1. Social Networks
Strong and Weak Ties
These slides are based on chapter 3 of Networks, Crowds and Markets,
by David Easley and Jon Kleinberg
Available online at:

2. How do people find jobs?
1960s, Granoveletter interviewed people who recently found jobs.
Many people found their jobs through personal contacts
Usually, through acquaintances, not through friends
Why?
We come back to this soon, after we explore social networks a bit more

3. How do new links develop in a social network
Principle of Triadic Closure: If 2 people have a friend in common, then there is increased likelihood that they will become friends in the future G B F E A C D G B F E A C D
Closing a triangle
Why do you think this naturally occurs?

4. Measuring Triadic Closure
The Clustering Coefficient of a node A is the probability that 2 random friends of A are connected
= the fraction of pairs of A’s friends that are connected G B F E A C D
What is A’s clustering coefficient here?

5. Bridges
An edge from A to B is called a bridge if deleting this edge will cause A and B to be in different components of the graph
Why are such social ties important in the real world?
Do you think that bridges are common? C A B D E

6. Is the edge (A,B) still a bridge?
Is the edge (A,B) still important? J G K F H C A B D E

7. Local Bridges
An edge from A to B is called a local bridge if A and B have no friends in common J G K
Can you find all bridges?
Can you find all local bridges? F H C A B D E

8. Strong and Weak Ties
In life (and also in Facebook) some friends are stronger and some are weaker
Suppose we color weak edges red and strong edges green J G K F H C A B D E

9. Strong Triadic Closure
Node A violates the strong triadic closure property if it has strong ties to nodes B and C and there is no edge at all between B and C.
Node A satisfies the strong triadic closure property if it does not violate it. J G K
Which nodes violate the strong triadic closure property? F H C A B D E

10. Local Bridges and Weak Ties
Local notion: edges can be strong or weak
Global notion: edges can be local bridges or not
Claim: If node A satisfies the strong triadic closure property and has at least 2 strong ties, then any local bridge involving A is a weak tie
Proof: ?

11. Are Local Bridges Usually Weak Ties in Real life?
Analyzed in a network of cell-phone calls
An edge between two cell-phone if they made calls in both directions over an 18-week period
Strength defined as a function of the number of minutes spend on calls
Real local bridges are not common. Consider “almost” local bridges
The neighborhood overlap of an edge (A,B) is the number of nodes who are neighbors of A and B, divided by the total number of neighbors of A, B (= Jaccard Coefficient)

12. Neighborhood Overlap: Example
What is the neighborhood overlap of (A,F)? J G K
When is an edge a local bridge? F H C A B D E

13. What do you expect…
Is the relationship between the strength of an edge and its neighborhood overlap?
Results on Cell-Phone Network

14. Tie Strength on Facebook
How strong are the friendships declared on Facebook?
Researchers at Facebook (Marlow, et. Al.) categorized edges:
An edge represents mutual communication if users sent messages to each other within observation period
An edge represents one-way communication if user sent message to other (regardless of whether reciprocated)
An edge represents maintained relationship if user followed information about friend
An edge can belong to several categories

15. Example Facebook User
Where is triadic closure more common?
Why?

16. Average number of edges
Even users with many friends communicate / maintain relationships, with few

17. Roles of Nodes So far we have discussed different types of edges
Now, we consider the roles that nodes can take
What advantages does A have in this network?
What advantages does B have in this network?

18. Regions
Can we recognize tightly knight regions (e.g., communities) in a social network?
This is a graph partitioning problem
Co-authorship graph

19. Approaches to Graph Partitioning
Ideas?

20. Approaches to Graph Partitioning
Divisive Methods: Identify and remove “spanning links”. Network will fall apart into large pieces. Continue on.
Agglomerative Methods: Find closely knit regions forming chunks of the graph. Combine closely related chunks. Continue on.
There are many algorithms. We will discuss the Girvan-Newman method.

21. Example

22. Betweenness
We discussed betweenness earlier as a measure of importance for nodes
Now, we define the betweenness of an edge
For every A,B connected by a path, imagine there being one unit of “flow” along the edges from A to B
The “flow” divides itself evenly along all shortest paths
If there are k shortest paths, then 1/k units of “flow” pass on each path
The betweenness of a node is the total amount of flow it carries, for all pairs of nodes using this edge

23. Example What is the betweenness of edge (7,8)?

24. The Givan-Newman Method
Find edge(s) with highest betweenness value and remove from graph
If this causes the graph to disconnect, this is the first level of regions in the partitioning
Recalculate betweenness and again remove edge(s) with highest betweenness value
If breaks components into smaller components, these are nested regions
Continue until all edges are removed

25. Example

26. Computing Betweenness
To compute the betweenness of an edge we consider the graph from the perspective of a single node A
We compute how the flow from A to all others is distributed over the edges
Afterwards we can simply add up flow computed for each edge from the perspective of each node
(Finally, divide by 2 since every pair of nodes is considered twice)

27. Computing Betweenness (2)
3 steps to computation:
Perform Breadth-First Search from A to all other nodes
Determine number of shortest paths from A to each other node
Based on (2) compute the flow from A to all other nodes that use each edge

28. Step 1: Breadth First Search

29. Step 2: Counting Number of Shortest Paths

30. Step 3: Computing F First consider K Only 1 unit of flow arrives at K
An equal number of shortest paths enter K from I and from J, so the flow is divided evenly

31. Step 3: Computing F Now consider I
There is 1 unit of flow for I and 0.5 a unit that flows through I to reach K
So, in total 1.5 units of flow reach I
Since 2/3 of the paths to I are via F, 2/3 * 1.5 = 1 unit of flow is on (F,I)
Similarly, 0.5 unit of flow on (G,I)
Continue on in the same manner

32. Networks in Their Surrounding Contexts
Social Networks
Networks in Their Surrounding Contexts
These slides are based on chapter 4 of Networks, Crowds and Markets,
by David Easley and Jon Kleinberg
Available online at:

33. Beyond Nodes and Edges
So far, we have considered a social network as only a set of nodes and edges
However, these nodes represent people (or other types of entities) that have many additional properties
How do these properties affect the nature and structure of a network?

34. Homophily
Principle of Homophily: People tend to be similar to their friends
Think about it:
Does triadic closure encourage or discourage homophily?

35. Measuring Homophily
How can we discern if a network exhibits homophily with respect to a given property, e.g.,
Do people tend to be friends with others of the same race in the network?
Do people tend to be friends with others of the same social status in the network?
Consider a property which has two values
Each person has exactly one of these two values, e.g.,
Male or female

36. Measuring Homophily
Suppose that a fraction p of people in the network are male, while a fraction q = 1-p are female
If nodes were randomly assigned “male” and “female” then with likelihood 2pq, an edge is cross-gender.
Homophily Test: If the faction of cross-gender edges is significantly less than 2pq, then there is evidence for homophily
(Significant = Statistically Significant)

37. Example: Does this network exhibit homophily?

38. What Causes Homophily? Is Obesity Contagious?
Selection:
People choose friends similar to them
People live in areas of homogenous nature
Etc
Social Influence:
People are influenced by their peers in their behavior
Join in friends interests (e.g., sports)
Join in friends behavior (e.g., taking drugs, drinking)
Why does it matter?

39. Positive and Negative Relationships
Social Networks
Positive and Negative Relationships
These slides are based on chapter 5 of Networks, Crowds and Markets,
by David Easley and Jon Kleinberg
Available online at:

40. Structural Balance
Up until now, we assumed all relationships in the network were positive
However, we may have a negative relationships in real life
Label edges with “+” and “-”
Do such networks exhibit certain characteristics?
Do local effects have global consequences?

41. Simple Model We will start by assuming that everyone knows everyone
Network is a clique (i.e., complete)
Every edge is labeled + or –
Motivating Examples:
Small groups of people
Countries

42. Relationships Between 3 People+ + + + B C B C + -
Balanced
Not Balanced A A - - + - B C B C - -
Which types of relationships are more likely?

43. Structural Balance Having A, B, C who are all friends is natural
Occurs also due to triadic closure
Having A, B friends and both dislike C is natural
Have a common enemy (e.g., Syria, Lebanon, Israel) A + + B C + A + - B C -

44. Structural Balance (cont)
Having A, B friends, and A, C friends, but B and C dislike each other is unstable
Either A will try to get B, C to be friends or
B will try to turn A against C or
C will try to turn A against B
Having A, B, C all enemies is unstable
Two will try to team up against the third A + + B C - A - - B C -

45. Structural Balance (cont).
What happens for larger networks?
Structural Balance Property: A complete graph is structurally balanced if, for every set of 3 nodes A, B, C, either
all edges between them are labeled “+” or
exactly one edge is labeled “-”

46. Example: Are these Balanced?
Network 1
Network 2

47. What do Balanced Networks Look Like?
All nodes are friends
Nodes can be divided into 2 sets X, Y, such that:
Everyone in X likes each other
Everyone in Y likes each other
All nodes from X dislike all nodes from Y (and vice-versa
Any other options?

48. Balance Theorem
The Balance Theorem: If a complete graph is balanced then either all nodes are friends, or they can be divided into two groups X, Y such that every pair of nodes in X likes each other, every pair of nodes in Y likes each other and everyone in X dislikes everyone in Y
Proof: Suppose that G is balanced. We will show it satisfies the required properties.
If G has only + edges, then is obvious
Suppose G has at least one - edge

49. Proof (cont) Must find sets X, Y Let A be a node in the graph.
Since the graph is complete, there is an edge from A to every other node
Define X = all friends of A
Define Y = all enemies of A
We must show that
Every 2 nodes in X are friends
Every 2 nodes in Y are friends
Every node in X is an enemy of every node in Y

50. Proof (cont)
Illustration of why all 3 required properties hold

51. Is All Imbalance Equally Unlikely?
Can be argued that while this arrangement is hard to sustain for a long time,
This arrangement is more sustainable A + + B C - A - - B C -

52. Weak Structural Balance
Weak Structural Balance Property: A complete graph is structurally balanced if, there is no set of three nodes A, B, C with exactly 2 positive edges and one negative edge
What do weakly balanced networks look like?

53. Any Number of Sets Possible

54. Weak Structural Balance Characterization
The Weak Balance Theorem: If a complete graph is weakly balanced then its nodes can be divided into groups such that every two nodes in the same group are friends and every two nodes in different groups are enemies
Proof: Pick a node A. Let X be the set containing A and all its friends. Observe that:
Every two nodes in A must be friends with each other
Every node in A must be an enemy of every node out of A
Remove the set X from the graph and continue in the same manner to find additional groups

55. Other Generalizations of Balance
Graphs that are not complete?
What if not everyone knows everyone?
Graphs that are approximately balanced?
What if most triangles are balanced, but not all?
(Not Discussed – See book)
Note that our discussions will use the original definition of balance, not the weak version!

56. General Networks Every edge is labeled “+” or “-”
Can be nodes A, B such that no edge exists between A and B
How should we define balance?

57. Local Definition
Graph is structurally balanced, if it is possible to “fill-in” missing values, and get a structurally balanced network

58. Global Definition
Graph is structurally balanced, if it is possible to divide up nodes into 2 sets X, Y such that
people in X who know each other are friends
people in Y who know each other are friends
people in X who know people in Y are enemies
Equivalent?
YES!

59. Complexity How can you prove that a graph is balanced?
How can you prove that it is not?
What makes a graph unbalanced? X
Oops! 1 + X - X 5 2 - - 4 3 Y Y +

60. What Makes a Graph Unbalanced?
Claim: A graph is unbalanced if and only if it contains a cycle with an odd number of negative edges
Proof:
We will try to search for sets X, Y that form a balanced division of the graph
We will show that either we succeed in finding X, Y or we find a cycle with an odd number of negative edges

61. Proof: Step 1
Divide the graph into components that must belong in the same sets – the connected components of the graph, if “-” edges were removed
Call these super-nodes
Only negative edges between super-nodes

62. Proof: Step 1 (cont)
If a super-node contains a negative edge between two of its nodes, then there is a cycle with an odd number of “-” edges
We can assume that there are only “+” edges within the supernode

63. Proof: Step 2 Now, collapse each super-node into a single node
The graph derived has only “-” edges
Then, choose a collapsed node arbitrarily and put it (and all the nodes it contains into X)
Put all its neighbors into Y
Put all neighbors of neighbors into X, etc
At the end, either we have divided the nodes into X, Y as required or we have found a cycle with an odd number of “-” edges in the collapsed graph.
Can expand to find a cycle in the original graph

64. Illustration of Step 2