. 446-449.[29] Malafeev O. A., Muraviev A. I. Modeling of conflict situations insocio-economic systems. Saint-Petersburg, 1998. P. 317.[30] Drozdov G. D., Malafeev O. A. Modeling of multi-agent interactionof insurance processes. Monograph. Ministry of Education and Scienceof the Russian Federation, St. Petersburg State University of Service andEconomics, Saint-Petersburg, 2010.[31] Zubova A. F., Malafeev O. A. The Lyapunov stability and oscil-lation in economic models. Saint-Petersburg, 2001. P. 101.[32] Bure V. M., Malafeev O. A. Agreed strategy in the repeatedfinal games of N persons. Bulletin of St. Petersburg University. Series1. Mathematics. Mechanics. Astronomy. Saint-Petersburg, 1995. № №
5, p. 25-36.[34] Malafeev O. A., Redinskikh N. D., Alferov G. V. Electriccircuits analogies in economics modeling: corruption networks. Proceed-ings Edited by: Egorov N. V., Ovsyannikov D. A., Veremey E.I. In proceeding: 2nd International Conference on Emission Electronics(ICEE) Selected papers, 2014. P. 28-32.835] Malafeev O. A. Conflict-driven processes with many participants.The dissertation author’s abstract on competition of a scientific degreeof physical and mathematical sciencesh, Leningrad, 1987.[36] Malafeev O. A., Neverova E. G., Nemnyugin S. A., AlferovG. V. Proceedings Edited by: Egorov N. V., Ovsyannikov D. A.,Veremey E. I. Multicriteria model of laser radiation control. In pro-ceedings: 2nd International Conference on Emission Electronics (ICEE)Selected papers, 2014. P. 33-37.[37] Kolokoltsov V. N., Malafeev O. A. Dynamic competitive systems ofmulti-agent interaction and their asymptotic behavior (Part II). Heraldof civil engineers, 2011. №
1, p. 134-145.[38] Malafeev O. A. Stability of solutions to multicriteria optimizationproblems and conflict-controlled dynamic processes. Saint-Petersburg,1990.[39] Malafeev O. A., Redinskikh N. D., Alferov G. V., SmirnovaT. E. Corruption in the models of the first price auction. In the col-lection: Management in marine and aerospace systems (UMAS-2014)7th Russian multiconference on problems of management: Conferencematerials. GNC RF OAO “Concern” CNII “Electrical Appliance”, 2014.P. 141-146.[40] Malafeev O. A., Nemnyugin S. A., Alferov G. V. Chargedperticles beam focusing with uncontrollable changing parameters. Pro-ceedings Edited by: Egorov N. V., Ovsyannikov D. A., Veremey E.I. In proceedings: 2nd International Conference on Emission Electronics(ICEE) Selected papers, 2014. P. 25-27.[41] Malafeev O. A., Redinskikh N. D., Smirnova T. E. Networkmodel of investing projects with corruption. Management processes andsustainability, 2015. V. 2, №
1, p. 659-664.[42] Pichugin Yu. A., Malafeev O. A. On assessing the risk of bankruptcyof a firm. In the book: Dynamic Systems: Steadiness, Management,Optimization, Theses of reports, 2015. P. 204-206.[43] Alferov G. V., Malafeev O. A., Maltseva A. S. Game-Theoreticmodel of inspection by anti-corruption group. In proceeding: AIP Con-ference Proceedings, 2015. P. 450009.[44] Malafeev O. A., Redinskikh N. D., Gerchiu A. L. Opti-mization model for the location of corrupt officials in the network. Inthe book: the construction and operation of energy-efficient buildings9theory and practice taking into account the corruption factor) (Pas-sivehouse) Kolchedantsev L. M., Legalov I. N., Badin G. M.,Malafeev O. A., Aleksandrov E. E., Gerchiu A. L., Vasilev U.G., collective monograph. Borovichi, 2015. P. 128-140.[45] Malafeev O. A., Redinskikh N. D., Gerchiu A. L. Projectinvestment model with possible corruption. In the book: the constructionand operation of energy-efficient buildings (theory and practice takinginto account the corruption factor) (Passivehouse) Kolchedantsev L.M., Legalov I. N., Badin G. M., Malafeev O. A., Aleksandrov E.E., Gerchiu A. L., Vasilev U. G., collective monograph. Borovichi,2015. P. 140-146.[46] Malafeev O. A., Chernych K. S. Mathematical modeling of thecompany’s development. Economic Revival of Russia, 2005. №№