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Networked Life and Social Networks

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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

28

29

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


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