Introduction to Networks Methods Measures James Moody jmoody

Introduction to Networks Methods & Measures James Moody jmoody 77@soc. duke. edu http: //www. soc. duke. edu/~jmoody 77/ Duke Network Analysis Center Department of Sociology

Introduction We live in a connected world: “To speak of social life is to speak of the association between people – their associating in work and in play, in love and in war, to trade or to worship, to help or to hinder. It is in the social relations men establish that their interests find expression and their desires become realized. ” Peter M. Blau Exchange and Power in Social Life, 1964

Introduction We live in a connected world: * "If we ever get to the point of charting a whole city or a whole nation, we would have … a picture of a vast solar system of intangible structures, powerfully influencing conduct, as gravitation does in space. Such an invisible structure underlies society and has its influence in determining the conduct of society as a whole. " J. L. Moreno, New York Times, April 13, 1933 *1934, NYTime. Moreno claims this work was covered in “all the major papers” but I can’t find any other clips…

Introduction But scientists are starting to take network seriously: “Networks”

Introduction But scientists are starting to take network seriously: why? “Obesity” “Networks”

Introduction …and NSF is investing heavily in it.

Introduction High Schools as Networks

Introduction Countryside High School, by grade

Introduction Countryside High School, by race

Introduction And yet, standard social science analysis methods do not take this space into account. “For the last thirty years, empirical social research has been dominated by the sample survey. But as usually practiced, …, the survey is a sociological meat grinder, tearing the individual from his social context and guaranteeing that nobody in the study interacts with anyone else in it. ” Allen Barton, 1968 (Quoted in Freeman 2004) Moreover, the complexity of the relational world makes it impossible to identify social connectivity using only our intuition. Social Network Analysis (SNA) provides a set of tools to empirically extend our theoretical intuition of the patterns that compose social structure.

Introduction Social network analysis is: • a set of relational methods for systematically understanding and identifying connections among actors. SNA • is motivated by a structural intuition based on ties linking social actors • is grounded in systematic empirical data • draws heavily on graphic imagery • relies on the use of mathematical and/or computational models. • Social Network Analysis embodies a range of theories relating types of observable social spaces and their relation to individual and group behavior.

Introduction Key Questions Social Network analysis lets us answer questions about social interdependence. These include: “Networks as Variables” approaches • Are kids with smoking peers more likely to smoke themselves? • Do unpopular kids get in more trouble than popular kids? • Do central actors control resources? “Networks as Structures” approaches • What generates hierarchy in social relations? • What network patterns spread diseases most quickly? • How do role sets evolve out of consistent relational activity? Both: Connectionist vs. Positional features of the network We don’t want to draw this line too sharply: emergent role positions can affect individual outcomes in a ‘variable’ way, and variable approaches constrain relational activity.

Introduction How do we best use these rapidly developing data and methods to promote health and wellbeing? Goals for today 1) Two mechanisms for networks & health: connections & positions 2) Basic network data structures • How are network data different from standard data • Volume measures 3) Measuring network properties • Reachability • Distance • Redundancy • Centrality • Triad Distributions/hierarchy • Structural Equivalence 4) Software sneak peak Connections Positions

X Y C P Why do networks matter? Two fundamental mechanisms: Problem space Networks As Cause Networks As Result Connectionist: Networks as pipes – networks matter because of what flows through them. Positional: Networks as roles – networks matter because of relational patterns This rubric is organized around social mechanisms – the reasons why networks matter, which ends up being loosely correlated with specific types of measures & analyses.

X Y Why do networks matter? Two fundamental mechanisms: Connections Connectionist network mechanisms : Networks matter because of the things that flow through them. Networks as pipes. C P

X Y Why do networks matter? Two fundamental mechanisms: Connections example The spread of any epidemic depends on the number of secondary cases per infected case, known as the reproductive rate (R 0). R 0 depends on the probability that a contact will be infected over the duration of contact (b), the likelihood of contact (c), and the duration of infectiousness (D). For STI, the trick is specifying c, which depends on the network. C P

X Y Why do networks matter? Two fundamental mechanisms: Connections example Isolated vision C P

X Y Why do networks matter? Two fundamental mechanisms: Connections example Connected vision C P

X Y C P Connections: Diffusion Example: Small local changes can create cohesion cascades Emergent Connectivity in “low-degree” networks Partner Distribution Component Size/Shape Based on work supported by R 21 -HD 072810 (NICHD, Moody PI), R 01 DA 012831 -05 (NIDA Morris, Martina PI)

X Y C P Why do networks matter? Two fundamental mechanisms: Positions Positional network mechanisms : Networks matter because of the way they capture role behavior and social exchange. Networks as Roles. Romantic Love Provides food for Bickers with

X Y C P Why do networks matter? Two fundamental mechanisms: Positions Positional network mechanisms : Networks matter because of the way they capture role behavior and social exchange. Networks as Roles. Parent Child Romantic Love Provides food for Bickers with

Why do networks matter? Two fundamental mechanisms: Problem space Networks As Cause Networks As Result Connectionist: Networks as pipes Positional: Networks as roles Diffusion Peer influence Social Capital “small worlds” Social integration Peer selection Homophily Network robustness Popularity Effects Role Behavior Network Constraint Group stability Network ecology “Structuration” This rubric is organized around social mechanisms – the reasons why networks matter, which ends up being loosely correlated with specific types of measures & analyses.

Network Methods & Measures Social network analysis is: • a set of relational methods for systematically understanding and identifying connections among actors. SNA • is motivated by a structural intuition based on ties linking social actors • is grounded in systematic empirical data • draws heavily on graphic imagery • relies on the use of mathematical and/or computational models. • Social Network Analysis embodies a range of theories relating types of observable social spaces and their relation to individual and group behavior.

Network Methods & Measures Books & Resources Hannaman’s online text: great, free, all done w. UCINET: http: //www. faculty. ucr. edu/~hanneman/nettext/ Wasserman & Faust 1994

Social Network Data The unit of interest in a network are the combined sets of actors and their relations. We represent actors with points and relations with lines. Actors are referred to variously as: Nodes, vertices or points Relations are referred to variously as: Edges, Arcs, Lines, Ties Example: b a d c e

Social Network Data In general, a relation can be: Binary or Valued Directed or Undirected b b d a c a e c 1 a 1 3 c Undirected, Valued e Directed, binary Undirected, binary b d d 4 b d 2 e a c Directed, Valued e

Social Network Data Basic Data Elements In general, a relation can be: (1) Binary or Valued (2) Directed or Undirected b a d c e Directed, Multiplex categorical edges The social process of interest will often determine what form your data take. Conceptually, almost all of the techniques and measures we describe can be generalized across data format, but you may have to do some of the coding work yourself….

Social Network Data Basic Data Elements: Levels of analysis We can examine networks across multiple levels: 1) Ego-network - Have data on a respondent (ego) and the people they are connected to (alters). Example: 1985 GSS module - May include estimates of connections among alters 2) Partial network - Ego networks plus some amount of tracing to reach contacts of contacts - Something less than full account of connections among all pairs of actors in the relevant population - Example: CDC Contact tracing data for STDs

Social Network Data Basic Data Elements: Levels of analysis We can examine networks across multiple levels: 3) Complete or “Global” data - Data on all actors within a particular (relevant) boundary - Never exactly complete (due to missing data), but boundaries are set -Example: Coauthorship data among all writers in the social sciences, friendships among all students in a classroom

Social Network Data Basic Data Elements: Levels of analysis Primary Group Global-Net Ego-Net Best Friend Dyad 2 -step Partial network

Social Network Data Graph Layout (teaser) A good network drawing allows viewers to come away from the image with an almost immediate intuition about the underlying structure of the network being displayed. However, because there are multiple ways to display the same information, and standards for doing so are few, the information content of a network display can be quite variable. Consider the 4 graphs drawn at right. After asking yourself what intuition you gain from each graph, click on the screen. Now trace the actual pattern of ties. You will see that these 4 graphs are exactly the same.

Social Network Data Graph Layout Network visualization helps build intuition, but you have to keep the drawing algorithm in mind. Here we show the same graphs with two different techniques: Spring embedder layouts Tree-Based layouts (Fair - poor) (good) Most effective for very sparse, regular graphs. Very useful when relations are strongly directed, such as organization charts or internet connections. Most effective with graphs that have a strong community structure (clustering, etc). Provides a very clear correspondence between social distance and plotted distance Two images of the same network

Social Network Data Graph Layout Another example: Spring embedder layouts Tree-Based layouts (poor) (good) Two layouts of the same network

Social Network Data Basic Data Structures In general, graphs are cumbersome to work with analytically, though there is a great deal of good work to be done on using visualization to build network intuition. I recommend using layouts that optimize on the feature you are most interested in. The two I use most are a hierarchical layout or a force-directed layout are best. We’ll go into much more detail in the visualization seminar.

Social Network Data Social network data are substantively divided by the number of modes in the data. 1 -mode data represents edges based on direct contact between actors in the network. All the nodes are of the same type (people, organization, ideas, etc). Examples: Communication, friendship, giving orders, sending email. This is commonly what people think about when thinking about networks: nodes having direct relations with each other.

Social Network Data Social network data are substantively divided by the number of modes in the data. 2 -mode data represents nodes from two separate classes, where all ties are across classes. Examples: People as members of groups People as authors on papers Words used often by people Events in the life history of people The two modes of the data represent a duality: you can project the data as people connected to people through joint membership in a group, or groups to each other through common membership There may be multiple relations of multiple types connecting your nodes.

Social Network Data Basic Data Elements: Modes Bipartite networks imply a constraint on the mixing, such that ties only cross classes. Here we see a tie connecting each woman with the party she attended (Davis data)

Social Network Data Basic Data Elements: Modes Bipartite networks imply a constraint on the mixing, such that ties only cross classes. Here we see a tie connecting each woman with the party she attended (Davis data)

Social Network Data Basic Data Elements: Modes By projecting the data, one can look at the shared between people or the common memberships in groups: this is the person-to-person projection of the 2 -mode data.

Social Network Data Basic Data Elements: Modes By projecting the data, one can look at the shared between people or the common memberships in groups: this is the group-to-group projection of the 2 -mode data.

Social Network Data Example of a 2 -mode network: faculty supervising students - Any list of what people do – meetings, clubs, activities, coauthorship, – that they do with others forms network data. Moody

Social Network Data Example of a 2 -mode network: Patients & Care Settings The Movement of Carbapenem-Resistant Klebsiella pneumoniae among Healthcare Facilities: A Network Analysis D van Duin, F Perez, E Cober, SS Richter, RC Kalayjian, RA Salata, N Scalera, R Watkins, Y Doi, S Evans, VG Fowler Jr, KS Kaye, SD Rudin, KM Hujer, AM Hujer, RA Bonomo, and J Moody for the Antibacterial Resistance Leadership Group

Social Network Data Example of a 2 -mode network: Patients & Care Settings Casalino, Lawrence P. , Michael F. Pesko, Andrew M. Ryan, David J. Nyweide, Theodore J. Iwashyna, Xuming Sun, Jayme Mendelsohn and James Moody. “Physician Networks and Ambulatory Care Admissions” Medical Care 53: 534 -41

Social Network Data Basic Data Structures From pictures to matrices Because network images are hard to work with, we often use an adjacency matrix to represent the network. The matrix (X) at right represents an undirected binary network. Each node (a-e) is listed on both the row and the column. The ith row and the jth column (X ij) records the value of a tie from node i to node j. For example, the line between nodes a and b is represented as an entry in the first row and second column (red at right). Because the graph is undirected the ties sent are the same as the ties receive, so every entry above the diagonal equals the entries below the diagonal. b d a c e Undirected, binary a a b 1 c d e 1 1 1 1 An undirected graph and the corresponding matrix is symmetric.

Social Network Data Basic Data Structures b Directed graphs, on the other hand, are asymmetrical. a c e Directed, binary We can see that Xab =1 and Xba =1, therefore a “sends” to b and b “sends” to a. However, Xbc=0 while Xcb=1; therefore, c “sends” to b, but b does not reciprocate. d a a b 1 c d e b 1 c 1 1 d e 1 1 1 A directed graph and the corresponding matrix is asymmetrical.

Social Network Data Basic Data Structures b Directed graphs, on the other hand, are asymmetrical. a d c e Directed, Valued We can see that Xab =1 and Xba =1, therefore a “sends” to b and b “sends” to a. However, Xbc=0 while Xcb=1; therefore, c “sends” to b, but b does not reciprocate. a a b 3 c d e b 1 c 1 2 d e 2 4 1 A directed graph and the corresponding matrix is asymmetrical.

Social Network Data Basic Data Structures From matrices to lists (binary) Social network analysts also use adjacency lists and arc lists to more efficiently store network data. a b a d c e a b 1 c d e 1 1 1 1 Arc List a b Adjacency List b a b c b b a c c d c b d e c e d c e c d d e e c e d

Social Network Data Basic Data Structures From matrices to lists (valued) Social network analysts also use adjacency lists and arc lists to more efficiently store network data. a 1 a b 2 3 c d 5 1 e a b 1 c d e 3 5 2 2 3 5 1 1 Arc List a b 1 Adjacency List b a 1 contact value b c 2 a b a 1 c b 2 b a c b 1 2 c d 3 c b d e c 2 3 1 c e 5 d c e d 3 1 d c 3 e c d e 5 1 d e 1 e c 5 e d 1

Social Network Data Basic Data Structures Working with two-mode data A person-to-group adjacency matrix is rectangular, with persons down rows and groups across columns A = A B C D E F 1 0 1 1 0 0 0 2 0 0 1 1 0 0 3 0 0 0 1 1 1 4 0 0 0 1 5 1 0 0 Each column is a group, each row a person, and the cell = 1 if the person in that row belongs to that group. You can tell how many groups two people both belong to by comparing the rows: Identify every place that both rows = 1, sum them, and you have the overlap.

Social Network Data Basic Data Structures Working with two-mode data One can get either projection easily with a little matrix multiplication. First define AT as the transpose of A (simply reverse the rows and columns). If A is of size P x G, then AT will be of size G x P. A = A B C D E F 1 0 1 1 0 0 0 2 0 0 1 1 0 0 3 0 0 0 1 1 1 4 0 0 0 1 5 1 0 0 AT = 1 2 3 4 5 A 0 0 1 B 1 0 0 C 1 1 0 0 0 D 0 1 1 E 0 0 1 0 0 F 0 0 1 1 0

Social Network Data Basic Data Structures A B A = C D E F 1 0 1 1 0 0 0 2 0 0 1 1 0 0 3 0 0 0 1 1 1 4 0 0 0 1 (6 x 5) P = A(AT) A * AT = P (6 x 5)(5 x 6) (6 x 6) A B C D E F A 1 0 0 B 0 1 1 0 0 0 P C 0 1 2 1 0 0 D 1 0 1 4 1 2 E 0 0 0 1 1 1 F 0 0 0 2 1 2 5 1 0 0 1 2 AT = 3 4 5 A 0 0 1 B 1 0 0 C 1 1 0 0 0 D 0 1 1 E 0 0 1 0 0 F 0 0 1 1 0 (5 x 6) G = AT(A) AT * A = P (5 x 6) 6 x 5) (5 x 5) 1 2 3 4 5 1 2 1 0 0 0 2 1 1 1 G 3 0 1 3 2 1 4 0 1 2 2 1 5 0 1 1 1 2

Social Network Data Basic Volume Measures Density: Mean of the adjacency matrix 1 b 2 a 3 c d 5 a a -- b 1 c 0 d 0 e 0 b 1 -- 2 0 0 c 0 2 -- 3 5 d 0 e 0 0 3 -- 0 5 1 1 -- b 1 c d a a -- 0 0 e 0 b 1 -- 1 0 0 c 0 1 -- 1 1 d 0 e 0 0 1 -- 1 0 1 1 -- 1 e

Social Network Data Basic Volume Measures Degree: Number of links adjacent to a node a a b 1 c d e 1 1 1 3 0 1 1 2 1 2 1 7

Social Network Data Basic Volume Measures Weighted Degree: Sum of links adjacent to a node a a b 1 c d e 1 1 3 2 1 6 0 5 1 6 1 3 5 4 1 14

Social Network Data Basic Volume Measures matter For example only 3. 9% of kids are in the most popular quintile all 5 waves, and a full 50% are in the top quintile at least once over the observation period. Similarly, only 1. 9% of kids are least popular all 5 waves, but 43% are least popular at least once. W. T. Grant Foundation 8316 & NIDA R 01 DA 018225 (Osgood)), The Impact of School-Based Prevention on Friendship Networks and Peer Influence

X Y Social Network Data This figure represents the distribution of cases across that space, with key points labeled. W. T. Grant Foundation 8316 & NIDA R 01 DA 018225 (Osgood)), The Impact of School-Based Prevention on Friendship Networks and Peer Influence C P

X Y Dynamics of networks Popularity Effects The regression estimates define a simple 2 -d space of slope (Y) and intercept (x). This region represents steady & sharp increases in popularity, from a low starting point to a high ending point. These kids are upwardly mobile W. T. Grant Foundation 8316 & NIDA R 01 DA 018225 (Osgood)), The Impact of School-Based Prevention on Friendship Networks and Peer Influence C P

X Y Dynamics of networks Popularity Effects The regression estimates define a simple 2 -d space of slope (Y) and intercept (x). This region represents steady & sharp increases in popularity, from a low starting point to a high ending point. These kids are upwardly mobile W. T. Grant Foundation 8316 & NIDA R 01 DA 018225 (Osgood)), The Impact of School-Based Prevention on Friendship Networks and Peer Influence C P

X Y Dynamics of networks Popularity Effects The regression estimates define a simple 2 -d space of slope (Y) and intercept (x). This region represents steady & sharp increases in popularity, from a low starting point to a high ending point. These kids are upwardly mobile W. T. Grant Foundation 8316 & NIDA R 01 DA 018225 (Osgood)), The Impact of School-Based Prevention on Friendship Networks and Peer Influence C P

X Y Dynamics of networks Popularity Effects At slope=0, we see the main effect of popularity as increasing along the X axis: There is a strong an steady increase in use as popularity goes up. W. T. Grant Foundation 8316 & NIDA R 01 DA 018225 (Osgood)), The Impact of School-Based Prevention on Friendship Networks and Peer Influence C P

X Y Dynamics of networks Popularity Effects When slope is positive, kids are becoming more popular, and we see an increase in use… W. T. Grant Foundation 8316 & NIDA R 01 DA 018225 (Osgood)), The Impact of School-Based Prevention on Friendship Networks and Peer Influence C P

X Y Dynamics of networks Popularity Effects But even stronger use among those with decreasing substance use. W. T. Grant Foundation 8316 & NIDA R 01 DA 018225 (Osgood)), The Impact of School-Based Prevention on Friendship Networks and Peer Influence C P

Mental health and network size Network size matters for depressive symptoms, but not linearly Falci, Christina, and Clea Mc. Neely. 2013. “Too Many Friends: Social Integration, Network Cohesion, and Adolescent Depressive Symptoms. ” Social Forces 87(4): 2031– 61.

Health benefits of network growth The risk of functional impairment among older adults decreases as they have more confidants in their personal networks Predicted probabilities of different degrees of functional impairment among older adults, by the number of confidants added Cornwell, Benjamin, and Edward O. Laumann. 2015. "The health benefits of network growth: New evidence from a national survey of older adults. " Social Science & Medicine 125: 94 -106.

Measuring Networks: Connectionist properties “Goods” flow through networks:

Measuring Networks: Connectionist properties Two features of the network’s topology are key to connectionist aspects of networks: connectivity and centrality Connectivity refers to how actors in one part of the network are connected to actors in another part of the network. • Reachability: Is it possible for actor i to reach actor j? This can only be true if there is a chain of contact from one actor to another. • Distance: Given they can be reached, how many steps are they from each other? • Redundancy: How many different paths connect each pair?

Measuring Networks: Connectionist properties These features combine in important ways to define characteristic features of networks : Degree x Reachability Emergent Connectivity Distance x (Local) Redundancy Small Worlds Reachability x redundancy Network Resilience

Measuring Networks: Connectivity Reachability Indirect connections are what make networks systems. One actor can reach another if there is a path in the graph connecting them. b a a d c b e f c f d e Paths can be directed, leading to a distinction between strong and weak components

Measuring Networks: Connectivity Reachability Basic elements in connectivity • A path is a sequence of nodes and edges starting with one node and ending with another, tracing the indirect connection between the two. On a path, you never go backwards or revisit the same node twice. Example: a b c d • A walk is any sequence of nodes and edges, and may go backwards. Example: a b c b c d • A cycle is a path that starts and ends with the same node. Example: a b c a

Measuring Networks: Connectivity Reachability If you can trace a sequence of relations from one actor to another, then the two are reachable. If there is at least one path connecting every pair of actors in the graph, the graph is connected and is called a component. Intuitively, a component is the set of people who are all connected by a chain of relations.

Measuring Networks: Connectivity Reachability This example contains many components.

Measuring Networks: Connectivity Reachability Because relations can be directed or undirected, components come in two flavors: For a graph with any directed edges, there are two types of components: Strong components consist of the set(s) of all nodes that are mutually reachable Weak components consist of the set(s) of all nodes where at least one node can reach the other.

Measuring Networks: Connectivity Reachability There are only 2 strong components with more than 1 person in this network. Components are the minimum requirement for social groups. As we will see later, they are necessary but not sufficient All of the major network analysis software identifies strong and weak components

Measuring Networks: Connectivity Reachability x Degree Distribution Scale Free Networks Many large networks are characterized by a highly skewed distribution of the number of partners (degree)

Measuring Networks: Connectivity Reachability x Degree Distribution Scale Free Networks Many large networks are characterized by a highly skewed distribution of the number of partners (degree)

Measuring Networks: Connectivity Reachability x Degree Distribution Scale Free Networks The scale-free model focuses on the distance-reducing capacity of high-degree nodes:

Measuring Networks: Connectivity Reachability x Degree Distribution Scale Free Networks The scale-free model focuses on the distance-reducing capacity of high-degree nodes: If a preferential attachment model is active, then high-degree nodes are hubs, that if removed can disconnect the network. Since hubs are rare, PA networks are robust to random attack, but very fragile to targeted attack.

Measuring Networks: Large-Scale Models Scale Free Networks Colorado Springs High-Risk (Sexual contact only) • Network is approximately scale-free, with l = -1. 3 • But connectivity does not depend on the hubs. • Pref. Attach Scale-free distribution, but scale-free distribution ^ preferential attachment.

Measuring Networks: Connectivity Distance Geodesic distance is measured by the smallest (weighted) number of relations separating a pair: Actor “a” is: 1 step from 4 2 steps from 5 3 steps from 4 4 steps from 3 5 steps from 1 a

Measuring Networks: Connectivity Distance When the graph is directed, distance is also directed (distance to vs distance from), following the direction of the tie. a b c d e f g h i j k l m ---------------------a. . 1 2. . . . 2 1 b. 3. 1. . . . 1 2 c. . . d. 4 3 1. 1 2 1. 2. . 2 3 e. 3 2 2 1. 1 2. 1. . 1 2 f. 4 3 3 2 1. 3. 2. . 2 3 g. 5 4 4 3 2 1. . 3 4 h. . 1. . i. . . j. . 1. . k. . 1. . l. 2 1 2. . 1 m. 1 2 3. . . . 1. a k m b l j i e c f d g h

Measuring Networks: Connectivity Distance Reachability in Colorado Springs (Sexual contact only) • High-risk actors over 4 years • 695 people represented • Longest path is 17 steps • Average distance is about 5 steps • Average person is within 3 steps of 75 other people (Node size = log of degree)

Measuring Networks: Connectivity Distance Calculating distance in global networks: Powers of the adjacency matrix Calculate reachability through matrix multiplication. (see p. 162 of W&F) 0 1 0 0 0 1 e d c b a f 1 0 0 0 X 0 0 1 1 1 0 0 0 1 1 0 0 0 2 0 1 1 2 X 2 2 0 0 1 4 1 1 2 1 1 0 1 Distance. 1 2 0 1. 1 2 2 1. 1 0 2 1 1 1 2 0 1 1 1 2 1 0 2 1 1. 2 4 0 6 1 1 0 X 3 0 2 6 1 2 5 5 2 5 3 6 1 0 2 0 1 1 2 0 4 0 2 2 4 2 1 5 3 2 1 4 0 6 1 1 0 1 2 2. Distance. 1 2 3 3 1. 1 2 2 2 1. 1 1 3 2 1 1. 1 2 1 2 2.

Measuring Networks: Connectivity Distance Calculating distance in global networks: Breadth-First Search In large networks, matrix multiplication is just too slow. A breadthfirst search algorithm works by walking through the graph, reaching all nodes from a particular start node. Distance is calculated directly in most SNA software packages.

Measuring Networks: Connectivity Distance As a graph statistic, the distribution of distance can tell you a good deal about how close people are to each other (we’ll see this more fully when we get to closeness centrality). The diameter of a graph is the longest geodesic, giving the maximum distance. We often use the l, or the mean distance between every pair to characterize the entire graph. For example, all else equal, we would expect rumors to travel faster through settings where the average distance is small.

Measuring Networks: Connectivity Redundancy Local For a real network, people’s friends are not random, but clustered. We can modify the random equation by adjusting a, such that some portion of the contacts are random, the rest not. This adjustment is a ‘bias’ - I. e. a non-random element in the model -- that gives rise to the notion of ‘biased networks’. People have studied (mathematically) biases associated with: • Race (and categorical homophily more generally) • Transitivity (Friends of friends are friends) • Reciprocity (i--> j, j--> i) There is still a great deal of work to be done in this area empirically, and it promises to be a good way of studying the structure of very large networks.

Measuring Networks: Connectivity Redundancy Local Transitivity No ego Density Transitivity 0 0. 4 0 0. 71 0 Local redundancy is known as “clustering” or “transitivity” - that one’s friends are friends with each other. 1. 0 Density is the proportion of pairs tied, excluding ego. Transitivity is the proportion of two-step ties that are closed (Friend of a Friend is a friend) 0. 78 . 64 1 1 1

X Y C P Measuring Networks: Connectivity Redundancy Global: Structural Cohesion: A network’s structural cohesion is equal to the minimum number of actors who, if removed from the network, would disconnect it. 0 2 1 Node Connectivity As size of cut-set 3

X Y C P Measuring Networks: Connectivity Redundancy Global: Structural Cohesion: A network’s structural cohesion is equal to the minimum number of actors who, if removed from the network, would disconnect it. 0 1 2 Node Connectivity As number of node-independent paths 3

X Y C P Measuring Networks: Connectivity Redundancy Global: Structural Cohesion Sociogram Nestedness Structure Cohesive Blocks Depth 1 2 3 4 5 Cohesive Blocking The arrangement of subsequently more connected sets by branches and depth uniquely characterize the connectivity structure of a network

Measuring Networks Centrality Distance & Connectivity measures “locate” a node based on particular features of the path structure, but there are many other ways of locating nodes in networks. Centrality refers to (one dimension of) location, identifying where an actor resides in a network. • For example, we can compare actors at the edge of the network to actors at the center. • In general, this is a way to formalize intuitive notions about the distinction between insiders and outsiders. As a terminology point, some authors distinguish centrality from prestige based on the directionality of the tie. Since the formulas are the same in every other respect, I stick with “centrality” for simplicity.

Measuring Networks Centrality Conceptually, centrality is fairly straight forward: we want to identify which nodes are in the ‘center’ of the network. In practice, identifying exactly what we mean by ‘center’ is somewhat complicated, but substantively we often have reason to believe that people at the center are very important. The standard centrality measures capture a wide range of “importance” in a network: • Degree • Closeness • Betweenness • Eigenvector / Power measures After discussing these, I will describe measures that combine features of each of them.

Measuring Networks Centrality The most intuitive notion of centrality focuses on degree. Degree is the number of direct contacts a person has. The ideas is that the actor with the most ties is the most important:

Measuring Networks Centrality Degree centrality, however, can be deceiving, because it is a purely local measure.

Measuring Networks Centrality If we want to measure the degree to which the graph as a whole is centralized, we look at the dispersion of centrality: Simple: variance of the individual centrality scores. Or, using Freeman’s general formula for centralization (which ranges from 0 to 1): UCINET, SPAN, PAJEK and most other network software will calculate these measures.

Measuring Networks Degree Centralization Scores Centrality Freeman: 1. 0 Variance: 3. 9 Freeman: . 02 Variance: . 17 Freeman: . 07 Variance: . 20 Freeman: 0. 0 Variance: 0. 0

Measuring Networks Centrality A second measure of centrality is closeness centrality. An actor is considered important if he/she is relatively close to all other actors. Closeness is based on the inverse of the distance of each actor to every other actor in the network. Closeness Centrality: Normalized Closeness Centrality

Measuring Networks Closeness Centrality in the examples Centrality Distance 0 1 1 1 1 0 2 2 2 1 2 2 0 2 2 1 2 2 2 0 2 2 2 1 2 2 0 2 2 Closeness 1 2 2 2 0 2 1 2 2 2 0 Distance 0 1 2 3 4 4 3 2 1 1 0 1 2 3 4 4 3 2 2 1 0 1 2 3 4 4 3 3 2 1 0 1 2 3 4 4 4 3 2 1 0 1 2 3 3 4 4 3 2 1 0 1 2 . 143. 077 Closeness 2 3 4 4 3 2 1 0 1 1 2 3 4 4 3 2 1 0 . 050 normalized 1. 00. 538 normalized. 400

Measuring Networks Centrality Closeness Centrality in the examples Distance 0 1 2 3 4 1 0 1 2 3 2 1 0 1 4 3 2 1 0 5 4 3 2 1 6 5 4 3 2 1 0 Closeness. 048. 063. 077. 083. 077. 063. 048 normalized. 286. 375. 462. 500. 462. 375. 286

Measuring Networks Closeness Centrality in the examples Centrality Distance 0 1 1 2 3 4 4 5 5 6 1 0 1 1 2 3 3 4 4 5 1 1 0 1 2 3 3 4 4 5 2 1 1 0 1 2 2 3 3 4 3 2 2 1 0 1 1 2 2 3 4 3 3 2 1 0 2 3 3 4 1 1 2 4 3 3 2 1 2 0 1 1 2 3 3 4 5 4 4 3 2 3 1 0 1 1 4 4 5 5 4 4 3 2 3 1 1 0 1 4 4 5 6 5 5 4 3 4 2 1 1 0 5 5 6 Closeness 5 4 4 3 2 1 3 4 4 5 0 1 1 5 4 4 3 2 1 3 4 4 5 1 0 1 6 5 5 4 3 2 4 5 5 6 1 1 0 . 021. 027. 034. 042. 034. 027. 021 normalized. 255. 324. 414. 500. 414. 324. 255

Measuring Networks Centrality Betweenness Centrality: Model based on communication flow: A person who lies on communication paths can control communication flow, and is thus important. Betweenness centrality counts the number of shortest paths between i and k that actor j resides on. b a C d e f g h

Measuring Networks Centrality Betweenness Centrality: Where gjk = the number of geodesics connecting jk, and gjk(ni) = the number that actor i is on. Usually normalized by:

Measuring Networks Centrality Betweenness Centrality: Centralization: 1. 0 Centralization: . 59 Centralization: . 31 Centralization: 0

Measuring Networks Centrality Betweenness Centrality: Centralization: . 183

Measuring Networks Centrality Information Centrality: It is quite likely that information can flow through paths other than the geodesic. The Information Centrality score uses all paths in the network, and weights them based on their length.

Measuring Networks Centrality Comparing across these 3 centrality values • Generally, the 3 centrality types will be positively correlated • When they are not (low) correlated, it probably tells you something interesting about the network. Low Degree Low Closeness Low Betweenness High Degree Embedded in cluster that is far from the rest of the network Ego's connections are redundant - communication bypasses him/her High Closeness Key player tied to important/active alters Probably multiple paths in the network, ego is near many people, but so are many others High Betweenness Ego's few ties are crucial for network flow Very rare cell. Would mean that ego monopolizes the ties from a small number of people to many others.

Measuring Networks Centrality Bonacich Power Centrality: Actor’s centrality (prestige) is equal to a function of the prestige of those they are connected to. Thus, actors who are tied to very central actors should have higher prestige/ centrality than those who are not. • a is a scaling vector, which is set to normalize the score. • b reflects the extent to which you weight the centrality of people ego is tied to. • R is the adjacency matrix (can be valued) • I is the identity matrix (1 s down the diagonal) • 1 is a matrix of all ones.

Measuring Networks Centrality Bonacich Power Centrality: The magnitude of b reflects the radius of power. Small values of b weight local structure, larger values weight global structure. If b is positive, then ego has higher centrality when tied to people who are central. If b is negative, then ego has higher centrality when tied to people who are not central. As b approaches zero, you get degree centrality.

Measuring Networks Centrality Bonacich Power Centrality: b = 0. 23

Measuring Networks Triads Network Sub-Structure: Triads (0) (1) (2) (3) (4) (5) 003 012 102 111 D 201 210 021 D 111 U 120 D (6) 300 Intransitive Transitive 021 U 030 T 120 U 021 C 030 C 120 C Mixed

Measuring Networks Triads An Example of the triad census Type Number of triads -------------------1 - 003 21 -------------------2 - 012 26 3 - 102 11 4 - 021 D 1 5 - 021 U 5 6 - 021 C 3 7 - 111 D 2 8 - 111 U 5 9 - 030 T 3 10 - 030 C 1 11 - 201 1 12 - 120 D 1 13 - 120 U 1 14 - 120 C 1 15 - 210 1 16 - 300 1 -------------------Sum (2 - 16): 63

Measuring Networks Triads: Estimating Macro Structure As with undirected graphs, you can use the type of triads allowed to characterize the total graph. But now the potential patterns are much more diverse 1) All triads are 030 T: A perfect linear hierarchy.

Measuring Networks Triads: Estimating Macro Structure Cluster Structure, allows triads: {003, 300, 102} N* Eugene Johnsen (1985, 1986) specifies a number of structures that result from various triad configurations M M N* N* 1 M 1 1 M

Measuring Networks Triads: Estimating Macro Structure PRC{300, 102, 003, 120 D, 120 U, 030 T, 021 D, 021 U} Ranked Cluster: M A* A* N* M M A* A* 1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 A* A* N* M M And many more. . .

Measuring Networks Triads: Estimating Macro Structure Prosper

Measuring Networks Structural Equivalence Positional network mechanisms : Networks matter because of the way they capture role behavior and social exchange. Networks as Roles. Parent Child Romantic Love Provides food for Bickers with

Measuring Networks Structural Equivalence White et al: From logical role systems to empirical social structures The key idea, is that we can express a role through a relation (or set of relations) and thus a social system by the inventory of roles. If roles equate to positions in an exchange system, then we need only identify particular aspects of a position. But what aspect? Block modeling focuses on equivalence positions. Structural Equivalence Two actors are structurally equivalent if they have the same types of ties to the same people. That is, they have the exact same ties.

Measuring Networks Structural Equivalence Structurally Equivalence: same ties to same people Automorphic Equivalence: graph theoretically identical – without labels, automorphic nodes are indistinguishable Regular Equivalence*: Similar pattern of ties to similar types of nodes *Formally: If actors i and j are regularly equivalent, and actor i has a tie to/from some actor, k, then actor j must have the same kind of tie to/from some actor l, and actors k and l must be regularly equivalent. This means it’s a recursive definition and not always unique.

Measuring Networks Structural Equivalence Blockmodeling is the process of identifying these types of positions. A block is a section of the adjacency matrix - a “group” of people. 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 Here I have blocked structurally equivalent actors 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0

Measuring Networks Structural Equivalence Blockmodeling is the process of identifying these types of positions. A block is a section of the adjacency matrix - a “group” of people. 1 2 3 4 5 6 1 2 1. 1 0 3 1 0 0 1 4 0 1 0 0 5 0 0 0 0 6 0 0 0 3 1 0 0 1 1 0 0 4 1 0 1. 0 0 1 1 0 0 0 1 1 5 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 6 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0. 1 2 3 4 5 6 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 Structural equivalence thus generates 6 positions in the network

Measuring Networks Structural Equivalence . 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1. 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0. 1 1 1 2 1 3 0 1 2 1 1 1 3 0 1 0 2 3 Regular equivalence (here I placed a one in the image matrix if there were any ties in the ij block)

Measuring Networks Structural Equivalence Operationally, one needs to measure the similarity between each pair. If two actors are structurally equivalent, then they will have exactly similar patterns of ties to other people. Consider the example again: 1 2 1. 1 0 3 1 0 0 1 4 0 1 0 0 5 0 0 0 0 6 0 0 0 3 1 0 0 1 1 0 0 4 1 0 1. 0 0 1 1 0 0 0 1 1 5 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 6 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0. C D Match 1 1 1 0 0 1. 1 0 1. 0 0 0 1 1 1 1 0 0 1 Sum: 12 C and D match on 12 other people

Measuring Networks Structural Equivalence If the model is going to be based on asymmetric or multiple relations, you simply stack the various relations: H Romance 0 1 0 0 0 0 0 0 W C C C Romantic Love Provides food for Bickers with 0 0 0 Feeds 0 1 1 0 0 0 0 Bicker 0 0 0 0 0 1 1 0 0 0 Stacked 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 0

Measuring Networks Structural Equivalence 0 8 7 7 5 5 11 11 7 7 8 7 7 5 5 11 11 11 0 5 5 7 7 7 5 0 12 0 0 8 8 8 5 12 0 0 0 8 8 8 7 0 0 0 12 4 4 4 7 0 0 12 0 4 4 4 7 8 8 4 4 0 12 12 7 8 8 4 4 12 0 12 7 8 8 4 4 12 12 0 7 8 8 4 4 12 12 12 11 4 4 8 8 8 8 8 (number of agreements for each ij pair) 11 7 8 8 4 4 12 12 12 0 8 8 7 11 4 4 8 8 8 0 12 12 12 7 11 4 4 8 8 8 12 0 12 12 7 11 4 4 8 8 8 12 12 0 12 7 11 4 4 8 8 8 12 12 12 0

Measuring Networks Structural Equivalence A common metric used to measure equivalence is the correlation between each node’s set of ties. For the example, this would be: 1. 00 -0. 20 0. 08 -0. 19 0. 77 -0. 26 -0. 20 1. 00 -0. 19 0. 08 -0. 26 0. 77 0. 08 -0. 19 1. 00 -1. 00 0. 36 0. 36 -0. 45 -0. 19 0. 08 -1. 00 1. 00 -0. 45 -0. 45 0. 36 0. 77 -0. 26 0. 36 -0. 45 1. 00 1. 00 -0. 20 -0. 20 0. 77 -0. 26 0. 36 -0. 45 1. 00 1. 00 -0. 20 -0. 26 0. 77 -0. 45 0. 36 -0. 20 -0. 20 1. 00 1. 00 -0. 26 0. 77 -0. 45 0. 36 -0. 20 1. 00 Another common metric is the Euclidean distance between pairs of actors, which you then use in a standard cluster analysis. -0. 26 0. 77 -0. 45 0. 36 -0. 20 1. 00

Measuring Networks Structural Equivalence Concor iteration 1: 1. 00 -. 77 0. 55 -. 57 0. 95 -. 75 -. 77 1. 00 -. 57 0. 55 -. 75 0. 95 0. 55 -. 57 1. 00 -1. 0 0. 73 0. 73 -. 75 -. 57 0. 55 -1. 0 1. 00 -. 75 -. 75 0. 73 0. 95 -. 75 0. 73 -. 75 1. 00 1. 00 -. 77 -. 77 0. 95 -. 75 0. 73 -. 75 1. 00 1. 00 -. 77 -. 75 0. 95 -. 75 -. 75 0. 73 0. 73 -. 77 -. 77 1. 00 1. 00 The initial method for finding structurally equivalent positions was CONCOR, the CONvergence of iterated CORrelations.

Measuring Networks Structural Equivalence Concor iteration 2: 1. 00 -. 99 0. 94 -. 94 0. 99 -. 99 1. 00 -. 94 0. 94 -. 99 0. 94 -. 94 1. 00 -1. 0 0. 97 0. 97 -. 94 0. 94 -1. 0 1. 00 -. 97 -. 97 0. 99 -. 99 0. 97 0. 97 -. 97 1. 00 1. 00 -. 99 -. 99 0. 99 -. 97 -. 97 0. 97 -. 99 -. 99 1. 00 1. 00 The initial method for finding structurally equivalent positions was CONCOR, the CONvergence of iterated CORrelations.

Measuring Networks Structural Equivalence Concor iteration 3: 1. 00 -1. 00 1. 00 -1. 0 -1. 0 1. 00 1. 00 -1. 0 1. 00 1. 00 -1. 0 1. 00 1. 00 -1. 0 -1. 0 1. 00 1. 00 -1. 0 -1. 0 1. 00 1. 00 -1. 0 1. 00 1. 00 The initial method for finding structurally equivalent positions was CONCOR, the CONvergence of iterated CORrelations.

Measuring Networks Structural Equivalence Concor iteration 3: 1 3 4 7 8 9 10 2 5 6 11 12 13 14 1. 00 1. 00 1. 00 -1. 0 -1. 0 -1. 0 -1. 0 1. 00 1. 00 1. 00 -1. 0 -1. 0 -1. 00 1. 00 1. 00 -1. 0 -1. 0 1. 00 1. 00 The initial method for finding structurally equivalent positions was CONCOR, the CONvergence of iterated CORrelations.

Measuring Networks Structural Equivalence Repeat the process on the resulting 1 -blocks until you have reached structural equivalent blocks Because CONCOR splits every sub-group into two groups, you get a partition tree that looks something like this:

Software Stand-alone tools PAJEK: A free program for drawing & analyzing networks, optimized for very large networks. Windows or Unix (not so much mac? ) http: //mrvar. fdv. uni-lj. si/pajek/

Software Stand-alone tools UCINET: A general program for analyzing networks. Point-and-Click. Has some very specialized tools.

Software Stand-alone tools UCINET: A general program for analyzing networks. Point-and-Click. Has some very specialized tools.

Software Stand-alone tools Gephi:

Software General tools R, RStudio Fast becoming the field standard, particularly for statistical modeling of networks, easy to construct measures and add to models if you work in R for the rest of your workflow. Not the best data management program…

Software General tools R, RStudio STATNET • Program designed to estimate statistical models on networks in R. Statnet Team http: //csde. washington. edu/statnet/ Other R Resources: Carter Butts (UC-Irvine, Sociology) – SNA & Perm. Net • Program for general network analysis in R • Does most of what we’ve discussed today…

Software General tools R, RStudio Other R Resources: i. Graph