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SOCIAL SOCIAL NET NETWORK ORK AN ANAL ALYSIS SIS USING ST USING STATA 10 June 2016 German Stata User Meeting GESIS, Cologne Thomas Grund University College Dublin thomas.u.grund@gmail.com www.grund.co.uk International art fairs

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SOCIAL SOCIAL NET NETWORK ORK AN ANAL ALYSIS SIS USING USING ST STATA

10 June 2016 German Stata User Meeting GESIS, Cologne Thomas Grund University College Dublin thomas.u.grund@gmail.com www.grund.co.uk

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NETWORK DYNAMICS

International art fairs Changes 2005 - 2006

Yogev, T. and Grund, T. (2012) Structural Dynamics and the Market for Contemporary Art: The Case

  • f International Art Fairs. Sociological Focus, 54(1), 23-40.
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CO-OFFENDING IN YOUTH GANG

Grund, T. and Densley, J. (2012) Ethnic Heterogeneity in the Activity and Structure of a Black Street

  • Gang. European Journal of Criminology, 9(3), 388-406.

Grund, T. and Densley, J. (2015). Ethnic homophily and triad closure: Mapping internal gang structure using exponential random graph models. Journal of Contemporary Criminal Justice, 31(3), 354–370

Caribbean East Africa UK West Africa

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MANCHESTER UTD – TOTTENHAM

9/9/2006, Old Trafford

Grund, T. (2012) Network Structure and Team Performance: The Case of English Premier League Soccer Teams. Social Networks, 34(4), 682-690.

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SOCIAL NETWORKS

  • Social
  • Friendship, kinship, romantic relationships
  • Government
  • Political alliances, government agencies
  • Markets
  • Trade: flow of goods, supply chains, auctions
  • Labor markets: vacancy chains, getting jobs
  • Organizations and teams
  • Interlocking directorates
  • Within-team communication, email exchange
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DEFINITION

  • Mathematically, a (binary) network is defined as 𝐻 = π‘Š, 𝐹

where π‘Š = 1,2, . . , π‘œ is a set of β€œvertices” (or β€œnodes”) and 𝐹 βŠ† 𝑗, π‘˜ | 𝑗, π‘˜ ∈ π‘Š is a set of β€œedges” (or β€œties”, β€œarcs”). Edges are simply pairs of vertices, e.g. 𝐹 βŠ† 1,2 , 2,5 … .

  • We write π‘§π‘—π‘˜ = 1 if actors 𝑗 and π‘˜ are related to each other (i.e.,

if 𝑗, π‘˜ ∈ 𝐹), and π‘§π‘—π‘˜ = 0 otherwise.

  • In digraphs (or directed networks) it is possible that π‘§π‘—π‘˜ β‰  π‘§π‘˜π‘—.
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ADJACENCY MATRIX

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ADJACENCY MATRIX

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ADJACENCY LIST

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ADJACENCY LIST

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NETWORK ANALYSIS

  • Simple description/characterization of networks
  • Calculation of node-level characteristics (e.g. centrality)
  • Components, blocks, cliques, equivalences…
  • Visualization of networks
  • Statistical modeling of networks, network dynamics
  • ….
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. findit nwcommands

http://nwcommands.org

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http://nwcommands.org

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Twitter: nwcommands GoogleGroup: nwcommands Search β€œnwcommands” to find a channel with video tutorials.

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NWCOMMANDS

  • Software package for Stata. Almost 100 new Stata commands

for handling, manipulating, plotting and analyzing networks.

  • Ideal for existing Stata users. Corresponds to the R packages

β€œnetwork”, β€œsna”, β€œigraph”, β€œnetworkDynamic”.

  • Designed for small to medium-sized networks (< 10000).
  • Almost all commands have menus. Can be used like Ucinet
  • r Pajek. Ideal for beginners and teaching.
  • Not just specialized commands, but whole infrastructure for

handling/dealing with networks in Stata.

  • Writing own network commands that build on the

nwcommands is very easy.

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LINES OF CODE

Type Files LoC .ado 94 14548 .dlg 57 5707 .sthlp 97 9954 Downloads Over 13 000 (since Jan 2015)

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. nwinstall, all

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. help nwcommands

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INTUITION

  • Software introduces netname and netlist.
  • Networks are dealt with like normal variables.
  • Many normal Stata commands have their network counterpart

that accept a netname, e.g. nwdrop, nwkeep, nwclear, nwtabulate, nwcorrelate, nwcollapse, nwexpand, nwreplace, nwrecode, nwunab and more.

  • Stata intuition just works.
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SETTING NETWORKS

  • β€œSetting” a network creates a network quasi-object that has a

netname.

  • After that you can refer to the network simply by its netname,

just like when refer to a variable with its varname.

Syntax:

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LIST ALL NETWORKS

Check out the return vector. Both commands populate it as well. These are the names of the networks in memory. You can refer to these networks by their name.

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LOAD NETWORK FROM THE INTERNET

. help netexample

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IMPORT NETWORK

  • A wide array of popular network file-formats are supported, e.g.

Pajek, Ucinet, by nwimport.

  • Files can be imported directly from the internet as well.
  • Similarly, networks can be exported to other formats with

nwexport.

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DROP/KEEP NETWORKS

  • Dropping and keeping networks works almost exactly like

dropping and keeping variables.

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DROP/KEEP NODES

You can also drop/keep nodes of a specific network.

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NODE ATTRIBUTES

  • Every node of a network has a nodeid, which is matched with the
  • bservation number in a normal dataset.
  • In this case, the node with nodeid == 1 is the β€œacciaiuoli” family and they

have a wealth of 10.

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nwset nwdrop nwds nwkeep nwcurrent nwimport webnwuse

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. webnwuse gang . nwplot gang, color(Birthplace) scheme(s2network)

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nwplot gang, color(Birthplace) symbol(Prison) size(Arrests)

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acciaiuoli albizzi barbadori bischeri castellani ginori guadagni lamberteschi medici pazzi peruzzi pucci ridolfi salviati strozzi tornabuoni

. webnwuse florentine . nwplot flomarriage, lab

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. nwplotmatrix flomarriage, lab

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. nwplotmatrix flomarriage, sortby(wealth) label(wealth)

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. webnwuse klas12 . nwmovie klas12_wave1-klas12_wave4

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. nwmovie _all, colors(col_t*) sizes(siz_t*) edgecolors(edge_t*)

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nwplot nwplotmatrix nwmovie

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SUMMARIZE

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SUMMARIZE

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TABULATE NETWORK

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TABULATE TWO NETWORKS

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TABULATE NETWORK AND ATTRIBUTE

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DYAD CENSUS

M: mutual A: asymmetric N: null

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nwsummarize nwtabulate nwdyads nwtriads

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TABULATE NETWORK

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RECODE TIE VALUES

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FLORENTINE FAMILIES

Marriage ties Business ties

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REPLACE TIE VALUES

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. help nwreplace

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GENERATE NETWORKS

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. help nwgen

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nwrecode nwreplace nwsync nwtranspose nwsym nwgen

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FLORENTINE FAMILIES

Who are the neighbors?

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NEIGHBORS

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NEIGHBORS

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CONTEXT

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CONTEXT

What is the average wealth of the β€œalbizzi’s” network neighbors?

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CONTEXT

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CONTEXT

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nwneighbor nwcontext

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DISTANCE

Length of a shortest connecting path defines the (geodesic) distance between two nodes.

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DISTANCE

π‘’π‘—π‘‘π‘’π‘π‘œπ‘‘π‘“π‘‘ = 1 1 1 2 1 2 2 2 1 1 3 3 1 2 2 2 1 3 3 1

3 1 2 4 5

𝑏𝑀𝑕𝑓𝑠𝑏𝑕𝑓 π‘‘β„Žπ‘π‘ π‘’π‘“π‘‘π‘’ π‘žπ‘π‘’β„Ž π‘šπ‘“π‘œπ‘•π‘’β„Ž = 1.8

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DISTANCE

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DISTANCE

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PATHS

How can one get from the β€œperuzzi” to the β€œmedici”?

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PATHS

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PATHS

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PATHS

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nwgeodesic nwpath nwplot

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CENTRALITY

Well connected actors are in a structurally advantageous position.

  • Getting jobs
  • Better informed
  • Higher status
  • …

What is β€œwell-connected?”

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DEGREE CENTRALITY

Degree centrality

  • Simply the number of incoming/outgoing ties => indegree

centrality, outdegree centrality

  • How many ties does an individual have?

𝐷𝑝𝑒𝑓𝑕𝑠𝑓𝑓 𝑗 = ෍

π‘˜=1 𝑂

π‘§π‘—π‘˜ 𝐷𝑗𝑒𝑓𝑕𝑠𝑓𝑓 𝑗 = ෍

π‘˜=1 𝑂

π‘§π‘˜π‘—

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BETWEENNESS CENTRALITY

Betweeness centrality

  • How many shortest paths go through an individual?

a e b c d 𝐷𝑐𝑓𝑒π‘₯π‘“π‘“π‘œπ‘œπ‘“π‘‘π‘‘ 𝑏 = 6

…

𝐷𝑐𝑓𝑒π‘₯π‘“π‘“π‘œπ‘œπ‘“π‘‘π‘‘ 𝑐 = 0

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BETWEENNESS CENTRALITY

Betweeness centrality

  • How many shortest paths go through an individual?

a e b c d e What about multiple shortest paths? E.g. there are two shortest paths from c to d (one via a and another

  • ne via e)

Give each shortest path a weight inverse to how many shortest paths there are between two nodes.

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nwdegree nwbetween nwevcent nwcloseness nwkatz

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RANDOM NETWORK

nwrandom 15, prob(.1)

Each tie has the same probability to exist, regardless of any other ties.

nwrandom 15, prob(.5)

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LATTICE RING LATTICE

nwlattice 5 5 nwring 15, k(2) undirected

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SMALL WORLD NETWORK

nwsmall 10, k(2) shortcuts(3) undirected

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PREFERENTIAL ATTACHMENT NETWORK

nwpref 10, prob(.5)

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HOMOPHILY NETWORK

nwhomophily gender, density(0.05) homophily(5)

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nwrandom nwlattice nwsmall nwpref nwring nwhomophily nwdyadprob

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Is a particular network pattern more (or less) prominent than expected?

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Question: Is there more or less correlation between these two networks than expected?

𝑑𝑝𝑠𝑠

𝑝𝑐𝑑 = 0.372

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Test-statistic

𝑑𝑝𝑠𝑠

𝑝𝑐𝑑 = 0.372

Distribution of test- statistic under null hypothesis

𝑑𝑝𝑠𝑠

π‘ π‘π‘œπ‘’π‘π‘› =? ?

1 2

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QUADRATIC ASSIGNMENT PROCEDURE

  • Scramble the network by permuting the actors

(randomly re-label the nodes), i.e. the actual network does not change, however, the position each node takes does.

  • Re-calculate the test-static on the

permuted networks and compare it with test-statistic on the unscrambled network.

Network structure is β€˜controlled’ for. Keeps dependencies.

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PERMUTATION TEST

permutation

  • 1

1 1

  • 1

1

  • 1

2 3 4 4 3 1 2

  • 1

1 1

  • 1

1

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GRAPH CORRELATION

nwcorrelate flobusiness flomarriage, permutations(100)

1 2 3 4

  • .2

.2 .4 correlation

based on 100 QAP permutations of network flobusiness

Corr(flobusiness, flomarriage)

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nwcorrelate nwpermute nwqap nwergm

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SOCIAL SOCIAL NET NETWORK ORK AN ANAL ALYSIS SIS USING USING ST STATA

10 June 2016 German Stata User Meeting GESIS, Cologne Thomas Grund University College Dublin thomas.u.grund@gmail.com www.grund.co.uk

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ERGM

logit 𝑄 𝑍

π‘—π‘˜ = 1 π‘œ 𝑏𝑑𝑒𝑝𝑠𝑑, 𝑍 π‘—π‘˜ 𝑑

= ෍

𝑙=1 𝐿

πœ„π‘™πœ€π‘‘π‘™ 𝒛

Probability that there is a tie from i to j. Given, n actors AND the rest

  • f the network, excluding the

dyad in question!

𝑍

π‘—π‘˜ 𝑑 = all dyads other than 𝑍 π‘—π‘˜

Amount by which the feature 𝑑𝑙 𝑧 changes when 𝑍

π‘—π‘˜ is

toggled from 0 to 1.

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ERGM

𝑄 𝒁 = 𝒛 πœ„ = 𝑓 πœ„π‘ˆπ‘‘ 𝒛 𝑑 πœ„

𝒁 = 𝒔𝒃𝒐𝒆𝒑𝒏 π’˜π’ƒπ’”π’‹π’ƒπ’„π’Žπ’‡, a randomly selected network from the pool of all potential networks 𝒛 = π’‘π’„π’•π’‡π’”π’˜π’‡π’† π’˜π’ƒπ’”π’‹π’ƒπ’„π’Žπ’‡, here observed network

Probability to draw β€˜our’ observed network y from all potential networks A score given to our network y using some parameters πœ„ and the network features s of y

𝜾 = 𝒒𝒃𝒔𝒃𝒏𝒇𝒖𝒇𝒔𝒕, to be estimated

A score given to all

  • ther networks we

could have observed

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ERGM: INTEPRETATION

ERGM’s ultimately give you an estimate for various parameters πœ„π‘™, which mean…

If a potential tie 𝑍

π‘—π‘˜ = 1

(between i and j) would change the network statistic 𝑑𝑙 by one unit. This changes the log-

  • dds for the tie 𝑍

π‘—π‘˜ to

actually exist by πœ„π‘™.

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EXAMPLE

Consider an ERGM for an undirected network with parameters for these three statistics: 𝑑𝑓𝑒𝑕𝑓𝑑 𝑧 = ෍ π‘§π‘—π‘˜ 𝑑2𝑑𝑒𝑏𝑠𝑑 𝑧 = ෍ π‘§π‘—π‘˜ 𝑧𝑗𝑙 π‘‘π‘’π‘ π‘—π‘π‘œπ‘•π‘šπ‘“π‘‘ 𝑧 = ෍ π‘§π‘—π‘˜ π‘§π‘˜π‘™π‘§π‘—π‘™

1) number of edges 2) number of 2-stars 3) number of triangles

𝑄 𝒁 = 𝒛 πœ„ ∝ 𝑓 πœ„π‘“π‘’π‘•π‘“π‘‘π‘‘π‘“π‘’π‘•π‘“π‘‘ 𝑧 + πœ„2𝑑𝑒𝑏𝑠𝑑𝑑2𝑑𝑒𝑏𝑠𝑑 𝑧 + πœ„π‘’π‘ π‘—π‘π‘œπ‘•π‘šπ‘“π‘‘π‘‘π‘’π‘ π‘—π‘π‘œπ‘•π‘šπ‘“π‘‘ 𝑧

Then the 3-parameter ERG distribution function is:

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SOCIAL SOCIAL NET NETWORK ORK AN ANAL ALYSIS SIS USING USING ST STATA

10 June 2016 German Stata User Meeting GESIS, Cologne Thomas Grund University College Dublin thomas.u.grund@gmail.com www.grund.co.uk