In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them (Bollobás 2001). From a mathematical perspective, random graphs are found to model and mirror the diverse types of complex networks encountered in different areas.
$G(n, p)$, the Erdos-Renyi Random Graph, defines a family of graphs, each of which starts with $n$ isolated nodes, and we place an edge between each distinct node pair with probability $p$. In $G(n, p)$ Model, the probability of obtaining any one particular random graph with $m$ edges is $p^{m}(1-p)^{N-m}$ with the notation $N=\binom{n}{2}$. As a result, $G(n, p)$ defines a bigger familiy than $G(n, m)$ since $n$ and $p$ do not uniquely determine the graph so number of possible graphs are larger.
Suppose we are given a graph data $G = (N, E)$, which contains $|N| = n$ nodes and $|E| = m$ edges. One big question we often ask is: “Which vertices are important?” Intuitively, we would consider a “star” (the central node that connects a lot of other nodes) as an obviously important case and also consider nodes in a “circle” are equivalently important. In general, we can use cantrality measures to rank the nodes in a graph. There are many centrality measures and page rank is currently the most prominent approach that deals with directed graphs.
The data reflecting our real world can be represented in networks sometimes. With network data, we can ask questions such as
To answer the questions, we need to understand network properties (e.g., diameter, scale-free / power law network, small-world behavior), network models that fit our observations (e.g., Erdos Renyi random graphs, Kleinberg’s model models, … etc), and algorithms that could unflod on our networks (e.g., page rank, decentralized search, label propagation, link prediction, community detection, … etc).
Classification is well so common in the area of machine learning and scikit-learn provides a comprehensive toolkit that can be easily used. Here I will share some common classification models and how to apply them on a dataset using this good toolkit, while the classification process will cover
In this document, I will build a predictive framework for predicting whether each flight in 2006 will be cancelled or not by using the data from 2000 to 2005 as training data.
Items to be delivered in this document includes:
The Wisconsin Diabetes Registry Study targeted all individuals $<30$ years of age diagnosed with Type I diabetes in southern Wisconsin, USA. Participants were requested to submit blood samples and were sent a questionnaire inquiring about hospitalizations and other events. The blood samples were used to determine glycosylated hemoglobin (GHb), and important indicator of glycemic control.
The data set diabetes.txt, which can be downloaded from here, contains the data from the Wisconsin Diabetes Registry Study. The data items are:
| Variable name | meaning |
|---|---|
| ID | an unique identification number |
| HEALTH | self-reported health status: 1=excellent, 2=good, 3=fair, 4=poor |
| BH | dichotomized health status: 1=excellent health, 0=worse than excellent health |
| GENDER | sex code: 1=female, 0=male |
| GHb | overal mean glycosylated hemoglobin value in study |
| AGE | age at diagnosis (years) |
This is a practice in R for realization of Monte Carlo Simulation, Central Limit Theorem (CLT), normal approximation, and so on.
This practice is based on Hans Rosling talks New Insights on Poverty and The Best Stats You’ve Ever Seen.
The assignment uses data to answer specific question about global health and economics. The data contradicts commonly held preconceived notions. For example, Hans Rosling starts his talk by asking: “for each of the six pairs of countries below, which country do you think had the highest child mortality in 2015?”
Most people get them wrong. Why is this? In part it is due to our preconceived notion that the world is divided into two groups: the Western world versus the third world, characterized by “long life,small family” and “short life, large family” respectively. In this homework we will use data visualization to gain insights on this topic.
The Universal Approximation Theorem states that a 2-layer network can approximate any function, given a complex enough architecture. That’s why we will create a neural network with two neurons in the hidden layer and we will later show how this can model the XOR function.
In this experiment, we will need to understand and write a simple neural network with backpropagation for “XOR” using only numpy and other python standard library.
The code here will allow the user to specify any number of layers and neurons in each layer. In addition, we are going to use the logistic function as the activity function for this network.
