1. Networked Life and Social Networks
Thanks to Michael Kearns, James Moody, Anna Nagurney

2. Networked Life Physical, social, biological, etc Static vs dynamic
Hybrids
Static vs dynamic
Local vs global
Measurable and reproducible

3. A purely technological network? “Points” are physical machines
“Links” are physical wires
Interaction is electronic
What more is there to say?
Internet, Router Level

4. Points: power stations Operated by companies
Connections embody business relationships
Food for thought:
2003 Northeast blackout
North American Power Grid

5. Points are still machines… but are associated with people
Links are still physical… but may depend on preferences
Interaction: content exchange
Food for thought: “free riding”
Gnutella Peers

6. Points: sovereign nations Links: exchange volume
A purely virtual network
Foreign Exchange

7. Purely biological network Links are physical Interaction is electrical
Food for thought:
Do neurons cooperate or compete?
The Human Brain

8. The Premise of Networked Life
It makes sense to study these diverse networks together.
The Commonalities:
Formation (distributed, bottom-up, “organic”,…)
Structure (individuals, groups, overall connectivity, robustness…)
Decentralization (control, administration, protection,…)
Strategic Behavior (economic, free riding, Tragedies of the Common)
An Emerging Science:
Examining apparent similarities between many human and technological systems & organizations
Importance of network effects in such systems
How things are connected matters greatly
Details of interaction matter greatly
The metaphor of viral spread
Dynamics of economic and strategic interaction
Qualitative and quantitative; can be very subtle
A revolution of measurement, theory, and breadth of vision

9. Who’s Doing All This? Computer & Information Scientists
Understand and design complex, distributed networks
View “competitive” decentralized systems as economies
Social Scientists, Behavioral Psychologists, Economists
Understand human behavior in “simple” settings
Revised views of economic rationality in humans
Theories and measurement of social networks
Physicists and Mathematicians
Interest and methods in complex systems
Theories of macroscopic behavior (phase transitions)
All parties are interacting and collaborating

10. Examples
Theories
Apps in all areas

11. The Networked Nature of Society
Networks as a collection of pairwise relations
Examples of (un)familiar and important networks
social networks
content networks
technological networks
biological networks
economic networks
The distinction between structure and dynamics
A network-centric overview of modern society.

12. Contagion, Tipping and Networks
Epidemic as metaphor
The three laws of Gladwell:
Law of the Few (connectors in a network)
Stickiness (power of the message)
Power of Context
The importance of psychology
Perceptions of others
Interdependence and tipping
Paul Revere, Sesame Street, Broken Windows, the Appeal of Smoking, and Suicide Epidemics

13. Graph & Network Theory Networks of vertices and edges
Graph properties:
cliques, independent sets, connected components, cuts, spanning trees,…
social interpretations and significance
Special graphs:
bipartite, planar, weighted, directed, regular,…
Computational issues at a high level

14. Social Network Theory Metrics of social importance in a network:
degree, closeness, between-ness, clustering…
Local and long-distance connections
SNT “universals”
small diameter
clustering
heavy-tailed distributions
Models of network formation
random graph models
preferential attachment
affiliation networks
Examples from society, technology and fantasy

15. The Web as a Network
Empirical web structure and components
Web and blog communities
Web search:
hubs and authorities
the PageRank algorithm
The Main Streets and “dark alleys” of the web
The algorithmic and social implications of network structure.

16. Towards Rationality: Emergence of Global from Local
Beyond the dynamics of transmission
Context, motivation and influence
The madness/wisdom of crowds:
thresholds and cascades
mathematical models of tipping
the market for lemons
private preferences and global segregation

17. Interdependent Security and Networks
Security investment and Tragedies of the Commons
Catastrophic events: you can only die once
Fire detectors, airline security, Arthur Anderson,…
Blending network, behavior and dynamics.

18. Network Economics
Buying and selling on a network
Modeling constraints on trading partners
Local imbalances of supply and demand
Preferential attachment, price variation, and the distribution of wealth
The effects of network structure on economic outcomes.

19. Modern Financial Markets
Stock market networks
correlation of returns
Market microstructure
limit and market orders
order books and electronic crossing networks
network, connectivity and data issues
Quantitative trading
VWAP trading, market making
limit order power laws
Herd behavior in trading
Economic theory and financial markets
Behavioral economics and finance
Impacts of the Internet on financial markets
A study of the network that runs the world.

20. Definition of Social Networks
“A social network is a set of actors that may have relationships with one another. Networks can have few or many actors (nodes), and one or more kinds of relations (edges) between pairs of actors.” (Hannemann, 2001)

21. History (based on Freeman, 2000)
17th century: Spinoza developed first model
1937: J.L. Moreno introduced sociometry; he also invented the sociogram
1948: A. Bavelas founded the group networks laboratory at MIT; he also specified centrality

22. History (based on Freeman, 2000)
1949: A. Rapaport developed a probability based model of information flow
50s and 60s: Distinct research by individual researchers
70s: Field of social network analysis emerged.
New features in graph theory – more general structural models
Better computer power – analysis of complex relational data sets

23. Integration of large-scale social systems
Foundations
Theory
“Structural Analysis: from method and metaphor to theory and substance.”
H. White: “The presently existing, largely categorical descriptions of social structure have no solid theoretical grounding; furthermore, network concepts may provide the only way to construct a theory of social structure.” (p.25)
Form Vs. Content
Integration of large-scale social systems

24. Social network analysis is:
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.

25. What are social relations?
Introduction
What are social relations?
A social relation is anything that links two actors. Examples include:
Kinship Co-membership
Friendship Talking with
Love Hate
Exchange Trust
Coauthorship Fighting

26. What properties relations are studied?
Introduction
What properties relations are studied?
The substantive topics cross all areas of sociology. But we can identify types of questions that social network researchers ask:
1) Social network analysts often study relations as systems. That is, what is of interest is how the pattern of relations among actors affects individual behavior or system properties.

27. High Schools as Networks
Introduction
High Schools as Networks

30. Representation of Social Networks
Matrices
Graphs
Ann
Sue
Nick
Rob

31. Graphs - Sociograms (based on Hanneman, 2001)
Labeled circles represent actors
Line segments represent ties
Graph may represent one or more types of relations
Each tie can be directed or show co-occurrence
Arrows represent directed ties

32. Graphs – Sociograms (based on Hanneman, 2001)
Strength of ties:
Nominal
Signed
Ordinal
Valued

33. Visualization Software: Krackplot
Sources:

34. Connections (based on Hanneman, 2001)
Size  
Number of nodes
Density
Number of ties that are present the amount of ties that could be present
Out-degree
Sum of connections from an actor to others
In-degree
Sum of connections to an actor

35. Distance (based on Hanneman, 2001)
Walk
A sequence of actors and relations that begins and ends with actors
Geodesic distance
The number of relations in the shortest possible walk from one actor to another
Maximum flow
The amount of different actors in the neighborhood of a source that lead to pathways to a target

36. Some Measures of Power (based on Hanneman, 2001)
Degree
Sum of connections from or to an actor
Closeness centrality
Distance of one actor to all others in the network
Betweenness centrality
Number that represents how frequently an actor is between other actors’ geodesic paths

37. Cliques and Social Roles (based on Hanneman, 2001)
Sub-set of actors
More closely tied to each other than to actors who are not part of the sub-set
Social roles
Defined by regularities in the patterns of relations among actors

38. Examples of Applications (based on Freeman, 2000)
Visualizing networks
Studying differences of cultures and how they can be changed
Intra- and interorganizational studies
Spread of illness, especially HIV

39. We represent actors with points and relations with lines.
Foundations
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, actors or points
Relations are referred to variously as:
Edges, Arcs, Lines, Ties
Example: b d a c e

40. Social Network data consists of two linked classes of data:
Foundations
Data
Social Network data consists of two linked classes of data:
Nodes: Information on the individuals (actors, nodes, points, vertices)
Network nodes are most often people, but can be any other unit capable of being linked to another (schools, countries, organizations, personalities, etc.)
The information about nodes is what we usually collect in standard social science research: demographics, attitudes, behaviors, etc.
Often includes dynamic information about when the node is active
b) Edges: Information on the relations among individuals (lines, edges, arcs)
Records a connection between the nodes in the network
Can be valued, directed (arcs), binary or undirected (edges)
One-mode (direct ties between actors) or two-mode (actors share membership in an organization)
Includes the times when the relation is active
Graph theory notation: G(V,E)

41. Foundations
Data
In general, a relation can be: (1) Binary or Valued (2) Directed or Undirected a b c e d
Undirected, binary
Directed, binary
Undirected, Valued
Directed, Valued 1 3 4 2
The social process of interest will often determine what form your data take. Almost all of the techniques and measures we describe can be generalized across data format.

42. Foundations Global-Net Data Primary Group Ego-Net Best Friend Dyad
2-step
Partial network
Primary
Group
Global-Net
Ego-Net
Best Friend
Dyad

43. We can examine networks across multiple levels:
Foundations
Data
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: terrorist networks
- 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

44. We can examine networks across multiple levels:
Foundations
Data
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

45. No standard way to draw a sociogram: each of these are equal:
Foundations
Graphs
Working with pictures.
No standard way to draw a sociogram: each of these are equal:

46. Foundations
Graphs
Network visualization helps build intuition, but you have to keep the drawing algorithm in mind:
Tree-Based layouts
Spring-embeder layouts
Most effective for very sparse, regular graphs. Very useful when relations are strongly directed, such as organization charts, 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

47. Foundations
Graphs
Network visualization helps build intuition, but you have to keep the drawing algorithm in mind:
Tree-Based layouts
Spring-embeder layouts
Two images of the same network

48. Foundations
Graphs

49. Foundations
Graphs
Using colors to code attributes makes it simpler to compare attributes to relations.
Here we can assess the effectiveness of two different clustering routines on a school friendship network.

50. Foundations
Graphs
As networks increase in size, the effectiveness of a point-and-line display diminishes - run out of plotting dimensions.
Insights from the ‘overlap’ that results in from a space-based layout as information.
Here you see the clustering evident in movie co-staring for about 8000 actors.

51. Foundations
Graphs
This figure contains over 29,000 social science authors. The two dense regions reflect different topics.

52. Foundations http://www.sociology.ohio-state.edu/jwm/NetMovies/ Graphs
Adding time to social networks is also complicated, run out of space to put time in most network figures.
One solution: animate the network.
Here we see streaming interaction in a classroom, where the teacher (yellow square) has trouble maintaining order.
The SONIA software program (McFarland and Bender-deMoll)

53. Foundations
Methods
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.
Recommendation: use layouts that optimize on the feature you are most interested in.
Start w. simply opening CNAT.
Then look at ZOOM, Refresh, etc.
Then re-draw the network based on Degree
Then redraw from a particular ego.

54. A graph is vertices and edges
A graph is vertices joined by edges
i.e. A set of vertices V and a set of edges E
A vertex is defined by its name or label
An edge is defined by the two vertices which it connects, plus optionally:
An order of the vertices (direction)
A weight (usually a number)
Two vertices are adjacent if they are connected by an edge
A vertex’s degree is the no. of its edges E L B M P
200 60
130
190
450
210

55. Directed graph (digraph)
Each edge is an ordered pair of vertices, to indicate direction
Lines become arrows
The indegree of a vertex is the number of incoming edges
The outdegree of a vertex is the number of outgoing edges E
210 M
450
190 60 B
200
130 L P

56. Traversing a graph (1)
A path between two vertices exists if you can traverse along edges from one vertex to another
A path is an ordered list of vertices
length: the number of edges in the path
cost: the sum of the weights on each edge in the path
cycle: a path that starts and finishes at the same vertex
An acyclic graph contains no cycles E L B M P
200 60
130
190
450
210

57. Traversing a graph (2)
Undirected graphs are connected if there is a path between any pair of vertices
Digraphs are usually either densely or sparsely connected
Densely: the ratio of number of edges to number of vertices is large
Sparsely: the above ratio is small E M B L P

58. Two graph representations: adjacency matrix and adjacency list
n vertices need a n x n matrix (where n = |V|, i.e. the number of vertices in the graph) - can store as an array
Each position in the matrix is 1 if the two vertices are connected, or 0 if they are not
For weighted graphs, the position in the matrix is the weight
Adjacency list
For each vertex, store a linked list of adjacent vertices
For weighted graphs, include the weight in the elements of the list

59. Representing an unweighted, undirected graph (example) 1 2 3 4
Adjacency
matrix
0:E
1:M
2:B 1 3 2 4
Adjacency
list
3:L
4:P

60. Representing a weighted, undirected graph (example) 1 2 3 4
Adjacency
matrix
0:E
210
1:M
450
190 60
2:B
3;450
2;60
3;190
3;130
4;200
1;190
2;130
1;210
0;210
1;60
0;450
2;200 1 2 3 4
Adjacency
list
200
130
3:L
4:P

61. Representing an unweighted, directed graph (example) 1 2 3 4
Adjacency
matrix
0:E
1:M
2:B 1 3 2 4
Adjacency
list
3:L
4:P

62. Comparing the two representations
Space complexity
Adjacency matrix is O(|V|2)
Adjacency list is O(|V| + |E|)
|E| is the number of edges in the graph
Static versus dynamic representation
An adjacency matrix is a static representation: the graph is built ‘in one go’, and is difficult to alter once built
An adjacency list is a dynamic representation: the graph is built incrementally, thus is more easily altered during run-time

63. Algorithms involving graphs
Graph traversal
Shortest path algorithms
In an unweighted graph: shortest length between two vertices
In a weighted graph: smallest cost between two vertices
Minimum Spanning Trees
Using a tree to connect all the vertices at lowest total cost

64. Graph traversal algorithms
When traversing a graph, we must be careful to avoid going round in circles!
We do this by marking the vertices which have already been visited
Breadth-first search uses a queue to keep track of which adjacent vertices might still be unprocessed
Depth-first search keeps trying to move forward in the graph, until reaching a vertex with no outgoing edges to unmarked vertices

65. Shortest path (unweighted)
The problem: Find the shortest path from a vertex v to every other vertex in a graph
The unweighted path measures the number of edges, ignoring the edge’s weights (if any)

66. Shortest unweighted path: simple algorithm
For a vertex v, dv is the distance between a starting vertex and v
1 Mark all vertices with dv = infinity
2 Select a starting vertex s, and set ds = 0, and set shortest = 0
3 For all vertices v with dv = shortest, scan their adjacency lists for vertices w where dw is infinity
For each such vertex w, set dw to shortest+1
4 Increment shortest and repeat step 3, until there are no vertices w

67. a b c d e 1 a b c d e 1 Foundations Build a socio-matrix a b c e d a b
From pictures to matrices a b c e d a b c e d
Undirected, binary
Directed, binary a b c d e 1 a b c d e 1

68. a b c d e 1 Arc List Adjacency List a b b a b c c b c d c e d c d e
Foundations
Methods
From matrices to lists a b c d e 1
Arc List
Adjacency List
a b
b a
b c
c b
c d
c e
d c
d e
e c
e d
a b
b a c
c b d e
d c e
e c d

69. å ) 1 ( - = D N X 10 Basic Measures For greater detail, see:
Foundations
Basic Measures
Basic Measures
For greater detail, see:
Volume
The first measure of interest is the simple volume of relations in the system, known as density, which is the average relational value over all dyads. Under most circumstances, it is calculated as: ) 1 ( - = D å N X
10

70. Node In-Degree Out-Degree a 1 1 b 2 1 c 1 3 d 2 0 e 1 2 Mean: 7/5 7/5
Foundations
Basic Measures
Volume
At the individual level, volume is the number of relations, sent or received, equal to the row and column sums of the adjacency matrix. a b c d e 1
Node In-Degree Out-Degree a b c d e
Mean: / /5

71. Basic Measures Reachability
Foundations
Data
Basic Measures
Reachability
Indirect connections are what make networks systems. One actor can reach another if there is a path in the graph connecting them. a b d b f a c e c d e f

72. Foundations
Basic Matrix Operations
One of the key advantages to storing networks as matrices is that we can use all of the tools from linear algebra on the socio-matrix.
Some of the basics matrix manipulations that we use are as follows:
Definition
A matrix is any rectangular array of numbers. We refer to the matrix dimension as the number of rows and columns a b c d e 1
W B
1 0
0 1
Age 13 10 7 8 16 11
(5 x 5)
(5x2)
(5x1)

73. Multiplication by a scalar: 3A =
Foundations
Basic Matrix Operations
Matrix operations work on the elements of the matrix in particular ways. To do so, the matrices must be conformable. That means the sizes allow the operation.
For addition (+), subtraction (-), or elementwise multiplication (#), both matrices must have the same number of rows and columns. For these operations, the matrix value is the operation applied to the corresponding cell values.
3 6
11 8
2 9
-1 0
-3 6
2 1
1 3
4 7
2 5
2 3
7 1
0 4 A= B=
A+B =
A-B =
2 9
28 7
0 20
3 9
12 21
6 15
A#B =
Multiplication by a scalar: 3A =

74. Matrix properties Addition contributes to the actors relations
Multiplication sums over a trait.
Negative values can occur
(friend, don’t care, enemy) = (1,0,-1)
Interpret operations carefully

75. So a M x N matrix becomes an N x M matrix.
Foundations
Basic Matrix Operations
The transpose (` or T) of a matrix reverses the row and column dimensions.
Atij=Aji
So a M x N matrix becomes an N x M matrix. T
a b
c d
e f
a c e
b d f =

76. A3x3 x B2x3 is not defined (actually a tensor)
Foundations
Basic Matrix Operations
The matrix multiplication (x) of two matrices involves all elements of the matrix, and will often result in a matrix of new dimensions. In general, to be conformable, the inner dimension of both matrices must match. So:
A3x2 x B2x3 = C3 x 3
But
A3x3 x B2x3 is not defined (actually a tensor)
Substantively, adding ‘names’ to the dimensions will help us keep track of what the resulting multiplications mean:
So multiplying (send x receive)x (send x receive) = (send x receive), giving us the two-step distances (the sender’s recipient's receivers).

77. The multiplication of two matrices Amxn and Bnxq results in Cmxq
Foundations
Basic Matrix Operations
The multiplication of two matrices Amxn and Bnxq results in Cmxq
a b
c d
e f
g h
ae+bg af+bh
ce+dg cf+dh =
a b
c d
e f
ag+bj ah+bk ai+bl
cg+dj ch+dk ci+dl
eg+fg eh+fk ei+fl
g h i
j k l =
(3x2) (2x3)
(3x3)

78. Foundations
Basic Matrix Operations
The powers (square, cube, etc) of a matrix are just the matrix times itself that many times.
A2 = AA or A3 = AAA
We often use matrix multiplication to find types of people one is tied to, since the ‘1’ in the adjacency matrix effectively captures just the people each row is connected to.

79. Basic Measures Reachability
Foundations
Data
Basic Measures
Reachability
The distance from one actor to another is the shortest path between them, known as the geodesic distance. If there is at least one path connecting every pair of actors in the graph, the graph is connected and is called a component.
Two paths are independent if they only have the two end-nodes in common. If a graph has two independent paths between every pair, it is biconnected, and called a bicomponent. Similarly for three paths, four, etc.

80. Calculate reachability through matrix multiplication.
Foundations
Data
Calculate reachability through matrix multiplication.
(see p.162 of W&F) X X2 X3 a b c e d f
Total # of directed walks for power n
Distance
Distance
Minimal distance from one node to another

81. Foundations
Data
Mixing patterns
Matrices make it easy to look at mixing patterns: connections among types of nodes. Simply multiply an indicator of category by the adjacency matrix. e X
Race
1 0
0 1
X(Race)
2 0
1 1
2 2
0 2 d c f
B 4 to selves
B 2 to G
G 2 to B
G 6 to selves b
Race`(X)Race=
4 2
2 6
B G B G a

82. Matrix manipulations allow you to look at direction of ties,
Foundations
Data
Matrix manipulations allow you to look at direction of ties,
and distinguish symmetric from asymmetric ties.
To transform an asymmetric graph to a symmetric graph, add it to its transpose. X XT
Max Sym MIN Sym
Interpretation?

83. Graphs / matrices Analysis of social structure Visualization tools
Other methods
Statistics
Power laws
Bayesian graphs

84. Global web: Winners take all
Pr(page has k inlinks)  k- [2.1]
Popular few receive disproportionate share of links
 traffic,  prob SE indexing,  SE ranking

85. Category-specific web: Winner’s don’t (quite) take all
All US company homepages
hist w/ exp  buckets (const on log scale)
Strong deviation from pure power law
Unimodal (l.n.) body, power law tail
Less skewed; many fare well against mode
Pennock, Giles, et.al PNAS 2002

86. Applications of Networked Life
Social structure in organizations
Economic and business behavior
Epidemiology
Information discovery
Design and robustness of networks …