### 数学代写|计算复杂度理论代写Computational complexity theory代考| Replicator Dynamics

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Replicator Dynamics

In the classical cases above mentioned, individuals may belong only to one species, e.g., a predator cannot become a prey. In doing so, a competition process can

lead to the extinction of a species, i.e., to the complete removal of its individuals, without to affect the amount of individuals belonging to the winning species (whose value might depend on other parameters, like the growth factor, etc.). Instead, in EGT, individuals might change their group of belonging, and the total size of the population is conserved over time. Actually, as we will see, in EGT we do not focus on the agents from a physical point of view (e.g., growth mechanisms), but on their strategies that constitute the parameter of the system that varies over time. Here, the benefits (or gains) deriving from interactions can be interpreted as fitness. So, again, what that reproduces/extincts in these dynamics are not the individuals but their strategies. The equivalence payoff-fitness, and the relation between the payoff of an individual and its strategy (and those of its neighbors), allows to introduce an analytical description of the system. For instance, in two-strategy games, i.e., with individuals that can adopt the strategy $C$ and the strategy $D$, we can write
$$\left{\begin{array}{l} \frac{d C}{d t}=C\left(\pi_{C}-\phi\right) \ \frac{d D}{d t}=D\left(\pi_{D}-\phi\right) \end{array}\right.$$
with $\phi=C \pi_{C}+D \pi_{D}$, and $C+D=1$. The solution of this system can lead to different equilibria, as the extinction of a strategy, as well as the coexistence of both. This approach leads to the so-called replicator dynamics, which considers a population with $n$ strategies, and a $n \times n$-matrix, named “payoff matrix,” with elements $a_{i j}$ representing the gain individuals receive according to their actions (i.e., strategies). The general differential equation, i.e., the replicator equation is defined as
$$\frac{d x_{i}}{d t}=x_{i}\left(\pi_{i}-\phi\right)$$
with $i$ going from 1 to $n, x_{i}$ representing the density of the $i$-th strategy in the population, $\pi_{i}$ payoff (or fitness) of the $i$-th strategy computed as $\pi_{i}=\sum_{j=1}^{n} a_{i j} x_{j}$, and $\phi$ average payoff equal to $\phi=\sum_{i} \pi_{i} x_{i}$. The Eq. (2.4) shows a deterministic dynamics, and, as reported in several previous studies, the behavior of the system strongly depends on the value of $n$, i.e., the number of strategies. Notably, in the most simple case with $n=2$, we can observe the prevalence of one strategy, their coexistence, and bistable behaviors. While for $n>2$, different behaviors can be observed as limit cycles and chaos.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|The Ising Model

Phase transitions are critical collective phenomena constituting one of the most important topics of Physics. Despite their underlying complexity, a number of models, for describing their dynamics, has been proposed. Among them, the Ising

model covers a special relevance, being at the same time both simple, powerful, and, in addition, useful for investigating also the behavior of nonphysical systems (e.g., see the area named Sociophysics). The Ising model considers a lattice of dimension $D$ composed of $N$ cells (see Fig. 2.1), each one provided with a spin $\sigma=\pm 1$. Since a cell may, in general, represent different kinds of objects (e.g., atoms, neurons, etc.), this simple model can be adopted in a wide variety of contexts. Accordingly, the variable $\sigma$ takes a meaning whose value depends on the related scenario, e.g., a magnetic moment, an opinion, an agent state, and so on. Here, it is worth to emphasize that the restricted range of $\sigma$, i.e., $\pm 1$ in the majority of cases, is reflected in the descriptive power of the model. However, a number of problems can be successfully faced by using this very simple modelization. Then, in the defined lattice, a pair of cells (e.g., $(i, j)$ ) forms a bond $J$, which represents their interaction. The whole set of bonds can be denoted as $B$, and for each element of the set, we have an energy of value $-J \sigma_{i} \sigma_{j}$. In doing so, the interaction energy is equal to $-J$ for $\sigma_{i}=\sigma_{j}$ and to $J$ in the opposite case. If $J$ is positive, the case $\sigma_{i}=\sigma_{j}$ has an energy smaller than the case $\sigma_{i}=-\sigma_{j}$, so the former is more stable. Positive interactions (i.e., $J>0$ ) are defined as “ferromagnetic,” while negative interactions as “antiferromagnetic.” In addition, some sites of the lattice can have an own energy of value $-h \sigma_{i}$ (here $h$ may represent an external field). So, the Hamiltonian of the Ising model reads
$$H=-J \sum_{(i j) \in B} \sigma_{i} \sigma_{j}-h \sum_{i=1}^{N} \sigma_{i}$$
Once defined the Hamiltonian function, it is possible to compute the expected value of a physical quantity, of the system under consideration, by using the

Gibbs-Boltzmann distribution. For instance, it is interesting to compute the average spin configuration $\Sigma$ at a given temperature $T$. To this end, we compute the distribution
$$P(\Sigma)=\frac{e^{-\beta H(\Sigma)}}{Z}$$
with $Z$ representing the partition function, and $\beta=\frac{1}{k_{b} T}$, i.e., inverse of the product between the Boltzmann constant $k_{b}$ and the system temperature $T$. As for the partition function $Z$, its role is normalizing the distribution (i.e., Eq. (2.6)) and, in general, takes the following form
$$Z=\sum_{i} e^{-\beta H\left(\Sigma_{i}\right)}$$
with the summation extended to all possible spin configurations. It is important to emphasize that, unfortunately, the explicit definition of $Z$ is not always trivial. Like previously mentioned, when spins $\sigma$ have values different from $\pm 1$, other models can be considered (e.g., the $X Y$ model). Among the quantities that can be measured in the Ising model, the parameter called magnetization is often particularly useful, allowing to have a high level overview on the system. In particular, the magnetization is defined as
$$m=\frac{1}{N}\left\langle\sum_{i=1}^{N} \sigma_{i}\right\rangle$$
In the thermodynamic limit, i.e., for $N \rightarrow \infty$, this parameter (i.e., $m$ ) measures the order of a system. Notably, the magnetization vanishes when the amount of positive spins is equal to that of negative ones, i.e., full disorder, and it is maximized when all spins are aligned in the same direction. It is then interesting to evaluate how the temperature affects the state of order of a system. Notably, at low temperatures (i.e., for $\beta \gg 1$ ), Eq. (2.6) suggests that low-energy configurations have a probability higher than high-energy configurations. Moreover, in absence of external fields (i.e., for $h=0$ ), low-energy states of the Ising model have all spins pointing in the same direction, so that the magnetization $m$ has a (absolute) value close to 1 . Now, increasing the temperature $T$, spin configurations with various energies emerge with equal probabilities. Accordingly, the macroscopic state of the Ising model becomes disordered, and its magnetization goes to zero. Therefore, it is possible to identify a relation between $m$ and $T$ and, most importantly, a critical temperature $T_{c}$. The latter entails that for $TT_{c}$ the magnetization reduces until its value goes to zero. The phenomenon here briefly described is known as “order-disorder phase transition,” and it has a deep relevance both in Physics and in the related applications to complex systems. Eventually.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Mean Field

In principle, the Gibbs-Boltzmann distribution defined in Eq. (2.6) indicates that it is possible to compute the expected value of any physical quantity. However, due to the huge amount of sums over $2^{N}$ terms in the partition function (i.e., Eq. (2.7)), sometimes this task is actually almost impossible (in a limited amount of time). Thus, in such cases, the utilization of opportune methods of approximation becomes mandatory, like, for instance, the mean-field approach now described. In few words, the mean-field approach neglects fluctuations of variables around their mean values. Notably, we assume $m=\frac{\sum_{i}\left(\sigma_{i}\right\rangle}{N}=\sum_{i}\left\langle\sigma_{i}\right\rangle$, and the deviation $\delta \sigma_{i}=\sigma_{i}-m$, in addition the second-order term with respect to the fluctuation $\delta \sigma_{i}$ is assumed to be small enough to be neglected. Accordingly, the Hamiltonian can be rewritten as
\begin{aligned} H &=-J \sum_{(i j) \in B}\left(m+\delta \sigma_{i}\right)\left(m+\delta \sigma_{j}\right)-h \sum_{i=1}^{N} \sigma_{i} \ & \sim-J m^{2} N_{B}-J m \sum_{(i j) \in B}\left(\delta \sigma_{i}+\delta \sigma_{j}\right)-h \sum_{i=1}^{N} \sigma_{i} \end{aligned}
with $N_{B}$ number of elements in the set $B$. Here $\delta \sigma_{i}$ and $\delta \sigma_{j}$, which refer to the extrema of each bond, are summed up $z$ times, with $z$ number of bonds starting from a site. In doing so, the Hamiltonian can be finally reduced to the following form:
$$H=N_{B} J m^{2}-(J m z+h) \sum_{i=1}^{N} \sigma_{i}$$
Becoming much more easy for analytical calculations.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Replicator Dynamics

$$\left{ dCd吨=C(圆周率C−φ) dDd吨=D(圆周率D−φ)\正确的。 在一世吨Hφ=C圆周率C+D圆周率D,一个ndC+D=1.吨H和s○l在吨一世○n○F吨H一世ss是s吨和米C一个nl和一个d吨○d一世FF和r和n吨和q在一世l一世br一世一个,一个s吨H和和X吨一世nC吨一世○n○F一个s吨r一个吨和G是,一个s在和ll一个s吨H和C○和X一世s吨和nC和○Fb○吨H.吨H一世s一个ppr○一个CHl和一个ds吨○吨H和s○−C一个ll和dr和pl一世C一个吨○rd是n一个米一世Cs,在H一世CHC○ns一世d和rs一个p○p在l一个吨一世○n在一世吨Hns吨r一个吨和G一世和s,一个nd一个n×n−米一个吨r一世X,n一个米和d“p一个是○FF米一个吨r一世X,”在一世吨H和l和米和n吨s一个一世jr和pr和s和n吨一世nG吨H和G一个一世n一世nd一世在一世d在一个lsr和C和一世在和一个CC○rd一世nG吨○吨H和一世r一个C吨一世○ns(一世.和.,s吨r一个吨和G一世和s).吨H和G和n和r一个ld一世FF和r和n吨一世一个l和q在一个吨一世○n,一世.和.,吨H和r和pl一世C一个吨○r和q在一个吨一世○n一世sd和F一世n和d一个s \frac{d x_{i}}{dt}=x_{i}\left(\pi_{i}-\phi\right)$$

## 数学代写|计算复杂度理论代写Computational complexity theory代考|The Ising Model

H=−Ĵ∑(一世j)∈乙σ一世σj−H∑一世=1ñσ一世

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Mean Field

H=−Ĵ∑(一世j)∈乙(米+dσ一世)(米+dσj)−H∑一世=1ñσ一世 ∼−Ĵ米2ñ乙−Ĵ米∑(一世j)∈乙(dσ一世+dσj)−H∑一世=1ñσ一世

H=ñ乙Ĵ米2−(Ĵ米和+H)∑一世=1ñσ一世

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## MATLAB代写

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