network

Exercises

1.Suppose that a team of anthropologists is studying a set of three small villages that neighbor one another. Each village has 30 people, consisting of 2-3 extended families.

Everyone in each village knows all the people in their own village, as well as the people in the other villages.

When the anthropologists build the social network on the people in all three villages taken together, they find that each person is friends with all the other people in their own village, and enemies with everyone in the two other villages. This gives them a network on 90 people (i.e., 30 in each village), with positive and negative signs on its edges.

According to the definitions in this chapter, is this network on 90 people balanced? Give a brief explanation for your answer.

2. Consider the network shown in Figure 5.18: there is an edge between each pair of nodes, with five of the edges corresponding to positive relationships, and the other five of the edges corresponding to negative relationships.

Each edge in this network participates in three triangles: one formed by each of the additional nodes who is not already an endpoint of the edge. (For example, the A-B edge participates in a triangle on A, B, and C, a triangle on A, B, and D, and a triangle on A, B, and E. We can list triangles for the other edges in a similar way.)

For each edge, how many of the triangles it participates in are balanced, and how many are unbalanced. (Notice that because of the symmetry of the network, the answer will be the same for each positive edge, and also for each negative edge; so it is enough to consider this for one of the positive edges and one of the negative edges.)

3. When we think about structural balance, we can ask what happens when a new node tries to join a network in which there is existing friendship and hostility. In Fig-ures 5.19–5.22, each pair of nodes is either friendly or hostile, as indicated by the + or−label on each edge.

First, consider the 3-node social network in Figure 5.19, in which all pairs of nodes know each other, and all pairs of nodes are friendly toward each other. Now, a fourthnode D wants to join this network, and establish either positive or negative relations with each existing node A,B, and C. It wants to do this in such a way that it doesn’t become involved in any unbalanced triangles. (I.e. so that after adding D and the labeled edges from D, there are no unbalanced triangles that contain D.) Is this possible? In fact, in this example, there are two ways for Dt o accomplish this, as indicated in Figure 5.20. First, D can become friends with all existing nodes; in this way, all the triangles containing it have three positive edges, and so are balanced. Alternately, it can become enemies with all existing nodes; in this way, each triangle containing it has exactly one positive edge, and again these triangles would be balanced .So for this network, it was possible for D to join without becoming involved in any unbalanced triangles. However, the same is not necessarily possible for other networks. We now consider this kind of question for some other networks.

a. Consider the 3-node social network in Figure 5.21, in which all pairs of nodes know each other, and each pair is either friendly or hostile as indicated by the+ or−label on each edge. A fourth node D wants to join this network, and establish either positive or negative relations with each existing node A,B, and C. Can node D do this in such a way that it doesn’t become involved in any unbalanced triangles?

i. If there is a way for D to do this, say how many different such ways there are, and give an explanation. (That is, how many different possible labelings of the edges out of D have the property that all triangles containing D are balanced?)

ii. If there is no such way for D to do this, give an explanation why not.

(In this and the subsequent questions, it possible to work out an answer by rea-soning about the new node’s options without having to check all possibilities.) Rubric: There is no way for Do to this because D BC triangle. Doesn’t work for the two before mentioned possibilities.

b. Same question, but for a different network. Consider the 3-node social network in Figure 5.22, in which all pairs of nodes know each other, and each pair is either friendly or hostile as indicated by the + or−label on each edge. A fourth node D wants to join this network, and establish either positive or negative relations with each existing node A,B, and C. Can node D do this in such a way that it doesn’t become involved in any unbalanced triangles?

i. If there is a way for D to do this, say how many different such ways there are, and give an explanation. (That is, how many different possible labeling of the edges out of D have the property that all triangles containing D are balanced?)

ii. If there is no such way for D to do this, give an explanation why not.

c. Using what you’ve worked out in Questions 2 and 3, consider the following ques-tion. Take any labeled complete graph — on any number of nodes — that is not balanced; i.e. it contains at least one unbalanced triangle. (Recall that a labeled complete graph is a graph in which there is an edge between each pair of nodes, and each edge is labeled with either + or−.) A new node X wants to join this network, by attaching to each node using a positive or negative edge. When, if ever, is it possible for X to do this in such a way that it does not become involved in any unbalanced triangles? Give an explanation for your answer. (Hint: Think about any unbalanced triangle in the network, and how X must attach to the nodes in it.)

Graph Theory

Recap from the last week
Importance and definition of a network
Two levels of understanding:
(1) at the level of structure  Graph theory
(2) at the level of behavior  Game theory

The next three weeks’ topic

What is a social network? Relations among People
https://ocw.mit.edu/courses/sloan-school-of-management/15-599-workshop-in-it-collaborative-innovation-networks-fall-2011/lecture-notes/MIT15_599F11_lec04

What is a Network? Relations among Institutions
https://ocw.mit.edu/courses/sloan-school-of-management/15-599-workshop-in-it-collaborative-innovation-networks-fall-2011/lecture-notes/MIT15_599F11_lec04

Basic concepts of the graph theory
Graph: A graph is a way of specifying relationships among a collection of items
Node: An object, an actor, a point
Computers, telephones
Persons, employees
Companies/business units
Articles/research projects
Neighbors: If the two nodes are connected

Edge
Edge: A link that connects a pair of nodes
Ties, edges, arcs, lines, and links
Types of social relations
Friendship
Acquaintance
Kindship
Advice
Hindrance
Sex
Allow different kind of flows
Messages
Money
Diseases
An edge may contain a direction when a direction is asymmetric

Paths and Cycles
Path: a sequence of nodes with the property that each consecutive pair in the sequence is connected by an edge.
containing not just the nodes but also the sequence of edges linking these nodes
Cycle: A “ring” structured path
A path with at least three edges, in which the first and last nodes are the same, but otherwise all nodes are distinct.
Q1. Why is this redundancy necessary?
Or what is the advantage of a ring structure as opposed to a star structure?

https://www.geeksforgeeks.org/difference-between-star-and-ring-topology/

Roles of Graph
Mathematical models of network structures
Represent how things are either physically or logically linked to one another in a network structure

Connectivity
Connectivity: The state of a graph in which every pair of nodes is connected via a path
The example (Figure 2.6) is built from the collaboration graph at a biological research center.
Nodes: researchers
Edge: co-authored publication
What is the most prominent node? (Centrality)
Who knows the most actors? (degree centrality)
Who has the shortest distance to the other actors?
Who controls knowledge flows?
What happens if this most prominent central node is removed?
Figure 2.6 Collaboration graph

Connected components (or just components)
A subset of the nodes such that: (i) every node in the subset has a path to every other; and (ii) the subset is not part of some larger set with the property that every node can reach every other.
(i) components are internally connected
(ii) a component is a free-standing
Q2. How many connected components do you see in Figure 2.6 in the previous slide and in Figure 2.5 on the right in this slide?

Figure 2.5

Giant Components
A connected component that contains a significant fraction of all the nodes.
In Figure 2.6, what is a giant component?
When a network contains a giant component, it almost always contains only one. Why?

Two giant components’ coexistence in history
What happens in human history when two giant components co-existed?
Guns, Germs, and Steel by Diamond
One in Americas and the other in Europe
(0:51-4:00)
When the two giant components clashed, what happened?
Q3: When the two giant components clashed, what happened? Did the two giant components continue to co-exist? Or did one dominate the other? Based on this video, explain why it is very rare, if not impossible, to see two giant components coexist.

Giant component and its implications
Figure 2.7 shows the romantic relationships in an American high school over an 18-month period
The fact that this graph contains such a large component is significant when one thinks about the spread of sexually transmitted diseases
A high-school student may have had a single partner over this time period and nevertheless — without realizing it — be part of this large component and hence part of many paths of potential transmission.

Distance
Length of a path: The number of steps it contains from beginning to end
Distance between the two nodes: The length of the shortest path between them
Breadth-First Search

Small World Phenomena
The idea that the world looks “small” when you think of how short a path of friends it takes to get from you to almost anyone else.
Six degrees of separation
All people are six, or fewer, social connections away from each other.
Kevin Bacon number

How do we measure “Real” social networks?

Interpersonal relationships, intensity of relationships, subtle acquaintance – impossible to measure
We need to define a link in an observable way (e.g., email exchanged, business transactions made, ”friends’ lists” on FB, Instant Messages exchanged)
What are some other limitations?
Lack of participation
Dishonest answers
The size and scope of network
Therefore, we need help from digital tracing and computational method to measure the networks
A partial reason why SNA has grown very fast in the past few years
Figure 2.12: Ron Graham’s hand-drawn picture of a part of the mathematics collaboration graph, centered on Paul Erd ̈os [189].

Growth of SNA since late 1990’s

Main sources of large-scale network data
Collaboration Graphs
Who talks to Whom graphs
Information linkage graphs
Technological networks
Q4: Pick one of the examples above and describe what problems, challenges, and limitations you would encounter if you were to generate a graph. For instance, if you were to draw a collaboration graph among all MSU faculty members, what problems would you encounter?

Exercises – Recitation Session on Thursday
A pivotal node
Gate-keepers
Do the exercise questions!
Be prepared to create some of these using NetLogo