Social Network Analysis 1

What is Network Analysis? Social network analysis is a method by which one can analyze the connections across individuals or groups or institutions. That is, it allows us to examine how political actors or institutions are interrelated. Focuses on interaction, not on individuals Examine how the configuration of networks influences how individuals and groups, organizations, or systems function. 2

What is Network Analysis? Applied across disciplines: Social networks political networks electrical networks transportation networks, and so on. 3

History of (Social) Network Analysis Early research in network analysis is found in educational psychology, and studies of child development. 1922: Almack asked children in a California elementary school to identify the classmates with whom they wanted as playmates. Correlated IQs and examined the hypothesis that choices were homophilous. 4

More Early History 1926, Wellman: Recorded pairs of individuals who were observed as being together frequently. Recorded data, including the student’s height, grades, IQ, score on a physical coordination test, and degree of introversion versus extraversion (based on teacher’s ratings). Examined whether interaction was homophilous. 5

More Early History In 1933, the New York Times reported on the new science of “psychological geography” which “aims to chart the emotional currents, cross-currents and under-currents of human relationships in a community”.new science Jacob Moreno analyzed the interconnections across 500 girls in the State Training School for Girls, and the interconnections of students within two NYC schools. Many relationships were non-reciprocal—and that many individuals were isolated. 6

Other Advances 1935, Theodore Newcomb: Bennington college women were exposed to the relatively liberal students and faculty, they became more liberal. 1950, Festinger: Influence of dorm room location on friendships Developments in the last few decades include much attention paid to several concepts, including “the strength of weak ties”, and “small worlds”. 7

Weak Ties Strong ties are edges between two people that have common friends Other edges are called weak ties 8

The Strength of Weak Ties Granovetter’s “The Strength of Weak Ties” argued that “weak ties” could actually be more advantageousThe Strength of Weak Ties The presence of weak ties often reduced path lengths (distance) between any two individuals—which led to quicker diffusion of information 9

NE MA Small world phenomenon: Milgram’s experiment Source: undetermined 10

“Six degrees of separation” Instructions: Given a target individual (stockbroker in Boston), pass the message to a person you correspond with who is “closest” to the target. Small world phenomenon: Milgram’s experiment Outcome: 20% of initiated chains reached target average chain length =

Why is Network Analysis Useful? (Some Examples) New product to market. Want to notify k people and have information disseminate. Which k people? Company wants to down-size. Who should not be fired? Friend Recommendation? John is looking for a dentist. Who should be recommended? Want to put together a committee to launch a conference. Who should we choose? 12

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Network basics 14

What are networks? Networks are collections of points joined by lines. “Network” ≡ “Graph” pointslines verticesedges, arcsmath nodeslinkscomputer science sitesbondsphysics actorsties, relationssociology node edge 15

Network elements: edges Directed A  B A likes B, A gave a gift to B, etc Undirected A  B or A – B A and B like each other, are siblings, are co- authors, etc. Edge attributes weight (e.g. frequency of communication) ranking (best friend, second best friend…) type (friend, relative, co-worker) 16

Edge weights can have positive or negative values One gene activates/inhibits another One person trusting/distrusting another (how to propagate)? Source: undetermined 17

Characterizing networks: Who is most central? ? ? ? 18

Characterizing networks: Is everything connected? 19

Network metrics: size of giant component if the largest component encompasses a significant fraction of the graph, it is called the giant component 20

Characterizing networks: How far apart are things? 21

Network metrics: shortest paths Shortest path (also called a geodesic path) The shortest sequence of links connecting two nodes Not always unique A – E – B – C A – E – D – C Diameter: the largest geodesic distance in the graph What is the diameter of this graph? 22 A B C D E

Characterizing networks: How dense are they? 23

Network metrics: graph density How many possible edges? Directed graph: emax = n * (n-1) Undirected graph: emax = n * (n-1)/2 Density = e/ emax What is the density of this graph? 24

Network Centrality 25

Node Centrality Which nodes are most important (central)? Is there one ultimate answer? Depends on context Measuring centrality Local measure: degree Relative to rest of network How evenly is centrality distributed among nodes? 26

indegreeoutdegreebetweennesscloseness centrality: who’s important based on their network position In each of the following networks, X has higher centrality than Y according to a particular measure 27

He who has many friends is most important. Degree centrality (undirected) When is the number of connections a good centrality measure? o people who will do favors for you o people you can talk to / have a beer with 28

degree: normalized degree centrality divide by the max. possible, i.e. (N-1) 29

Freeman’s general formula for centralization: Centralization: how equal are the nodes? How much variation is there in the centrality scores among the nodes? maximum value in the network 30

degree centralization examples C D = C D = 1.0 ? 31

when degree isn’t everything In what ways does degree fail to capture centrality in the following graphs? 32

In what contexts may degree be insufficient to describe centrality? ability to broker between groups likelihood that information originating anywhere in the network reaches you… 33

betweenness: another centrality measure Intuition: how many pairs of individuals would have to go through you in order to reach one another in the minimum number of hops? who has higher betweenness, X or Y? XY 34

g jk = the number of geodesics connecting jk g jk (i) = the number of them that node i is on. Usually normalized by: number of pairs of vertices excluding the vertex itself Betweenness centrality: definition adapted from a slide by James Moody 35

Lada’s facebook network: nodes are sized by degree, and colored by betweenness. Example 36

Can you spot nodes with high betweenness but relatively low degree? Explain how this might arise. Betweenness example (continued) What about high degree but relatively low betweenness? 37

Betweenness on toy networks Non-normalized version: ABCED A lies between no two other vertices B lies between A and 3 other vertices: C, D, and E C lies between 4 pairs of vertices (A,D),(A,E),(B,D),(B,E) note that there are no alternate paths for these pairs to take, so C gets full credit 38

betweenness on toy networks non-normalized version: 39

betweenness on toy networks non-normalized version: ? 40

betweenness on toy networks non-normalized version: A B C E D why do C and D each have betweenness 1? They are both on shortest paths for pairs (A,E), and (B,E), and so must share credit: ½+½ = 1 Can you figure out why E has betweenness 0.5? What is the betweenness of B? ? 41

Closeness: another centrality measure What if it’s not so important to have many direct friends? Or be “between” others But one still wants to be in the “middle” of things, not too far from the center 42

Closeness is based on the length of the average shortest path between a vertex and all vertices in the graph Closeness Centrality: Normalized Closeness Centrality Closeness centrality: definition 43

closeness centrality: toy example ABCED ? 44

closeness centrality: more toy examples 45

Complexity Degree: Degree of node / N-1 Betweenness Closeness 46

Efficiently Computing Betweenness How do you compute g jk (i)? LEMMA 1 (Bellman criterion): A vertex i lies on a shortest path between vertices j, k, if and only if d(j, k) = d(j, i) + j(i, k) Therefore: g jk (i) = 0if d(j, k) < d(j, i) + j(i, k) g ji * g ik otherwise 47

Efficiently Computing Betweenness How do you compute g jk ? Define P j (i) to be the set of all predecessors of i on a shortest path from j Then, g jk = Sum i in P j (k) g ji We can BFS to count paths Left as exercise We do something like this later when we discuss betweenness of edges 48