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# 数据科学代写|复杂网络代写Complex Network代考|Model Description

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## 数据科学代写|复杂网络代写Complex Network代考|Model Description

Consider a social network composed of $N$ agents whose connections are given by an adjacency matrix $A=\left(a_{i j}\right)$, where $a_{i j}=1$ if there is a link between $i$ and $j$ and $a_{i j}=0$ otherwise. Agents in our model society are interested in finding out the truth about some external event about which rumours are propagated along the social network. For this purpose, an agent $i$ is further characterised by a strategy $p_i \in[0,1]$, which models the probability with which the agent will spend effort to verify the truth. Alternatively, when on its turn to update its beliefs, with probability $1-p_i$ an agent will seek information from a randomly selected neighbour by just copying the current belief of that neighbour. We will also assume that agents in our model society have some propensity to forget about the truth, hence they constantly need to update their beliefs either by verification at some cost or copying the beliefs of their social peers (which we assume to not be costly).

In more detail, we consider the following updating process: (i) with probability $1-q$ an agent will forget about the truth; alternatively with probability $q$ the agent will update its beliefs. (ii) When updating beliefs, with probability $p$ the agent will spend some cost $c>0$ to learn the truth; alternatively, with probability $1-p$ the agent will update its belief by copying the belief of a randomly selected network neighbour (at no cost). We assume that steps (i) and (ii) are iterated at a fairly fast timescale and agents will derive utility from knowledge of the truth (which bestows a benefit $b>0$ ). Utilities are diminished by the potential cost $c$ agents expend on verifying the truth. Suppose $P_i$ gives the stationary probability that agent $i$ knows the truth. Agent $i$ is then supposed to derive utility $U_i=-q p_i c+P_i b$. As, in the following, only relative utilities are of interest, we set
$$U_i=-r p_i+P_i$$
where $r=q c / b$ quantifies a cost-benefit ratio of knowing the truth relative to average costs of learning the truth.

Note, that the updating of agent probabilities to know the truth corresponds to a Markov process. Accordingly, the probability $P_i^{t+1}$ that agent $i$ knows the truth at time $t+1$ can be obtained from
$$P_i^{(t+1)}=q p_i+q\left(1-p_i\right) 1 / k_i \sum_j a_{i j} P_j^t,$$
where the first term represents learning the truth directly, and the second term reflects copying of beliefs of randomly chosen neighbours. Stationary state probabilities can be obtained by solving the linear system
$$\left(I-\operatorname{diag}\left(q \frac{1-p_i}{k_i}\right) A\right) \boldsymbol{P}=q \boldsymbol{p}$$
where $\operatorname{diag}\left(q \frac{1-p_i}{k_i}\right)$ is the diagonal matrix with entries $q \frac{1-p_i}{k_i}$ along its diagonal. Note, that the left hand matrix in the equation is diagonally dominant. Solutions of the linear system can thus be conveniently obtained through Jacobi iteration. Further, seeding Jacobi iteration with stationary states from previous rounds of the evolutionary game described below, allows for considerable speedup of the numerics of determining stationary states, enabling simulations to be performed on fairly large social networks.

## 数据科学代写|复杂网络代写Complex Network代考|Mean-Field Solution

In this section we derive an approximate solution to equilibrium states of the evolutionary dynamics based on a mean-field approximation to Eq. (3). There, we assume that agents are typically exposed to neighbours with a mean probability $\langle P\rangle$ of knowing the truth. Averaging over Eq. (3), one obtains
$$\langle P\rangle=q\langle p\rangle+q(1-\langle p\rangle)\langle P\rangle$$
where $\langle\cdot\rangle$ stands for averages over the whole population. One easily obtains
$$\langle P\rangle=\frac{q\langle p\rangle}{1-q(1-\langle p\rangle)}$$
To proceed, from Eq. (1) we can then obtain the utility of agent $i$ as
$$U_i=p_i(-r+q(1-\langle P\rangle))+q\langle P\rangle$$
which gives an average utility of
$$\langle U\rangle=\langle p\rangle(-r+q(1-\langle P\rangle))+q\langle P\rangle$$
Let us consider the noiseless case of an evolutionary dynamics with $p_{\text {mut }}=0$ and $K \rightarrow 0$, in which an agent $i$ strictly copies a strategy of $j$ if $U_j>U_i$. In the latter case frequencies of strategies evolve according to the differences of their utilities relative to the average utility in the population and hence a stationary state for strategy $i$ is reached if $U_i=\langle U\rangle$. Combining Eqs. (6) and (7) and inserting the (approximate) result for $\langle P\rangle$ from Eq. (5), one obtains the equilibrium condition
$$\langle p\rangle=(1-q)\left(\frac{1}{r}-\frac{1}{q}\right),$$
which also entails that the average probability of knowing the truth is given by $\langle P\rangle=q$ if $r<r_0,\langle P\rangle=1-r / q$ if $r_0<r<r_1$, and $\langle P\rangle=0$ otherwise, with $r_0=q(1-q)$ giving a lower bound when $p(r)=1$ and $r_1=q$ an upper bound when $p(r)=0$.

## 数据科学代写|复杂网络代写Complex Network代考|Model Description

$$U_i=-r p_i+P_i$$

$$P_i^{(t+1)}=q p_i+q\left(1-p_i\right) 1 / k_i \sum_j a_{i j} P_j^t,$$

$$\left(I-\operatorname{diag}\left(q \frac{1-p_i}{k_i}\right) A\right) \boldsymbol{P}=q \boldsymbol{p}$$

## 数据科学代写|复杂网络代写Complex Network代考|Mean-Field Solution

$$\langle P\rangle=q\langle p\rangle+q(1-\langle p\rangle)\langle P\rangle$$

$$\langle P\rangle=\frac{q\langle p\rangle}{1-q(1-\langle p\rangle)}$$

$$U_i=p_i(-r+q(1-\langle P\rangle))+q\langle P\rangle$$

$$\langle U\rangle=\langle p\rangle(-r+q(1-\langle P\rangle))+q\langle P\rangle$$

$$\langle p\rangle=(1-q)\left(\frac{1}{r}-\frac{1}{q}\right),$$

## MATLAB代写

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