In this study, a financial network of globally operating stock markets are modelled by a simple undirected graph *G* = (*V, E*), where *V* is the set of stock markets and *E* are the edges determined by the Pearson Correlation amongst the markets of Holland (AEX), Austria (ATX), Turkey (BIST), France (CAC), Germany (DAX), USA (DOW, NASDAQ, SP500), European Union (EUROSTOXX), UK (FTSE), Mexica (IPC), South Korea (KOSPI), Argentina (MERVAL), Japan (NIKKEI), Switzerland (SMI), Israel (TELAVIV), and Taiwan (TSEC). The data we used is obtained from the daily logarithmic return of the closure price of each market between the dates from 02.01.2006 to 15.02.2016.

For the daily closure price *C l*_{i} of the *i*-th stock market, the daily logarithmic return *R*_{i} is calculated as
$${R}_{i}=\mathrm{log}(\frac{C{l}_{i+1}}{C{l}_{i}})$$

and to obtain links between the stock markets, we use the Pearson correlation of each stock market as
$${\rho}_{ij}=\frac{<{R}_{i}{R}_{j}>-<{R}_{i}><{R}_{j}>}{\sqrt{(<{R}_{j}^{2}>-<{R}_{i}{>}^{2})(<{R}_{j}^{2}>-<{R}_{i}{>}^{2})}}$$

where < .. > is a temporal average performed on the trading days.

To avoid the non-positive weights on edges, we introduce the distance function involving correlations as $CorrDist:=\sqrt{2(1-{\rho}_{ij})}/2.$ While correlation coefficients vary between —1 to 1, the values of *CorrDist* vary 0 to 1.

To determine the links, we use the following formation rule:
$$({v}_{i},{v}_{j})\in E\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathbf{i}\mathbf{f}\mathbf{f}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}CorrDist({v}_{i},{v}_{j})\le ThV.$$

Here, *ThV* is an empirical threshold value. Our method to determine the network initially starts with a complete graph. The spectrum of this complete graph involves only one 0 eigenvalue. Then, we subdivide [0,1] closed interval with *1*/*h* step. The natural candidate to threshold value is *i*/*h* for *i* = 0, . . . , *h*. Hence the lower boundary for threshold value is 0; i.e., the initial complete graph. At certain point between 0 and 1, the graphs formed by the forementioned formation rule will have more than one component. To check this, counting the 0 eigenvalues of each graph spectrum will reduce the computational complexity.

In this study, we let *h =* 100 and determined the boundaries for *ThV* as [0,0.86]. The multiplicity of the 0 eigenvalues of each graph for *i*/100, *i* = 0, . . . , 100 is given in Figure 1.

Fig. 1 The multiplicity of the 0 eigenvalues of each graph with respect to threshold values. For the threshold values bigger than 0.86, the number of components of graphs exponentially grows.

To determine the convenient *ThV* in boundary, we need to introduce a concept called the disparity measure. For each graph with respect to *ThV ∈* [0, 0.86], it is possible to obtain PMFGs. An analysis on all of the 4-cliques in the PMFG reveals a high degree of homogeneity with respect to the 17 globally operating stock markets. The mean of disparity measure < *y >* can be defined analogously to [42] as the mean of
$$y(i)=\sum _{j\ne i,j\in clique}(\frac{CorrDist(i,j)}{{s}_{i}}{)}^{2}$$

over the clique, where i is a generic element of the clique and
$$y(i)=\sum _{j\ne i,j\in clique}CorrDist(i,j)$$

It should be noted that this measure is only valid for the connected graphs [42], henceforth we computed it only for *TvH ∈* [0, 86]. More interestingly, local boundaries occur for the mean disparity measure. For *ThV ∈* [0.81, 0.86], the formed graphs yield infinite disparity measures since the *s*_{i} values tend to 0. Therefore, we choose the *ThV* as 0.8 to model the network more efficiently. In Figure 2, the values of the disparity measure for the *ThV ∈* [0, 0.8] are given.

Fig. 2 The mean disparity measures *< y >* of each graph with respect to threshold values.

The network under consideration is given in Figure 3 as the triple *G* = (*V, E, w*). Each vertices represent the corresponding stock market. The set of edges are formed with respect to *CorrDist ≤* 0:8. The natural weight defined on network is the *CorrDist* value. The central stock markets, that have the minimal greatest graph distance to other vertices, are DOW, KOSPI, NIKKEI, and SMI. The rest serve as peripheral nodes. Also DOW, EUROSTOXX, KOSPI and ATX, DOW, MERVAL form 3-cliques, respectively.

Fig. 3 The network amongst the 17 stock markets that build with the formation rule *CorrDist ≤* 0.8. The markets DOW and NIKKEI have the highest vertex degree 7, and the markets with degree 6 are EUROSTOXX, KOSPI, SMI.

In our study, the vertices of the network are considered as the infectious components of the virus epidemic model. The spreading of the virus is through the noninfectious components, that are the edges of the network. Hence, the components of the network are assumed to be nonhomogeneous. This infection can be seen as the global economic crisis or stress analogously in financial networks. Once an economic crisis occurs in a stock market, the crisis replicate itself into other stock markets that are the adjacent to infectious one in the network. Progressively, more and more stock markets will be influenced by the crisis exponentially, and then the vertices of whole connected network will be under crisis immediately. In most cases, all vertices would be affected before the recovering process of an individual vertex. Hence, a two-state model is not sufficient to model the crisis phenomena in the financial network.

An advantageous alternative to a two-state model is the continuous-state model where the state of each vertex takes real values to represent the crisis state. The continuous state model is defined on the state sp
ace Ω *∈* [0, 1]. The “0” state is representing the perfect functioning that is there is no crisis, and "1" is the complete crisis state. *μ*(*t*) denotes the state index of vertex *i* at time *t*. The fractional process plays key role to capture long-range memory and self-similarity behaviour in complex systems. Therefore, we describe our model for financial crisis spreading as an epidemic fractional differential equation. By the aforementioned assumptions, we set *μ*_{i} (*t*) = 0 if the vertex *v*_{i} is at healthy state. The fractional deviation from the normal state represents the level of crisis occuring in the vertex. The weight *w*_{ij} is the strength of the interaction between the vertices *v*_{i} and *v*_{j}. As we represent our network with the weighted graph *G* = (*V, E, w*), the weights are computed by *w*_{ij} = *CorrDist*(*v*_{i}, *v*_{j}). For the vertex *v*_{i}* ∈* *G*, its state is affected by the total impact of neighboring vertices *v*_{j} ∈ G. Hence the ability of the vertex *v*_{i} to overcome a crisis is proportional to its vertex degree *d*_{vi}. Taking all these assumptions into consideration, the status change of the vertex *v*_{i} can be expressed as the following epidemic fractional differential equation:
$${D}_{{t}_{0}}^{\alpha}({\displaystyle {\mu}_{j}(t))=\sum _{j\in {N}_{i}}{w}_{ij}\mu j(t)-{d}_{{v}_{i}}{\mu}_{j}(t).}$$(3)

where *N*_{i} is the set of the neighboring vertices of the vertex *v*_{i} and the *α* is the fractional dimension.

In our model we use the Hurst exponent *(H)* to determine the fractional dimension empirically. *H* is a statistical measure of long-term memory of time series. Basically, it relates the autocorrelations of the time series. The measurement can be done by
$E\left[{\displaystyle \frac{R(n)}{S(n)}}\right]=C{n}^{H}as\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n\to \mathrm{\infty}$

where *E*[*x*] is the expected value, *R*(*n*) is the range of the first *n* values, *S*(*n*) is their standard deviation, *C* is a constant. The value of *H* varies on (0, 1) by the definition [44]. *H* can be referred to as the index of dependence and quantifies the relative tendency of a time series either to regress strongly to the mean or to a cluster in a direction. If 0.5 < *H <* 1 the time series has a long-term positive autocorrelation while for 0 < *H <* 0.5 it has a long-term negative autocorrelation. For *H* = 0.5 can indicate a completely uncorrelated series [44]. The computed *H* values of the stock market time series used in this study are given in .

Table 1 The H values of each stock market. TELAVIV and SP500 have the strongest positive and negative autocorrelations, respectively. AEX, DAX, IPC, KOSPI, and NIKKEI are almost uncorrelated.

In this study, we examine two cases for the fractional dimension *α*. First case is for *α* = 0.5, that is the model discarding the autocorrelation of each stock market. Then, the commensurate linear fractional order system can be expressed as
$${D}_{{t}_{0}}^{\alpha}-(t)=(\mathbf{W}-{\mathbf{D}}_{\mathrm{v}})-(t),-(0)={\mathbf{R}}_{0},$$(4)

where *t ∈* Ω, — = [*μ*_{1}, . . . , *μ*_{N}], **W** is the correlation distance matrix, **D**_{v} is the diagonal matrix with $Tr({\mathbf{D}}_{\mathrm{v}})={\displaystyle {\sum}_{i=1}^{N}{d}_{{v}_{i}},}$ and **R**_{0} ⊂ ℝ^{N} is the vector whose *i*-th entry is *R*_{i} (0), and N is the number of vertices in the network.

The stability of such system is studied in [45] where necessary and sufficient conditions are derived. The following theorem is an analogues to similar results.

*The autonomous fractional order linear system (4) is asymptotically stable iff |* arg(spec(**W — D**_{v}) > *απ*/*2. In this case the components of the state decay towards 0 like t*^{–α}

For the case *α =* 1 ; if no poles of the system (4) lie in the closed right-half plane, then the stability occurs. Hence, this is consistent with the results for ordinary systems [46].

The second case is *α*_{i} *= H*_{i}, that is the fractional rate of change is equal to Hurst exponent of each stock market. In this case our model includes the autocorrelation, positive or negative, into consideration. Then, the incommensurate linear fractional order system can be expressed as
$${\mathbf{D}}^{\alpha}-(t)=(\mathbf{W}-{\mathbf{D}}_{\mathbf{v}})-(t),-(0)={\mathbf{R}}_{0}.$$(5)

In this system, only the matrix **D**^{α} differs from the system (4). Here ${\mathrm{D}}^{\alpha}=[{D}_{{t}_{0}}^{{\alpha}_{1}},...,{D}_{{r}_{0}}^{{\alpha}_{N}}]\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{D}_{{t}_{0}}^{{\alpha}_{i}}$ indicates the Caputo fractional derivative of *μ*_{i} of order *α*_{i}. For the positive rational numbers *α*_{i}, i = 1, . . . , *N*, the following stability result is proposed in [47], and similar results can be derived as follows:

*Suppose that ?i are rational numbers in* (0,1) *for i =* 1, . . . , *N. Let γ = 1*/*m where m is the least common multiple of the denominators m*_{i} of α_{i} ‘s where α_{i} = k_{i}/*m*_{i}, k_{i},m_{i} ∈ ℕ. *Then the system* (*5*)* is asymptotically stable if all roots λ of the equation*
$$det(\mathit{d}\mathit{i}\mathit{a}\mathit{g}({\lambda}^{m{\alpha}_{1}},{\lambda}^{m{\alpha}_{2}},\dots ,{\lambda}^{m{\alpha}_{N}})+{\mathbf{D}}_{\mathbf{v}}-\mathbf{W})=0$$

*satisfy |* arg(*λ*)| > *γπ*/*2*.

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