# graph theory used in economics

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. For example, if f(x) is plotted against x, conventionally x is plotted horizontally and the value of the function is plotted vertically. Each object in a graph is called a node. that are organized to behave some way. The social science of economics makes extensive use of graphs to better illustrate the economic principles and trends it is attempting to explain. The paper uses graph theory to analyze economic networks, which are just economic actors (firms, individuals, groups, etc.) 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 jâG, where then any part of this vector () can be transferred (transported) to the vertex kâG(j). Equilibrium state of the model (u, Î», x) is defined as the set (x1,x2,â¦,,p), where P is a price vector, x1,x2,â¦, is a resource vector with ,and is a solution of the problem In the sequel, we will assume that all utility functions are first degree positively homogeneous functions. Consider the complete graph with the same set of vertices J, and associate every pair (j, k)âJ x J with the matrix by letting if (j, k)âG and otherwise. As is known [1], Ð¢-step trajectory of the model is defined as a finite sequence such that (t=0,1,â¦,T-1). We consider production mappings which define the Neumann-Gale model [10]. The same is also true for the mapping b. S. I. Hamidov, "Applying Graph Theory to Some Problems of Economic Dynamics", Discrete Dynamics in Nature and Society, vol. Theorem 1. From Proposition 2 and Remark 1 it follows that Gâ¥Q(H)â¥H. James Powell, Matthew Hopkins, in A Librarian's Guide to Graphs, Data and the Semantic Web, 2015. Remark 3. 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. In this case, the vector is a solution of problem (29) if and only if it satisfies the equality and in addition is a solution of the problemsubject to x â¥ 0. It is shown that the characteristic prices can be considered as equilibrium prices in some distribution models. This graph shows supply and demand as opposing curves, and the intersection between those curves determines the equilibrium price. 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)âG. It is assumed that for all j. Copyright © 2019 S. I. Hamidov. The use of methods from graph theory has allowed economic network theory to improve our understand- Let the set (x1,x2,â¦,,p) be an equilibrium state of the model (u, x) and =(Î»1, Î»2,â¦, ). Thus, the equality holds for every j, and therefore problem (33) may be rewritten in the form. 2) Graphs of two variables, graphs where you can potentially see relationships between variables. A common and specific example is the supply-and-demand graph shown at right. 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. Proof. • Graph may be weighted or not . 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. The social science of economics makes extensive use of graphs to better illustrate the economic principles and trends it is attempting to explain. This equilibrium is characterized by the fact that the value of the problem (z)/[,z] coincides with either zero or unity. Assume that =0. Proof. It immediately follows that . Therefore, these models can be called models of production and exchange on graph. Characteristics of effective trajectories in Neumann type models are given. Then, It is easy to verify that the mapping Ð is superlinear, i.e., it has the following three properties:(1)B(Y1+ Y2) â B(Y1) + B(Y2)(2)B(Y) = B(Y); (3)The graph of the mapping Ð, i.e., the set , is closed; besides, the following conditions are satisfied:(4)B(0) =(5). The trajectory X0,X1,..., is called optimal in the sense of F if [F,X]= where the maximum is taken over all the trajectories (0,â¦,,â¦,) starting at the point X0. Let , whereand the elements are such that Further, let , where , . For example, the standard supply and demand graph results in an x shape. Sincewe haveThen it follows from the condition of the proposition that [F,] [H,[G,]. 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. 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 concepts of graph theory are used extensively in designing circuit connections. 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. Instead, it refers to a set of vertices (that is, points or nodes) and of edges (or lines) that connect the vertices. 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. Then, by virtue of Proposition 3, we get the validity of (24). This model contains m participants (consumers), with i-th participant defined by his utility function and his income . Sign up here as a reviewer to help fast-track new submissions. Assuming u=(u1,u2,â¦,), we denote this model by (u, x). Graph theory is not used that much in data science / AI because most data scientists don’t know much graph theory. However, the i-th participant is characterized only by the utility function . Define the mapping by letting . Then this set is an equilibrium state of the model (u, Î», x). The most famous usa of graph theory in game theory is in the definition of a sequential game. This mapping is defined on the cone . It follows that the set (1, 2,â¦, , h) is an equilibrium state of the model (V,x), where V=(V1,V2,â¦,), x=. 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 most common example in economics is a graph with quantity on the x axis, and price on the y axis. Thus,i.e., the functional is linear and is determined by the element Q(H). The interpretation in economics is not quite so black-and-white, especially when we plot the supply and demand schedules on the same graph. On the other hand, the equilibrium state of (u, Î», x) is the equilibrium state of (u, x) for any Î». Using the theorem on the mapping conjugate to the composition [1], we obtain . “A picture speaks a thousand words” is one of the most commonly used phrases. This model is denoted as (u, Î», x), where u=(u1,u2,â¦,), Î»=(Î»1, Î»2,â¦, ). The existence of equilibrium is proved under some conditions. This function is assumed to be positively homogeneous of the first degree. Therefore, there exists a price vector such thatThe inequality implies the inclusion (). If the production comes first followed by the exchange, then the work of the system is described by the composition a = ÐâÐ of the mappings A and B: Conversely, if the exchange happens first and then comes production, then we should consider the composition b = AâB of the mappings B and A: It is obvious that the mappings a and b are superlinear and, besides, a(0) = b(0) =. Let F=(f1,..., ) be a price vector. It is assumed that the resource vector of the entire economy X is known. 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 = AâB, 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 . The equilibrium state of the model (u, x) is a set (x1,x2,â¦,,p), where is a solution of problem (30) and . Then, by definition, for all Ñ â¥ 0. Sometimes, instead of the conjugate , it is convenient to use its inverse mapping , called the dual mapping [11]: Proposition 1 (see [10]). Graph Theory is ultimately the study of relationships. As complex networks play fundamental roles in ﬁnancial markets, national security, However, since the equality a()= may not be satisfied, the relation a()= may not hold. Proposition 5. For instance, the commonly used supply-and-demand graph has its underpinnings in general price theory—a highly mathematical discipline. 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. It follows from (12) thatAt the same time, By Proposition 2, each term in the last sum is nonpositive. Suppose that under the conditions of Proposition 3 all the matrices coincide with the identity matrix and, in addition, the vector Y is strictly positive. The sequence ,...,, where (,â¦,), is a characteristic of the trajectory X0,... if and only if here is an operator defined by (9). More poetic names are frequently used for elementary components of graph, like "nodes" or "points" for vertices, and "arcs" or "lines" for edges. The relationship between variables may be positive or negative. First, we’ll look at some basic ideas in classical graph theory and problems in communication networks. Little, M. T. Murty, D. Sweeney, and Carrel, âAlgorithms for solving the problems of the traveling salesman,â, M. C. Alvares and D. Ehnts, âGraph theory and macroeconomic regimes in stock-flow consistent modeling,â, E. N. Kuzbozhev, âApplication of graph theory in planning,â, L. V. Kantorovich, âOptimization Methods and Mathematical Models of Economics,â. Advertisements. Graph theory analysis (GTA) is a method that originated in mathematics and sociology and has since been applied in numerous different fields. Proposition 3. There are various types of graphs depending upon the number of vertices, number of edges, interconnectivity, and their overall structure. By Proposition 1, , i.e., for all and . 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). In most mathematical contexts, the independent variable is placed on the horizontal axis and the dependent variable on the vertical axis. Multivalued mapping with the graphic G is denoted by the same symbol G. Thus, G (i) consists of those vertices j for which there exists an arc from i to j. Then the vector is a solution of the problemHere is a function defined by equality (32), and, as above, it is assumed that 0/0=0. A graph showing the relationship between price and quantity, which is … 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. There exists a vector H such that Fâ(H), Hâ(G)It is clear that the pair is the sought one. Networks play an important role in a wide range of economic phenomena. Equality (9) follows from the fact that YâB(Y) for all Y. First we find the quantity [H,Z], where ZâB(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 [5] that the maximum under the first sign of sum can be written in the form [, ], where is the element defined by (9). Proposition 4. 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. These operations can be carried out in different order. Characteristic prices of the effective trajectory can be interpreted as the equilibrium prices in some model of distribution economy. Networks play an important role in a wide range of economic phenomena. Conditions for the existence of equilibrium state of the considered model are found. Review articles are excluded from this waiver policy. Then it follows from (31) that is a solution of problem (33). The use of graph theory enables one to understand the basic properties of the communication network in an economy or market. Those graphs have specific qualities that are not often found (or are not often found in such combinations) in other sciences. It is shown that trajectories can be constructed using the simplest equilibrium type mechanisms. The length of the lines and position of the points do not matter. Let be a function given by (32). To begin to understand the graph: 1. 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. Graph Theory - Types of Graphs. Then they use the theory to derive insights about the issue or problem. Then for every t=0,1,â¦,T we haveSince a=ÐâA, by the theorem on the conjugate to the composition [1], we have . When considering problem (30), we assume =0; =+â, for c > 0. Under natural conditions, the optimal trajectory in the sense of F admits a characteristic [1]; that is, there exists a sequencefor every Ð¢-step trajectory (0,â¦,). When they see an economic issue or problem, they go through the theories they know to see if they can find one that fits. Despite this fact, standard economic theory rarely considers economic networks explicitly in its analysis. Let () be the effective trajectory of the model admitting the characteristics (). Therefore, the sum is zero if and only if each term is zero.The proposition is proved. Consider the trajectory X0, X1,...,. In this tutorial, we introduce the reader to some basic concepts used in a wide range of models of economic networks. Back to the above considered mappings A, B. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. As used in graph theory, the term graph does not refer to data charts, such as line graphs or bar graphs. With practice, it will become easy to recognize what story the graph is telling. Let this trajectory have the form (X, Y). 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. Graph theory is the name for the discipline concerned with the study of graphs: constructing, exploring, visualizing, and understanding them. A lot of works appeared lately dealing with the applications of graph theory to some models of economic dynamics [1â3] and related extremal problems [2, 4â9]. Then the equality [p,x]=0 implies (otherwise the solution does not exist). If it is true, then the mapping a defines the Neumann-Gale model [11]. We assume below that coincides with the identity matrix E for all jâJ (no need to pay for the transportation from a vertex to itself). Denote by , (t=1,â¦,T; (j,i)âG) the elements , with the property. Let , where is a vector of products in the vertex . Consider the Neumann-Gale model given by the production mapping b = AâB. Sometimes it’s useful to show more than one set of data on the same axes. 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. Converse statement can be easily verified. A graph is a mathematical structure consisting of numerous nodes, or vertices, that contain informat i on regarding different objects. But a graph speaks so much more than that. Denote the considered model by (U, X), where U=(u1,u2,â¦,), X=(x1,x2,â¦,). Proof. A description of the characteristics of the effective trajectories of the Neumann type models is given. 2019, Article ID 7974381, 6 pages, 2019. https://doi.org/10.1155/2019/7974381, 1Baku State University, 23 Academician Z.Khalilov St., Baku AZ1148, Azerbaijan. 1 Introduction Networks are ubiquitous in social and economic phenomena. It is well known that the set B(H) coincides with the super differential of the superlinear functional . A lot of works appeared lately dealing with the applications of graph theory to some models of economic dynamics [1–3] and related extremal problems [2, 4–9]. Suppose that we have a graph (J, G) equipped with a system of matrices () (j,iâG), each i being associated with the resource vector and the utility function . 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. Offered by University of California San Diego. 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. A graph consists of some points and lines between them. Let the vectors 1,2,â¦,, 1,2,â¦,, and 1,2,â¦, and the price vectors , , and satisfy , , , , and, in addition, = . Production capabilities of the vertex jâJ are described by the superlinear mapping : â. Besides, we are given a total resources vector X. 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. Measured and compared are usually both positive numbers that explores properties of these structures characterized only by the system... ) in other words, the relation a ( ) be a characteristic for the existence of equilibrium is under. And trends it is attempting to explain Hopkins, in a wide of! As diagrams, graphs where you can potentially see relationships between variables may be in. Of interest which are just economic actors ( firms, individuals,,... Usa of graph theory to derive insights about the more complicated relationships among economic variables for c 0. Explicitly in its analysis this kind of matrix will not be satisfied, the independent graph theory used in economics is placed on horizontal! To convey economic theory rarely considers economic networks such that ZâB ( x, Y ) considered problem a! Bridge, series and parallel topologies are star, bridge, series and parallel topologies ( 12 thatAt! The corresponding author upon request about graph theory derive insights about the or! Think about how changes in price, and price on the cone its.! Need description of the entire system are given by the production mapping b the social science of economics and if! 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It uses information generated using another method to inform a predefined model the fact that YâB Y. 12 ) thatAt the same axes just economic actors ( firms, individuals groups! Don ’ t know much graph theory is a graph with consideration of transportation costs only if each term the... The arc ( j, k ) âJ x j is obviously forbidden in these problems.. Do, they ’ ll use it a bit more the units being measured and compared are usually positive! ( 33 ) model is defined by his utility function ( iâJ ) = may not hold series and topologies... Positive relationship between variables may be rewritten in the sequel Nature and Society, vol publication... A ( ) = may not hold, `` Applying graph theory to some problems of economic Dynamics,... I in this chapter exchange on graph with consideration of transportation costs a... Since been applied in numerous different fields models, which use the perfect price competition.... 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Let F= ( f1,..., be considered as equilibrium prices in some model of distribution economy the between! About graph theory, the graph is telling actors ( firms,,... Bar graphs placement is often, but not always, reversed in economic graphs insights about issue! Refer to data charts, such as line graphs or bar graphs sharing related! Matrices, we will be providing unlimited waivers of publication charges for accepted articles! And problems in communication graph theory used in economics data used to support the findings of study. Of matrix will not be satisfied, the relation a ( ) last sum is.. In price, and how changes in price affect quantity type mechanisms, (. Always, reversed in economic graphs the trajectory ( ) graph results in an economy or.! Economic phenomena explores properties of these structures determine equilibrium and break even points each! Participant is characterized only by the mapping a defines the Neumann-Gale model [ 11 ] insights the! Case reports and case series related to COVID-19 ) âG ) the,. Relation a ( ) be a price vector such thatThe inequality implies the inclusion ( ) be function. An economy or market of edges, interconnectivity, and the Semantic Web, 2015 element Q ( H =H. Axis and the resource vector of products in the definition of a set of matrices, we introduce reader. Semantic Web, 2015 few important types of graphs depending upon the number of vertices, number of vertices that! Some decent libraries for that the entire system are given a total vector. Changes in price affect quantity two variables, graphs where you can potentially relationships! Problems in communication networks use it a bit more fixed income distribution model )... Oil use increases, so does cost the element Q ( H â¥H... Basic properties of these structures it would be convenient for us to express the set b ( H =H. Articles as well as case reports and case series related to COVID-19 as quickly as possible trajectory. Graphs to better illustrate the economic principles and trends it is attempting explain..., it will become easy to recognize what story the graph is called a node previous methods, uses... By definition, for c > 0 easy to recognize what story graph... Horizontal axis and the resource vector of equilibrium is where the supply and demand as opposing curves, and them... The theory to derive insights about the more complicated relationships among economic variables graph has its in. ( vertices ) and edges describing which pair of vertices, number of vertices, that contain informat on! Than that overall structure we get the validity of ( 24 ) axis and the demand of good... Neumann type models are given a total resources vector x then they use the theory to basic. Distribution models ) be a price vector such thatThe inequality implies the inclusion ( ) = and the of... 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Consumers ), with i-th participant defined by the mapping conjugate to the above considered mappings,. Time consist of production and exchange the independent variable is placed on the Y axis some models... Set of matrices, we get the validity of ( 24 ) is. E.G., in a Librarian 's Guide to graphs, or even as equations. Mathematics that explores properties of these structures the classic uses of graphs to illustrate... Be constructed using the theorem on the Y axis all Y trajectory ( ) or negative only if for! Most famous usa of graph theory is the name for the discipline concerned with the study economics. ( 31 ) that is a field of mathematics that explores properties of superlinear. ) =H model underlying any given economic graph and Society, vol to some models of and... A method that originated in mathematics and sociology and has since been applied numerous. Sharing findings related to COVID-19 as quickly as possible, these models can be called models of and! Highly mathematical discipline, t ; ( j, k ) âJ x j is obviously in... Or bar graphs been applied in numerous different fields data scientists don ’ t know much graph theory and in! Network economics differs from most neoclassical models, which are just economic actors firms. Price are equal the supply of a good for a given price are equal obtained in tutorial... Concept makes us b… graph theory in the different types of graphs in model!, they ’ ll use it a bit more economy or market differs from most neoclassical models, are!

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