Graph Algorithms SOlution

$35.00

Description

Introduction

In this project we will explore graph theory theorems and algorithms, by applying them on real data. In the first part of the project, we con-sider a particular graph which models correlations between stock price time series. In the second part, we analyse traffic data on a dataset provided by Uber.

  • Stock Market

In this part of the project, we study data from stock market. The data is available on this Dropbox Link. The goal of this part is to study correlation structures among fluctuation patterns of stock prices using tools from graph theory. The intuition is that investors will have sim-ilar strategies of investment for stocks that are effected by the same economic factors. For example, the stocks belonging the transporta-tion sector may have different absolute prices, but if for example fuel prices change or are expected to change significantly in the near future,

then you would expect the investors to buy or sell all stocks similarly and maximize their returns. Towards that goal, we construct different graphs based on similarities among the time series of returns on differ-ent stocks at different time scales (day vs a week). Then, we study prop-erties of such graphs. The data is obtained from Yahoo Finance website for 3 years. You’re provided with a number of csv tables, each contain-ing several fields: Date, Open, High, Low, Close, Volume, and Adj Close price. The files are named according to Ticker Symbol of each stock. You may find the market sector for each company in Name_sector.csv.

1.1 Return correlation

In this part of the project, we will compute the correlation among log-normalized stock-return time series data. Before giving the expression for correlation, we introduce the following notation:

pi(t) is the closing price of stock i at the tth day

qi(t) is the return of stock i over a period of [t 1; t]

qi(t) = pi(t) pi(t 1)

pi(t 1)

ri(t) is the log-normalized return stock i over a period of [t 1; t] ri(t) = log(1 + qi(t))

Then with the above notation, we define the correlation between the log-normalized stock-return time series data of stocks i and j as

ij = p

hri(t)rj(t)i hri(t)ihrj(t)i

(hri(t)2i hri(t)i2)(hrj(t)2i hrj(t)i2)

where h i is a temporal average on the investigated time regime (for our data set it is over 3 years).

Question 1: Provide an upper and lower bound on ij. Also, provide a justification for using log-normalized return (ri(t)) instead of regular return (qi(t)).

2

1.2 Constructing correlation graphs

In this part,we construct a correlation graph using the correlation co-efficient computed in the previous section. The correlation graph has the stocks as the nodes and the edge weights are given by the following expression

q

wij = 2(1 ij)

Compute the edge weights using the above expression and construct the correlation graph.

Question 2: Plot the degree distribution of the correlation graph and a histogram showing the un-normalized distribution of edge weights.

1.3 Minimum spanning tree (MST)

In this part of the project, we will extract the MST of the correlation graph and interpret it.

Question 3: Extract the MST of the correlation graph. Each stock can be categorized into a sector, which can be found in Name_sector.csv file. Plot the MST and color-code the nodes based on sectors. Do you see any pattern in the MST? The structures that you find in MST are called Vine clusters. Provide a detailed explanation about the pattern you observe.

1.4 Sector clustering in MST’s

In this part, we want to predict the market sector of an unknown stock. We will explore two methods for performing the task. In order to eval-uate the performance of the methods we define the following metric

    • X

= jV j vi2V P (vi 2 Si)

where Si is the sector of node i. Define

jQij

P (vi 2 Si) =

jNij

where Qi is the set of neighbors of node i that belong to the same sector as node i and Ni is the set of neighbors of node i. Compare with the case where

jSij

P (vi 2 Si) = jV j

Question 4: Report the value of for the above two cases and provide an interpretation for the difference.

1.5 Correlation graphs for weekly data

In the previous parts, we constructed the correlation graph based on daily data. In this part of the project, we will construct a correlation graph based on weekly data. To create the graph, sample the stock data weekly on Mondays and then calculate ij using the sampled data. If there is a holiday on a Monday, we ignore that week. Create the corre-lation graph based on weekly data.

Question 5: Extract the MST from the correlation graph based on weekly data. Compare the pattern of this MST with the pattern of the MST found in question 3.

  • Let’s Help Santa!

Companies like Google and Uber have a vast amount of statistics about transportation dynamics. Santa has decided to use network theory to

facilitate his gift delivery for the next christmas. When we learned about his decision, we designed this part of the project to help him. We will send him your results for this part!

2.1 Download the Data

Go to “Uber Movement” website and download data of Monthly Ag-gregate (all days), 2017 Quarter 4, for San Francisco area 1. The dataset contains pairwise traveling time statistics between most pairs of points in San Francisco area. Points on the map are represented by unique IDs. To understand the correspondence between map IDs and areas, download Geo Boundaries file from the same website 2. This file contains latitudes and longitudes of the corners of the polygons cir-cumscribing each area. In addition, it contains one street address in-side each area, referred to as DISPLAY_NAME. To be specific, if an area is represented by a polygon with 5 corners, then you have a 5 2 matrix of the latitudes and longitudes, each row of which represents latitude and longitude of one corner.

2.2 Build Your Graph

Read the dataset at hand, and build a graph in which nodes correspond to locations, and undirected weighted edges correspond to the mean traveling times between each pair of locations (only December). Add the following attributes to the vertices:

  1. Display name: the street address

  1. Location: mean of the coordinates of the polygon’s corners (a 2-D vector)

1If you download the dataset correctly, it should be named as

san_francisco-censustracts-2017-4-All-MonthlyAggregate.csv 2The file should be named SAN_FRANCISCO_CENSUSTRACTS.JSON

5

The graph will contain some isolated nodes (extra nodes existing in the Geo Boundaries JSON file) and a few small connected components. Re-move such nodes and just keep the giant connected component of the graph. In addition, merge duplicate edges by averaging their weights

  • . We will refer to this cleaned graph as G afterwards. Question 6: Report the number of nodes and edges in G.

2.3 Traveling Salesman Problem

Question 7: Build a minimum spanning tree (MST) of graph G. Report the street addresses of the two endpoints of a few edges. Are the results intuitive?

Question 8: Determine what percentage of triangles in the graph (sets of 3 points on the map) satisfy the triangle inequality. You do not need to inspect all triangles, you can just estimate by random sampling of 1000 triangles.

Now, we want to find an approximation solution for the traveling sales-man problem (TSP) on G. Apply the 2-approximate algorithm described in the class4. Inspect the sequence of street addresses visited on the map and see if the results are intuitive.

Question 9: Find the empirical performance of the approximate algo-rithm:

= Approximate TSP Cost Optimal TSP Cost

Question 10: Plot the trajectory that Santa has to travel!

  • Duplicate edges may exist when the dataset provides you with the statistic of a road in both direc-tions. We remove duplicate edges for the sake of simplicity.

4You can find the algorithm in: Papadimitriou and Steiglitz, “Combinatorial optimization: algo-rithms and complexity”, Chapter 17, page 414

  • Analysing the Traffic Flow

Next December, there is going to be a sport event between Stanford University and University of California, Santa Cruz (UCSC). A large number of students are enthusiastic about the event, which is going to be held in UCSC. Stanford fans want to drive from their campus to the rival’s. We would like to analyse the maximum traffic that can flow from Stanford to UCSC.

3.1 Estimate the Roads

We want to estimate the map of roads without using actual road datasets. Educate yourself about Delaunay triangulation algorithm and then ap-ply it to the nodes coordinates5.

Question 11: Plot the road mesh that you obtain and explain the result.

Create a subgraph G induced by the edges produced by triangulation.

3.2 Calculate Road Traffic Flows

Question 12: Using simple math, calculate the traffic flow for each road in terms of cars/hour.

Hint: Consider the following assumptions:

Each degree of latitude and longitude 69 miles Car length 5 m = 0.003 mile

Cars maintain a safety distance of 2 seconds to the next car Each road has 2 lanes in each direction

5You can use scipy.spatial.Delaunay in python

Assuming no traffic jam, consider the calculated traffic flow as the max capacity of each road.

3.3 Calculate the Max Flow

Consider the following addresses:

Source address: 100 Campus Drive, Stanford

Destination address: 700 Meder Street, Santa Cruz

Question 13: Calculate the maximum number of cars that can com-mute per hour from Stanford to UCSC. Also calculate the number of edge-disjoint paths between the two spots. Does the number of edge-disjoint paths match what you see on your road map?

3.4 Defoliate Your Graph

In G , there are a number of unreal roads that could be removed. For instance, there are many fake bridges crossing the bay. Apply a thresh-old on the travel time of the roads in G to remove the fake edges. Trim

the fake edges and call the resulting graph ~ . GQuestion 14: Plot ~ on real map coordinates. Are real bridges pre-G

served?

Hint: You can consider the following coordinates:

Golden Gate Bridge: [[-122.475, 37.806], [-122.479, 37.83]]

Richmond, San Rafael Bridge: [[-122.501, 37.956], [-122.387, 37.93]]

San Mateo Bridge: [[-122.273, 37.563], [-122.122, 37.627]]

Dambarton Bridge: [[-122.142, 37.486], [-122.067, 37.54]]

San Francisco – Oakland Bay Bridge: [[-122.388, 37.788], [-122.302, 37.825]]

Question 15: Now, repeat question 8 for ~ and report the results. Do G

you see any significant changes?

9