Network Analysis of Financial Markets

(Karthick Venkatesan)

1. Abstract

In this project we have analysed the dynamics of the Financial Markets through Network Analysis.We have built networks of the equities that are part of the S and P 500 index over a range Time Periods between 2007 and 2017 based on Winner Take All and the Minimum Spanning Tree method .Both these methods utilise the correlation coefficient computed between the attributes of these stocks such as Price,Volume,Returns etc . Community detection techniques were then applied to the constructed networks. The resulting communities were compared for consistency with the identified market sections using Standard Industrial Classification code.We also studied the evolution of the network and the communities over the study period and found interesting behaviors. We have compared our results from both the methods for each of our analysis.We created a GEXF file for this dynamic network and visuvalised the same in Gephi a open source network visuvalisation software.The visualization results offer a very intuitive way to look at the overall correlation structure of different the equities in the S and P 500 and evolution of these networks over a period of time .

2. Introduction

Network analysis of Equities is a extensively researched topic and in section 3 we have detailed current literature which was utilised as part of this project.In all the current studies the focus has been on studying the properties of the market in a stationary view for a fixed time period.By leveraging the techniques noted in these current literature we have as part of this study built multiple networks of the stocks in the S and P 500 index for mutiple non overlapping windows of Time Period (T) between 2007 and 2017 and studied how the network evolves and how the communities in the network behave with the changing dynamics of the market.

Below are key Objectives of the project.For each of these items we have compared the results we got for for the networks built based on both these methods.

1. Build network for the stocks in  S and P 500 index based on the correlations between Prices/Volume for Multiple Time Periods using Winner Take All method and Minimum Spanning Tree Method
2. Analyse the topology of the networks in multiple time periods.Does the network of stocks exhibit scale free properties at each of the time period?
3. Detect communities in these networks and find out if the stocks actually trade in groups based on the SIC(Standard Industry Classification) Code.
4. Studied the evolution of these communities 
5. Find important stocks and sectors they belong to based on the Network Properties at Different time periods.
6. Visuvalize the network and also the dynamic evolution of network by building dynamic graphs using Gephi

3. Literature Review

Current Studies about network analysis for stock market can be classified into below categories :

(1) Applying network analysis techniques for different markets and analyze the topological characteristics of each market Statistical Analysis of Financial Markets, Hierarchical structure in financial markets

(2) Propose different correlation metric analysis among various stock markets to suggest different definitions of edges between stocks and study the impact on the network using different edge definitions Network analysis of a financial market based on genuine correlation and threshold method, Network of Equities in Financial Markets, A network perspective of the stock market

3.1. Edge Definition

Approach to construct the edges of stock market network is not unique. In the current literature, multiple measures were investigated to construct the edges between nodes namely Zero-lag correlation,Detrended covariance,Time-lag correlations of prices changes over a certain period of time

3.2. Network Properties

Studies have covered both emerging and mature markets. Authors claim that understanding the topological properties can help to understand correlation patterns among stocks, thus providing guidance for risk management. Topological properties often of interest include degree distribution, clustering and component structure. In this subcategory study, usually only one correlation measure is proposed to establish the connections between nodes. In the introduction session of Statistical Analysis of Financial Markets, the author covered a wide range of previous studies in this category

4. Build Network

4.1 Data Collection

We collected the prices for the stocks that trade in both NASDAQ and the NYSE stock exchange from Eod Data . The data consisted of Opening , Closing prices , Volume information for each trading day for the period of 2007 to 2017.From this data we filtered and selected only the prices that are a part of S and P 500 . We chose the S and P 500 since the index had a well balanced portfolio of stocks from different industry segments .

## Read S and P 500 list
import pandas as pd
import numpy as np
dfsp500 = pd.read_csv('data/SANDP500.csv')
companies=dfsp500['Symbol'].tolist()
companies=np.random.choice(companies, size=500, replace=False)

import glob
import os
path = r'data/NASDAQ'                     
all_files = glob.glob(os.path.join(path, "*.txt"))     
df_from_each_file = (pd.read_csv(f) for f in all_files)
concatenated_df_NAS   = pd.concat(df_from_each_file, ignore_index=True)
concatenated_df_NAS=concatenated_df_NAS[concatenated_df_NAS['<ticker>'].isin(companies)]

path = r'data/NYSE'                     
all_files = glob.glob(os.path.join(path, "*.txt"))     
df_from_each_file = (pd.read_csv(f) for f in all_files)
concatenated_df_NYS   = pd.concat(df_from_each_file, ignore_index=True)
concatenated_df_NYS=concatenated_df_NYS[concatenated_df_NYS['<ticker>'].isin(companies)]

concatenated_df = pd.concat([concatenated_df_NAS,concatenated_df_NYS])

col_p = 'close'
concatenated_df.columns = ['ticker','date','open','high','low','close','vol']
concatenated_df=concatenated_df[concatenated_df['ticker'].isin(companies)]
concatenated_df=concatenated_df.merge(dfsp500,left_on='ticker',right_on='Symbol')
concatenated_df['ticker'] = concatenated_df['ticker']
df_price = concatenated_df[['ticker','date',col_p]]
df_price=df_price.drop_duplicates( keep='last')
df_price['date'] = pd.to_datetime(df_price['date'], format='%Y%m%d', errors='ignore')
df_price.set_index(['date','ticker'],inplace=True)
df_price=df_price.unstack()[col_p]
df_price.reset_index(inplace=True)
df_price.fillna(method='bfill',inplace=True)
df_price.fillna(method='ffill',inplace=True)

4.2 Detrend data - Compute log returns

Of the methods in current literature number we choose Time-lag correlations of prices changes over a certain period of time .One of the keys challenges in computing the correlation on stock prices is that the values are moving time series and have inherent trends which can lead to spurious correlations if the data is not properly normalised.

If we think about a time series of prices, you could write it out as

[P0,P1,P2,...,PN], or [P0,P0+R1,P0+R1+R2,...,P0+R1+...+RN], where Ri = Pi-P(i-1). 

Written this way we can see that the first return R1, contributes to every entry in the series, whereas the last only contributes to one. This gives the early values in the correlation of prices more weight than they should have.

So for computing the correlation we take the difference between the prices for each day giving us the returns for each .We computed the log returns between two days since it has a key benefit of being additive over multiple time periods .

Though log returns can be computed over multiple time periods of 7 , 30 , 60 , 100 days for the sake of simplicity we kept the return window to 1 day.

import scipy.signal
t = 1
for key in df_price.columns:
    if key not in companies:
        continue
    try:
        df_price[key] = np.log(df_price[key]) - np.log(df_price[key].shift(t))
    except:
        print (key)
df_price.set_index('date',inplace=True)

## A quick visualization: detrended data
import matplotlib.pyplot as plt
%matplotlib inline
import random as rn
NUM_COLORS = len(companies)
cm = plt.get_cmap('gist_rainbow')
colors = [cm(i/NUM_COLORS) for i in range(NUM_COLORS)]
rn.seed = len(companies)  # for choosing random colors
fig, ax = plt.subplots(nrows=5,ncols=2,figsize=(20, 20))
y=2007
for row in ax:
    for col in row:
        yfs = str(y) + '0101'
        yfe = str(y) + '1231'
        n = 0
        col.set_ylim([0.5, -0.5])
        for i in df_price.columns:
            df_price.loc[yfs:yfe][i].plot(ax=col,color=colors[n])
            n = n + 1
        y = y + 1
plt.tight_layout()
plt.show()

4.3 Compute Correlation matrix for multiple windows

Next we computed the pearson correlation between the log returns.The data is divided into windows of width (T) in order to uncover dynamic characteristics of the networks. The window width corresponds to the number of daily returns included in the computation of the correlation between Stocks. The method of time windows division to construct asset graphs can be found in the literature Asset trees and asset graphs in financial markets

To determine the ideal length of the window we computed the mean correlation for window values of 21,42,63,84 and 105 and plotted the variations in the correlation.As seen in the plot the window of 63 captures the fluctuations of the market well.Values less than this are too noisy and higher than this we lose the sensitivity in the changes in the market .Also from a market perspective 63 days ideally falls into the Quarterly reporting cycle of these companies so we felt it would be appropriate choice

The correlation matrix is then computed based on this window length of 63 by dividing the period between 2007 and 2017 into multiple windows.

import matplotlib.pyplot as plt
%matplotlib inline
corr_dict = {}
T = 1
for w in range(21,126,21):
    x = []
    y = []
    W = w
    for i in range(t,len(df_price),W):
            dkey = i
            corr_dict[dkey]=df_price.iloc[i:(i+W)].corr(method='pearson')
            corr_dict[dkey].fillna(0,inplace=True)
            x.append(dkey)
            y.append(np.mean([abs(j) for j in corr_dict[dkey].values.flatten().tolist()]))
    plt.plot(x,y)
    plt.xlabel('Days')
    plt.ylabel('Mean Correlation')
    plt.legend(list(range(21,126,21)), loc='center left', bbox_to_anchor=(1, 0.5),ncol=1)
plt.show()
W = 63
corr_dict = {}
for i in range(t,len(df_price),W):
      dkey = i
      corr_dict[dkey]=df_price.iloc[i:(i+W)].corr(method='pearson')
      corr_dict[dkey].fillna(0,inplace=True)

4.4 Build network - (Winner take all method)

Literature A network perspective of the stock market details the winner take all method where we build a network based on the correlation matrix if the value of the Correlation index is greater than a threshold.

What is the ideal value of threshold ?.Since one the key objectives of the study is to find if the stocks behave in groups we wanted to select a threshold which maximises the modularity .However we were also consious not overfit the data in which case we might loose some of the underlying dynamics at work in the market.

In the next step we built the network based on different thresholds ranging from 0.6 to 0.99 for the windows identified .We then picked the value of the threshold from the values where the modularity is maximum.On analysis of this result we noted that most networks tended to have a high modularity for threshold value between 0.75 and 0.85 .There were certains windows where the threshold for the best modularity was seen to be greater than 0.9 however at this threshold the number edges was very less so in such cases we set the threshold to 0.8 and built the network.We treated both positive and negative correlations the same and looked at the absolute value.

import networkx as nx
import community
def get_modularity(y,threshold):
    df_price_corr = corr_dict[y]
    elist = []
    outdict=df_price_corr.to_dict()
    for i in outdict.keys():
        for j in outdict[i].keys():    
            if abs(outdict[i][j]) > threshold :
                if i == j :
                    continue
                if i < j:
                    elist.append([i,j,dict(weight=abs(outdict[i][j]),start=y,end=y+W)])
                    #elist.append([i,j,dict(start=y,end=y+1)])
                else:
                    None
    #print (len(elist))
    G=nx.Graph()
    G.add_edges_from(elist)
    #print (nx.info(G))
    partition = community.best_partition(G)
    try:
        m = community.modularity(partition, G)
    except:
        m = 0 
    return m
# This will be our list of fractions to run the simulation over
fractions = np.linspace(0.6, 0.99, 20)
M_list = {}
for y in corr_dict.keys():
    M_list[y] = [ get_modularity(y, frac)  for frac in fractions ]

The below is the plot between the threshold value and the computed markdown in the various windows

import matplotlib.pyplot as plt
%matplotlib inline
plt.figure(figsize=(20,10))
for y in corr_dict.keys():
    plt.plot(fractions, M_list[y], lw=2)
plt.legend(list(M_list.keys()), loc='center left', bbox_to_anchor=(1, 0.5),ncol=2)
plt.xlabel('Threshold')
plt.ylabel('Modularity')
plt.show()

# Pick value of threshold for each Window (T)
T_val = {}
for y in M_list.keys():
    val, idx = max((val, idx) for (idx, val) in enumerate(M_list[y]))
    if fractions[idx] > 0.8:
        T_val[y] = 0.8
    else:
        T_val[y] = fractions[idx]
    #print (str(y) + ":" + str(T_val[y]))

# Create Edge List
elist_dict={}
for y in corr_dict.keys():
    df_price_corr = corr_dict[y]
    threshold = T_val[y]
    elist = []
    outdict=df_price_corr.to_dict()
    for i in outdict.keys():
        for j in outdict[i].keys():
            if abs(outdict[i][j]) > threshold :
                if i == j :
                    continue
                if i < j:
                    elist.append([i,j,dict(weight=1,start=y,end=y+W-1)])
                else:
                    None
    elist_dict[y] = elist

# Constructing the graph different windows
import networkx as nx
import community
G_dict = {}
for y in elist_dict.keys():
    G=nx.Graph()
    elist = elist_dict[y]
    G.add_edges_from(elist)
    values = dfsp500.set_index('Symbol').to_dict(orient='dict')['Sector']
    for node, value in values.items():
        try:
            G.node[node]['Sector'] = value
        except:
            #name = (value[0:3] + '-' + node)
            #G.add_node(name,Sector=value)
            None
    partition = community.best_partition(G)
    
    deg_cent=dict((k,float(v)) for k,v in nx.degree_centrality(G).items())
    degree = dict((k,float(v)) for k,v in nx.degree(G).items())
    #katz_cent=nx.katz_centrality(G)
    #eigen_cent= dict((k,float(v)) for k,v in nx.eigenvector_centrality(G).items()) 
    close_cent= dict((k,float(v)) for k,v in nx.closeness_centrality(G).items())  
    betw_cent= dict((k,float(v)) for k,v in nx.betweenness_centrality(G).items()) 
    nx.set_node_attributes(G, "community", partition)  
    nx.set_node_attributes(G, "degreecent", deg_cent)
    nx.set_node_attributes(G, "degree", degree)
    #nx.set_node_attributes(G, "katz", katz_cent)
    #nx.set_node_attributes(G, "eigenvector", eigen_cent)
    nx.set_node_attributes(G, "closeness", close_cent)
    nx.set_node_attributes(G, "betweenness", betw_cent)
    nx.set_node_attributes(G, 'start',y)
    nx.set_node_attributes(G, 'end',y+W-1)
    #G.remove_nodes_from(nx.isolates(G)) 
    #T = nx.minimum_spanning_tree(G)
    T = G
    G_dict[y] = T

# Collect the node level attributes for the nodes for all the windows
df_list = []
for k in G_dict.keys():
    G = G_dict[k]
    a = G.node
    df_list.append(pd.DataFrame(a).T.reset_index())
attrib_df = pd.concat(df_list)
attrib_df.fillna(0,inplace=True)
attrib_df1=attrib_df.merge(dfsp500,left_on='index',right_on='Symbol')
attrib_df = attrib_df1[['index','Sector_x','betweenness','closeness','community','degree','degreecent','start','Name']]
attrib_df.columns=['ticker','Sector','Betweeness','Closeness','Community','Degree','DegreeCent','start','Name']

# Collect the graph level  attributes for all the windows
from scipy.stats import linregress
G_val_dict = {}
for Y in G_dict.keys():
    G_val = {}
    G= G_dict[Y]
    G_val['nodes'] =  int(nx.number_of_nodes(G))
    G_val['edges'] =  int(nx.number_of_edges(G))
    #G_val['AvgDegree'] =  nx.average_degree(G)
    G_val['AvgClustering'] = nx.average_clustering(G)
    try:
        G_val['AvgShortestPathLength'] = nx.average_shortest_path_length(G)
    except:
        G_val['AvgShortestPathLength'] = 99999
    try:
        G_val['Diameter'] = nx.diameter(G)
    except:
        G_val['Diameter'] = 99999
    degs = {}
    for n in G.nodes() :
        deg = G.degree(n)
        if deg not in degs.keys() :
            degs[deg] = 0
        degs[deg] += 1
    items = sorted(degs.items())
    x= [k for (k , v ) in items ]
    y= [ v for (k ,v ) in items ]
    xlog= np.array([np.log(k) for (k , v ) in items ])
    ylog= np.array([np.log(v) for (k ,v ) in items ])
    slope,intercept,rvalue,pvalue,stderr=linregress(xlog,ylog)
    G_val['Slope'] = slope
    G_val['No of Communities'] = attrib_df.groupby(by=['start'])['Community'].nunique().ix[Y]
    G_val_dict[Y] = G_val
Gvaldf=pd.DataFrame(G_val_dict).T

/opt/conda/lib/python3.6/site-packages/ipykernel_launcher.py:32: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#deprecate_ix

The below table lists the network properties for the networks created using the Winner Take All Method Method over multiple Time Periods.

pd.options.display.max_rows = 999
Gvaldf

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.dataframe thead th {
    text-align: left;
}

.dataframe tbody tr th {
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</style>

	AvgClustering	AvgShortestPathLength	Diameter	No of Communities	Slope	edges	nodes
1	0.437301	99999.000000	99999.0	17.0	-0.734717	462.0	139.0
64	0.583921	99999.000000	99999.0	19.0	-0.555489	897.0	178.0
127	0.538656	99999.000000	99999.0	16.0	-1.081694	307.0	112.0
190	0.464502	99999.000000	99999.0	17.0	-0.917059	222.0	84.0
253	0.343108	99999.000000	99999.0	21.0	-1.244365	379.0	149.0
316	0.337280	99999.000000	99999.0	20.0	-1.183432	253.0	121.0
379	0.472535	99999.000000	99999.0	22.0	-0.892980	781.0	171.0
442	0.497699	99999.000000	99999.0	11.0	-0.829045	3798.0	344.0
505	0.439454	99999.000000	99999.0	18.0	-1.018640	696.0	192.0
568	0.429914	99999.000000	99999.0	20.0	-0.967200	1108.0	215.0
631	0.397613	99999.000000	99999.0	18.0	-1.110171	303.0	120.0
694	0.489646	99999.000000	99999.0	20.0	-1.072235	266.0	110.0
757	0.351498	99999.000000	99999.0	14.0	-1.209373	142.0	83.0
820	0.378896	99999.000000	99999.0	22.0	-1.131378	470.0	173.0
883	0.618682	99999.000000	99999.0	14.0	-0.514376	6146.0	309.0
946	0.340429	99999.000000	99999.0	18.0	-1.257725	219.0	106.0
1009	0.111712	99999.000000	99999.0	11.0	-1.735520	33.0	37.0
1072	0.128993	99999.000000	99999.0	22.0	-1.509773	59.0	64.0
1135	0.371843	99999.000000	99999.0	22.0	-1.279903	381.0	173.0
1198	0.732433	1.884846	5.0	4.0	-0.251248	22994.0	409.0
1261	0.586980	99999.000000	99999.0	13.0	-0.751500	2997.0	271.0
1324	0.151453	99999.000000	99999.0	17.0	-1.969121	53.0	59.0
1387	0.374351	99999.000000	99999.0	20.0	-1.112039	464.0	172.0
1450	0.123333	99999.000000	99999.0	15.0	-2.144086	39.0	50.0
1513	0.235170	99999.000000	99999.0	22.0	-1.584047	70.0	70.0
1576	0.050292	99999.000000	99999.0	24.0	-2.596231	35.0	57.0
1639	0.369880	99999.000000	99999.0	26.0	-1.521337	301.0	165.0
1702	0.422181	99999.000000	99999.0	16.0	-1.106065	129.0	71.0
1765	0.203037	99999.000000	99999.0	20.0	-2.030411	60.0	69.0
1828	0.260765	99999.000000	99999.0	20.0	-1.401260	93.0	72.0
1891	0.304440	99999.000000	99999.0	22.0	-1.711807	95.0	79.0
1954	0.448140	99999.000000	99999.0	21.0	-0.996135	234.0	100.0
2017	0.344473	99999.000000	99999.0	36.0	-1.140130	514.0	206.0
2080	0.515361	99999.000000	99999.0	32.0	-0.840034	706.0	187.0
2143	0.323645	99999.000000	99999.0	27.0	-0.978314	244.0	116.0
2206	0.537092	99999.000000	99999.0	26.0	-0.815566	3269.0	342.0
2269	0.443097	99999.000000	99999.0	37.0	-1.063193	521.0	178.0
2332	0.301949	99999.000000	99999.0	43.0	-1.113032	557.0	223.0
2395	0.469023	99999.000000	99999.0	31.0	-0.826418	472.0	136.0
2458	0.454574	99999.000000	99999.0	32.0	-0.914775	1348.0	246.0
2521	0.446138	99999.000000	99999.0	37.0	-1.093756	1307.0	313.0
2584	0.470094	99999.000000	99999.0	23.0	-0.797446	306.0	111.0
2647	0.439164	99999.000000	99999.0	28.0	-0.794508	328.0	109.0
2710	0.352363	99999.000000	99999.0	40.0	-1.442425	297.0	190.0

4.5. Build network - Minimum Spanning Tree Method

One of the negatives of the Winner Take all method is that in certain period the threshold gave us highly noisy data .There were two many edges in the graph and results during this period are difficult to visuvalise .In this section we built the same network based on the Minimum Spanning tree method as noted in Network of Equities in Financial Markets .

In case of the minimum spanning tree method a metric distance dij is calculated using the cross correlation matrix.

		dij = (2(1-Cij))^(0.5)

Where dij is the edge distance between stock i and stock j.

To find the ideal window size for constructing the network we computed the mean distance metric for mutiple windows ranging from 21 to 105 .The plot of the results is below.The Results largely indicate similar pattern to mean correlation we saw in the Winner take all method.Here to we can see that the window size of 63 resonable captures the fluctuations in the market.So we used the window width of 63 to compute the correlation and the corresponding distance metric and built the networks using the Minimum Spanning Tree Method.

import math
import matplotlib.pyplot as plt
%matplotlib inline
def calc_d(x):
    x = round(x,3)
    d = math.sqrt(2 * (1 - x))
    return d
corr_dict = {}
corr_dist_dict = {}
T = 1
W = 63
x = []
y = []
for w in range(21,126,21):
    x = []
    y = []
    W = w
    for i in range(t,len(df_price),W):
            dkey = i
            corr_dict[dkey]=df_price.iloc[i:(i+W)].corr(method='pearson')
            corr_dict[dkey].fillna(1,inplace=True)
            corr_dist_dict[dkey] = corr_dict[dkey].applymap(calc_d)
            x.append(dkey)
            y.append(np.mean([abs(j) for j in corr_dist_dict[dkey].values.flatten().tolist()]))
    plt.plot(x,y)
    plt.xlabel('Days')
    plt.ylabel('Mean Distance')
    plt.legend(list(range(21,126,21)), loc='center left', bbox_to_anchor=(1, 0.5),ncol=1)
W = 63
corr_dict = {}
corr_dist_dict = {}
for i in range(t,len(df_price),W):
      dkey = i
      corr_dict[dkey]=df_price.iloc[i:(i+W)].corr(method='pearson')
      corr_dict[dkey].fillna(1,inplace=True)
      corr_dist_dict[dkey] = corr_dict[dkey].applymap(calc_d)

#MST Start 
elistmst_dict={}
for y in corr_dist_dict.keys():
    df_price_corr = corr_dist_dict[y]
    elistmst = []
    outdict=df_price_corr.to_dict()
    for i in outdict.keys():
        for j in outdict[i].keys():
            if (abs(outdict[i][j]) > 0 and (i>j)):
                elistmst.append([i,j,dict(weight=abs(outdict[i][j]),start=y,end=y+W-1)])
    elistmst_dict[y] = elistmst

import networkx as nx
import community
GMST_dict = {}
for y in elistmst_dict.keys():
    G=nx.Graph()
    elist = elistmst_dict[y]
    G.add_edges_from(elist)
    T = nx.minimum_spanning_tree(G)
    G = T
    values = dfsp500.set_index('Symbol').to_dict(orient='dict')['Sector']
    for node, value in values.items():
        try:
            G.node[node]['Sector'] = value
        except:
            #name = (value[0:3] + '-' + node)
            #G.add_node(name,Sector=value)
            None
    partition = community.best_partition(G)
    
    deg_cent=dict((k,float(v)) for k,v in nx.degree_centrality(G).items())
    degree = dict((k,float(v)) for k,v in nx.degree(G).items())
    #katz_cent=nx.katz_centrality(G)
    #eigen_cent= dict((k,float(v)) for k,v in nx.eigenvector_centrality(G).items()) 
    close_cent= dict((k,float(v)) for k,v in nx.closeness_centrality(G).items())  
    betw_cent= dict((k,float(v)) for k,v in nx.betweenness_centrality(G).items()) 
    nx.set_node_attributes(G, "community", partition)  
    nx.set_node_attributes(G, "degreecent", deg_cent)
    nx.set_node_attributes(G, "degree", degree)
    #nx.set_node_attributes(G, "katz", katz_cent)
    #nx.set_node_attributes(G, "eigenvector", eigen_cent)
    nx.set_node_attributes(G, "closeness", close_cent)
    nx.set_node_attributes(G, "betweenness", betw_cent)
    nx.set_node_attributes(G, 'start',y)
    nx.set_node_attributes(G, 'end',y+W)
    T = G
    GMST_dict[y] = T

df_list = []
for k in GMST_dict.keys():
    G = GMST_dict[k]
    a = G.node
    df_list.append(pd.DataFrame(a).T.reset_index())
attribMST_df = pd.concat(df_list)
attribMST_df.fillna(0,inplace=True)
attribMST_df1=attribMST_df.merge(dfsp500,left_on='index',right_on='Symbol')
attribMST_df = attribMST_df1[['index','Sector_x','betweenness','closeness','community','degree','degreecent','start','Name']]
attribMST_df.columns=['ticker','Sector','Betweeness','Closeness','Community','Degree','DegreeCent','start','Name']

from scipy.stats import linregress
G_valMST_dict = {}
for Y in GMST_dict.keys():
    G_val = {}
    G= GMST_dict[Y]
    G_val['nodes'] =  nx.number_of_nodes(G)
    G_val['edges'] =  nx.number_of_edges(G)
    #G_val['AvgDegree'] =  nx.average_degree(G)
    G_val['AvgClustering'] = nx.average_clustering(G)
    try:
        G_val['AvgShortestPathLength'] = nx.average_shortest_path_length(G)
    except:
        G_val['AvgShortestPathLength'] = 99999
    try:
        G_val['Diameter'] = nx.diameter(G)
    except:
        G_val['Diameter'] = 99999
    degs = {}
    for n in G.nodes() :
        deg = G.degree(n)
        if deg not in degs.keys() :
            degs[deg] = 0
        degs[deg] += 1
    items = sorted(degs.items())
    x= [k for (k , v ) in items ]
    y= [ v for (k ,v ) in items ]
    xlog= np.array([np.log(k) for (k , v ) in items ])
    ylog= np.array([np.log(v) for (k ,v ) in items ])
    slope,intercept,rvalue,pvalue,stderr=linregress(xlog,ylog)
    G_val['Slope'] = slope
    G_val['No of Communities'] = attribMST_df.groupby(by=['start'])['Community'].nunique().ix[Y]
    G_valMST_dict[Y] = G_val
GMST_df=pd.DataFrame(G_valMST_dict).T

/opt/conda/lib/python3.6/site-packages/ipykernel_launcher.py:31: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#deprecate_ix

The below table lists the network properties for the networks created using the Minimum Spanning Tree Method over multiple Time Periods.

GMST_df

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	AvgClustering	AvgShortestPathLength	Diameter	No of Communities	Slope	edges	nodes
1	0.0	10.006982	25.0	23.0	-2.302370	429.0	430.0
64	0.0	11.656817	30.0	24.0	-2.511209	431.0	432.0
127	0.0	11.685365	25.0	24.0	-2.414082	434.0	435.0
190	0.0	12.705015	33.0	23.0	-2.094925	436.0	437.0
253	0.0	13.100791	30.0	24.0	-2.414212	436.0	437.0
316	0.0	12.929396	34.0	25.0	-2.226890	440.0	441.0
379	0.0	11.736004	37.0	23.0	-2.397362	442.0	443.0
442	0.0	9.870432	24.0	23.0	-2.176792	444.0	445.0
505	0.0	11.700028	29.0	26.0	-2.423807	448.0	449.0
568	0.0	11.991151	28.0	25.0	-2.337993	448.0	449.0
631	0.0	12.206835	32.0	25.0	-2.094288	447.0	448.0
694	0.0	11.966274	28.0	23.0	-2.401395	448.0	449.0
757	0.0	10.225780	26.0	25.0	-2.400149	448.0	449.0
820	0.0	10.720473	25.0	22.0	-2.286885	450.0	451.0
883	0.0	9.268146	24.0	25.0	-2.270480	451.0	452.0
946	0.0	7.232595	17.0	24.0	-2.094576	453.0	454.0
1009	0.0	11.460324	27.0	23.0	-2.404244	455.0	456.0
1072	0.0	9.898856	27.0	23.0	-2.202451	457.0	458.0
1135	0.0	11.048054	26.0	24.0	-2.306812	458.0	459.0
1198	0.0	10.048669	24.0	25.0	-2.201338	459.0	460.0
1261	0.0	9.375795	21.0	26.0	-2.121953	468.0	469.0
1324	0.0	10.258025	25.0	22.0	-2.509338	466.0	467.0
1387	0.0	10.286589	26.0	26.0	-2.438308	468.0	469.0
1450	0.0	9.377163	25.0	22.0	-2.249755	469.0	470.0
1513	0.0	11.060595	28.0	24.0	-2.509590	470.0	471.0
1576	0.0	9.406258	27.0	23.0	-2.332476	471.0	472.0
1639	0.0	9.620221	21.0	23.0	-2.281254	475.0	476.0
1702	0.0	10.191933	27.0	27.0	-2.419526	475.0	476.0
1765	0.0	9.666815	26.0	25.0	-2.572109	479.0	480.0
1828	0.0	10.022886	25.0	23.0	-2.278903	479.0	480.0
1891	0.0	13.378051	41.0	27.0	-2.429328	481.0	482.0
1954	0.0	12.746888	38.0	26.0	-2.497661	482.0	483.0
2017	0.0	12.801533	33.0	25.0	-2.468109	483.0	484.0
2080	0.0	11.445382	30.0	26.0	-2.523927	486.0	487.0
2143	0.0	11.845514	36.0	25.0	-2.380275	485.0	486.0
2206	0.0	11.853195	31.0	26.0	-2.531080	491.0	492.0
2269	0.0	14.269999	38.0	25.0	-2.344246	493.0	494.0
2332	0.0	11.389670	27.0	24.0	-2.401746	494.0	495.0
2395	0.0	15.049847	33.0	25.0	-2.560258	496.0	497.0
2458	0.0	12.346634	32.0	26.0	-2.450851	497.0	498.0
2521	0.0	16.771448	47.0	25.0	-2.231318	497.0	498.0
2584	0.0	15.656228	36.0	25.0	-2.619341	498.0	499.0
2647	0.0	14.443294	41.0	25.0	-2.464404	497.0	498.0
2710	0.0	14.880367	34.0	26.0	-2.511511	497.0	498.0

5. Result

5.1 Degree distribution and Scale Free Properties

We plotted the degree distribution histogram and also plotted the degree distribution on a log log plotted and regression fitted a line whose slope will give as the Power law exponent.The plot show that network shows scale free properties in most of the windows .The scale free nature is more evident in the Networks generated based on the Minimum Spanning Tree Method .

Winner Take All Method

## Explore graph properties
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.stats import linregress
fig, ax = plt.subplots(nrows=10,ncols=4,figsize=(20, 30))
Y = 1
for row in ax:
    for col in row:
        degs = {}
        for n in G_dict[Y].nodes() :
            deg = G_dict[Y].degree(n)
            if deg not in degs.keys() :
                degs[deg] = 0
            degs[deg] += 1
        items = sorted(degs.items())
        x= [k for (k , v ) in items ]
        y= [ v for (k ,v ) in items ]
        xlog= np.array([np.log(k) for (k , v ) in items ])
        ylog= np.array([np.log(v) for (k ,v ) in items ])
        col.scatter(xlog, ylog)
        slope,intercept,rvalue,pvalue,stderr=linregress(xlog,ylog)
        col.plot(xlog, (slope * xlog + intercept), color='red')
        #ax.set_xscale( 'log' )
        #ax.set_yscale( 'log' )
        col.set_title ( " Day :" + str(Y) + " - Slope :" + str(round(slope,2) ),fontsize=8)
        Y = Y + W
plt.tight_layout()
plt.show()

## Explore graph properties
import matplotlib.pyplot as plt
fig, ax = plt.subplots(nrows=10,ncols=4,figsize=(20, 30))
y = 1
for row in ax:
    for col in row:
        deg_dist = [v for k,v in nx.degree(G_dict[y]).items()]
        deg_dist.sort(reverse=True)
        pdf, bins, patch = col.hist(deg_dist, bins=10)
        col.set_title ( " Day :" + str(y),fontsize=8 )
        y = y + W
plt.tight_layout()
plt.show()

Minimum Spanning Tree Method

## Explore graph properties
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.stats import linregress
fig, ax = plt.subplots(nrows=10,ncols=4,figsize=(20, 30))
Y = 1
for row in ax:
    for col in row:
        degs = {}
        for n in GMST_dict[Y].nodes() :
            deg = GMST_dict[Y].degree(n)
            if deg not in degs.keys() :
                degs[deg] = 0
            degs[deg] += 1
        items = sorted(degs.items())
        x= [k for (k , v ) in items ]
        y= [ v for (k ,v ) in items ]
        xlog= np.array([np.log(k) for (k , v ) in items ])
        ylog= np.array([np.log(v) for (k ,v ) in items ])
        col.scatter(xlog, ylog)
        slope,intercept,rvalue,pvalue,stderr=linregress(xlog,ylog)
        col.plot(xlog, (slope * xlog + intercept), color='red')
        #ax.set_xscale( 'log' )
        #ax.set_yscale( 'log' )
        col.set_title ( " Day :" + str(Y) + " - Slope :" + str(round(slope,2) ),fontsize=8)
        Y = Y + W
plt.tight_layout()
plt.show()

## Explore graph properties
import matplotlib.pyplot as plt
%matplotlib inline
fig, ax = plt.subplots(nrows=10,ncols=4,figsize=(20, 30))
y = 1
for row in ax:
    for col in row:
        deg_dist = [v for k,v in nx.degree(GMST_dict[y]).items()]
        deg_dist.sort(reverse=True)
        pdf, bins, patch = col.hist(deg_dist, bins=10)
        col.set_title ( " Degree Dist Day :" + str(y),fontsize=8 )
        y = y + W
plt.tight_layout()
plt.show()

5.2 Average degree of the network over time

We looked at the change in the average degree degree over time .For the networks from the Winner Take All method we can see that this varies widely overtime indicating the dynamic nature of the stock market.The peaks in the graph correspond well to major events in the market such as the 2008 - 2009 subprime crisis.However for the networks based on the MST method the Average degree is constant this is one of the key drawbacks of the MST method were major fluctuations in the market are not well represented in the network.We can also see in the plot that during normal time periods the Average Degree between the networks from both the methods is the same.

import matplotlib.pyplot as plt
%matplotlib inline
avgdf=attrib_df.groupby(by=['start'])['Degree'].mean()
avgdf.plot()
avgdf=attribMST_df.groupby(by=['start'])['Degree'].mean()
avgdf.plot()
plt.xlabel('Days')
plt.ylabel('Average Degree')
plt.legend(['Winner Take All','MST'], loc='center left', bbox_to_anchor=(1, 0.5),ncol=1)
plt.show()

5.3 High degree stocks in the network

We looked at the high degree stocks in the network at different windows to get out the important stocks which have high influence or which are a good indicator of how the stock market as whole is moving.The results are below.As we can see the stocks from financials sector have the highest degree in a number of windows .Is there a pattern here?

Winner Take All Method

attrib_df.sort_values(['start','Degree'],ascending=False).groupby(['start']).head(3)[['start','ticker','Name','Degree','Sector']]

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	start	ticker	Name	Degree	Sector
995	2710	DVN	Devon Energy Corp.	14.0	Energy
2774	2710	RIG	Transocean	14.0	Energy
2026	2710	MRO	Marathon Oil Corp.	12.0	Energy
2576	2647	PRU	Prudential Financial	24.0	Financials
3202	2647	UNM	Unum Group	24.0	Financials
2516	2647	PNC	PNC Financial Services	23.0	Financials
3572	2584	AEE	Ameren Corp	18.0	Utilities
4092	2584	LNT	Alliant Energy Corp	18.0	Utilities
4257	2584	PNW	Pinnacle West Capital	18.0	Utilities
2058	2521	MS	Morgan Stanley	39.0	Financials
4819	2521	DFS	Discover Financial Services	38.0	Financials
2271	2521	O	Realty Income Corporation	36.0	Real Estate
172	2458	AMP	Ameriprise Financial	60.0	Financials
4514	2458	ETN	Eaton Corporation	56.0	Industrials
5307	2458	MCO	Moody's Corp	56.0	Financials
2572	2395	PRU	Prudential Financial	29.0	Financials
546	2395	C	Citigroup Inc.	28.0	Financials
2056	2395	MS	Morgan Stanley	28.0	Financials
2055	2332	MS	Morgan Stanley	35.0	Financials
2511	2332	PNC	PNC Financial Services	33.0	Financials
475	2332	BK	The Bank of New York Mellon Corp.	32.0	Financials
413	2269	BBT	BB&T Corporation	27.0	Financials
2510	2269	PNC	PNC Financial Services	26.0	Financials
2570	2269	PRU	Prudential Financial	26.0	Financials
5261	2206	HON	Honeywell Int'l Inc.	105.0	Industrials
3391	2206	WFC	Wells Fargo	96.0	Financials
6474	2206	BRK.B	Berkshire Hathaway	96.0	Financials
509	2143	BXP	Boston Properties	18.0	Real Estate
2716	2143	REG	Regency Centers Corporation	17.0	Real Estate
1263	2143	FRT	Federal Realty Investment Trust	16.0	Real Estate
167	2080	AMP	Ameriprise Financial	27.0	Financials
3021	2080	STI	SunTrust Banks	25.0	Financials
3390	2080	WFC	Wells Fargo	25.0	Financials
166	2017	AMP	Ameriprise Financial	25.0	Financials
1989	2017	MET	MetLife Inc.	25.0	Financials
1503	2017	HES	Hess Corporation	21.0	Energy
845	1954	D	Dominion Resources	20.0	Utilities
3353	1954	WEC	Wec Energy Group Inc	18.0	Utilities
3468	1954	XEL	Xcel Energy Inc	18.0	Utilities
649	1891	CMS	CMS Energy	10.0	Utilities
3467	1891	XEL	Xcel Energy Inc	9.0	Utilities
3078	1891	TMK	Torchmark Corp.	8.0	Financials
163	1828	AMP	Ameriprise Financial	11.0	Financials
2563	1828	PRU	Prudential Financial	8.0	Financials
1862	1828	LNC	Lincoln National	7.0	Financials
3388	1765	WFC	Wells Fargo	7.0	Financials
5812	1765	L	Loews Corp.	6.0	Financials
162	1765	AMP	Ameriprise Financial	4.0	Financials
2342	1702	PEG	Public Serv. Enterprise Inc.	12.0	Utilities
3153	1702	UDR	UDR Inc	12.0	Real Estate
3349	1702	WEC	Wec Energy Group Inc	12.0	Utilities
2371	1639	PFG	Principal Financial Group	21.0	Financials
3348	1639	WEC	Wec Energy Group Inc	17.0	Utilities
4077	1639	LNT	Alliant Energy Corp	16.0	Utilities
1858	1576	LNC	Lincoln National	5.0	Financials
2560	1576	PRU	Prudential Financial	3.0	Financials
130	1576	AMG	Affiliated Managers Group Inc	2.0	Financials
3462	1513	XEL	Xcel Energy Inc	8.0	Utilities
39	1513	AEP	American Electric Power	6.0	Utilities
943	1513	DTE	DTE Energy Co.	6.0	Utilities
2291	1450	OXY	Occidental Petroleum	5.0	Energy
3461	1450	XEL	Xcel Energy Inc	4.0	Utilities
5136	1450	IVZ	Invesco Ltd.	4.0	Financials
465	1387	BK	The Bank of New York Mellon Corp.	20.0	Financials
1257	1387	FRT	Federal Realty Investment Trust	19.0	Real Estate
1855	1387	LNC	Lincoln National	19.0	Financials
2230	1324	NTRS	Northern Trust Corp.	8.0	Financials
532	1324	C	Citigroup Inc.	6.0	Financials
3385	1324	WFC	Wells Fargo	5.0	Financials
5808	1261	L	Loews Corp.	119.0	Financials
5255	1261	HON	Honeywell Int'l Inc.	99.0	Industrials
127	1261	AMG	Affiliated Managers Group Inc	98.0	Financials
2905	1198	SNA	Snap-On Inc.	294.0	Consumer Discretionary
5197	1198	CTAS	Cintas Corporation	289.0	Industrials
5624	1198	EFX	Equifax Inc.	280.0	Industrials
1852	1135	LNC	Lincoln National	18.0	Financials
205	1135	APA	Apache Corporation	17.0	Energy
2961	1135	SPG	Simon Property Group Inc	16.0	Real Estate
3274	1072	VNO	Vornado Realty Trust	9.0	Real Estate
98	1072	AIV	Apartment Investment & Mgmt	6.0	Real Estate
498	1072	BXP	Boston Properties	6.0	Real Estate
3181	1009	UNM	Unum Group	5.0	Financials
280	1009	AVB	AvalonBay Communities, Inc.	4.0	Real Estate
1099	1009	EQR	Equity Residential	4.0	Real Estate
4208	946	PCAR	PACCAR Inc.	22.0	Industrials
1913	946	MAA	Mid-America Apartments	17.0	Real Estate
2902	946	SNA	Snap-On Inc.	16.0	Consumer Discretionary
5251	883	HON	Honeywell Int'l Inc.	169.0	Industrials
5194	883	CTAS	Cintas Corporation	152.0	Industrials
730	883	COL	Rockwell Collins	149.0	Industrials
3270	820	VNO	Vornado Realty Trust	22.0	Real Estate
2590	820	PSA	Public Storage	21.0	Real Estate
1096	820	EQR	Equity Residential	20.0	Real Estate
276	757	AVB	AvalonBay Communities, Inc.	12.0	Real Estate
493	757	BXP	Boston Properties	12.0	Real Estate
3269	757	VNO	Vornado Realty Trust	11.0	Real Estate
2466	694	PLD	Prologis	20.0	Real Estate
492	694	BXP	Boston Properties	17.0	Real Estate
1094	694	EQR	Equity Residential	17.0	Real Estate
2170	631	NOV	National Oilwell Varco Inc.	20.0	Energy
2283	631	OXY	Occidental Petroleum	20.0	Energy
2640	631	PXD	Pioneer Natural Resources	19.0	Energy
1628	568	JPM	JPMorgan Chase & Co.	50.0	Financials
2493	568	PNC	PNC Financial Services	44.0	Financials
3098	568	TROW	T. Rowe Price Group	42.0	Financials
489	505	BXP	Boston Properties	29.0	Real Estate
2463	505	PLD	Prologis	28.0	Real Estate
3848	505	EMN	Eastman Chemical	28.0	Materials
806	442	CVX	Chevron Corp.	110.0	Energy
919	442	DIS	The Walt Disney Company	109.0	Consumer Discretionary
5366	442	A	Agilent Technologies Inc	104.0	Health Care
3095	379	TROW	T. Rowe Price Group	39.0	Financials
2949	379	SPG	Simon Property Group Inc	38.0	Real Estate
1243	379	FRT	Federal Realty Investment Trust	37.0	Real Estate
523	316	C	Citigroup Inc.	18.0	Financials
1088	316	EQR	Equity Residential	15.0	Real Estate
1697	316	KIM	Kimco Realty	15.0	Real Estate
1696	253	KIM	Kimco Realty	23.0	Real Estate
3261	253	VNO	Vornado Realty Trust	22.0	Real Estate
2691	253	REG	Regency Centers Corporation	21.0	Real Estate
1695	190	KIM	Kimco Realty	19.0	Real Estate
2581	190	PSA	Public Storage	19.0	Real Estate
267	190	AVB	AvalonBay Communities, Inc.	18.0	Real Estate
520	127	C	Citigroup Inc.	22.0	Financials
2728	127	RF	Regions Financial Corp.	16.0	Financials
2063	127	MTB	M&T Bank Corp.	14.0	Financials
997	64	ED	Consolidated Edison	24.0	Utilities
931	64	DTE	DTE Energy Co.	23.0	Utilities
2916	64	SO	Southern Co.	23.0	Utilities
142	1	AMP	Ameriprise Financial	21.0	Financials
387	1	BBT	BB&T Corporation	21.0	Financials
2687	1	REG	Regency Centers Corporation	20.0	Real Estate

Minimum Spanning Tree Method

attribMST_df.sort_values(['start','Degree'],ascending=False).groupby(['start']).head(3)[['start','ticker','Name','Degree','Sector']]

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	start	ticker	Name	Degree	Sector
8578	2710	HON	Honeywell Int'l Inc.	9.0	Industrials
8314	2710	HCP	HCP Inc.	7.0	Real Estate
14900	2710	RIG	Transocean	7.0	Energy
3996	2647	CMS	CMS Energy	11.0	Utilities
1757	2647	APH	Amphenol Corp	8.0	Information Technology
14987	2647	ROK	Rockwell Automation Inc.	8.0	Industrials
5395	2584	DOV	Dover Corp.	9.0	Industrials
11918	2584	MRO	Marathon Oil Corp.	9.0	Energy
13622	2584	PEP	PepsiCo Inc.	9.0	Consumer Staples
2767	2521	BRK.B	Berkshire Hathaway	8.0	Financials
3774	2521	CL	Colgate-Palmolive	8.0	Consumer Staples
18505	2521	XEL	Xcel Energy Inc	8.0	Utilities
6361	2458	ETN	Eaton Corporation	12.0	Industrials
3949	2458	CMI	Cummins Inc.	10.0	Industrials
11388	2458	MCO	Moody's Corp	10.0	Financials
1313	2395	AMG	Affiliated Managers Group Inc	9.0	Financials
9585	2395	ITW	Illinois Tool Works	8.0	Industrials
3992	2395	CMS	CMS Energy	7.0	Utilities
37	2332	A	Agilent Technologies Inc	12.0	Health Care
11958	2332	MS	Morgan Stanley	9.0	Financials
14146	2332	PPG	PPG Industries	9.0	Materials
4342	2269	COP	ConocoPhillips	9.0	Energy
10111	2269	KMB	Kimberly-Clark	9.0	Consumer Staples
8571	2269	HON	Honeywell Int'l Inc.	8.0	Industrials
8570	2206	HON	Honeywell Int'l Inc.	12.0	Industrials
11384	2206	MCO	Moody's Corp	11.0	Financials
9846	2206	JPM	JPMorgan Chase & Co.	10.0	Financials
430	2143	ADP	Automatic Data Processing	14.0	Information Technology
17399	2143	UTX	United Technologies	12.0	Industrials
11647	2143	MMC	Marsh & McLennan	11.0	Financials
1396	2080	AMP	Ameriprise Financial	14.0	Financials
9580	2080	ITW	Illinois Tool Works	11.0	Industrials
11690	2080	MMM	3M Company	10.0	Industrials
1395	2017	AMP	Ameriprise Financial	14.0	Financials
6926	2017	FIS	Fidelity National Information Services	12.0	Information Technology
9579	2017	ITW	Illinois Tool Works	9.0	Industrials
11688	1954	MMM	3M Company	10.0	Industrials
1394	1954	AMP	Ameriprise Financial	9.0	Financials
2626	1954	BLK	BlackRock	8.0	Financials
8565	1891	HON	Honeywell Int'l Inc.	14.0	Industrials
13831	1891	PH	Parker-Hannifin	9.0	Industrials
16471	1891	TMK	Torchmark Corp.	9.0	Financials
16470	1828	TMK	Torchmark Corp.	18.0	Financials
1260	1828	AME	AMETEK Inc	13.0	Industrials
1392	1828	AMP	Ameriprise Financial	9.0	Financials
13697	1765	PFG	Principal Financial Group	12.0	Financials
18009	1765	WFC	Wells Fargo	11.0	Financials
1391	1765	AMP	Ameriprise Financial	10.0	Financials
1390	1702	AMP	Ameriprise Financial	13.0	Financials
8562	1702	HON	Honeywell Int'l Inc.	12.0	Industrials
13564	1702	PEG	Public Serv. Enterprise Inc.	9.0	Utilities
13695	1639	PFG	Principal Financial Group	18.0	Financials
17963	1639	WEC	Wec Energy Group Inc	11.0	Utilities
1653	1639	APC	Anadarko Petroleum Corp	10.0	Energy
1388	1576	AMP	Ameriprise Financial	17.0	Financials
16554	1576	TROW	T. Rowe Price Group	11.0	Financials
10670	1576	LNC	Lincoln National	10.0	Financials
18621	1513	XOM	Exxon Mobil Corp.	11.0	Energy
10669	1513	LNC	Lincoln National	10.0	Financials
1299	1513	AMG	Affiliated Managers Group Inc	9.0	Financials
18488	1450	XEL	Xcel Energy Inc	13.0	Utilities
18926	1450	IVZ	Invesco Ltd.	13.0	Financials
1386	1450	AMP	Ameriprise Financial	12.0	Financials
15011	1387	ROP	Roper Industries	14.0	Industrials
5948	1387	EMN	Eastman Chemical	12.0	Materials
10667	1387	LNC	Lincoln National	10.0	Financials
11678	1324	MMM	3M Company	11.0	Industrials
12854	1324	NTRS	Northern Trust Corp.	10.0	Financials
19399	1324	L	Loews Corp.	10.0	Financials
8555	1261	HON	Honeywell Int'l Inc.	14.0	Industrials
19398	1261	L	Loews Corp.	14.0	Financials
416	1261	ADP	Automatic Data Processing	9.0	Information Technology
15536	1198	SNA	Snap-On Inc.	19.0	Consumer Discretionary
4589	1198	CTAS	Cintas Corporation	13.0	Industrials
14216	1198	PRU	Prudential Financial	11.0	Financials
2393	1135	BEN	Franklin Resources	11.0	Financials
1601	1135	APA	Apache Corporation	9.0	Energy
4544	1135	CSX	CSX Corp.	9.0	Industrials
18920	1072	IVZ	Invesco Ltd.	12.0	Financials
17382	1072	UTX	United Technologies	11.0	Industrials
18482	1072	XEL	Xcel Energy Inc	11.0	Utilities
412	1009	ADP	Automatic Data Processing	11.0	Information Technology
16545	1009	TROW	T. Rowe Price Group	9.0	Financials
17161	1009	UNM	Unum Group	9.0	Financials
13376	946	PCAR	PACCAR Inc.	21.0	Industrials
15532	946	SNA	Snap-On Inc.	13.0	Consumer Discretionary
16544	946	TROW	T. Rowe Price Group	13.0	Financials
8549	883	HON	Honeywell Int'l Inc.	19.0	Industrials
4232	883	COL	Rockwell Collins	11.0	Industrials
3704	883	CINF	Cincinnati Financial	9.0	Financials
2388	820	BEN	Franklin Resources	10.0	Financials
8548	820	HON	Honeywell Int'l Inc.	10.0	Industrials
19391	820	L	Loews Corp.	10.0	Financials
5850	757	EIX	Edison Int'l	12.0	Utilities
6950	757	FISV	Fiserv Inc	11.0	Information Technology
8547	757	HON	Honeywell Int'l Inc.	11.0	Industrials
11888	694	MRO	Marathon Oil Corp.	13.0	Energy
14120	694	PPG	PPG Industries	12.0	Materials
19389	694	L	Loews Corp.	10.0	Financials
6332	631	ETN	Eaton Corporation	8.0	Industrials
8545	631	HON	Honeywell Int'l Inc.	8.0	Industrials
11447	631	MET	MetLife Inc.	8.0	Financials
4491	568	CSCO	Cisco Systems	12.0	Information Technology
6331	568	ETN	Eaton Corporation	12.0	Industrials
15922	568	SWK	Stanley Black & Decker	11.0	Consumer Discretionary
5934	505	EMN	Eastman Chemical	16.0	Materials
5362	505	DOV	Dover Corp.	10.0	Industrials
19386	505	L	Loews Corp.	9.0	Financials
4885	442	DD	Du Pont (E.I.)	12.0	Materials
15920	442	SWK	Stanley Black & Decker	12.0	Consumer Discretionary
535	442	AEE	Ameren Corp	10.0	Utilities
12663	379	NOV	National Oilwell Varco Inc.	10.0	Energy
3305	379	CCL	Carnival Corp.	9.0	Consumer Discretionary
10651	379	LNC	Lincoln National	9.0	Financials
1280	316	AMG	Affiliated Managers Group Inc	9.0	Financials
6503	316	EXPD	Expeditors Int'l	9.0	Industrials
1676	316	APD	Air Products & Chemicals Inc	8.0	Materials
1631	253	APC	Anadarko Petroleum Corp	12.0	Energy
2379	253	BEN	Franklin Resources	9.0	Financials
11661	253	MMM	3M Company	9.0	Industrials
2070	190	BAC	Bank of America Corp	12.0	Financials
16532	190	TROW	T. Rowe Price Group	12.0	Financials
575	190	AEP	American Electric Power	8.0	Utilities
15915	127	SWK	Stanley Black & Decker	12.0	Consumer Discretionary
10647	127	LNC	Lincoln National	9.0	Financials
2905	127	C	Citigroup Inc.	8.0	Financials
12834	64	NTRS	Northern Trust Corp.	12.0	Financials
15826	64	STT	State Street Corp.	10.0	Financials
15562	64	SO	Southern Co.	9.0	Utilities
13669	1	PFG	Principal Financial Group	13.0	Financials
2199	1	BBT	BB&T Corporation	12.0	Financials
18465	1	XEL	Xcel Energy Inc	12.0	Utilities

We plotted the count of the sector of the high degree stock in the windows and we can see that Finance stocks definetly are the center of the market network.This kind of makes sense since Finance stocks are structurally dependent on what happens in the other sectors and we can expect them to be the important stocks which are correlated to many of the other stocks in the market.

df1=attrib_df.sort_values(['start','Degree'],ascending=False).groupby(['start']).head(3).groupby(['Sector']).count()['ticker']
df2=attribMST_df.sort_values(['start','Degree'],ascending=False).groupby(['start']).head(3).groupby(['Sector']).count()['ticker']
df3=pd.concat([df1, df2], axis=1).fillna(0)
df3.columns = ['t1','t2']

import matplotlib.pyplot as plt
%matplotlib inline
fig = plt.figure(figsize=(10,10))
df3['t1'].plot(kind='bar',color='red',  position=0, width=0.25)
df3['t2'].plot(kind='bar',color='blue',  position=1, width=0.25)
plt.xticks(rotation=90,fontsize=8)
plt.ylabel('Count')
plt.legend(['Winner Take All','MST'], loc='center left', bbox_to_anchor=(1, 0.5),ncol=1)
plt.plot()

[]

5.4 Stocks With High Betweenness Centrality

We looked at the stocks with high betweeness Centrality in the network at different windows to get out the important stocks which because of the position in the network will be good predictors in the movement of prices of the stocks.The results are below.As we can see financials stocks still lead in most periods .

Winner Take All Method

attrib_df.sort_values(['start','Betweeness'],ascending=False).groupby(['start']).head(3)[['start','ticker','Name','Betweeness','Sector']]

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	start	ticker	Name	Betweeness	Sector
6727	2710	FOXA	Twenty-First Century Fox Class A	0.007932	Consumer Discretionary
5244	2710	HOLX	Hologic	0.006344	Health Care
6673	2710	MDLZ	Mondelez International	0.004696	Consumer Staples
3202	2647	UNM	Unum Group	0.011359	Financials
2576	2647	PRU	Prudential Financial	0.007036	Financials
1653	2647	JPM	JPMorgan Chase & Co.	0.005982	Financials
1874	2584	LNC	Lincoln National	0.006935	Financials
1363	2584	GS	Goldman Sachs Group	0.006339	Financials
3547	2584	ZION	Zions Bancorp	0.005442	Financials
6454	2521	MMC	Marsh & McLennan	0.260341	Financials
5493	2521	BDX	Becton Dickinson	0.246517	Health Care
5376	2521	A	Agilent Technologies Inc	0.240836	Health Care
5221	2458	DOV	Dover Corp.	0.098698	Industrials
4514	2458	ETN	Eaton Corporation	0.052089	Industrials
2182	2458	NOV	National Oilwell Varco Inc.	0.051739	Energy
2572	2395	PRU	Prudential Financial	0.016133	Financials
171	2395	AMP	Ameriprise Financial	0.008358	Financials
2056	2395	MS	Morgan Stanley	0.005031	Financials
3792	2332	DHR	Danaher Corp.	0.114964	Health Care
6731	2332	XYL	Xylem Inc.	0.111858	Industrials
2414	2332	PH	Parker-Hannifin	0.101198	Industrials
544	2269	C	Citigroup Inc.	0.015129	Financials
689	2269	COF	Capital One Financial	0.012840	Financials
4141	2269	MSFT	Microsoft Corp.	0.010529	Information Technology
5261	2206	HON	Honeywell Int'l Inc.	0.178959	Industrials
5790	2206	KMB	Kimberly-Clark	0.099971	Consumer Staples
3504	2206	XOM	Exxon Mobil Corp.	0.096947	Energy
4569	2143	NOC	Northrop Grumman Corp.	0.003051	Industrials
4615	2143	UTX	United Technologies	0.003051	Industrials
4085	2143	LNT	Alliant Energy Corp	0.002730	Utilities
1614	2080	ITW	Illinois Tool Works	0.024743	Industrials
442	2080	BEN	Franklin Resources	0.021179	Financials
2411	2080	PH	Parker-Hannifin	0.017146	Industrials
1047	2017	EMR	Emerson Electric Company	0.043283	Industrials
441	2017	BEN	Franklin Resources	0.043158	Financials
6016	2017	ROP	Roper Industries	0.036402	Industrials
845	1954	D	Dominion Resources	0.010613	Utilities
1146	1954	ESS	Essex Property Trust, Inc.	0.008675	Real Estate
2343	1954	PEG	Public Serv. Enterprise Inc.	0.006047	Utilities
5258	1891	HON	Honeywell Int'l Inc.	0.068598	Industrials
3078	1891	TMK	Torchmark Corp.	0.047453	Financials
2504	1891	PNC	PNC Financial Services	0.039960	Financials
3077	1828	TMK	Torchmark Corp.	0.077264	Financials
6446	1828	MMC	Marsh & McLennan	0.076123	Financials
163	1828	AMP	Ameriprise Financial	0.072502	Financials
3388	1765	WFC	Wells Fargo	0.031168	Financials
5812	1765	L	Loews Corp.	0.024583	Financials
3755	1765	CINF	Cincinnati Financial	0.018437	Financials
2342	1702	PEG	Public Serv. Enterprise Inc.	0.013637	Utilities
104	1702	AIV	Apartment Investment & Mgmt	0.013070	Real Estate
3349	1702	WEC	Wec Energy Group Inc	0.012457	Utilities
2371	1639	PFG	Principal Financial Group	0.111754	Financials
2790	1639	ROK	Rockwell Automation Inc.	0.061661	Industrials
2013	1639	MRO	Marathon Oil Corp.	0.043269	Energy
1858	1576	LNC	Lincoln National	0.010065	Financials
159	1576	AMP	Ameriprise Financial	0.006494	Financials
130	1576	AMG	Affiliated Managers Group Inc	0.003896	Financials
1857	1513	LNC	Lincoln National	0.052643	Financials
1348	1513	GS	Goldman Sachs Group	0.040921	Financials
3387	1513	WFC	Wells Fargo	0.020460	Financials
2291	1450	OXY	Occidental Petroleum	0.040816	Energy
158	1450	AMP	Ameriprise Financial	0.029762	Financials
5136	1450	IVZ	Invesco Ltd.	0.017857	Financials
3735	1387	CBG	CBRE Group	0.174827	Real Estate
6013	1387	ROP	Roper Industries	0.113051	Industrials
3106	1387	TROW	T. Rowe Price Group	0.111419	Financials
2230	1324	NTRS	Northern Trust Corp.	0.059286	Financials
532	1324	C	Citigroup Inc.	0.039322	Financials
3385	1324	WFC	Wells Fargo	0.035995	Financials
4237	1261	PNW	Pinnacle West Capital	0.119275	Utilities
3498	1261	XOM	Exxon Mobil Corp.	0.096802	Energy
5808	1261	L	Loews Corp.	0.051756	Financials
5358	1198	WYN	Wyndham Worldwide	0.025580	Consumer Discretionary
2905	1198	SNA	Snap-On Inc.	0.020653	Consumer Discretionary
755	1198	COP	ConocoPhillips	0.018428	Energy
4539	1135	GE	General Electric	0.092994	Industrials
5211	1135	DOV	Dover Corp.	0.051663	Industrials
1852	1135	LNC	Lincoln National	0.049190	Financials
3274	1072	VNO	Vornado Realty Trust	0.014088	Real Estate
498	1072	BXP	Boston Properties	0.009269	Real Estate
2960	1072	SPG	Simon Property Group Inc	0.007442	Real Estate
280	1009	AVB	AvalonBay Communities, Inc.	0.020106	Real Estate
2593	1009	PSA	Public Storage	0.019841	Real Estate
1137	1009	ESS	Essex Property Trust, Inc.	0.016667	Real Estate
1579	946	HST	Host Hotels & Resorts	0.204877	Real Estate
4208	946	PCAR	PACCAR Inc.	0.191795	Industrials
3101	946	TROW	T. Rowe Price Group	0.117985	Financials
5194	883	CTAS	Cintas Corporation	0.065674	Industrials
730	883	COL	Rockwell Collins	0.043076	Industrials
5251	883	HON	Honeywell Int'l Inc.	0.033122	Industrials
5803	820	L	Loews Corp.	0.147791	Financials
6323	820	NWS	News Corp. Class B	0.132988	Consumer Discretionary
2590	820	PSA	Public Storage	0.127557	Real Estate
2096	757	MUR	Murphy Oil	0.073472	Energy
2007	757	MRO	Marathon Oil Corp.	0.061427	Energy
2642	757	PXD	Pioneer Natural Resources	0.061126	Energy
4997	694	PPG	PPG Industries	0.256966	Materials
2362	694	PFG	Principal Financial Group	0.255081	Financials
864	694	DD	Du Pont (E.I.)	0.243555	Materials
3407	631	WMB	Williams Cos.	0.152520	Energy
5800	631	L	Loews Corp.	0.145706	Financials
1969	631	MET	MetLife Inc.	0.145504	Financials
5682	568	FLS	Flowserve Corporation	0.138415	Industrials
4205	568	PCAR	PACCAR Inc.	0.136017	Industrials
4119	568	MMM	3M Company	0.102029	Industrials
3848	505	EMN	Eastman Chemical	0.228678	Materials
5798	505	L	Loews Corp.	0.121144	Financials
3097	505	TROW	T. Rowe Price Group	0.120766	Financials
5124	442	IVZ	Invesco Ltd.	0.064671	Financials
4632	442	CSCO	Cisco Systems	0.055969	Information Technology
919	442	DIS	The Walt Disney Company	0.055591	Consumer Discretionary
1966	379	MET	MetLife Inc.	0.063732	Financials
4535	379	GE	General Electric	0.047302	Industrials
1840	379	LNC	Lincoln National	0.045903	Financials
523	316	C	Citigroup Inc.	0.140048	Financials
1624	316	JPM	JPMorgan Chase & Co.	0.051430	Financials
1839	316	LNC	Lincoln National	0.048165	Financials
348	253	BAC	Bank of America Corp	0.104403	Financials
2065	253	MTB	M&T Bank Corp.	0.075769	Financials
1470	253	HD	Home Depot	0.065637	Consumer Discretionary
2581	190	PSA	Public Storage	0.115885	Real Estate
2729	190	RF	Regions Financial Corp.	0.074904	Financials
320	190	AXP	American Express Co	0.031737	Financials
389	127	BBT	BB&T Corporation	0.184087	Financials
4576	127	TRV	The Travelers Companies Inc.	0.183383	Financials
3446	127	XEL	Xcel Energy Inc	0.176740	Utilities
4301	64	STT	State Street Corp.	0.076092	Financials
3482	64	XOM	Exxon Mobil Corp.	0.066149	Energy
2539	64	PRU	Prudential Financial	0.046668	Financials
142	1	AMP	Ameriprise Financial	0.026139	Financials
116	1	AMG	Affiliated Managers Group Inc	0.025558	Financials
1032	1	EMR	Emerson Electric Company	0.022638	Industrials

Minimum Spanning Tree Method

attribMST_df.sort_values(['start','Betweeness'],ascending=False).groupby(['start']).head(3)[['start','ticker','Name','Betweeness','Sector']]

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	start	ticker	Name	Betweeness	Sector
2418	2710	BEN	Franklin Resources	0.649713	Financials
18946	2710	IVZ	Invesco Ltd.	0.581805	Financials
8578	2710	HON	Honeywell Int'l Inc.	0.477113	Industrials
17187	2647	UNM	Unum Group	0.570723	Financials
2109	2647	BAC	Bank of America Corp	0.565530	Financials
9589	2647	ITW	Illinois Tool Works	0.548996	Industrials
10686	2584	LNC	Lincoln National	0.623492	Financials
11038	2584	MA	Mastercard Inc.	0.599686	Information Technology
13710	2584	PFG	Principal Financial Group	0.509717	Financials
2767	2521	BRK.B	Berkshire Hathaway	0.618615	Financials
11653	2521	MMC	Marsh & McLennan	0.543698	Financials
2371	2521	BDX	Becton Dickinson	0.524445	Health Care
1402	2458	AMP	Ameriprise Financial	0.654670	Financials
11388	2458	MCO	Moody's Corp	0.632002	Financials
6361	2458	ETN	Eaton Corporation	0.462152	Industrials
9453	2395	IR	Ingersoll-Rand PLC	0.625261	Industrials
18941	2395	IVZ	Invesco Ltd.	0.606101	Financials
1313	2395	AMG	Affiliated Managers Group Inc	0.553152	Financials
2104	2332	BAC	Bank of America Corp	0.720245	Financials
11958	2332	MS	Morgan Stanley	0.605990	Financials
2940	2332	C	Citigroup Inc.	0.493196	Financials
18017	2269	WFC	Wells Fargo	0.675696	Financials
3726	2269	CINF	Cincinnati Financial	0.547420	Financials
2763	2269	BRK.B	Berkshire Hathaway	0.489289	Financials
9846	2206	JPM	JPMorgan Chase & Co.	0.680760	Financials
14232	2206	PRU	Prudential Financial	0.601696	Financials
15816	2206	STI	SunTrust Banks	0.509497	Financials
430	2143	ADP	Automatic Data Processing	0.642566	Information Technology
11647	2143	MMC	Marsh & McLennan	0.627954	Financials
17399	2143	UTX	United Technologies	0.569319	Industrials
1396	2080	AMP	Ameriprise Financial	0.721395	Financials
9580	2080	ITW	Illinois Tool Works	0.666124	Industrials
11558	2080	MKC	McCormick & Co.	0.279403	Consumer Staples
1395	2017	AMP	Ameriprise Financial	0.750213	Financials
15021	2017	ROP	Roper Industries	0.543654	Industrials
15857	2017	STT	State Street Corp.	0.501954	Financials
2626	1954	BLK	BlackRock	0.657465	Financials
11468	1954	MET	MetLife Inc.	0.614263	Financials
14228	1954	PRU	Prudential Financial	0.511184	Financials
8565	1891	HON	Honeywell Int'l Inc.	0.641060	Industrials
17395	1891	UTX	United Technologies	0.638150	Industrials
9577	1891	ITW	Illinois Tool Works	0.495383	Industrials
16470	1828	TMK	Torchmark Corp.	0.720504	Financials
11642	1828	MMC	Marsh & McLennan	0.528795	Financials
10674	1828	LNC	Lincoln National	0.479425	Financials
1391	1765	AMP	Ameriprise Financial	0.751050	Financials
19406	1765	L	Loews Corp.	0.505638	Financials
13697	1765	PFG	Principal Financial Group	0.450503	Financials
1390	1702	AMP	Ameriprise Financial	0.836793	Financials
14532	1702	R	Ryder System	0.409247	Industrials
19014	1702	DFS	Discover Financial Services	0.320586	Financials
13695	1639	PFG	Principal Financial Group	0.847559	Financials
2577	1639	BK	The Bank of New York Mellon Corp.	0.352938	Financials
15543	1639	SNA	Snap-On Inc.	0.340600	Consumer Discretionary
1388	1576	AMP	Ameriprise Financial	0.802485	Financials
10670	1576	LNC	Lincoln National	0.483489	Financials
16554	1576	TROW	T. Rowe Price Group	0.348268	Financials
10669	1513	LNC	Lincoln National	0.680996	Financials
18621	1513	XOM	Exxon Mobil Corp.	0.601379	Energy
4330	1513	COP	ConocoPhillips	0.507735	Energy
1386	1450	AMP	Ameriprise Financial	0.752720	Financials
18926	1450	IVZ	Invesco Ltd.	0.627248	Financials
13252	1450	OXY	Occidental Petroleum	0.332039	Energy
15011	1387	ROP	Roper Industries	0.660618	Industrials
16551	1387	TROW	T. Rowe Price Group	0.586394	Financials
5948	1387	EMN	Eastman Chemical	0.555885	Materials
12854	1324	NTRS	Northern Trust Corp.	0.663861	Financials
18924	1324	IVZ	Invesco Ltd.	0.633735	Financials
11678	1324	MMM	3M Company	0.539462	Industrials
1383	1261	AMP	Ameriprise Financial	0.703993	Financials
19398	1261	L	Loews Corp.	0.545069	Financials
17253	1261	UPS	United Parcel Service	0.449057	Industrials
15536	1198	SNA	Snap-On Inc.	0.704655	Consumer Discretionary
5813	1198	EFX	Equifax Inc.	0.546698	Industrials
4589	1198	CTAS	Cintas Corporation	0.530249	Industrials
2393	1135	BEN	Franklin Resources	0.666307	Financials
14127	1135	PPG	PPG Industries	0.571020	Materials
4544	1135	CSX	CSX Corp.	0.549951	Industrials
18920	1072	IVZ	Invesco Ltd.	0.684374	Financials
17382	1072	UTX	United Technologies	0.618229	Industrials
2392	1072	BEN	Franklin Resources	0.519780	Financials
16545	1009	TROW	T. Rowe Price Group	0.659050	Financials
5414	1009	DOW	Dow Chemical	0.563267	Materials
17161	1009	UNM	Unum Group	0.558852	Financials
13376	946	PCAR	PACCAR Inc.	0.769189	Industrials
15532	946	SNA	Snap-On Inc.	0.619479	Consumer Discretionary
16544	946	TROW	T. Rowe Price Group	0.426928	Financials
8549	883	HON	Honeywell Int'l Inc.	0.775314	Industrials
4584	883	CTAS	Cintas Corporation	0.551220	Industrials
18171	883	WMB	Williams Cos.	0.499975	Energy
11626	820	MMC	Marsh & McLennan	0.647800	Financials
8548	820	HON	Honeywell Int'l Inc.	0.545726	Industrials
16542	820	TROW	T. Rowe Price Group	0.537194	Financials
19390	757	L	Loews Corp.	0.685513	Financials
11889	757	MRO	Marathon Oil Corp.	0.625609	Energy
8547	757	HON	Honeywell Int'l Inc.	0.413511	Industrials
14120	694	PPG	PPG Industries	0.644525	Materials
13680	694	PFG	Principal Financial Group	0.556997	Financials
16012	694	SWN	Southwestern Energy	0.511855	Energy
18167	631	WMB	Williams Cos.	0.654137	Energy
12667	631	NOV	National Oilwell Varco Inc.	0.574091	Energy
13239	631	OXY	Occidental Petroleum	0.520811	Energy
13370	568	PCAR	PACCAR Inc.	0.598444	Industrials
6331	568	ETN	Eaton Corporation	0.580038	Industrials
5363	568	DOV	Dover Corp.	0.558415	Industrials
5934	505	EMN	Eastman Chemical	0.743458	Materials
1283	505	AMG	Affiliated Managers Group Inc	0.526646	Financials
16537	505	TROW	T. Rowe Price Group	0.470168	Financials
17372	442	UTX	United Technologies	0.709607	Industrials
7	442	A	Agilent Technologies Inc	0.599770	Health Care
5141	442	DIS	The Walt Disney Company	0.457273	Consumer Discretionary
16535	379	TROW	T. Rowe Price Group	0.627205	Financials
17635	379	VNO	Vornado Realty Trust	0.544936	Real Estate
15611	379	SPG	Simon Property Group Inc	0.532274	Real Estate
6503	316	EXPD	Expeditors Int'l	0.630099	Industrials
13674	316	PFG	Principal Financial Group	0.579633	Financials
10650	316	LNC	Lincoln National	0.545993	Financials
17193	253	UNP	Union Pacific	0.610661	Industrials
12057	253	MTB	M&T Bank Corp.	0.568301	Financials
2071	253	BAC	Bank of America Corp	0.532743	Financials
14728	190	RF	Regions Financial Corp.	0.664726	Financials
2070	190	BAC	Bank of America Corp	0.533955	Financials
2862	190	BXP	Boston Properties	0.514363	Real Estate
2905	127	C	Citigroup Inc.	0.668533	Financials
793	127	AIG	American International Group, Inc.	0.544492	Financials
16575	127	TRV	The Travelers Companies Inc.	0.536425	Financials
12834	64	NTRS	Northern Trust Corp.	0.600108	Financials
15826	64	STT	State Street Corp.	0.581946	Financials
9812	64	JPM	JPMorgan Chase & Co.	0.549021	Financials
13669	1	PFG	Principal Financial Group	0.658857	Financials
2067	1	BAC	Bank of America Corp	0.626898	Financials
1363	1	AMP	Ameriprise Financial	0.535597	Financials

We plotted the count of the sector of the high betweenness stocks in the windows and we can see that Finance stocks still lead however industrials come in number two.

df1=attrib_df.sort_values(['start','Betweeness'],ascending=False).groupby(['start']).head(3).groupby(['Sector']).count()['ticker']
df2=attribMST_df.sort_values(['start','Betweeness'],ascending=False).groupby(['start']).head(3).groupby(['Sector']).count()['ticker']
df3=pd.concat([df1, df2], axis=1).fillna(0)
df3.columns = ['t1','t2']

import matplotlib.pyplot as plt
%matplotlib inline
fig = plt.figure(figsize=(10,10))
df3['t1'].plot(kind='bar',color='red',  position=0, width=0.25)
df3['t2'].plot(kind='bar',color='blue',  position=1, width=0.25)
plt.xticks(rotation=90,fontsize=8)
plt.ylabel('Count')
plt.legend(['Winner Take All','MST'], loc='center left', bbox_to_anchor=(1, 0.5),ncol=1)
plt.plot()

[]

5.5. Communities detected over time

We looked at the number of communities detected over different periods.The variation in the number of communities denotes the dynamic nature of the market where a number of new communities of stocks are formed in each window and each of which can die or continue into subsequent periods

import matplotlib.pyplot as plt
%matplotlib inline
cdf=attrib_df.groupby(by=['start'])['Community'].nunique()
cdf.plot()
cdf=attribMST_df.groupby(by=['start'])['Community'].nunique()
cdf.plot()
plt.legend(['Winner Take All','MST'], loc='center left', bbox_to_anchor=(1, 0.5),ncol=1)
plt.xlabel('Days')
plt.ylabel('No of Communities')
plt.show()

5.6 Jaccard Similarity of Communities Detected with Sector Code

One the key points we wanted to look as part of the project was too see if the network created based on the stock prizes has any relationship to the SIC(Standard Industry classification Code) of these stocks .That is do the Financial,Industrial and IT stocks trade in a group ?.So to study the similarity of the communities detected with the SIC code of these stocks in the SP500 we computed the Jaccard similarity coefficient of these communities in each of the time windows with the stock list grouped by industry code.

We converted the stock list in the communities detected and the stock list as per the SIC code into a binary matrix.The presence of the stock in the community/group is denoted by 1 and the absence by 0.

Given two objects, A and B, each with n binary attributes, the Jaccard coefficient is a useful measure of the overlap that A and B share with their attributes. Each attribute of A and B can either be 0 or 1. The total number of each combination of attributes for both A and B are specified as follows:

- M_11 represents the total number of attributes where A and B both have a value of 1.
- M_01 represents the total number of attributes where the attribute of A is 0 and the attribute of B is 1.
- M_10 represents the total number of attributes where the attribute of A is 1 and the attribute of B is 0.
- M_00 represents the total number of attributes where A and B both have a value of 0.

Each attribute must fall into one of these four categories, meaning that M_11 + M_01 + M_10 + M_00 = n.

The Jaccard similarity coefficient, J, is given as

$$J = {M_{11} \over M_{01} + M_{10} + M_{11}}$$

So if a particular community has a Jaccard coefficient of more than 0.25 with the the group of stocks classified by the SIC code we classified the community as that particular SIC code .The plot is below and provides a interesting result.

So as we can visuvalize some sectors are well correlated or connected and the communities detected clearly indicate these stocks trade in groups over different periods

Finance
Real Estate
Utilities
Energy
Telecommunication
Industrials

However , the interesting result is that there are certain sectors which don't trade as groups .We don't see any communities being detected in these sectors.This seems to indicate that these sectors has various other stock specific factors at play in the market and they don't generally trade as a group

Information Technology
Materials
Consumer Staples
Consumer Discretionary
Health Care

cmatrix = attrib_df.groupby(by=['start','Community','ticker'])['ticker'].count().unstack()
cmatrix.fillna(0,inplace=True)
smatrix = attrib_df.groupby(by=['Sector','ticker'])['ticker'].count().unstack()
smatrix.fillna(0,inplace=True)
smatrix[smatrix != 0] = 1
csmatrix=pd.concat([smatrix,cmatrix])
jval = {}
for idxs,rows  in smatrix.iterrows():
    xlist = []
    ylist = []
    for idxc,rowc in cmatrix.iterrows():
        x = [int(i) for i in rows]
        y = [int(i) for i in rowc]
        jc = (np.double(np.bitwise_and(x, y).sum()) / np.double(np.bitwise_or(x, y).sum()))
        if jc > 0.25:
            xlist.append(idxc[0])
            ylist.append(jc)
    jval[idxs] = [xlist,ylist]

import matplotlib.pyplot as plt
%matplotlib inline
plt.figure(figsize=(20,10))
cm = plt.get_cmap('gist_rainbow')
NUM_COLORS = 11
colors = [cm(i) for i in np.linspace(0,1,NUM_COLORS)]
color_dict=dict(zip(list(jval.keys()), colors))
for key in jval:
    x = jval[key][0]
    y = jval[key][1]
    plt.scatter(x,y,color=color_dict[key],label=key)
    
#plt.legend(list(jval.keys()), loc='center left', bbox_to_anchor=(1, 0.5))
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.xlabel('Days')
plt.ylabel('Jaccard Simmilarity')
plt.title('Community Evolution-Winner Take All')
plt.show()

cmatrix = attribMST_df.groupby(by=['start','Community','ticker'])['ticker'].count().unstack()
cmatrix.fillna(0,inplace=True)
smatrix = attribMST_df.groupby(by=['Sector','ticker'])['ticker'].count().unstack()
smatrix.fillna(0,inplace=True)
smatrix[smatrix != 0] = 1
csmatrix=pd.concat([smatrix,cmatrix])
jval = {}
for idxs,rows  in smatrix.iterrows():
    xlist = []
    ylist = []
    for idxc,rowc in cmatrix.iterrows():
        x = [int(i) for i in rows]
        y = [int(i) for i in rowc]
        jc = (np.double(np.bitwise_and(x, y).sum()) / np.double(np.bitwise_or(x, y).sum()))
        if jc > 0.25:
            xlist.append(idxc[0])
            ylist.append(jc)
    jval[idxs] = [xlist,ylist]

import matplotlib.pyplot as plt
%matplotlib inline
plt.figure(figsize=(20,10))
cm = plt.get_cmap('gist_rainbow')
NUM_COLORS = 11
colors = [cm(i) for i in np.linspace(0,1,NUM_COLORS)]
color_dict=dict(zip(list(jval.keys()), colors))
for key in jval:
    x = jval[key][0]
    y = jval[key][1]
    plt.scatter(x,y,color=color_dict[key],label=key)
    
#plt.legend(list(jval.keys()), loc='center left', bbox_to_anchor=(1, 0.5))
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.xlabel('Days')
plt.ylabel('Jaccard Simmilarity')
plt.title('Community Evolution-MST')
plt.show()

5.7 Visuvalize the network and the evolution

We then merged the graphs from different windows into one graph assigned and exported the output to gexf format to visuvalise the dynamic evolution of the network.Since the networkx gexf export doesn't support dynamic attributes we uses XML parsing to convert the GEXF file to include the dynamic attributes and visuvalised the network and its dynamic evolution in Gephi

The video is available here

https://www.youtube.com/watch?v=KGpEs97YWJ4

# Constructing a graph
import networkx as nx
M=nx.MultiGraph()
for y in G_dict.keys():
    M.add_nodes_from(G_dict[y].nodes(data=True))
    M.add_edges_from(G_dict[y].edges(data=True))
nx.write_gexf(M,'data/price-sp500.gexf')

import xml.etree.ElementTree as ET
tree = ET.parse('data/price-sp500.gexf')
root = tree.getroot()
for element in root.iter("{http://www.gexf.net/1.1draft}node"):
    #print (element.attrib)
    noden = element.attrib['id']
    for child in element:
        for i in range(len(child)):
            child.remove(child[0])
        for y in G_dict.keys():
            T = G_dict[y]
            n = 0
            try:
                for att in T.node[noden]:
                    new = ET.Element("{http://www.gexf.net/1.1draft}attvalue")
                    new.set('for',str(n))
                    new.set('value',str(T.node[noden][att]))
                    new.set('start',str(y))
                    new.set('end',str(y+W-1))
                    child.append(new)
                    #print (T.node[noden][att])
                    n = n + 1
            except:
                continue
for element in root.iter("{http://www.gexf.net/1.1draft}edge"):
    #print (element.attrib)
    noden = element.attrib['id']
    for child in element:
        for i in range(len(child)):
            child[i].set('start',str(element.attrib['start']))
            child[i].set('end',str(element.attrib['end']))
tree.write('data/price-sp500-o.gexf')

# Constructing a graph
import networkx as nx
M=nx.MultiGraph()
for y in GMST_dict.keys():
    M.add_nodes_from(GMST_dict[y].nodes(data=True))
    M.add_edges_from(GMST_dict[y].edges(data=True))
nx.write_gexf(M,'data/price-MST-sp500.gexf')

import xml.etree.ElementTree as ET
tree = ET.parse('data/price-MST-sp500.gexf')
root = tree.getroot()
for element in root.iter("{http://www.gexf.net/1.1draft}node"):
    #print (element.attrib)
    noden = element.attrib['id']
    for child in element:
        for i in range(len(child)):
            child.remove(child[0])
        for y in GMST_dict.keys():
            T = GMST_dict[y]
            n = 0
            try:
                for att in T.node[noden]:
                    new = ET.Element("{http://www.gexf.net/1.1draft}attvalue")
                    new.set('for',str(n))
                    new.set('value',str(T.node[noden][att]))
                    new.set('start',str(y))
                    new.set('end',str(y+W-1))
                    child.append(new)
                    #print (T.node[noden][att])
                    n = n + 1
            except:
                continue
for element in root.iter("{http://www.gexf.net/1.1draft}edge"):
    #print (element.attrib)
    noden = element.attrib['id']
    for child in element:
        for i in range(len(child)):
            child[i].set('start',str(element.attrib['start']))
            child[i].set('end',str(element.attrib['end']))
tree.write('data/price-MST-sp500-o.gexf')

Conclusion

In conclusion we have been able to find answers to our questions and find some interesting results on the dynamics of the Equities in the S and P 500 index through Network Analysis.

We have been able to

1. Build networks for the various phases of the stock market between 2007 and 2017 split based on time periods,  for the stocks in the S and P 500 based  on the Winner Take All and the Minimum Spanning tree Method
2. Analyse the network and found that the network indeed exhibits scale free properties
3. Detect communities in these networks and compare these communities with the communities of the stocks based on the SIC code using Jaccard Similarity.
4. Study the evolution of these communities and noted the essential result that there are some sectors which essentially trade together as a group and some that don't.
5. Leverage the properties of the network to find the important stocks and the sectors which reflect movement of other stocks in the network
6. Identify stocks/sectors which based on their Betweeness cetrality are ideally placed to predict the movement of the prices in the market
7. Visuvalize the dynamic evolution of network by building dynamic graphs  using Gephi

The Analysis and the results provide a interesting insight into the stock market and its dynamic nature.The results from this study and further research can be used in areas of Portfolio optimisation, Risk Mitigation etc.Further analysis can also be done by building networks between different asset classes such as currency , commodities and studying their evolution and behavior over time.

Appendix

MST-Network - Period Starting 1702

from IPython.display import Image
Image(filename='data/MST-Period-1702.png') 

Winner Take All -Network - Period Starting 1702

from IPython.display import Image
Image(filename='data/WTA-period-1702.png') 

Winner Take All -Network - Period Starting 568

from IPython.display import Image
Image(filename='data/WTA-period-568.png')