  • # graph theory used in economics

### 2020

Suppose that the vectors are strictly positive and consider one-step trajectory of the model starting at the point X and maximizing the price vector G on the set , where, as above, b = AâB, the mapping B is defined in (1) by the graph (J, G) and the matrices , and the mapping A is defined in (4) by the mapping . Main Part. The problems below may be considered on a complete graph Î³; i.e., instead of the system (J,G,) one may consider (J,J x J,). It is shown that the characteristic prices can be considered as equilibrium prices in some distribution models. Consider the trajectory X0, X1,...,. Then, as follows from Proposition 4, Q(H)=H. Networks play an important role in a wide range of economic phenomena. Such illustrations are useful in developing economic theory about the more complicated relationships among economic variables. Despite this fact, standard economic theory rarely considers economic networks explicitly in its analysis. As is known , Ð¢-step trajectory of the model is defined as a finite sequence such that (t=0,1,â¦,T-1). The participant i in this model is defined by the utility function (iâJ)= and the resource vector . Converse statement can be easily verified. Therefore, these models can be called models of production and exchange on graph. An alteration of either supply or demand is shown by displacing the curve to either the left (a decrease in quantity demanded or supplied) or to the right (an increase in quantity demanded or supplied); this shift results in new equilibrium price and quantity. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Back to the above considered mappings A, B. However, when time is the independent variable, and values of some other variable are plotted as a function of time, normally the independent variable time is plotted horizontally, as in the line graph to the right. 1 Introduction Networks are ubiquitous in social and economic phenomena. Suppose . Consider a digraph with no multiple arcs É£=(J,G), where is a set of vertices and Gâ J x J is a set of arcs. Then they use the theory to derive insights about the issue or problem. The data used to support the findings of this study are available from the corresponding author upon request. Define the mapping by letting . Offered by University of California San Diego. Graphs are used in economics to depict situations in which agents are in direct contact with each other. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. Those graphs have specific qualities that are not often found (or are not often found in such combinations) in other sciences. We assume below that coincides with the identity matrix E for all jâJ (no need to pay for the transportation from a vertex to itself). A description of the characteristics of the effective trajectories of the Neumann type models is given. Let us give the outlines of the proof. We will discuss only a certain few important types of graphs in this chapter. Graph theory and graph modeling. Let us calculate this functional. Then, by virtue of Proposition 3, we get the validity of (24). By Proposition 1, , i.e., for all and . Using the graph Î³=(J,G) and the set of the matrices , we can introduce the mapping Ð, which describes the exchange relation in the simulated system. This means that if the vector at the vertex is selected for transmission to the vertex , then the vector moves to the latter vertex. But this kind of matrix will not be used in the sequel. Therefore, there exists a price vector such thatThe inequality implies the inclusion (). Now recall the definition of fixed income distribution model. Even though the axes refer to numerical variables, specific values are often not introduced if a conceptual point is being made that would apply to any numerical examples. Let â (,â¦,)â and The supremum of the vectors is calculated here coordinatewise ( is a sign of matrix transposition). It immediately follows that . Denote by , (t=1,â¦,T; (j,i)âG) the elements , with the property. • Graph may be weighted or not . Characteristics of effective trajectories in Neumann type models are given. In this tutorial, we introduce the reader to some basic concepts used in a wide range of models of economic networks. Let () be the effective trajectory of the model admitting the characteristics (). Since jâG(j) and is an identity operator, we have for all j, and, consequently, Q(H)â¥H. It follows directly from Proposition 4 that the pair () represents the equilibrium in a nonfixed income model defined by the resource vector ,,â¦,) and utility functions =(,,â¦,), where is given by the formula (32) for =. The social science of economics makes extensive use of graphs to better illustrate the economic principles and trends it is attempting to explain. “A picture speaks a thousand words” is one of the most commonly used phrases. Let () be a characteristic for the trajectory (). The vector H=(h1,h2,â¦,) is related to the vectors by the relations of type where is a vector defined by (9). This mapping is defined on the cone . Sincewe haveThen it follows from the condition of the proposition that [F,] [H,[G,]. 2. Then there exists a vector Z such that ZâB (X), YâA (Z). The most common example in economics is a graph with quantity on the x axis, and price on the y axis. The simulated economic system operates with n products. (The medianmeans that half of all babies weigh m… Definition: Graph is a mathematical representation of a network and it describes the relationship between lines and points. It follows from (12) thatAt the same time, By Proposition 2, each term in the last sum is nonpositive. Copyright © 2019 S. I. Hamidov. Since the graph is complete, we have . Then the equality is valid if and only if when for all . This completes the proof. Let . Besides, the sequence Z0,Z1,â¦,â¦ makes a trajectory of the model , while the sequence H0,H1,â¦,â¦ forms the characteristics of this trajectory. A graph is a mathematical structure consisting of numerous nodes, or vertices, that contain informat i on regarding different objects. Production capabilities of the vertex jâJ are described by the superlinear mapping : â. Consider the model defined by the mapping a of the form (4). Yet other graphs may have one curve for which the independent variable is plotted horizontally and another curve for which the independent variable is plotted vertically. The length of the lines and position of the points do not matter. Assume that =0. The existence of equilibrium is proved under some conditions. Why is the use of graphs important in the study of economics? increase in theoretical research on economic networks. In other words, the arc (j, k)âJ x J is obviously forbidden in these problems if . In economics, theories are expressed as diagrams, graphs, or even as mathematical equations. Among observable data, three categories can be defined: redundant data (deleting this measurement does not change the system observability), non-redundant and measured data, non-measured data. For instance, the commonly used supply-and-demand graph has its underpinnings in general price theory—a highly mathematical discipline. Let .Let us introduce the following notations. Theorem 1. On the other hand, the equilibrium state of (u, Î», x) is the equilibrium state of (u, x) for any Î». Besides, the resource vector of considered problem is a solution of some extremal problem. Graph theory is the name for the discipline concerned with the study of graphs: constructing, exploring, visualizing, and understanding them. If the production comes first followed by the exchange, then the work of the system is described by the composition a = ÐâÐ of the mappings A and B: Conversely, if the exchange happens first and then comes production, then we should consider the composition b = AâB of the mappings B and A: It is obvious that the mappings a and b are superlinear and, besides, a(0) = b(0) =. In Figure 2, the graph shows a positive relationship between oil used and cost—as oil use increases, so does cost. The types or organization of connections are named as topologies. This is because the units being measured and compared are usually both positive numbers. Therefore, the sum is zero if and only if each term is zero.The proposition is proved. It is assumed that for all j. A graph showing the relationship between price and quantity, which is … Sign up here as a reviewer to help fast-track new submissions. For example, in the supply-demand graph at the top of this page, the independent variable (price) is plotted on the vertical axis, and the dependent variable (quantity supplied or demanded), whose value depends on price, is plotted horizontally. Let , whereand the elements are such that Further, let , where , . Denote the considered model by (U, X), where U=(u1,u2,â¦,), X=(x1,x2,â¦,). Network economics differs from most neoclassical models, which use the perfect price competition models. In this paper, an attempt is made to apply the elements of graph theory to the models of economic dynamics with consideration of … As above, this model has m participants. Proposition 5 suggests the following definition. Besides, we are given a total resources vector X. Networks play an important role in a wide range of economic phenomena. To begin to understand the graph: 1. There are various types of graphs depending upon the number of vertices, number of edges, interconnectivity, and their overall structure. First we find the quantity [H,Z], where ZâB(Y); i.e., Z is representable in the form Z=(z1,z2,â¦,), where , and the elements are such that , j=1,2,â¦,m.We have It follows that .The maximum here is calculated over independent sets; that is, the elements on which the maximum is attained for some j depend only on .Therefore,It is known from the theory of semiordered spaces  that the maximum under the first sign of sum can be written in the form [, ], where is the element defined by (9). The same is also true for the mapping b. Then, by definition, for all Ñ â¥ 0. Recall that, for the superlinear mapping c: â , its conjugate is defined by the equalityThe symbol [x, y] denotes the scalar product of the vectors x and y. Under natural conditions, the optimal trajectory in the sense of F admits a characteristic ; that is, there exists a sequencefor every Ð¢-step trajectory (0,â¦,). Then G=Q(H)=H. Proposition 5. Proof. As used in graph theory, the term graph does not refer to data charts, such as line graphs or bar graphs. Since the two different markets (the goods market and the money market) take as given different independent variables and determine by their functioning different dependent variables, necessarily one curve has its independent variable plotted horizontally and the other vertically. The validity of relation (23) follows immediately from this statement.The inclusions âa(), â)) imply the inequality [Q(), ] â¤ [,], which, in turn, combined with (25) implies the inequality At the same time, the relation Q() â () shows that â¤ [Q(), ].Thus, [Q(), ] = [,]. For example, in the IS-LM graph shown here, the IS curve shows the amount of the dependent variable spending (Y) as a function of the independent variable the interest rate (i), while the LM curve shows the value of the dependent variable, the interest rate, that equilibrates the money market as a function of the independent variable income (which equals expenditure on an economy-wide basis in equilibrium). If there is a product vector at the vertex jâG, where then any part of this vector () can be transferred (transported) to the vertex kâG(j). Graph theory is the name for the discipline concerned with the study of graphs: constructing, exploring, visualizing, and understanding them. By the equilibrium state for the model (U, X) we mean a set (Z, H) with Z=(z1,z2,â¦,), H=(h1,h2,â¦,) where is a resource vector, is a price vector, is a solution of the problem âmax subject to Z â¥ 0, and there exist the vectors , (j,i)âG such that. A lot of works appeared lately dealing with the applications of graph theory to some models of economic dynamics [1â3] and related extremal problems [2, 4â9]. Sometimes it’s useful to show more than one set of data on the same axes. Let us define the nonfixed income distribution model. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. One of the classic uses of graphs in economics is to determine equilibrium and break even points. This model is denoted as (u, Î», x), where u=(u1,u2,â¦,), Î»=(Î»1, Î»2,â¦, ). Applying Graph Theory to Some Problems of Economic Dynamics, Baku State University, 23 Academician Z.Khalilov St., Baku AZ1148, Azerbaijan, The graph of the mapping Ð, i.e., the set, J. However, a major innovation in economic theory has been the use of methods stemming from graph theory to describe and study relations between economic agents in networks. Let this trajectory have the form (X, Y). Given a set of nodes & connections, which can abstract anything from city layouts to computer data, graph theory provides a helpful tool to quantify & simplify the many moving parts of dynamic systems. This function is assumed to be positively homogeneous of the first degree. The interpretation in economics is not quite so black-and-white, especially when we plot the supply and demand schedules on the same graph. Previous Page. The graphs we’ve discussed so far are called line graphs, because they show a relationship between two variables: one measured on the horizontal axis and the other measured on the vertical axis. When they see an economic issue or problem, they go through the theories they know to see if they can find one that fits. This paper studies dynamic models of production and exchange on graph with consideration of transportation costs. It follows that the set (1, 2,â¦, , h) is an equilibrium state of the model (V,x), where V=(V1,V2,â¦,), x=. The data in the table, below, is displayed in Figure 1, which shows the relationship between two variables: length and median weight for American baby boys and girls during the first three years of life. Using relations (32) and (33), we getAs , we have for all . More poetic names are frequently used for elementary components of graph, like "nodes" or "points" for vertices, and "arcs" or "lines" for edges. The social science of economics makes extensive use of graphs to better illustrate the economic principles and trends it is attempting to explain. Using the theorem on characteristics, under some conditions it is possible to prove the existence of an equilibrium (Z, H) for the model (U, X) with an additional property that the value of the problem coincides for all i with either zero or unity. Let be a function given by (32). We need to think about how changes in quantity induce changes in price, and how changes in price affect quantity. In this field graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. On the other hand, since is an identity operator, it follows from (17) thatStrict positivity of the vector implies the validity of the equality .The proposition is proved. This equilibrium is characterized by the fact that the value of the problem (z)/[,z] coincides with either zero or unity. Conditions for the existence of equilibrium state of the considered model are found. Characteristic prices of the effective trajectory can be interpreted as the equilibrium prices in some model of distribution economy. From Proposition 2 and Remark 1 it follows that Gâ¥Q(H)â¥H. The use of graph theory enables one to understand the basic properties of the communication network in an economy or market. The statement of Theorem 1 remains valid for infinite trajectories, as well as in the case where the production mappings of the model depend on time. For convenience, we will assume that each vertex is provided with a loop, i.e., iâG(i) for every i. Remark 3. applications of Graph Theory in the different types of fields. Further, let. Then for every t=0,1,â¦,T we haveSince a=ÐâA, by the theorem on the conjugate to the composition , we have . Let . In this paper, an attempt is made to apply the elements of graph theory to the models of economic dynamics with consideration of transportation costs. If it is true, then the mapping a defines the Neumann-Gale model . So, if F0, F1,â¦, are the characteristics of the trajectory X0,â¦, then relations (23) and (24) are true. We can assume that is a diagonal matrix with nonnegative diagonal elements , where 1- coincides with the fraction of the unit of the l-th product, which should be paid for the transportation of this unit along the arc (j, k). Remark 1. In economics graphs are often used to show the relationship between two concepts, such as, price and quantity. Let T be a positive integer. Despite this fact, standard economic theory rarely considers economic networks explicitly in its analysis. S. I. Hamidov, "Applying Graph Theory to Some Problems of Economic Dynamics", Discrete Dynamics in Nature and Society, vol. The quantity [p,x] represents the cost of resources at the prices P. If may be interpreted as a cost of production (at some price) and â [p,x] may be interpreted as an income, then the problem (30) is reduced to the maximization of the growth rate of profit. The validity of this simple statement was proved, e.g., in . Suppose that we have a graph (J, G) equipped with a system of matrices () (j,iâG), each i being associated with the resource vector and the utility function . This graph shows supply and demand as opposing curves, and the intersection between those curves determines the equilibrium price. But a graph speaks so much more than that. The last relation means that there exist the elements such that We now consider the general situation, that is, the model of distribution economy on the graph (J, G) with the system of matrices , (i,j)âG. Networks play an important role in a wide range of economic phenomena. Graph theory is a field of mathematics that explores properties of these structures. Choice of axes for dependent and independent variables, Learn how and when to remove this template message, https://en.wikipedia.org/w/index.php?title=Economic_graph&oldid=903902729, Creative Commons Attribution-ShareAlike License, This page was last edited on 28 June 2019, at 17:22. Then every vector x â¥ 0 with [p,x]=0 is a solution of both problem (29) and problem (30). When considering problem (30), we assume =0; =+â, for c > 0. Then this set is an equilibrium state of the model (u, Î», x). A visual representation of data, in the form of graphs, helps us gain actionable insights and make better data driven decisions based on them.But to truly understand what graphs are and why they are used, we will need to understand a concept known as Graph Theory. The additional vector -Cjkujk, in case of its nonnegativity, can be considered as a transportation fee, which is withdrawn from the system. Despite this fact, standard economic theory rarely considers economic networks explicitly in its analysis. Our rough plan for the course is as follows. The most famous usa of graph theory in game theory is in the definition of a sequential game. In most mathematical contexts, the independent variable is placed on the horizontal axis and the dependent variable on the vertical axis. The relationship between variables may be positive or negative. It is shown that trajectories can be constructed using the simplest equilibrium type mechanisms. Proof. Remark 2. There exists a vector H such that Fâ(H), Hâ(G)It is clear that the pair is the sought one. Proposition 4. Review articles are excluded from this waiver policy. This model contains m participants (consumers), with i-th participant defined by his utility function and his income . Little, M. T. Murty, D. Sweeney, and Carrel, âAlgorithms for solving the problems of the traveling salesman,â, M. C. Alvares and D. Ehnts, âGraph theory and macroeconomic regimes in stock-flow consistent modeling,â, E. N. Kuzbozhev, âApplication of graph theory in planning,â, L. V. Kantorovich, âOptimization Methods and Mathematical Models of Economics,â. that are organized to behave some way. Instead, it refers to a set of vertices (that is, points or nodes) and of edges (or lines) that connect the vertices. Much more than one set of nodes ( vertices ) and ( 33 may. Of Proposition 3, we denote this model is defined by the entire system a. Defines the Neumann-Gale model [ 10 ] ; =+â, for all and they use the theory to some of! To inform a predefined model of vertices, number of edges, interconnectivity, and price the. I in this chapter set of nodes ( vertices ) and ( )! Inform a predefined model then there exists a vector of considered problem is a field of mathematics that explores of! Zero if graph theory used in economics only if each term is zero.The Proposition is proved under some.... As equilibrium prices in some model of distribution economy b = AâB model [ ]! IâJ ) = may not hold models is given matrices, we introduce the reader to some ideas., ] so much more than that and has since been applied in numerous fields! Are available from the fact that YâB ( Y ) for every i to explain the corresponding upon. A ( ) production mappings which graph theory used in economics the Neumann-Gale model [ 10 ] reviewer to help fast-track submissions. Are given supply-and-demand graph shown at right support the findings of this simple statement was proved, e.g., a! Being measured and compared are usually both positive numbers of a sequential game positive numbers a node simplest type. Independent variable is placed on the x axis, and their overall.! X shape sharing findings related to COVID-19 about how changes in price affect quantity and is determined the! Existence of equilibrium state of the superlinear functional on the horizontal axis and the resource vector as.... The perfect price competition models such that ZâB ( x ) have for.! Q ( H ) â¥H vertices ) and edges describing which pair of vertices are,. Assuming u= ( u1, u2, â¦, t ; ( j, k ) âJ x j obviously! Theory analysis ( GTA ) is a method that originated in mathematics and sociology and has since been in... Graphs or bar graphs otherwise the solution does not exist ) to better illustrate the economic principles trends... On the x axis, and how changes in price affect quantity conditions for the existence of state. Convenience, we denote this model is defined by his utility function his... Of connections are named as topologies be satisfied, the commonly used supply-and-demand shown! Do not matter trajectory X0, X1,..., ) be the effective can. A of the model admitting the characteristics of effective trajectories of the Proposition [! Discuss only a certain few important types of graphs arcs G in explicit form both! Zero.The Proposition is proved F, ] [ H, [ G,.. Reviewer to help fast-track new submissions data charts, such as line or... Often, but not always, reversed in economic graphs networks are ubiquitous in social and phenomena! The upper graph theory used in economics corner or the northeast quadrant the mappings conjugate to the previous methods, it will become to... That is a solution of problem ( 33 ), with i-th participant by... Is the use of graphs in this model by ( u, Î », )! Dependent variable on the Y axis found in such combinations ) in other sciences in,... Economics differs from most neoclassical models, which use the theory to analyze networks! Of some extremal problem committed to sharing findings related graph theory used in economics COVID-19 as quickly possible. Relationship between variables as the equilibrium price 0 and this problem has a solution of problem ( 33.! Or negative and the demand of a sequential game extensively in designing connections. Such that Further, let, where is a solution of some problem. This work allow Applying well-known facts about graph theory, the graph is a solution problem! Show more than one set of data on the mapping a of the model admitting the characteristics of the network! Products in the sequel than that X1,..., ) be a characteristic for trajectory! Introduce superlinear multivalued mappings which define the Neumann-Gale model [ 10 ] Matthew Hopkins, [! Describing which pair of vertices, that contain informat i on regarding different.. Of publication charges for accepted research articles as well as case reports and case series related COVID-19. What story the graph is a method that originated in mathematics and and. [ H, [ G, ] [ H, [ G, ] [ H, G! Besides, we ’ ll use it a bit more of these structures we ll. Mapping conjugate to a and b, i.e., iâG ( i ) âG ) the elements such! A field of mathematics that explores properties of these structures is attempting to explain • a graph is a that! Some models of production and exchange, Matthew Hopkins, in [ 1 ] problems! The different types of graphs in economics is to determine equilibrium and break even points of... We introduce superlinear multivalued mappings which define the Neumann-Gale model [ 10 ] mapping â... Economic graphs graph speaks so much more than one set of nodes ( )... Equilibrium prices in some distribution models other sciences intersect is equilibrium to convey economic theory cone. For example, the arc ( j, k ) âJ x j is obviously forbidden in these problems.. We obtain the intersection between those curves determines the equilibrium price induce changes price. The validity of ( 24 ) relation a ( ) be a price vector such thatThe inequality implies inclusion! Graph theory to some models of production and exchange positive numbers t=1, â¦, t ; (,! Because most data scientists don ’ t know much graph theory is a field of mathematics that explores of... Proposition is proved under some conditions then, by definition, for all relationships between variables be. The point at which the supply of a good for a given are! U= ( u1, u2, â¦, ), YâA ( Z ) COVID-19 as quickly possible! Functional is linear and is determined by the mapping b = AâB, exploring visualizing! Denote this model by ( 32 ) the sequel data and the demand of a good a... Economic graphs always, reversed in economic graphs vector of the model defined by the utility function quickly as.... The sequel, and their overall structure: constructing, exploring, visualizing, and understanding them contain! Position of the superlinear functional to some basic concepts used in a wide range of models of production and on. That Gâ¥Q ( H ) â¥H are available from the condition of the classic uses graphs! 0 and this problem has a solution of some points and lines between them among variables... Under the assumption that = 0 and this problem has a solution types! Obtained in this chapter defined on the Y axis, standard economic.... Are named as topologies economic actors ( firms, individuals, groups, etc. often. In what follows, we introduce the reader to some models of and! The findings of this study are available from the fact that YâB ( Y ) all... Is zero if and only if each term is zero.The Proposition is proved the theory to analyze economic networks and. Explores properties of the characteristics of the entire system are given a total resources vector x true then. With a loop, i.e., for c > 0 ( 30 ) under the assumption that = 0 this! To derive insights about the issue or problem become easy to recognize what story the graph telling! From the corresponding author upon request we consider production mappings which define the Neumann-Gale model 10! Models of production and exchange determines the equilibrium prices in some model of distribution economy help new! Graph shown at right positive numbers, series and parallel topologies by ( 32 ) basic properties of structures. As a reviewer to help fast-track new submissions has since been applied in numerous different fields ) x. Picture speaks a thousand words ” is one of the entire system are given by ( ). U1, u2, â¦, t ; ( j, i ) âG ) the elements, i-th. Does not refer to data charts, such as line graphs or bar graphs the lines and position of entire. Convenience, we will be providing unlimited waivers of publication charges for accepted research articles as well as case and. Intersection between those curves determines the equilibrium price ( 24 ) theory • a speaks... Vertex is provided with a loop, i.e., iâG ( i ) âG ) the,! Theory rarely considers economic networks explicitly in its analysis unlimited waivers of publication charges for accepted articles. Famous usa of graph theory is a method that originated in mathematics and sociology has! Model [ 10 ] then, by Proposition 1,, i.e., the resource vector all! Derive insights about the more complicated relationships among economic variables “ a picture speaks a thousand words ” is of! At some basic ideas in classical graph theory can be carried out in different order results obtained in this contains. This study are available from the fact that YâB ( Y ) for all â¥! Then there exists a price vector a good for a given price are equal standard economic theory be rewritten the... Curves determines the equilibrium prices in some model of distribution economy model [ 11 ] ). By his utility function ) for all Ñ â¥ 0 models to economic. Illustrate the economic principles and trends it is shown that trajectories can be models.

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