A Gentle Introduction to Social Network Analysis From a Sociologist’s Perspective Reuben (Jack) Thomas reubenjthomas@unm.edu April 2017

The Structure of Relationships Relationships as the unit of analysis, but not as isolated units: their interconnection, their collective structure

Ties and Nodes Ties (Edges, Arcs) indicate relationships Nodes (Vertices) indicate social entities that form the relationships

Ties can be… Nodes can be… Exchanges & Transactions Alliances Friendships Enemies Sexual Encounters Murders Joke Telling Collaborations Co-sponsorship Co-attendance Lending a hand People Places Boats Organizations Social Movements Works of Art Events Dogs Countries Schools of Thought Memes

Ties can vary by… Strength (Valence) – binary ties vs weighted ties Multiplex: multiple types\roles\exchanges\etc in the relationship (vs Simplex) Directed vs Undirected Other variables about the relationship

Network Data can be Complete (Sociocentric): a sampling of ties within a predefined group Egocentric: a sampling of ties from unconnected nodes. (e.g. survey data from a national sample)

Network Data: Arc List Ego Alter Tie Strength Bob Maria 1 Bob Weiwei 1 Maria Weiwei 1 Weiwei Maria 1

Network Data: Adjacency Matrix Bob Maria Weiwei Bob 1 1 1 Maria 0 1 1 Weiwei 0 1 1

Network Data: Affiliation Matrix Karate Spanish Swim Bob 1 1 0 Maria 1 1 1 Weiwei 1 0 1

Multi-Mode Networks People and Groups Corporations and Governments Scientists and Papers and Journals and Disciplines Ships and Captains and Ports

Social Network Visualization

Density What proportion of dyads are connected? Number of Ties / Number of Possible Ties Density = 2(Edges) / Nodes(Nodes-1)

Network Distances Shortest Paths (Geodesics) The Diameter of a network is the length of its longest Geodesic Reachability or Connectedness: Can every node reach every other node? All geodesics are1 in perfectly dense networks

Six Degrees of Separation

Centrality How central is a node to the network, or how peripheral?

Degree-Centrality Having a lot of connections This is just the number of ties a node has In-degree vs Out- degree in directed networks

Eigenvector Centrality Having a lot of connections to people who have a lot of connections to people who... (recursive) Google’s PageRank is based on this idea

Closeness Centrality Being close to everyone in the network. The Farness of a node is the average distance to all other nodes in the network The Closeness of a node is the inverse of its Farness

Betweenness Centrality Being between a lot of people in the network. Being on many geodesics. The Betweenness of a node is the proportion of all the geodesics in the network that the node is a part of.

Clustering & Transitivity “My friends are friends” Transitivity Triangle completion rate The Clustering Coefficient: Actual links between alters / Possible links between alters Actual triangles / Possible triangles

Lack of diamond-clustering can create spanning trees in hetero-sexual networks

Community Structure & Cohesion Cliques, the Idea: sub-sets of the network in which nodes are more closely\strongly tied to each other. Formal definitions vary. A Maximal Clique is as big as possible while remaining perfectly dense. Less strict Clique definitions can be tricky (N-Cliques, N-Clans)

Cliques, Clans & Cores N-Cliques: All members must be mutually reachable at distance N. N-Clans: …but only through other members K-Cores: subgraphs within which all nodes have at least K ties (e.g. in a 4-core all nodes have at least 4 ties within it) Also see K-plexes, F-Groups, etc.

K-Connectivity The minimum number of actors you need to remove to break the network in two. You can talk about K-edge-connected vs K-vertex-connected. K-cutsets: the cliques formed by cutting K nodes/edges out

Betweenness Partitioning Remove edges that are the most “between” to reveal nested networks

Segregation \ Assortativity \ Homophily Are ties disproportionately between similar nodes? Can refer to any characteristic, not just groups: age, centrality, GPA, GDP, delinquency, height, hygiene, etc.

Equivalence Structural Equivalence Automorphic Equivalence same ties to same nodes Automorphic Equivalence indistinguishable structural position (but maybe not same alters) Regular Equivalence connected to equivalent alters (but maybe not same number of alters)

Statistical Models for Network Data Approaches vary by discipline, a lot in Sociology, ERGMS and SENIA are popular now, QAP models used to be

Quadratic Assignment Procedure (QAP) Regression Or, “Why can’t I just use dyadic data in a regular regression?” Answer: Auto-correlation, big time Solution: randomly switch who is tied to whom in the data many many times, re-estimating the model each time, to create an unbiased distribution of possible samples for calculating the standard errors. Appropriate when your cases are dyads within a network, and your dependent variable is information about the dyad.

Exponential Random Graph Models (ERGMs) Or, “Can’t we just run a logit on this?” The goal is to calculate the odds of a tie, controlling for aspects of network structure and dyadic variables. (earlier versions were called p* models)

Dynamic Networks An awful lot of SNA has treated networks as static things, but they rarely are. The cutting edge of SNA is all about modeling dynamic change in networks.

SIENA Simulation Investigation for Empirical Network Analysis Similar to ERGMs, but specifically developed for dynamic network data. I don’t really know it, ask Dan Ragan and Brian Soller.

Free Software! NodeXL: The easiest visualization tool I’ve used, but 1) it only works with Microsoft Excel in Windows, and 2) it is no longer free for advanced features. Also calculates some metrics. https://nodexl.codeplex.com Gephi: Another nice visualization tool, works with Macs, Linux and Windows alike, and its open source. https://gephi.org Pajek: Once the favorite visualization tool in SNA, and still popular http://vlado.fmf.uni-lj.si/pub/networks/pajek/ SoNIA: a visualization software package for dynamic data (network movies!) http://web.stanford.edu/group/sonia/

Not Free but Useful UCINET: relatively easy to use and versatile SNA package, free trial https://sites.google.com/site/ucinetsoftware/home

Free but Advanced R Packages: statnet: for Exponential Random Graphs (ERGMs), by UW-Seattle igraph: nice visualization tools, also in Python and C\C++ sna: a package of various SNA tools developed by Carter Butts at UC-Irvine RSiena: just like it sounds, run SIENA models in R

Good Intro to SNA Books John Scott’s Social Network Analysis David Easley & Jon Kleinberg’s Networks, Crowds, and Markets: Reasoning About a Highly Connected World Duncan Watts’s Six Degrees: The Science of a Connected Age Stanley Wasserman & Katherine Faust’s Social Network Analysis: Methods and Applications (this is more of a reference book) (find these slides on my website: www.unm.edu/~reubenjthomas)