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Spatial price dynamics: From complex network perspective
Physica A 387 (2008) 5852–5856
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Spatial price dynamics: From complex network perspective Y.L. Li a , J.T. Bi b , H.J. Sun a,∗ a
School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, PR China
b
Institute of Remote Sensing Applications, Chinese Academy of Sciences, Beijing 100101, PR China
article
info
Article history: Received 13 November 2007 Received in revised form 2 April 2008 Available online 14 June 2008 PACS: 89.75.Hc 89.75.Fb 87.23.Ge
a b s t r a c t The spatial price problem means that if the supply price plus the transportation cost is less than the demand price, there exists a trade. Thus, after an amount of exchange, the demand price will decrease. This process is continuous until an equilibrium state is obtained. However, how the trade network structure affects this process has received little attention. In this paper, we give a evolving model to describe the levels of spatial price on different complex network structures. The simulation results show that the network with shorter path length is sensitive to the variation of prices. © 2008 Elsevier B.V. All rights reserved.
Keywords: Spatial price Complex networks Dynamics
1. Introduction Physical networks are pervasive in today’s society, they can be in the form of transportation networks, telecommunication networks, energy pipelines, electric power networks, etc. Mathematical networks, on the other hand, may be used to represent not only physical networks but also interactions among economic agents. In many applications, the network representation of an economic equilibrium problem may be abstract in that the nodes of the network need not correspond to locations in space and the links of the network to trade or travel routes. The structure and dynamics on complex networks have attracted a tremendous amount of recent interest [1–6] since the seminal works on scale-free networks by Barabási and Albert [3] and on the small-world phenomenon by Watts and Strogatz [2]. Mathematically, a way to characterize a complex network is to examine the degree distribution P (k), where k is used to measure the number of links at a node. Scale-free networks are characterized by P (k) ∼ k−λ , where k is the algebraic scaling exponent [3]. Structure characteristics have great effects on network dynamics, hence, it is important to study dynamics phenomena on different network structures. In a network economics age, analyzing economic phenomena from a network point of view is now well-recognized and receiving growing attention. One of the remarkable features of a market is its ability to process information. Even with the myriad factors that can influence economic phenomena, in a well-functioning market economy agents need only pay attention to one piece of information for each good, namely its price. While network or communication structures under which agents operate and transmit or exchange information have received little attention. The details of who is connected to whom will clearly affect the price and the trade behaviors. In this paper, we study the dynamics of spatial price on trade networks. The spatial price problem means that if the supply price plus the transportation cost is less than the demand price, there exists a trade, and after an amount of exchange, the
∗
Corresponding author. Tel.: +86 1051684265; fax: +86 1051684265. E-mail address: [email protected] (H.J. Sun).
0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.06.008
Y.L. Li et al. / Physica A 387 (2008) 5852–5856
5853
demand price will decrease. This process is continuous until the equilibrium state is obtained. An equilibrium condition satisfies that the demand price is equal to the supply price plus the cost of transportation, if there is trade between the pair of supply and demand markets; if the demand price is less than the supply price plus the transportation cost, then there will be no trade [7]. i.e., Ci + Uij
if Qij > 0, if Qij = 0,
= Cj ≥ Cj
(a) (b)
(1)
where Cj is the price of node i, Uij is the cost on link ij, Qij is amount of the trade between node i and j. The distinguishing characteristic of spatial price equilibrium models lies in their recognition of the importance of space and transportation costs associated with shipping a commodity from a supply market to a demand market.These models are perfectly competitive partial equilibrium models, in that one assumes that there are many producers and consumers involved in the production and consumption respectively.Hence, the spatial price equilibrium problem seeks to compute the commodity supply prices, demand prices, and trade flows satisfying the equilibrium condition. In our paper, the price dynamics is paid attention to especially, and the trade flow is not considered. There are many researches on mathematical programming problems of spatial price equilibrium problems [7,8]. Nagurney [7] has studied the optimization model of the spatial price problem under equilibrium conditions. Sun et al. [8] proposed a variational inequality model of spatial price problem in logistics networks. However, the above work has the following two shortcomings: (1) The effects of topologies on price dynamics are not considered. (2) The community effects on the price propagation are not researched. In this paper, we intend to fill this gap by studying the evolution of space price in complex networks by considering the two cases mentioned above. 2. A model of traffic flow assignment The model we propose explores how the topology of agents’ influences the variation of the market price, and how such effects will propagate in the network. Cowan et al. [9] have modeled knowledge diffusion as a barter process in small-world networks, in which agents exchange different type of knowledge. We assume that the variation of price takes place through a trade exchange. In this model when two agents meet, the decision is simply whether to trade or to walk away [9]. For each agent this decision is based only on whether or not the other agent has something to offer, and not on the amount he has to offer. For a spatial price problem, each node acts as not only the supplier but also the demander. What role it plays is mainly dependent on the market price of the commodities. if the price (supply price) of node i, Ci , plus the cost (transportation cost) on link ij, Uij , is less than the price (demand price) of node j, Cj , there will exist a trade, and after this trade, the supply price will decrease. After the exchange price levels have become Cj (t + 1) =
Cj (t ) − α(Cj (t ) − Ci (t ))β Ci (t )
if Ci (t ) + Uij < Cj (t ), if Ci (t ) + Uij ≥ Cj (t ),
(a) (b)
(2)
where Ci (t ) indicates the price level of node i at time t. α and β are parameters. The trading process continues until all trading possibilities are exhausted, that is, there are no further profit of trading behaviors. 3. Price dynamics in different network topologies We construct manmade complex networks to describe the above dynamics of the price. Each supplier or demander is treated as a node. If there exists a possible trade between two nodes, there is a edge. Networks can be represented as graphs G = (V , E ) where V and E are the sets of vertices and edges. G is described by the N × N adjacency matrix eij . Define N as the size of the network. To facilitate comparison, we create regular, small-world, scale-free and random networks methods with the same size and total edges. The average degree of networks hki ≈ 3, and α = 0.4, β = 1. Transportation cost Uij is given randomly (0, 1]. In this paper we illustrate how our model works in practice by considering the artificially created regular lattice (RN), random graph (ER), small-world (SW) and scale-free (SF) networks. These networks are generated according to the procedure in Ref. [10], where self- and repeated links are forbidden. The small world network is generated by a RE rewired with probability 0.5. We simulate the evolution of a spatial price problem on the above networks. Each period a node, i and one of his neighbours j are chosen uniformly at random. This process continues until all possible trades have been made. Since with real numbers of price, levels never become identical, we consider them identical if they differ by less than one percent [9]: 0.99 <
Ci + Uij Cj
< 1.01,
∀i, j.
(3)
This condition guarantees that the process eventually stops. For the longer evolution, we give the step is 2000. Each curve corresponds to the average over 25 realizations of the network.
5854
Y.L. Li et al. / Physica A 387 (2008) 5852–5856
Fig. 1. The time series of average price level with different network structures.
Fig. 2. The time series of price variance with different network structures.
We use the average and variance of price levels to evaluate theqefficiency of the system in price propagation. Average
price level of systems is u = 1/N ×
P
Ci , and the variance is σ =
P
Ci2 /N − u2 . At the steady state we remove the time
index [9]. Different network structure has different effects on the distribution of the average price level and variance. Figs. 1 and 2 give the time series of the average price level and variance for different structures. The network size N = 100 (while being quite small, this does provide a sufficiently sized network to gain statistically significant attributes). From the plots we can see that the regular network decreases slowly and other three kinds of networks (small-world, scale-free and random networks) decrease quickly at first. The reason is that the average path length of regular network is long, but its clustering coefficient is higher. Therefore propagation in the regular network is slower than in other ones, but its progress continues longer. Small-world and scale-free networks have relatively short path lengths and propagation in the early periods is relatively fast. Random networks have the shorter path lengths and smaller clustering coefficient, therefore the price level decreases quickly. Because of the quick decrease of the average price level in early periods on regular networks, the variance of price will increase first and then fall. When the average price level remains steady, the variance starts to decline. We choose the typical network topology, SF, as the object to investigate the effects on the price dynamics with different network size. Selecting this particular structure is based on two reasons: (1) Scale-free is a most popular structure in the real world. A
Y.L. Li et al. / Physica A 387 (2008) 5852–5856
5855
Fig. 3. The time series of average price level with different network size N = 100, 200, 300, 400, 500, 600.
larger number of previous literature has concluded that most real economic networks display the scale-free characteristics; (2) In our previous work [8], we find that the supply chain distribution network exhibits the power-law form. The network size also has large influence on the price evolution. We report the time series of the average price level on scale-free networks for different network size in Fig. 3. From the plot, we can see that the average price level falls quickly for the small networks size. However, for a larger network the effects of price variation could only propagate a small part of nodes, therefore the average price level decrease slowly. We can also find that the time of phase transition is longer for the larger network. It is because that, with the increase of network size, the clustering becomes more and more strong.
4. Price dynamics in networks with community structure In the continuing flurry of research activity within physics and mathematics on the properties of networks, a particular recent focus has been the analysis of communities within networks [11]. Many networks in nature or social networks can be divided into some groups such that the connections within each group are dense, while connections between groups are sparse, which is also called community structure. There are many real-world networks which exhibit community structure, and community structures are supposed to play an important role in many real networks [12]. For example, communities in a citation network might represent related papers on a single topic [13]; communities on the web might represent pages on related topics [14]; communities in a biochemical network or neuronal system might correspond to functional units of some kinds [15]; communities also play an important role in information networks [16]. In trade networks some exchanges can occur in part of a tightly connected group, others can be completely isolated, while some others may act as bridges between groups [17,18]. We generate the scale-free network with community structure using our previous method [18]. Based on a previous measure of assortative mixing [17], Newman and Girvan have proposed of the quality P the measure Pof a particular division 2 of a network, which they called the modularity as follows [19]: Q = r (hrr − ar ), where ar = w hr w denotes the row (or column) sums which represent the fraction of edges that connect to nodes in community r and hr w is the fraction of edges in the original network that connect nodes in subset r with nodes in subset w . In a given network in which edges fall between nodes without regard for the communities they belong to, hr w = ar aw can be obtained. In our network model, we can adjust c-value to get networks with various strength Q of community structures. The larger the value of Q is, the most accurate a partition into communities will be. If the number of within-community edges is no better than random, we will get Q = 0. Values approaching Q = 1, which is the maximum, indicate strong community structure [12]. In this paper, the edge whose two nodes are in a same community is defined as the local edge and the bridge edge represents the edge whose two nodes belong to different communities. Different modularity can display different structures and functions of many real-world networks. Here, we analyze the effects of community modular Q on the spatial price dynamics. In Figs. 4 and 5, we analyze the variation of price levels with different modularity. The network and its parameters come from the reference [12]. From Fig. 4, we can see that the average price level will decrease sharply for a larger modularity Q , which means that for a strong community structure, the effects of price will spread quickly and easily. We can see that the time of phase transition is about 170, 300, and 400 for Q = 0.14, Q = 0.15 and Q = 0.16. This is because the strong community structure network has the relatively short path length.
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Y.L. Li et al. / Physica A 387 (2008) 5852–5856
Fig. 4. The time series of average price level in complex network with different community structure.
Fig. 5. The time series of price variance in complex network with different community structure.
5. Conclusion In conclusion, in this paper, the dynamic of the spatial price problem is studied. By simulating the variation of price in trade networks, we have found that network structures with short path lengths will suffer quick variations of price levels. However, the quantity of commodity flows are not considered in the exchange process, which may be our further work. Acknowledgments This paper is partly supported by NSFC of China (70501005), FANEDD (200763) and FYETF (111083). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
P. Erdös, A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17. D.J. Watts, S.H. Strogatz, Nature (London) 393 (1998) 440. A.-L. Barabási, R. Albert, Science 286 (1999) 509. M.J. Newman, e-print arXiv:cond-mat/0201433, 2002. Z. Dezso, A.-L. Barabási, Phys. Rev. E 65 (2002) 055103. L. Zhao, Y.-C. Lai, K. Park, N. Ye, Phys. Rev. E 71 (2005) 026125. A. Nagurney, A Variational Inequality Approach, Kluwer Academic Publishers, London, 1999. H.J. Sun, Z.Y. Gao, Proce. 2004 ICTTS, 555, Dalian, China. R. Cowan, N. Jonard, J. Econom. Dynam. Control 28 (2004) 1557. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Phys. Rep. 424 (2006) 175. M. Girvan, M.E.J. Newman, Proc. Natl. Acad. Sci. USA 99 (2002) 7821. C.G. Li, P.K. Maini, arXiv:physics/0510239, 2005, 26. S. Redner, Eur. Phys. J. B 4 (1998) 131. G.W. Flake, S.R. Lawrence, C.L. Giles, F.M. Coetzee, IEEE Comput. 35 (2002) 66. P. Holme, M. Huss, H. Jeong, Bioinformatics 19 (2003) 532. H. Xie, K.-K. Yan, S. Maslov, cond-mat/0409087. M.E.J. Newman, Phys. Rev. E 67 (2003) 026126. J.J. Wu, Z.Y. Gao, H.J. Sun, Phys. Rev. E 74 (2006) 066111. M.E.J. Newman, M. Girvan, Phys. Rev. E 69 (2004) 026113.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Spatial price dynamics: From complex network perspective Y.L. Li a , J.T. Bi b , H.J. Sun a,∗ a
School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, PR China
b
Institute of Remote Sensing Applications, Chinese Academy of Sciences, Beijing 100101, PR China
article
info
Article history: Received 13 November 2007 Received in revised form 2 April 2008 Available online 14 June 2008 PACS: 89.75.Hc 89.75.Fb 87.23.Ge
a b s t r a c t The spatial price problem means that if the supply price plus the transportation cost is less than the demand price, there exists a trade. Thus, after an amount of exchange, the demand price will decrease. This process is continuous until an equilibrium state is obtained. However, how the trade network structure affects this process has received little attention. In this paper, we give a evolving model to describe the levels of spatial price on different complex network structures. The simulation results show that the network with shorter path length is sensitive to the variation of prices. © 2008 Elsevier B.V. All rights reserved.
Keywords: Spatial price Complex networks Dynamics
1. Introduction Physical networks are pervasive in today’s society, they can be in the form of transportation networks, telecommunication networks, energy pipelines, electric power networks, etc. Mathematical networks, on the other hand, may be used to represent not only physical networks but also interactions among economic agents. In many applications, the network representation of an economic equilibrium problem may be abstract in that the nodes of the network need not correspond to locations in space and the links of the network to trade or travel routes. The structure and dynamics on complex networks have attracted a tremendous amount of recent interest [1–6] since the seminal works on scale-free networks by Barabási and Albert [3] and on the small-world phenomenon by Watts and Strogatz [2]. Mathematically, a way to characterize a complex network is to examine the degree distribution P (k), where k is used to measure the number of links at a node. Scale-free networks are characterized by P (k) ∼ k−λ , where k is the algebraic scaling exponent [3]. Structure characteristics have great effects on network dynamics, hence, it is important to study dynamics phenomena on different network structures. In a network economics age, analyzing economic phenomena from a network point of view is now well-recognized and receiving growing attention. One of the remarkable features of a market is its ability to process information. Even with the myriad factors that can influence economic phenomena, in a well-functioning market economy agents need only pay attention to one piece of information for each good, namely its price. While network or communication structures under which agents operate and transmit or exchange information have received little attention. The details of who is connected to whom will clearly affect the price and the trade behaviors. In this paper, we study the dynamics of spatial price on trade networks. The spatial price problem means that if the supply price plus the transportation cost is less than the demand price, there exists a trade, and after an amount of exchange, the
∗
Corresponding author. Tel.: +86 1051684265; fax: +86 1051684265. E-mail address: [email protected] (H.J. Sun).
0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.06.008
Y.L. Li et al. / Physica A 387 (2008) 5852–5856
5853
demand price will decrease. This process is continuous until the equilibrium state is obtained. An equilibrium condition satisfies that the demand price is equal to the supply price plus the cost of transportation, if there is trade between the pair of supply and demand markets; if the demand price is less than the supply price plus the transportation cost, then there will be no trade [7]. i.e., Ci + Uij
if Qij > 0, if Qij = 0,
= Cj ≥ Cj
(a) (b)
(1)
where Cj is the price of node i, Uij is the cost on link ij, Qij is amount of the trade between node i and j. The distinguishing characteristic of spatial price equilibrium models lies in their recognition of the importance of space and transportation costs associated with shipping a commodity from a supply market to a demand market.These models are perfectly competitive partial equilibrium models, in that one assumes that there are many producers and consumers involved in the production and consumption respectively.Hence, the spatial price equilibrium problem seeks to compute the commodity supply prices, demand prices, and trade flows satisfying the equilibrium condition. In our paper, the price dynamics is paid attention to especially, and the trade flow is not considered. There are many researches on mathematical programming problems of spatial price equilibrium problems [7,8]. Nagurney [7] has studied the optimization model of the spatial price problem under equilibrium conditions. Sun et al. [8] proposed a variational inequality model of spatial price problem in logistics networks. However, the above work has the following two shortcomings: (1) The effects of topologies on price dynamics are not considered. (2) The community effects on the price propagation are not researched. In this paper, we intend to fill this gap by studying the evolution of space price in complex networks by considering the two cases mentioned above. 2. A model of traffic flow assignment The model we propose explores how the topology of agents’ influences the variation of the market price, and how such effects will propagate in the network. Cowan et al. [9] have modeled knowledge diffusion as a barter process in small-world networks, in which agents exchange different type of knowledge. We assume that the variation of price takes place through a trade exchange. In this model when two agents meet, the decision is simply whether to trade or to walk away [9]. For each agent this decision is based only on whether or not the other agent has something to offer, and not on the amount he has to offer. For a spatial price problem, each node acts as not only the supplier but also the demander. What role it plays is mainly dependent on the market price of the commodities. if the price (supply price) of node i, Ci , plus the cost (transportation cost) on link ij, Uij , is less than the price (demand price) of node j, Cj , there will exist a trade, and after this trade, the supply price will decrease. After the exchange price levels have become Cj (t + 1) =
Cj (t ) − α(Cj (t ) − Ci (t ))β Ci (t )
if Ci (t ) + Uij < Cj (t ), if Ci (t ) + Uij ≥ Cj (t ),
(a) (b)
(2)
where Ci (t ) indicates the price level of node i at time t. α and β are parameters. The trading process continues until all trading possibilities are exhausted, that is, there are no further profit of trading behaviors. 3. Price dynamics in different network topologies We construct manmade complex networks to describe the above dynamics of the price. Each supplier or demander is treated as a node. If there exists a possible trade between two nodes, there is a edge. Networks can be represented as graphs G = (V , E ) where V and E are the sets of vertices and edges. G is described by the N × N adjacency matrix eij . Define N as the size of the network. To facilitate comparison, we create regular, small-world, scale-free and random networks methods with the same size and total edges. The average degree of networks hki ≈ 3, and α = 0.4, β = 1. Transportation cost Uij is given randomly (0, 1]. In this paper we illustrate how our model works in practice by considering the artificially created regular lattice (RN), random graph (ER), small-world (SW) and scale-free (SF) networks. These networks are generated according to the procedure in Ref. [10], where self- and repeated links are forbidden. The small world network is generated by a RE rewired with probability 0.5. We simulate the evolution of a spatial price problem on the above networks. Each period a node, i and one of his neighbours j are chosen uniformly at random. This process continues until all possible trades have been made. Since with real numbers of price, levels never become identical, we consider them identical if they differ by less than one percent [9]: 0.99 <
Ci + Uij Cj
< 1.01,
∀i, j.
(3)
This condition guarantees that the process eventually stops. For the longer evolution, we give the step is 2000. Each curve corresponds to the average over 25 realizations of the network.
5854
Y.L. Li et al. / Physica A 387 (2008) 5852–5856
Fig. 1. The time series of average price level with different network structures.
Fig. 2. The time series of price variance with different network structures.
We use the average and variance of price levels to evaluate theqefficiency of the system in price propagation. Average
price level of systems is u = 1/N ×
P
Ci , and the variance is σ =
P
Ci2 /N − u2 . At the steady state we remove the time
index [9]. Different network structure has different effects on the distribution of the average price level and variance. Figs. 1 and 2 give the time series of the average price level and variance for different structures. The network size N = 100 (while being quite small, this does provide a sufficiently sized network to gain statistically significant attributes). From the plots we can see that the regular network decreases slowly and other three kinds of networks (small-world, scale-free and random networks) decrease quickly at first. The reason is that the average path length of regular network is long, but its clustering coefficient is higher. Therefore propagation in the regular network is slower than in other ones, but its progress continues longer. Small-world and scale-free networks have relatively short path lengths and propagation in the early periods is relatively fast. Random networks have the shorter path lengths and smaller clustering coefficient, therefore the price level decreases quickly. Because of the quick decrease of the average price level in early periods on regular networks, the variance of price will increase first and then fall. When the average price level remains steady, the variance starts to decline. We choose the typical network topology, SF, as the object to investigate the effects on the price dynamics with different network size. Selecting this particular structure is based on two reasons: (1) Scale-free is a most popular structure in the real world. A
Y.L. Li et al. / Physica A 387 (2008) 5852–5856
5855
Fig. 3. The time series of average price level with different network size N = 100, 200, 300, 400, 500, 600.
larger number of previous literature has concluded that most real economic networks display the scale-free characteristics; (2) In our previous work [8], we find that the supply chain distribution network exhibits the power-law form. The network size also has large influence on the price evolution. We report the time series of the average price level on scale-free networks for different network size in Fig. 3. From the plot, we can see that the average price level falls quickly for the small networks size. However, for a larger network the effects of price variation could only propagate a small part of nodes, therefore the average price level decrease slowly. We can also find that the time of phase transition is longer for the larger network. It is because that, with the increase of network size, the clustering becomes more and more strong.
4. Price dynamics in networks with community structure In the continuing flurry of research activity within physics and mathematics on the properties of networks, a particular recent focus has been the analysis of communities within networks [11]. Many networks in nature or social networks can be divided into some groups such that the connections within each group are dense, while connections between groups are sparse, which is also called community structure. There are many real-world networks which exhibit community structure, and community structures are supposed to play an important role in many real networks [12]. For example, communities in a citation network might represent related papers on a single topic [13]; communities on the web might represent pages on related topics [14]; communities in a biochemical network or neuronal system might correspond to functional units of some kinds [15]; communities also play an important role in information networks [16]. In trade networks some exchanges can occur in part of a tightly connected group, others can be completely isolated, while some others may act as bridges between groups [17,18]. We generate the scale-free network with community structure using our previous method [18]. Based on a previous measure of assortative mixing [17], Newman and Girvan have proposed of the quality P the measure Pof a particular division 2 of a network, which they called the modularity as follows [19]: Q = r (hrr − ar ), where ar = w hr w denotes the row (or column) sums which represent the fraction of edges that connect to nodes in community r and hr w is the fraction of edges in the original network that connect nodes in subset r with nodes in subset w . In a given network in which edges fall between nodes without regard for the communities they belong to, hr w = ar aw can be obtained. In our network model, we can adjust c-value to get networks with various strength Q of community structures. The larger the value of Q is, the most accurate a partition into communities will be. If the number of within-community edges is no better than random, we will get Q = 0. Values approaching Q = 1, which is the maximum, indicate strong community structure [12]. In this paper, the edge whose two nodes are in a same community is defined as the local edge and the bridge edge represents the edge whose two nodes belong to different communities. Different modularity can display different structures and functions of many real-world networks. Here, we analyze the effects of community modular Q on the spatial price dynamics. In Figs. 4 and 5, we analyze the variation of price levels with different modularity. The network and its parameters come from the reference [12]. From Fig. 4, we can see that the average price level will decrease sharply for a larger modularity Q , which means that for a strong community structure, the effects of price will spread quickly and easily. We can see that the time of phase transition is about 170, 300, and 400 for Q = 0.14, Q = 0.15 and Q = 0.16. This is because the strong community structure network has the relatively short path length.
5856
Y.L. Li et al. / Physica A 387 (2008) 5852–5856
Fig. 4. The time series of average price level in complex network with different community structure.
Fig. 5. The time series of price variance in complex network with different community structure.
5. Conclusion In conclusion, in this paper, the dynamic of the spatial price problem is studied. By simulating the variation of price in trade networks, we have found that network structures with short path lengths will suffer quick variations of price levels. However, the quantity of commodity flows are not considered in the exchange process, which may be our further work. Acknowledgments This paper is partly supported by NSFC of China (70501005), FANEDD (200763) and FYETF (111083). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
P. Erdös, A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17. D.J. Watts, S.H. Strogatz, Nature (London) 393 (1998) 440. A.-L. Barabási, R. Albert, Science 286 (1999) 509. M.J. Newman, e-print arXiv:cond-mat/0201433, 2002. Z. Dezso, A.-L. Barabási, Phys. Rev. E 65 (2002) 055103. L. Zhao, Y.-C. Lai, K. Park, N. Ye, Phys. Rev. E 71 (2005) 026125. A. Nagurney, A Variational Inequality Approach, Kluwer Academic Publishers, London, 1999. H.J. Sun, Z.Y. Gao, Proce. 2004 ICTTS, 555, Dalian, China. R. Cowan, N. Jonard, J. Econom. Dynam. Control 28 (2004) 1557. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Phys. Rep. 424 (2006) 175. M. Girvan, M.E.J. Newman, Proc. Natl. Acad. Sci. USA 99 (2002) 7821. C.G. Li, P.K. Maini, arXiv:physics/0510239, 2005, 26. S. Redner, Eur. Phys. J. B 4 (1998) 131. G.W. Flake, S.R. Lawrence, C.L. Giles, F.M. Coetzee, IEEE Comput. 35 (2002) 66. P. Holme, M. Huss, H. Jeong, Bioinformatics 19 (2003) 532. H. Xie, K.-K. Yan, S. Maslov, cond-mat/0409087. M.E.J. Newman, Phys. Rev. E 67 (2003) 026126. J.J. Wu, Z.Y. Gao, H.J. Sun, Phys. Rev. E 74 (2006) 066111. M.E.J. Newman, M. Girvan, Phys. Rev. E 69 (2004) 026113.