Networked Life
===
[TOC]
# Classes 2 (4 Feb 2020)
$SIR = \frac{Useful \ singal}{interference + noise}$
$SIR_{2} = \frac{G_{22}p_{2}}{G_{21}p_{1}+n_{2}}$

==Increasing one's power leads to decrease of SIR for others==
Goals: __Minimum Power to get minimum SIR for each channel__ $SIR_{1} \geq \gamma_{1}$

## Distributed Power Control(DPC)
$$p_{i}[t+1] = \frac{\gamma_{i}}{SIR_{i}[t]}p_{i}[t], \forall i$$

- How well would they do, as compared to a global presence of social planner (full information).

Reach an equilibrium, is the equilibrium good for society?
Distributed algorithm only works with local information

$$
\begin{pmatrix}
p_{1}[t+1]\\
p_{2}[t+1]\\
p_{3}[t+1]\\
\end{pmatrix} =
\begin{pmatrix}
\gamma_{1}&0&0\\
0&\gamma_{2}&0\\
0&0&\gamma_{3}\\
\end{pmatrix}
\begin{pmatrix}
0&\frac{G_{12}}{G_{11}}&\frac{G_{13}}{G_{11}}\\
\frac{G_{21}}{G_{22}}&0&\frac{G_{23}}{G_{22}}\\
\frac{G_{31}}{G_{33}}&\frac{G_{32}}{G_{33}}&0
\end{pmatrix}\begin{pmatrix}
p_{1}[t]\\
p_{2}[t]\\
p_{3}[t]\\
\end{pmatrix}+\begin{pmatrix}
\frac{n_{1}\gamma{1}}{G_{11}}\\
\frac{n_{2}\gamma{2}}{G_{22}}\\
\frac{n_{3}\gamma{3}}{G_{33}}\\
\end{pmatrix}
$$

$p[n] = (DF)^{n}p[0]+((DF)^{n-1}+..+I)v$

if p(A) <1 , $A^{n} \approx$ (0) p() [spectral radius] is the largest eigenvalue

Thus $p[n] \Rightarrow 0\cdot p[0] + (I-DF)^{-1}v$

## Game
- Players
- Strategy space per player
- Payoff function per player

Strategic equilibrium

Dynamic Game vs One-shot Game

- Best response strategy
- Dominant strategy (always bettter)
- Socially optimal strategies (deny,deny)
- Pareto optimal strategies

Case for 2 stations
$$u_{i}(p_{i},p_{-i})= \left\{
\begin{array}
\ -\infty \ if\ SIR_{i} < \gamma_{i} \\
-p_{i} \ if\ SIR_{i} \geq \gamma_{i}
\end{array}
\right.
$$

# Class 3(6 Feb)
## Recap
Mechanism Design -> Changing the rule of the game to reach a new equilibrium

We prove that $p[\infty] = (I-D(\gamma)F)^{-1}v$ from Distributed system
Social Optium p* := min $1^{T}p \ s.t.(I-D(\gamma)F)p\geq v$

$I-D(\gamma)F$ is always invertable as eigenvalue of I is 1 and eigenvalue of $D(\gamma)F<1$, as such the eigenvalue of the outcome $\neq$ 0

$\rho(DF) = \sqrt{\gamma_{1}\gamma_{2}\frac{G_{12}G_{21}}{G_{11}G_{22}}} > 1$

## Recommendation system
- Collaborative filtering
How to measure accuracy or predicition score?
SSE and MSE for testing on actual data
- Latent-factor model
key factor, i.e category, length , etc

# Class 4 (7 Feb)

==$min ||Ab-y||_{2}$
$b^{*} = (A^{T}A)^{-1}A^{T}y$==


One hot encoding for row and columns to make recommendation bias similar to that of a linear regression

## Neighbourhood system
positive|negative|uncorrelated
consine similarity in errors

$\tilde{R} = R - \hat{R}$

# Class5 (11 Feb)
## Auction
Truthful players?
Use to allocate resources to the player who value it the most, (i.e he will then use it more efficiently)
Paying as a reprication to ensure the efficient usage after the purchase.

CTR -> click through rate. (percentage of people clikcing on your link)
CTR = no. of clicks / impression
CPC -> cost per click

Paying for the opportunity cost incurred to others.

In a 1 advertise situation, with max bid 4 and quality 4, every click is valued at 1.6..
Thus if another advertiser with max bid 2 and quality 10 enters the systems, he would pay the opportunity cost of 1.6 by taking away the position.

Vicktry auction (2nd price auction)
No gain for lying

## eBay auction
bids $b_{(1)}^{t},b_{(2)}^{t}...$
winning price= $b_{(1)}^{t}$
Asking price(AP): $t=b_{(1)}^{t} + \delta$
a valid bid is $\geq AP$

# Class 7 (18 Feb) [Google Page Rank]
$$\pi_{i} = \sum_{j\rightarrow i}\frac{1}{outdegree(j)}\pi_{j}$$

$$\pi^{T} = \pi^{T}H$$
$\pi$ is the eigenvector of H
random walk interpretation for $\pi$ as steady state probability

transition matrix converges if it is irreducable and aperiodic

if have dangling node (absorb state):
$$\hat{H} = H + \frac{1}{N}(w1^{T})

if not well mixing (i.e 2 seperate groups):
$$G= \theta\hat{H} + (1-\theta)(11^{T})\frac{1}{N}$$

In G, the spectral radius = 1, rate of convergence depends on second largest eigenvalue. $\lambda_{2}$