DH3501: Advanced Social Networks

Class 19: Bargaining, Stability, and Balance in Networks

Western University
Department of Modern Languages and Literatures
Digital Humanities – DH 3501

Instructor: David Brown
E-mail: dbrow52@uwo.ca
Office: AHB 1R14

What determines an individual's power?

  • Is it an individual characteristic?

  • A network property?

"Indeed, as Richard Emerson has observed in his fundamental work on this subject, power is not so much a property of an individual as it is a property of a relation between two individuals -- it makes more sense to study the conditions under which one person has power over another, rather than simply asserting that a particular person is "powerful". E & K, 340

In [1]:
%matplotlib inline
import networkx as nx
import matplotlib.pyplot as plt
g = nx.Graph([("A", "B")])

Value in relationship

If we assume that a relationsip holds some sort of value, how is that value divided?

  • Think about it...what kinds of value could a relationship hold?

  • If we think about power in terms of an imbalance in social exchange, how is the value of a relationship distributed based on the power of the individuals of the network?

  • Where does power come from?

Network Exchange Theory addresses questions of social imbalance and its relation to network structure.

Principles of power

In [2]:
g.add_edges_from([("B", "C"), ("B", "D"), ("D", "E")])

  • Dependence - if relationships confer value, nodes A and C are completely dependent on node B for value.

  • Exclusion - node B can easily exclude node A or C from the value conferred by the network.

  • Satiation - at a certain point, nodes like B begin to see diminishing returns and only maintains relations from which they can receive an unequal share of the value.

  • Betweenness - can confer power, this sort of centrality allows nodes like B to take advantages of structural holes and also control the flow of information througout the network. Note: high betweenness does not always confer an advantage in bargaining situations (as we will soon see).

Experimental methodology: Riddle me this...

Recall the experimental methodology typically used to study power and exchange? Get together with your pods and refresh your memories...there are five steps.

  • Are the results of these experiments considered to be robust?

  • Why or why not (according to E & K)?

Application: The following visualizations show 4 commonly tested paths. What were the experimental results for each path?

In [3]:
g = nx.Graph([("A", "B")])
plt.title("2-Node Path")

<matplotlib.text.Text at 0x7fd67ca1a690>

In [4]:
g = nx.Graph([("A", "B"), ("B", "C")])
plt.title("3-Node Path")

<matplotlib.text.Text at 0x7fd67c957690>

In [5]:
g = nx.Graph([("A", "B"), ("B", "C"), ("C", "D")])
plt.title("4-Node Path")

<matplotlib.text.Text at 0x7fd67c8aea90>

In [6]:
g = nx.Graph([("A", "B"), ("B", "C"), ("C", "D"), ("D", "E")])
plt.title("5-Node Path")

<matplotlib.text.Text at 0x7fd67c7ee350>

How about power in a network that looks like this?

In [7]:
g = nx.Graph([("A", "B"), ("B", "C"), ("B", "D"), ("C", "D")])
plt.title("Triangle with outlier")

<matplotlib.text.Text at 0x7fd67c6cdb50>

Or this?

In [8]:
g = nx.Graph([("A", "B"), ("B", "C"), ("C", "A")])

<matplotlib.text.Text at 0x7fd67c621810>

The Nash bargaining solution

  • What happens when exchanges take place on arbitrary networks?

  • What happens if we try to formalize a mathematical framework?

Ever see A Beautiful Mind?

Don't worry! We won't start hallucinating Russian conspiracies!

In order to draw distinction between equal and asymetrical power distribution over an edge; between strong power and weak power, and between networks that stabilize and networks that don't, we can use a solution proposed by John Nash, the Nash bargaining solution.

Assume that

  • $value\ of\ edge \gt 1$


  • $x + y \leq 1$

The negotiation (in terms of competition) becomes about how to split the surpluss:

  • $1 - x - y$

If A and B have equal power, we expect that each individual receives half of the surplus, so the outcome is equidependent:

  • $x + \frac{1}{2}s = \frac{x + 1 - y}{2}\ to\ A,\ and\\ y + \frac{1}{2}s = \frac{y + 1 - x}{2}\ to\ B$

How did perception of status effect the outcomes of empirical studies of the Nash bargaining soluiton?

The ultimatum game

  • (i) Person A is givin a dollar and told to propose a division of it to person B. That is, A should propose how much he keeps for herself, and how much she gives to B.

  • (ii) Person B is then given the option of approving or regecting the proposed division.

  • (iii) If B approves, each person keeps the proposed amount. If B rejects, then each person gets nothing.

What are the experimental results of the ultimatum game? Would you walk away with only 1% if you were person B?

Balance and stability

How do we predict the outcomes of network exchanges on an arbitrary graph?

Well...what is an outcome?

  • A matching on a the network that specifies who exchanges with whom. A node may at most complete one exchanges, others may be left out and complete none.

  • A number that is associated with each node that indicates how much value the node receives from its exchange. If two nodes are matched in the outcome, the sum of their values must equal 1. An unmatched node has the value of 0.


  • Stability is a state in which no node X can propose an offer to some other node Y that makes both X and Y better off -- thus "stealing" node Y away from an existing agreement.

Based on the above definition of stability, why are (a) and (c) not stable outcomes?

An instability can be defined as follows:

  • Given an outcome consisting of a matching and values for the nodes, and instability in this outcome is an edge not in the matching, joining two nodes X and Y, such that the sum of X's value and Y's value is less than 1.

Therefore, an outcome of network exchange is stable if and only if it contains no instablities.

This sort of prediction is limited in that humans do not usually acheive the kinds of extremes presented by this model, furthermore, it creates ambiguity in more complex situations as in (c) and (d) above.

Balanced outcomes

To account for these limitations, we can consider outcomes from the perspective of the Nash bargaining solution. Here we will consider related nodes in the network as the providers of outside options.

We can define a balanced outcome as follows:

  • An outcome is balanced if, for each edge in the matching, the split of the money represents the Nash bargaining outcome for the two nodes involved, given the best outside options for each node provided by the values in the rest of the network.

Application: Calculate the Nash bargaining outcome for each node in the image below

Remember all balanced outcomes are stable, but not all stable outcomes are balanced!