Financial Risk Information Spreading on Metapopulation Networks

*e financial risk information diffuses through various kinds of social networks, such as Twitter and Facebook. Individuals transmit the financial risk information which can migrate among different platforms or forums. In this paper, we propose a financial risk information spreading model on metapopulation networks. *e subpopulation represents a platform or forum, and individuals migrate among them to transmit the information. We use a discrete-time Markov chain approach to describe the spreading dynamics’ evolution and deduce the outbreak threshold point. We perform numerical simulation on artificial networks and discover that the financial risk information can be promoted once increasing the information transmission probability and active subpopulation fraction. *e weight variance and migration probability cannot significantly affect the financial risk spreading size. *e discrete-time Markov chain approach can reasonably predict the above phenomena.


Introduction
Many real-world systems in society, economy, and biological systems can be described as complex networks [1][2][3]. e nodes represent the element, and edges stand for the relationships among nodes. For instance, in financial network, the nodes stand for the financial institution, e.g., banks, and edges means the loan relationships among those financial institutions [4][5][6]. With such description framework, the dynamics of financial behavior, risk spreading can be mapped into studying the dynamics on financial networks [7][8][9][10][11][12][13]. Gai et al. [7] studied the risk contagion on directed financial networks and assumed that a bank's failure could trigger its lenders and further induce the cascading failures. Garas et al. [14] used a susceptible-infected-recovered (SIR) model to describe the economic crisis on the financial network and revealed that Belgium could initiate a global crisis.
In reality, the information about the financial risk is always accompanied by the financial risk [15]. at information widely spreads on various kinds of social platforms and forums, such as Twitter and Facebook. To describe the information spreading dynamics, some successful models have been proposed, such as the epidemic susceptible-infected-susceptible [16,17], susceptible-infected-recovered [18,19], Watts threshold model [20], and independent cascading [21]. Researchers have demonstrated that the network topologies markedly affect the spreading dynamics [22][23][24][25][26][27][28]. While spreading dynamics on networks with heterogeneous degree distribution, the strong heavy-tail may induce the outbreak threshold to disappear. When the information spreads on different platforms and forums, scholars are modeled as multiplex networks [29][30][31][32][33] or metapopulation networks [34][35][36]. Compared with spreading dynamics on single networks, multiplex networks can promote or suppress the spreading, which depends on the evolutionary mechanisms [37][38][39].
Recently, Gomez et al. [40] studied the phase diagram of the information spreading on metapopulation networks and revealed that mobility of individuals in large-size subpopulations towards smaller ones suppress the spreading.
Real-data analysis has revealed that different platforms and forums usually have a different attitude on the same piece of information. Some exhibit enthusiasm in the information, while others demonstrate depression. To include this factor, Wang et al. [41] proposed an informationspreading model on heterogeneous multiplex networks and discussed the system's spreading size and critical points.
ere is still a mathematical model to describe the financial risk information-spreading dynamics on heterogeneous metapopulation networks to our vast knowledge. We, therefore, propose a model in Section 2. en, we develop a discrete-time Markov chain approach to study the evolution of financial risk information spreading on metapopulation in Section 3. e numerical simulations of the spreading dynamics are performed in Section 4. Finally, we make conclusions in Section 5.

Model Descriptions
In this section, we propose information spreading model on metapopulation, which includes N subpopulation. In each subpopulation i, there are n i aboriginal (nodes). An edge between two subpopulations represents a physical or cybernetic connection that exists. e edge's wight stands for the strength between the two subpopulations, which can be used to describe the interaction strength between different platforms or forums. In this paper, we build the interconnections among subpopulation according to a given degree distribution: where 〈k〉 is the average neighbor subpopulation. Assume the edge weight distribution where 〈w〉 is the average weight and σ is the standard variance. An illustration of the metapopulation is shown in Figure 1. Mathematically, the metapopulation's network topology can be described by using the weighted adjacency matrix W, in which W ij � 1 means that subpopulation i and j are connected by an edge with strength W ij . In reality, each subpopulation has its characters in spreading dynamics. To include this character, we classify the subpopulation into two types: active subpopulation (AS) and depressing subpopulation (DS). For the active subpopulation, the individuals actively participate and transmit the information. On the contrary, individuals in depressing subpopulation are less active in transmitting the information. To obtain a mathematical result, we assume a fraction α of (AS) and 1 − α fraction of (DS). At every time step, each individual will prefer to move to a neighboring subpopulation with probability p d . e preference probability of an individual in subpopulation i moving to subpopulation j is as follows: At the end of each time step, each travelling individual returns to his/her place of residence.
To describe the financial risk information spreading dynamics, we adopt an unaware-aware-unaware (UAU) model [42,43]. e unaware individual means that an individual does not know the financial risk information and maybe obtain the financial risk information in the following steps. An aware individual who stands for himself/herself has obtained the information and is willing to share it with others in the same subpopulation. e financial risk information spreading dynamics evolves as follows. Initially, we randomly select ϱn i individuals as the seeds of the spreading, i.e., set those ϱn i individuals in the aware state. At every time step, we perform the following two processes sequentially. (i) Each individual in each subpopulation moves with probability p d , and his/her determination selection probability is determined according to equation (3).
is migration can be regarded as an individual move from one forum (or communication group) to another. Individuals in the same subpopulation are sharing financial risk information. In (AS), e.g., subpopulation i, each active individual will randomly contact k individuals and try to transmit the financial risk information to each unaware individuals of the k individuals with probability λ. For the unaware individual, he/she becomes aware if he/she receives at least one piece of financial risk information from neighbors successfully. e contagion process in the depressing subpopulation (DS) is similar to that in the AS. e only difference is that an unaware individual becoming aware should receive at least θ pieces of information from aware neighbors successfully. (ii) e travelling individuals return to his/her residence, and each aware individual becomes unaware with probability μ. e spreading dynamics evolve until the system reaches a dynamic steady state.

Theoretical Analysis
Inspired by [40,44,45], we develop a discrete-time Markov chain approach to study the evolution of financial risk information spreading on metapopulation. In theory, we assume that the infection probability of unaware node is independent, that is to say, there are no dynamical Figure 1: Illustration of financial risk information spreading on metapopulation network. Orange and blue circles represent the depressing subpopulation (DS) and active subpopulation (AS), respectively. Green and blue circles stand for the unaware and aware individuals. e directed edges R ij means the preference travelling probability from subpopulation i to j.

Complexity
correlations. Denote ρ i (t) (i � 1, . . . , N) as the fraction of active individuals in subpopulation i at time t. e evolution of ρ i (t) can be expressed as On the right-hand side of equation (4), the first expression represents the fraction of active individuals who do not recover. e second expression stands for the probability that the unaware individuals become aware. e expression Θ i (t) means an unaware individual in subpopulation i is infected by active individuals.
Once the expression of Θ i (t) is obtained, we know the evolution of ρ i (t). In the following, we solve the expression of Θ i (t). An unaware individual b i in subpopulation i becoming aware has two situations. (i) If individual b i does not travel to neighboring subpopulation, he/she will be infected at time t in his own aboriginal subpopulation i with probability: e first expression on the right hand of equation (5) stands for the contagion probability when subpopulation i is AS.
e second expression represents the contagion probability when the subpopulation i is (DS). When the subpopulation i is (AS), the contagion probability is where x i (t) stands for the fraction of active individuals in subpopulation i after the migration, and the expression of where n eff i � j n j⟶i is the number of effective individuals in subpopulation i after migration. e term n j⟶i represents the number of individuals travelling from subpopulation j to subpopulation i and can be expressed as where δ ij � 1 when i � j; otherwise, δ ij � 0. In equation (8), the first second term stands for the number of individuals that do not travel to neighboring subpopulation, and the second term represents the number of individuals in which individuals in neighbors' subpopulation travelling to subpopulation i. (ii) If individual b i travels to neighboring subpopulation and is infected at his/her determination, the contagion probability is p d N j�1 h i⟶j P j (t). Combining the above stated two situations, we obtain the expression of Θ i (t) as e above equations describe the financial risk information spreading dynamics on metapopulation. Iterating equation (4) from an initial value of ρ i (t), we obtain the dynamical steady state of ρ i (t ⟶ ∞). When the number of subpopulation is large, numerically solving the equations is hard, and we may use the dimensionality reduction approach. For the sake of simplicity, we denote ρ i (t ⟶ ∞) as ρ * i . e average fraction of individuals in the active state in the steady state is as e values of 〈ρ〉 can be regarded as the order parameter of the system in statistical physics. at is to say, the global financial risk information outbreaks, i.e., 〈ρ〉 > 0, when the information transmission probability λ is above a critical value λ c . Otherwise, only a vanishingly small fraction of individuals is in the active state, i.e., 〈ρ〉 � 0. In the following, we solve the expression of λ c . In the steady state, we know ρ i (t + 1) � ρ i (t) � ρ * i ; therefore, equation (4) can be rewritten as where Near the critical value λ c , the information spreading size is very small, i.e., ρ * i ≪ 1; thus, we neglect the high orders of ρ * i . In addition, we assume that λ is small enough and obtain (1 − λ) n ≃1 − nλ. Linearizing equation (12), we have P i ≃αx i kλ. Using this term, we further obtain Inserting equations (7) and (8) into the above equation, we have During the computing process, we used the relation An element of matrix M 1 is e global financial risk information outbreak condition is where Λ max (M) is the largest eigenvalue of matrix M. Numerically, solving equation (20), we obtain the value of λ c .

Numerical Results' Analysis
To systemically investigate the financial risk information spreading dynamics on metapopulation networks, we perform extensive numerical simulations. We generate N � 50 subpopulation. For each subpopulation i, we set n i � 100 number of residents. We randomly connect the 50 subpopulations with a given probability of u � 0.3. As a result, the average connection of a subpopulation is 〈k〉 � 15. e edge weight distribution is G(w) with average weight 〈w〉 � 10. For the DS, we set the θ � 3. at is to say, an unaware individual in a DS becoming aware must receive three pieces of information successfully at a given time step. rough extensive numerical simulations, we found that other values of θ do not affect the results qualitatively. We perform numerical simulations at least 200 times when the system reaches a steady state for a given parameter set. We found that the spreading dynamics reached a steady state after 500 time steps through performing extensive numerical simulations. Mathematically, we verify that the system reaches a steady state when is small than ε � 10 − 5 , where ρ(t) � t t′�t− 100 ρ(t ′ ).
In Figure 2, we investigate the effects of four key parameters on the financial risk information spreading on metapopulation networks. We first discuss the impact of information transmission probability. As shown in Figures 2(a) and 2(b), we present the average fraction 〈ρ〉 of individuals in the aware state as a function of information transmission probability λ. We reveal that increasing the values of λ is beneficial for information spreading, i.e., 〈ρ〉 with λ. When λ ≤ λ c , which can be obtained by numerically solving equation (20), there are no individuals who received the financial risk information, that is, 〈ρ〉 � 0. When λ > λ c , we find that 〈ρ〉 is a finite value, which indicates the global financial risk information outbreak.
is phenomenon demonstrates that we can contain the financial risk information spreading by reducing the information transmission probability.
We now discuss the effects of the second key parameter, fraction of AD, on the spreading dynamics. As shown in Figure 2(a), we find that increasing the fraction of AD promotes the information spreading.
at is to say, 〈ρ〉 increases with α. For large values of α, there are more AD.
erefore, more unaware individuals becoming aware only need to receive one piece of information from others. As a result, the global financial risk information spreading dynamics is promoted. e third key parameter is the variance of the weight distribution. In Figure 1, we set σ 2 � 0, 4, and 9 for different values of α and p d . We reveal that the variance does not markedly affect the financial risk information spreading size and outbreak threshold. is phenomenon indicates that the varying connection strength among subpopulations cannot significantly affect the spreading of financial risk. Lastly, we revealed that the travelling probability does not significantly affect the spreading dynamics. Our proposed discrete-time Markov chain approach can predict the above phenomena.
Finally, we study the financial risk information spreading size as a function of travelling probability p d and information transmission probability λ in detail in Figure 3.

Conclusions
In this paper, we investigated the financial risk information spreading on social networks. To describe the different social platforms and forums in the spreading dynamics, we adopt the metapopulation network, in which a forum is a subpopulation and mixed randomly. At every time step, each individual may travel from one subpopulation to another. We used a discrete-time Markov chain approach to describe the dynamical process and critical point. Our theory can reasonably predict numerical simulations. We noted that the financial risk information spreading size depends on the  information transmission probability, a fraction of active subpopulation, and independent on the variance of the weight distribution and travelling probability. Our results may shed some light on studying the financial risk information spreading dynamics and contain the financial risk. On the one hand, our model and theoretical approach maybe used to study other dynamics on metapopulation networks. On the other hand, to contain the financial risk information spreading, we can reduce the information transmission probability. For more realistical situations, the fraction of DS is not simply following a binomial distribution. erefore, other forms of distribution need further studies.
Data Availability e data that support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.