Fractional Interaction of Financial Agents in a Stock Market Network

Mehmet Ali Balcı 1
  • 1 Muğla Sıtkı Koçman University, 48000, Muğla, Turkey
Mehmet Ali Balcı
  • Corresponding author
  • Muğla Sıtkı Koçman University, Faculty of Science, Department of Mathematics, 48000, Muğla, Turkey
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Abstract

In this study, we present a model which represents the interaction of financial companies in their network. Since the long time series have a global memory effect, we present our model in the terms of fractional integro-differential equations. This model characterize the behavior of the complex network where vertices are the financial companies operating in XU100 and edges are formed by distance based on Pearson correlation coefficient. This behavior can be seen as the financial interactions of the agents. Hence, we first cluster the complex network in the terms of high modularity of the edges. Then, we give a system of fractional integro-differential equation model with two parameters. First parameter defines the strength of the connection of agents to their cluster. Hence, to estimate this parameter we use vibrational potential of each agent in their cluster. The second parameter in our model defines how much agents in a cluster affect each other. Therefore, we use the disparity measure of PMFGs of each cluster to estimate second parameter. To solve model numerically we use an efficient algorithmic decomposition method and concluded that those solutions are consistent with real world data. The model and the solutions we present with fractional derivative show that the real data of Borsa Istanbul Stock Exchange Market always seek for an equilibrium state.

1 Introduction

Complex systems are mathematical structures involving interacting agents at different levels. These interactions emerge from the financial, chemical, social, and computer system entities. In the realm of computational finance, a financial market can be viewed as interacting group of boundedly-rational agents and its fluctuation represent strong nonlinearity and persistent memory. The mathematical tools such as network and graph theories can be used to understand and analyze these systems [1, 2]. There are several models expressed in the terms of differential equations in biological complex systems. For instance, the virus models that classify individuals and hosts can be used to analyze spread of a contagion [3, 4, 5]. Besides, bursting electrical activity in the pancreatic β-cell, population models, and unilingual-bilingual interactions [6], and the interaction of biological species living together [7] can also be modelled by differential equations. However, these models are not only restricted to biological systems. Recent studies show that complex systems involving financial agents have similar structures as systems involving biological agents [8, 9]. Therefore, it is reasonable to model the interaction of financial agents as we model the interaction of biological agents.

In a financial market, heterogeneous agents interact through simple investment strategies driven by the investors. In a perfect rational market, information is transmitted continuously and agents adopt their behavior accordingly. Besides, asset prices reflect economic fundamentals. Agents are considered as they only interact though price system. Hence, a complex network where agents are expressed as vertices and edges are formed by the correlation of price fluctuations emerges as a powerful mathematical tool to model such a financial system in the traditional way. In contrast to Keynesian approach, such traditional way takes account of prices of assets as they are only driven by market fundamentals and the role of market psychology is neglected. Even though we use the traditional way to express our model in this study, we need to point out two important classes of investors which are called chartists and fundamentalists in the traditional way of interaction of agents [10]. Chartists tend to look for simple patterns such as trends, past prices, and base and make their investment upon those patterns. Conversely, fundamentalists make their decisions upon the expectation of asset price as moving towards its fundamental value. The fundamentalist investors buy or sell assets that are under or overvalued. The market tends to be dominated by one of fundamentalists or chartists. However, since the behavior of the agents is persistent, the majority of agents switches to the other view at certain point [11, 12].

Our approach in this study aims to model interaction of agents in a stock market network in the traditional way. We first use a threshold method to construct a network model where vertices are the companies operating in a stock market and edges are formed by the correlation distance of daily logarithmic returns of stock prices. The dimensionality of the resulting network model would be really high and the patterns that yield power law of degree distributions would be disappeared, however it would involve optimally many edges to characterize community structures. By maximization of the modularity of edges in the network, we can cluster agents into densely connected vertex sets. Then, each cluster has its subdominant ultra-metric structure that is a hierarchical structure with at least one leading actor. We set the number of cluster to two, then assume the investors, even chartists or fundamentalists, start to invest one cluster regarding to factors such that merging, capital augmentation, public flotation, etc. Then, the investors in the other cluster start to sell assets to get the capital to invest increasing valued assets. Therefore, the price fluctuation spread within each cluster by conducting leading actors. However, at certain time, the profit realizations start within the asset price increasing cluster, and then the capital emergent by the profit realization is used to invest assets in price decreasing cluster. Eventually, the two clusters find an equilibrium state.

In the complex network model of financial agents the interactions are modelled by the correlations of long time series [13, 14, 15, 16, 17, 18, 19]. Beside the other types of complex systems, financial systems have the strong memory and heredity properties. Therefore, while using differential equations in financial models it is much more useful to get fractional calculus involved. Fractional calculus is the extension of the integer order differential and integral operators to fractional orders [20,21]. The dynamic memory in a financial process can be defined as the averaged characteristic that describes the dependence of a process in the past. Such memory assumes withitness of financial agents about the history of the process. In formal way, the information on the state of the process {t, χ(t)} does not only affect the behavior of financial agents, but also the information about the process state {τ, χ(τ)} also has effect at τ ∈ [0,t]. This effect is related with the fact that the change of the factors can leads to different amount of change in indicators that is there exist multivalent dependencies among variables. One type of such memory of financial agents is called the fading memory and have range application area in physical sciences [22, 23, 24, 25, 26, 27, 28, 29, 30]. In this study, we assume that financial agents can remember the previous changes of investments and the impact of these changes on the output by following fading memory by using Caputo’s definition of fractional derivative.

In this study, we present our model with fractional derivative as similar as the model describing biological species living together. This model of biological species is first given in [7] as the following usual integro-differential equations:

dμ1dt=μ1(t)[k1γ1μ2(t)tT0tf1(tτ)μ2(τ)dτ],k1>0
dμ2dt=μ2(t)[k2+γ2μ1(t)+tT0tf2(tτ)μ1(τ)dτ],k2>0.

Several solution methods are also presented to study this model [31, 32, 33]. The characterization of the fractional order of the model is also studied in [34].

The rest of the paper is organized as follow: In Section 2, we present the preliminaries about to fractional calculus and graph theoretical concepts that we use throughout the paper. We start our analysis by first determining the financial agents in Section 3. The stock market we choose to study is Borsa Istanbul Stock Exchange Market (XU100). The agents are the companies operating in XU100 and expressed with the time series of the time span of working days from 2013 to 2015. Afterwards, we determine the clusters of financial agents that have strong correlations by using high modularity method. Afterwards we introduce the Adomian decomposition method for solving the system numerically. In Section 5, we present the results by solving the model with Adomian decomposition method. And finally, in Section 6, we deeply discuss the computational results.

2 Preliminaries

Fractional calculus is an efficient mathematical tool to express complex system phenomenon which involve memory effect. Hence, we use fractional derivatives and integrals to study the model we present in this study. In this section, we give some basics about the fractional calculus in Caputo sense. We also introduce some basics about the graph theory which is the fundamental tool for network modelling. Throughout the paper we let Γ to represent Gamma function that is an extension of the factorial function.

2.1 Fractional Calculus

The generalization of the integer order differentiation and integration to the fractional order is called fractional calculus [20, 21]. The basic definitions and properties of fractional calculus theory is given as follows:

Definition 1

[20] For f (x) ∈ 𝒞 (a,b) and n − 1 < α ≤ n, the Caputo fractional derivative operator of order α is given as

aCDtαf(t)=1Γ(n1)atf(n)(τ)(tτ)α+1ndτ.

Throughout this paper, we denote the Caputo fractional derivative operator as aCDtα=Daα . We also let a = 0 since our formulation only involves the initial conditions as t = 0.

Definition 2

[20] The Riemann-Liouville fractional integral operator of order α ≥ 0 of a function f is defined as

Jαf(t)=1Γ(α)0t(xτ)α1f(τ)dτ,α>0,t>0J0f(t)=f(t).

The several properties of the Riemann-Liouville fractional integral operator can be found in [35, 36, 37]. Since the Caputo fractional derivative allows traditional initial and boundary conditions to be included in the formulation of the problem [38], we present our model in the sense of Caputo fractional derivative. By the introduction of the Jα, the Daα can also be expressed as

D0αf(t)=JmαDmf(t),
where m − 1 < αm, m ∈ ℕ,t > 0.

Also, the following two basic properties of the entwined relations among Caputo and Riemann-Liouville fractional operators are needed to present the solution of the fractional differential equations.

Lemma 1

[35] If m − 1 < αm, m ∈ ℕ, then

D0αJαf(t)=f(t)
and
JαD0αf(t)=f(t)k=0m1f(k)(0+)tkk!,t>0.

2.2 Graph Theory

The real–world problems are often expressed with the relations of interacting individuals. One of the efficient mathematical tools to represent such relations is the simple graphs. In the Stock Market Networks, interactions of financial agents can be modelled by simple graphs. Let V be the set of the interacting individuals and E be the set of relations, then a simple graph is a tuple G = (V,E). Here we call V as the vertex set and E as the set of edges. The each element of E is the unordered pair of vertices as ek = (vi,vj), where vi,vjV for all i, j,k ∈ ℕ. The number of edges incident to a vertex v is called as the degree of v and we denote the degree as dv.

A sequence of edges between the vertices vi and vj is called a path, and if there is a path between any vertices of the graph G, then G is called as connected. If there is an edge between all elements of V, then G is called as a complete graph. The k-clique of the graph G is the complete subgraph of G which involves k-many vertices of G.

For the simple graph G = (V,E) with the unordered edges, a binary matrix which has the entities as

AG(i,j)={1,if(vi,vj)E0,otherwise
is used to represent the relations and called as adjacency matrix. AG is symmetric by definition.

Now let |V| = n and DG be the diagonal degree matrix of G defined as DG = diag[dv1,...,dvn]. The matrix LG = AG − DG is called the Laplacian matrix of G. The spectrum of the LG encodes structural properties of G. The one that we use in this study is helpful to construct a threshold network of the financial agents. All eigenvalues of LG are positive semi-definite with the least one 0. The multiplicity of the 0 eigenvalue equals the connected components number of G [39].

Several types of subgraphs also involve the information about the network which is expressed as a simple graph G = (V,E). One of them is the tree structure that has minimum weight. Such subgraphs are called as Minimum Spanning Tree (MST) and involve the junction vertices which are dominant in the flow of information and comes up with subdominant ultra-metric structure [40, 41]. In the case of financial agents are the vertices of the network, MST gives the hierarchical structure of the financial network [42, 43, 44]. A planar graph is a simple graph that can be embedded on the plane, which is none of the graph edges intersect. Trees like planar graph that involve cliques are also useful to extract information about the network. Such tree like planar graphs have the same hierarchical structure as MST but they contain larger amount of information about the relation among the interacting agents [45, 46, 47]. In [45], authors present a method to obtain a planar graph with maximum non-crossing edges among the agents of a network and called it Planar Maximally Filtered Graph (PMFG).

3 Model

This study involves 93 companies that have been operating in Borsa Istanbul 100 Stock Exchange Index (XU100) from January 2013 to January 2015. The Pearson correlation coefficient of time series assumes the equality of the length of time series. Hence, even though XU100 has 100 operating companies, we only consider 93 of them which have the equal time length. Trading hours for the stocks are held by two sessions on business days with mid-day break, and one session in some official holidays [44, 48]. The tickers of the companies operating in XU100 and that are considered in this study is given in Table 1. The more details on the data can be found in [44].

Table 1

Sectors of each considered stock

Financials

AKBNK, SKBNK, SNGYO, TSKB, TEKST, TRGYO, VKGYO, ALGYO, ISGYO, GARAN, ALBRK, GLYHO, ISCTR, YKBNK, SAHOL, GOZDE, HALKB, VAKBN, ECZYT, SAFGY, EKGYO, SAHOL, GSDHO
Industrials

ASELS, TAVHL, TKFEN, TTRAK, CLEBI
Consumer Discretionary

ASUZU, TKNSA, TOASO, YAZIC, AKSA, ARCLK, GSRAY, KARSN, THYAO, BRISA, DOAS, FENER, MNDRS, METRO, VESBE, ADEL, BJKAS, NTTUR, GOODY, OTKAR, TMSN, EGEEN, FROTO, IHLAS
Energy

AYGAZ, TUPRS, IPEKE, KCHOL
Technology

NETAS, VESTL
Materials

SASA, AFYON, ANACM, BAGFS, CIMSA, KONYA, KOZAA, ERBOS, KRDMD, PRKME, SISE, ALKIM, TRKCM, GUBRF, KOZAL, BRSAN, KARTN, PETKM, GOLTS, EREGL
Communications

TTKOM, TCELL, DOHOL, HURGZ
Consumer Staples

AEFES, CCOLA, BIZIM, ECILC, BIMAS, MGROS, SODA, ULKER
Utilities

AKSEN, ALARK, TRCAS, ZOREN, ENKAI

The data we use is available with sessional closure price, therefore we calculate sessional closure price logarithmic return as

Cli=logPi(t+1)logPi(t),
where Pi(t) is the closure price of the i-th stock at the session t. To represent the relation between stock pairs, we use the Pearson correlation coefficient of stocks as
ρij=<CliClj><Cli><Clj>(<Cli2><Cli>2)(<Clj2><Clj>2),
where < ··· > is the temporal average performed on the trading days.

Pearson correlation coefficient varies between −1 and 1. ρij = 1 indicates that the stocks i and j are completely correlated whilst ρij indicates that the stocks i and j are completely uncorrelated [42]. Hence, it is also possible to introduce a new distance dCorr:=2(1ρij)/2 as in [13,44]. We can conclude that if dCorr(i, j) = 0, then the stocks i and j are completely correlated, and if dCorr(i, j) = 1, then the stocks i and j are completely uncorrelated.

This distances based on Pearson correlation is helpful to us for edge determination on the network. By using an empirical threshold value among the stocks, it is possible to determine edges representing strong relations as

AG(i,j)=1iffdCorr(i,j)ThV,
where T hV is the threshold value. The threshold value can be determined by the subdivision of the interval [0,1], where the boundaries are the extremal values of dCorr, into h many subintervals. The details on the algorithm of network construction and computational complexities can be found in [13, 44].

The model we proposed in this study first deal with the two clusters of financial agents of the network. In the literature, the cluster of densely connected vertices of a network is called graph communities [49, 50]. This densely connection is internal and can be used to analyze the relations that are represented by edges on the network. There are several methods to detect communities in a network [51, 52, 53, 54]. To find the graph communities in the network; we use the Modularity Maximization Method which is based on the maximizing the Newman modularity index [51] defined as

Q=i=1NC[Ekm14m2(jVkdj)2],
where Ek is the number of edges in the k-th module, NC is the total number of modules, m is the total number of edges and dj is the vertex degree. Since the resulting communities are non-overlapping and this method let us to determine final number of the communities, we choose it as an efficient tool.

Now let us consider the two communities of financial agents with the total number of investments µ1(t) and µ2(t), respectively, at time t. Let us assume the investment in the first community is increasing with the coefficient of increase k1 and in the second community is decreasing with the coefficient of decrease k2. Both coefficients k1 and k2 are positive reals. If the two communities are left separate; i.e. they are non-overlapping, then the fractional growth of the first can be represented by

D0αμ1(t)=k1μ1(t)
and the decline of the second community can be represented by
D0αμ2(t)=k2μ2(t).

The neoclassical liberal economy states that markets always look for the equilibrium state. Hence, if we put these two communities together in the corresponding stock market environment, the decrease of the rate of the increase of first community is proportional to µ2(t) and vice versa. Therefore, it is reasonable to assume the increase and decrease coefficient as

k1=k1γ1μ2(t)
and
k2=k2+γ2μ1(t),
where γ1 and γ2 are the proportionality constants which depend on other investor behavior, respectively. The actual decrease and increase of the investments in the communities are due not only to the presence of the other community but also to all previous presences for the whole time interval tT0 < τ < t, where T0 is the finite heredity duration of both communities. In addition to the present γ1 and γ2 factors, we may have the record of decrease as f1(τ) and increase as f2(τ). Therefore, by considering the heredity duration of both communities, the total decrease in k1 in the time interval T0 is
δk1=tT0tf1(ts)μ2(s)ds
the total fractional increase in k2 is
δk2=tT0tf2(ts)μ1(s)ds.

Now, by considering effective values of the k1 and k2 and the equations 38, the fractional model with fading memory of the equilibrium state of the two communities of financial agents in same stock market can be given as the system of fractional integro-differential equations as follows:

D0αμ1(t)=μ1(t)[k1γ1μ2(t)tT0tf1(ts)μ2(s)ds],
D0αμ2(t)=μ2(t)[k2+γ2μ1(t)+tT0tf2(ts)μ1(s)ds],
μ1(0)=N1,  μ2(0)=N2,
where N1 and N2 are the initial conditions.

4 Method

In this section, we present the graph theoretical methods to determine parameters k1, k2, γ1, and γ2 of the fractional integro-differential equation model in the initial value problem 911 and the numerical solution that is based on Adomian Decomposition Method.

4.1 Parameter Estimation

The parameters in network models can be estimated by using graph theoretical concepts. For the system 911, to determine the coefficients of increase and decrease, we use an interpretation of the displacement of a vertex in a network from its equilibrium state while the network is under a thermal bath. This thermal bath can be seen as the change of investment strategies on given network. This procedure is called vibrational potential and first presented in [55]. The later studies consider vibrational potential as an efficient measure to vertex centrality [56, 57, 58]. The main idea to compute the vibrational potential of a network is embedding vertices to n-dimensional Euclidean space by using the Moore-Penrose pseudo-inverse of the Laplacian, where n = |V|. Within the hierarchical structure of each community, each stock market tends to be adjacent to junction vertices. Therefore, the change of investment on the junction vertices directly affect the corresponding leaves. Therefore, we correspond the increase/decrease coefficients with vibrational potential of the network. However, instead of direct computing vibrational potential of the network, we compute vibrational potential of each vertex respect to its neighborhood graph.

For this purpose we present the vertex displacement in vibrational potential of a vertex respect to its neighboring vertices with

V(xv)=k2xvTLNxv,
where k is the spring constant, LN is the Laplacian of the neighboring graph GN of the vertex v in G, and xv is the vector whose i-th entry is the displacement of v. The mean displacement of the vertex v can be computed with the reverse temperature β as
Δxi=xi2P(xv)dxv,
where the probability distribution P(xv) is
P(xv)=exp(βk2)/exp(βk2)dxv.

Similarly the displacement correlation of the vertices in same neighborhood can be defined as

<xi,xj>=xixjP(xv)dxv,
where < ··· > is the thermal average.

Let 0=λ1N<λ2NλnN be the spectrum of LN respect to eigenvalues λμN . Since the quantity respect to 0 eigenvalue is the center of mass, the 0 eigenvalue does not affect the vertex displacement. Then the integral measure can be transformed by

dxv=i=1ndxi=|detUN|i=1ndξi=dξv
where UN is the matrix formed by the orthogonal eigenvectors of LN. By the introduction of this transform the new probability distribution can be obtained as
P(ξv)=exp(βk2ξvTΛNξv)/exp(βk2ξvTΛNξv)dξv=exp(βk2ξvTΛNξv)/μ=1nexp(βk2λμNξμ2),
where the diagonal matrix ΛN involves the eigenvalues λμN .

Since 0 eigenvalue does not effect the vertex displacement, we can remove the component µ = 1 from the Equation 17, and the probability distribution can be computed as

P(ξv)=exp(βk2ξvTΛNξv)/μ=2nexp(βk2λμNξμ2)=exp(βk2ξvTΛNξv)/μ=2n2πβkλμ.

Hence, by using the probability distribution obtained in Equation 18, it is possible to compute the Equation 13 as

Δxi=<xi2>=i/μ=2n2πβkλμ,
where
i=j=2n(Uijξj)2exp(βk2λjξj2)dξj×μ=2,μjnexp(βk2λμξμ2)dξμ=μ=2n2πβkλμ×j=2nUij2βkλj.

Therefore, the mean displacement of a vertex from its neighborhood can be computed as

Δxi=j=2nUij2βkλj.

By the introduction of the Moore-Penrose pseudo inverse LijN+ of LN as in [59, 60], it is also possible to compute the mean displacement of a vertex from its neighborhood as

Δxi=1βk(LiiN+).

We also note that the displacement correlation of the vertices in the same neighborhood that is given in the Equation 15 can be computed in the terms of Moore-Penrose pseudo inverse as

<xi,xj>=1βk(LijN+).

Another parameters we need to estimate in the system 911 are γ1 and γ2 which are the proportionality values. The proportionality values control how much financial agent in the same community affect each other. Hence, they can be measured as how strong each agents are connected internally. This measurement is naturally arise from the PMFG of each communities. PMFG structure allows cliques which are the topological subgraph structure representing strong relations. Since PMFG also has the information about the hierarchical structure, it is reasonable to measure internal connectedness of communities by using PMFG. For this measurement we follow the way presented in [45]. The mean disparity measurement < y > of a PMFG can be defined as the mean of

y(i)=ij,jClique(dCorr(i,j)si)2,
where i is the generic element of the clique and
si=ij,jCliquedCorr(i,j).

4.2 Adomian Decomposition Method

It is well known that many nonlinear differential equations exhibit strange attractors and their solutions have been discovered to move toward strange attractors. If these strange attractors are examined deeply, it can be seen that these are fractals. Therefore, we aim to deal with fractal nonlinear differential equations rather than classical forms of them. Hence we shall extend the Adomian decomposition method to be used for solving fractional nonlinear equations. For the solution of the system 911, we use an efficient decomposition method for approximating the solution of systems of fractional integro-differential equation that are given in Caputo sense. The approximate solutions are calculated in the terms of a convergent series as in [34].

Now let us consider the system 911 with 0 < α 1. By following the decomposition idea we may state that

D0αμ1(t)=m1(t),
D0αμ2(t)=m2(t).

This equations lead us to the integral equations

μ1(t)=μ1(0)+Jα(m1(t))
=N1+J0αm1(s)ds,
μ2(t)=μ2(0)+Jα(m2(t))
=N2+J0αm2(s)ds,
m1(t)=μ1(t)(k1γ1μ2(t)tT0tf1(ts)μ2(s)ds)=k1(N1+0tm1(s)ds)μ1(t)(γ1μ2(t)tT0tf1(ts)μ2(s)ds),
m2(t)=μ2(t)(k2+γ2μ1(t)tT0tf2(ts)μ1(s)ds)=k2(N2+0tm2(s)ds)+μ2(t)(γ2μ1(t)+tT0tf2(ts)μ1(s)ds).

Afterwards, the Adomian process will be as follows:

μ1,0=N1,μ2,0=N2,
m1,0=k1N1,m2,0=k2N2,
μ1,j+1=J0αm1,j(s)ds,μ2,j+1=J0αm2,j(s)ds,
m1,j+1=k10tm1,j(s)dsγ1k=0jμ1,k(t)μ2,jk(t)tT0tf1(ts)(k=0jμ1,k(t)μ2,jk(t))ds
m2,j+1=k20tm2,j(s)dsγ2k=0jμ1,k(t)μ2,jk(t)+tT0tf2(ts)(k=0jμ1,k(t)μ2,jk(t))ds.

5 Results

In order to study the proposed model in the Borsa Istanbul Stock Exchange, we first apply our algorithm to the data set to obtain stock market network. For the fraction size h = 10000, the algorithm determines the control parameter as 0.6854. The vertices are sorted from 1 to 93 by the alphabetical order in Table 1. The formed network is presented in Figure 1. The vertices with maximum vertex degree are ADEL, AKBNK, AKSEN, ALBRK, ALGYO, ALKIM, ARCLK, ASELS, BRISA, DOAS, GARAN, GOLTS, HALKB, ISCTR, KARTN, KCHOL, KONYA, KRDMD, MGROS, OTKAR, PRKME, SAHOL, SISE, SNGYO, TKFEN, TKNSA, TMSN, TOASO, TRCAS, VAKBN, and YKBNK with the degree number 92, and the vertex with minimum vertex degree is VKGYO. The maximum and mean correlation distances among the companies are 0.7211 and 0.5798, respectively. Hence it can also be concluded this network has strong internal connectedness. The correlation distance matrix of the agents are presented in Figure 2 with temperature mapping.

Fig. 1
Fig. 1

The network of XU 100 with the threshold value 0.6854.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 2
Fig. 2

The matrix of correlation distance among the financial agents of XU 100. The darker points are closer to 1 while the lighter ones are closer to 0.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

To determine the two non-overlapping clusters of financial actors we use the high modularity method. The resulting communities are presented in Figure 3. The agent numbers of each community give us the initial conditions as N1 = 66 and N = 27.

Fig. 3
Fig. 3

The community plot of the resulting network.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

As aforementioned, parameters of the model described by the system 911 are obtained by vibrational potential respect to neighborhood graphs and mean disparity measures of each communities. We need to note that vibrational potentials of each vertices are tend to form internal clusters; i.e., some of them have higher values and some of them have lower values. Therefore, while determining k1 and k2 values, we choose the mean value of each vibrational potentials. Afterwards forming the PMFG of each community, it becomes possible to obtain disparity measures respect to 4-cliques, which are the representation of the strongest connections. The resulting parameters γ1 = 0.3342 and γ2 = 0.3388. The parameters are close to the value 1/3 which also states that the clustering method we choose is reasonable [45]. To interpret the results, we present MSTs and PMFGs of both two clusters in Figures 45

Fig. 4
Fig. 4

MST (above) and PMFG (below) filtering of Community 1.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 5
Fig. 5

MST (above) and PMFG (below) filtering of Community 2.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

In the light of these computed parameters we may now state the system 911 with T0=100, f1(t) = f2(t) = et as

D0αμ1(t)=μ1(t)[5.950.3342μ2(t)t100te(ts)μ2(s)ds],
D0αμ2(t)=μ2(t)[4.39+0.3388μ1(t)+t100te(ts)μ1(s)ds],
μ1(0)=66,  μ2(0)=27.

By applying the Adomian process obtained in Section 4.2, the solution of the initial value problem 3739 can be obtained as

μ1(t)=N1+k1N1tαΓ(1+α)+N1tα((1+eT0)N2+4αk12πtα/Γ(0.5+α)N2γ1)Γ(1+α),
μ2(t)=N2+k2N2tαΓ(1+α)+N2tα((1eT0)N1+4αk22πtα/Γ(0.5+α)N1γ2)Γ(1+α).
with a three-term approximation.

The plots of the solution functions (4041) are presented in Figures 615 for the different α values.

Fig. 6
Fig. 6

The solutions of the initial value problem 4041 for α = 0.1

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 7
Fig. 7

The solutions of the initial value problem 4041 for α = 0.2

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 8
Fig. 8

The solutions of the initial value problem 4041 for α = 0.3

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 9
Fig. 9

The solutions of the initial value problem 4041 for α = 0.4

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 10
Fig. 10

The solutions of the initial value problem 4041 for α = 0.5

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 11
Fig. 11

The solutions of the initial value problem 4041 for α = 0.6

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 12
Fig. 12

The solutions of the initial value problem 4041 for α = 0.7

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 13
Fig. 13

The solutions of the initial value problem 4041 for α = 0.8

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 14
Fig. 14

The solutions of the initial value problem 4041 for α = 0.9

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

Fig. 15
Fig. 15

The solutions of the initial value problem 4041 for α = 0.10

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00030

6 Conclusions

Ordinary differential equations are the most common mathematical tool to represent real world problems. But ordinary differential equations become less effective whenever the problem involves memory effect. Complex systems that representing financial agents have the memory effect, hence it is reasonable to model such systems by using the idea of fractional differential.

In this study, we propose a model which is represents the fractional interaction of financial agents. The interaction of the agents is determined within a complex network of a stock market. We express the model as a system of fractional integro-differential equations in Caputo sense. Hence, we keep the fading memory of the financial interaction. Our model considers two clusters of agents where one cluster tends to get investment flow. To determine the clusters we use maximization of the edge modularity in stock market network. The resulting clusters are consistent with the structure of Borsa Istanbul. Both MST and PMFG filtration of the clusters involve agents of Financials sector as the leading elements. To estimate the parameters of the model, we use graph theoretical concepts such as vibrational potentials and disparity measure of respected PMFGs.

By the computed parameters, we use Adomian decomposition method to obtain a solution of the model. This solution show us that for different fractional dimensions α, the model always reaches to an equilibrium state. For the lesser values of fraction rate α, agents reach to an equilibrium state relatively slower. Besides, the flows of investments tend to be in same characteristics. For the greater α values, agents reach to an equilibrium state faster and similarly the flows of investments tend to be in same characteristics. The model keeps the memory of the investment in best for 0.4 ≤ α ≤ 0.6. This results shows us that the fractional interaction of financial agents is consistent with reality when autocorrelations are discarded.

As neoclassical liberal theory of economics states, markets always seek for an equilibrium state. Hence, the model we present with fractional derivative is consistent with the real data of Borsa Istanbul Stock Exchange Market. We also believe that these kind of models can provide useful information for understanding and prediction of the global economic crisis.

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    Babolian E., Biazar J. (2002) Solving the problem of biological species living together by Adomian decomposition method. Applied Mathematics and Computation, 129(2): 339–343

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    Yousefi S. A. (2011) Numerical solution of a model describing biological species living together by using Legendre multiwavelet method. International Journal of Nonlinear Science, 11(1): 109–113

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    Van Dam E. R., Haemers W. H. (2003) Which graphs are determined by their spectrum?, Linear Algebra Appl., 373: 241–272

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    Cyman J., Lemańska M., Raczek J. (2006) On the doubly connected domination number of a graph, Open Mathematics, 4(1): 34–45

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    Estrada E., Hatano N. (2010) A vibrational approach to node centrality and vulnerability in complex networks. Physica A: Statistical Mechanics and its Applications, 389(17): 3648–3660

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    Estrada E., Hatano N., Benzi M. (2012) The physics of communicability in complex networks. Physics reports, 514(3): 89–119

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    Ranjan G., Zhang Z. L. (2013) Geometry of complex networks and topological centrality. Physica A: Statistical Mechanics and its Applications, 392(17): 3833–3845

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    Feng L., Bhan B. (2015) Understanding dynamic social grouping behaviors of pedestrians. IEEE Journal of Selected Topics in Signal Processing, 9(2): 317–329

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    Davidson J., Savaliya S., Shah J. J. (2013) Least-squares fit of measured points for square line-profiles, Procedia CIRP, 10: 203–210

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    Grant W. S., Voorhies R. C., Itti L. (2013) Finding planes in LiDAR point clouds for real-time registration, In Intelligent Robots and Systems (IROS), 2013 IEEE/RSJ International Conference, 4347–4354

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Haken H., Jumarie G. (2006) A macroscopic approach to complex system, Springer-Verlag, Berlin, Heidelberg, New York, 3th Edition

  • [2]

    Milo R., Shen-Orr S. (2002) Itzkovitz S., Kashtan N., Chklovskii D., Alon U., Network motifs: simple building blocks of complex networks. Science, 298(5594): 824–827

  • [3]

    Castellano C., Pastor-Storras R. (2010) Thresholds for epidemic spreading in networks, Phys. Rev. Kett., 105(21): 218701

  • [4]

    Read J. M., Eames K. T., Edmunds W. J. (2008) Dynamic social networks and the implications for the spread of infectious disease, J. R. Soc. Interface, 6–6(26): 1001–1007

  • [5]

    Keeling M. J. (2008) Rohani P, Modelling Infectious Diaseases in Human and Animals, Princeton University Press

  • [6]

    Busenberg S., Martelli M. (Eds.) (1990) Differential Equations Models in Biology, Epidemiology and Ecology, Proceedings of a Conference Held in Claremont California, January 13–16 (Vol. 92). Springer Science and Business Media, 2002

  • [7]

    Jerri A. J. (1999) Introduction to integral equations with applications

  • [8]

    Rosenthal J., Gilliam D. S. (Eds.) (2012) Mathematical systems theory in biology, communications, computation and finance (Vol. 134). Springer Science and Business Media

  • [9]

    Capasso V., Bakstein D. (2015) An introduction to continuous-time stochastic processes: theory, models, and applications to finance, biology, and medicine. Birkhäuser

  • [10]

    Hens T., Schenk-Hoppé K. R. (Eds.) (2009) Handbook of financial markets: dynamics and evolution. Elsevier

  • [11]

    Deffuant G., Neau D., Amblard F. (2000) Weisbuch G, Mixing beliefs among interacting agents. Advances in Complex Systems, 3(01n04): 87–98

  • [12]

    Samanidou E., Zschischang E., Stauffer D., Lux T. (2007) Agent-based models of financial markets. Reports on Progress in Physics, 70(3): 409.

  • [13]

    Balcı M. A. (2016) Fractional virus epidemic model on financial networks. Open Mathematics, 4(1): 1074–1086

  • [14]

    Onnela J. P., Kaski K., Kertész J. (2004) Clustering and information in correlation based financial networks. The European Physical Journal B, 38(2): 353–362

  • [15]

    Kumar S., Deo N. (2012) Correlation and network analysis of global financial indices. Physical Review E, 86(2): 026101

  • [16]

    Mizuno T., Takayasu H., Takayasu M. (2006) Correlation networks among currencies. Physica A: Statistical Mechanics and its Applications, 364:336–342

  • [17]

    Balcı M. A., Akgüller, Ö. (2016) Soft Vibrational Force on Stock Market Networks. Library Journal, 3, e3050.

  • [18]

    Akgüller Ö., Balcı, M.A. (2018) Geodetic convex boundary curvatures of the communities in stock market networks. Physica A: Statistical Mechanics and its Applications, 505, pp.569–581.

  • [19]

    Akgüller, Ö (2017) Geometric Soft Sets. Hittite Journal of Science & Engineering, 4(2).

  • [20]

    Podlubny I. (1999) Fractional Differential Equation, Academic Press, New York

  • [21]

    Gorenflo R., Mainardi F. (1997) Fractional calculus: integral and differential equations of fractional order, Springer, New York

  • [22]

    Wang C. C. (1965) The principle of fading memory. Archive for Rational Mechanics and Analysis, 18(5): 343–366

  • [23]

    Tarasova V. V., Tarasov V. E. (2018) Concept of dynamic memory in economics. Communications in Nonlinear Science and Numerical Simulations, 55:127–145

  • [24]

    David S. A., Fischer C., Machado J. T. (2018) Fractional electronic circuit simulation of a nonlinear macroeconomic model. AEU-International Journal of Electronics and Communications, 84: 210–220.

  • [25]

    Babenkov M. B., Vitokhin E. Y. (2017) Thermoelastic Waves in a Medium with Heat-Flux Relaxation. Encyclopedia of Continuum Mechanics, 1–10

  • [26]

    Bas E., Acay B., Ozarslan R. (2019). The price adjustment equation with different types of conformable derivatives in market equilibrium. AIMS Mathematics, 4(3): 805–820.

  • [27]

    Ozarslan R., Ercan A., Bas E. (2019). Novel Fractional Models Compatible with Real World Problems. Fractal and Fractional, 3(2), 15.

  • [28]

    Bas E., Acay B., Ozarslan R. (2019). Fractional models with singular and non-singular kernels for energy efficient buildings. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(2), 023110.

  • [29]

    Bas E., Ozarslan R., Baleanu D., Ercan A. (2018). Comparative simulations for solutions of fractional Sturm–Liouville problems with non-singular operators. Advances in Difference Equations, 2018(1), 350.

  • [30]

    Bas E., Ozarslan R. (2018). Real world applications of fractional models by Atangana aleanu fractional derivative. Chaos, Solitons & Fractals, 116, 121–125.

  • [31]

    Shakeri F., Dehghan M. (2008) Solution of a model describing biological species living together using the variational iteration method. Mathematical and Computer Modelling, 48(5): 685–699

  • [32]

    Babolian E., Biazar J. (2002) Solving the problem of biological species living together by Adomian decomposition method. Applied Mathematics and Computation, 129(2): 339–343

  • [33]

    Yousefi S. A. (2011) Numerical solution of a model describing biological species living together by using Legendre multiwavelet method. International Journal of Nonlinear Science, 11(1): 109–113

  • [34]

    Momani S., Qaralleh R. (2006) An Efficient Method for Solving Sysyems of Fractional Integro-Differential Equations, Computers and Mathematics with Applications, 52: 459–470

  • [35]

    Luchko A., Groneflo R. (1998) The initial value problem for some fractional differential equations with the Capoto derivative, Preprint Series A08-98, FAchbreich Mathematik und Informatik, Freie Universitat Berlin

  • [36]

    Miller K. S., Ross B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York

  • [37]

    Oldhami K. B., Spanier J. (1974) The Fractional Calculus, Academic Press, New York

  • [38]

    Ren R. F., Li H. B., Jiang W., Song M. Y. (2013) An efficient Chebyshev-tau method for solving the space fractional diffusion equations. Applied Mathematics and Computation, 224: 259–267

  • [39]

    Van Dam E. R., Haemers W. H. (2003) Which graphs are determined by their spectrum?, Linear Algebra Appl., 373: 241–272

  • [40]

    Cyman J., Lemańska M., Raczek J. (2006) On the doubly connected domination number of a graph, Open Mathematics, 4(1): 34–45

  • [41]

    Graham R. L., Hell P. (1985) On the history of the minimum spanning tree problem, Annals of the History of Computing, 7(1): 43–57

  • [42]

    Mantegna R. N. (1999) Hierarchical structure in financial markets. The European Physical Journal B-Condensed Matter and Complex Systems, 11(1):193–197

  • [43]

    Naylor M. J., Rose L. C., Moyle B. J. (2007) Topology of foreign exchange markets using hierarchical structure methods. Physica A: Statistical Mechanics and its Applications, 382(1): 199–208

  • [44]

    Balcı, M. A. (2018). Hierarchies in communities of Borsa Istanbul Stock Exchange. Hacettepe Journal of Mathematics and Statistics, 47(4), 921–936.

  • [45]

    Tumminello M., Aste T., Di Matteo T., Mantegna R. N. (2005) A tool for filtering information in complex systems. Proceedings of the National Academy of Sciences of the United States of America, 102(30): 10421–10426

  • [46]

    Barfuss W., Massara G. P., Di Matteo T., Aste T. (2016) Parsimonious modeling with information filtering networks. Physical Review E, 94(6): 062306

  • [47]

    Wang G. J., Xie C., He K., Stanley H. E. (2017) Extreme risk spillover network: application to financial institutions. Quantitative Finance, 1–17

  • [48]

    Aksu M., Kosedag A. (2006) Transparency and disclosure scores and their determinants in the Istanbul Stock Exchange. Corporate Governance: An International Review, 14(4): 277–296.

  • [49]

    Clauset A., Moore C., Newman M. E. J. (2008) Hierarchical structure and the prediction of missing links in networks. Nature, 453(7191): 98–101

  • [50]

    Lü L., Zhou T. (2011) Link prediction in complex networks: A survey. Physica A: statistical mechanics and its applications, 390(6): 1150–1170

  • [51]

    Newman M. E. J. (2004) Detecting community structure in network, The European Physical Journal B-Condensed Matter and Complex Systems, 38(2): 321–330

  • [52]

    Lancichinetti A., Fortunato S., Kertész J. (2009) Detecting the overlapping and hierarchical community structure in complex networks. New Journal of Physics, 11(3): 033015

  • [53]

    Newman M. E. J., Girvan M. (2004) Finding and evaluating community structure in networks. Physical review E, 69(2): 026113

  • [54]

    Agarwal G., Kempe D. (2008) Modularity-maximizing graph communities via mathematical programming. The European Physical Journal B, 66(3): 409–418

  • [55]

    Estrada E., Hatano N. (2010) A vibrational approach to node centrality and vulnerability in complex networks. Physica A: Statistical Mechanics and its Applications, 389(17): 3648–3660

  • [56]

    Estrada E., Hatano N., Benzi M. (2012) The physics of communicability in complex networks. Physics reports, 514(3): 89–119

  • [57]

    Ranjan G., Zhang Z. L. (2013) Geometry of complex networks and topological centrality. Physica A: Statistical Mechanics and its Applications, 392(17): 3833–3845

  • [58]

    Feng L., Bhan B. (2015) Understanding dynamic social grouping behaviors of pedestrians. IEEE Journal of Selected Topics in Signal Processing, 9(2): 317–329

  • [59]

    Davidson J., Savaliya S., Shah J. J. (2013) Least-squares fit of measured points for square line-profiles, Procedia CIRP, 10: 203–210

  • [60]

    Grant W. S., Voorhies R. C., Itti L. (2013) Finding planes in LiDAR point clouds for real-time registration, In Intelligent Robots and Systems (IROS), 2013 IEEE/RSJ International Conference, 4347–4354

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