Summer 2008 REU at Oberlin College.

1. FINDING BALANCE IN SOCIAL NETWORKS
Michael Brooks Katie Kuksenok
Supervisor: Alexa M. Sharp, Computer Science Department
BACKGROUND EXAMPLE
Family Tension
Balance Theory Social Networks
Imagine a family with a mother, father, and son. The Three outcomes would bring balance: (1) the parents and
Initially formalized by Fritz Heider in 1958 [2], balance These are collections of nodes and edges representing
mother, father, and son are friendly with one another. son become estranged, (2) the son makes his parents
theory is an idea from sociology that describes the people and the relationships between them. An
change their minds about the bride, or (3) the parents
attitudes of groups of people toward each other. ASSIGNMENT is a function mapping each edge to its The son announces that he is getting married. However,
makes the son break the engagement.
Relationships can either be positive (+) or negative (-), label: (+) or (-). A particular assignment on a social the mother and father do not get along with the bride. This
denoting friendship or enmity. network is considered balanced when every triangle of situation is illustrated in Figure 6. Of these possible outcomes, the third only requires one
nodes is in a balanced state, as described below. relationship change (Fig. 7), while the others require two.
This is the optimal transition to stability.
M Figure 6. The triad of
a a a a M
family members was Figure 7. The son has
+ + + - + + - - S B balanced, but the bride S B broken his engagement to
has caused imbalance. the bride.
F F
b c b c b c b c
+ - - -
Figure 1. Balanced triads Figure 2. Unbalanced triads
These are BALANCED triads. Of the four possible These are UNBALANCED or FRUSTRATED triads.
arrangements of positive and negative labels, these They tend to morph into either of the balanced states.
are considered less stressful. The first is balanced In the first example, it is stressful for a to stay friends GOAL
because of the principle “the friend of my friend is my with two people who are enemies of one another. In
friend.” The second is balanced because “the enemy of the second example, it is better to ally with one of your
my enemy is my friend.” enemies against the third than to remain enemies with Finding Balance Quickly
both.
Our objective was to find a way to balance an unbalanced We developed and analyzed several different algorithms
network while flipping as few edges as possible. that attempt to solve or approximate this problem.
Properties of Complete Networks
A COMPLETE NETWORK is one where every person has
a relationship with every other person. Figure 3 is an
example of a complete network. NP-COMPLETENESS
An important property of complete social networks is that,
given a balanced assignment, the nodes can be divided
into two groups called CLIQUES (Figure 4) with the Correlation Clustering
following two properties: Figure 3. A complete network A version of the correlation clustering problem, called Finding the optimal division into k = 2 groups is known to
1. all relationships within each clique are friendly MinDisagree[k], is defined as follows: be NP-Complete. This means that the problem is
2. all relationships between the two cliques are Given a graph where each edge is labeled positive or probably not solvable in a polynomial number of steps.
unfriendly negative, divide the nodes into k groups such that as few However, there are fast algorithms to find approximate
positive edges are between groups and as few negative solutions.
Given an unbalanced complete network, balance can be
achieved by forcing the people into two groups and edges are within groups as possible. Since the problem of dividing up nodes into 2 groups
flipping some labels to satisfy the two balance properties. This problem has been analyzed by Giotis and while minimizing “bad” edges is NP-complete, and this is
Guruswami [1]. exactly our problem, we know that our problem is NP-
complete.
Figure 4. A balanced network divided into two cliques
2. Algebraic Aggregation 3. Add/Remove Method
SOLUTIONS
What is the best solution that can be achieved in What is the best solution that can be achieved on
a single step of decision-making? specific kinds of social networks?
1. The Physics Simulation This algorithm is intended to translate the principles of the This algorithm is an attempt to improve on the following
physical approach described above to a single, non- game-theoretic procedure:
What is the best solution that can be generated
Method iterative procedure. While the physical algorithm uses Repeatedly, every player chooses to join one clique or the
We wrote a simulator that iteratively calculates the effects
by allowing people to interact over time? repeated interaction between the 'people' in the network, other to minimize the number of edges he must flip.
of the attractive and repulsive forces on the positions of
algebraic aggregation is not iterative. Unlike the physical
Given that balance theory is supposed to reflect changing This process has the following properties:
the n nodes in an n-dimensional space. Points are pulled
solution, this algorithm has a definite point of termination.
degrees of attraction between people, it seems natural to 1. Guaranteed termination,
together and pushed apart until they tend toward some
define the input as a physical simulation. configuration. We allow the nodes to move in n Method 2. Quick (polynomial) running time, and
3. At most half of the edges must be changed (m / 2).
The positive edges exert an attractive force on their two dimensions so that it was possible to start the nodes at Under this method, we say each person only cares about
nodes and negative edges cause a repulsive force. The equidistant locations. The figures show the positions minimizing the number of relationships they have to To improve the quality of the solution, the Add/Remove
idea is that over time the nodes diverge into two groups projected into a plane. (Fig. 2) change. There are four steps: algorithm attempts to identify the most troublesome
that can be separated easily. Hopefully, making that 1. Every person makes a proposed division of all people nodes. It then uses this procedure on this subset of
division stable would require a minimum or at least low
Results into two groups, placing itself and its friends into one nodes, reducing the potential for bad outcomes.
In many cases, this method results in a good grouping. In
number of relationship changes. group and all others into the other group.
graphs with little symmetry, the points tend into two
2. Each such proposal is given a weight based on the
Results
groups such that we are able to find an appropriate This algorithm terminates quickly and produces a solution
quality of the proposal—the inverse of the total
division between them. However, it can be unclear when that flips at most half of all edges (m / 2). If there are few
SYMMETRY number of edges that must be flipped in order for that
the simulation should terminate, making the quality of this malicious nodes, we will not change more than half of the
partition to be adopted.
Problems sometimes arise in networks where all or method difficult to analyze. edges incident to those nodes, so the bound, depending
3. Each edge is assigned a score—the sum of the
many of the points are interchangeable, or symmetric. on the actual structure of the network, can be between
weights of proposed partitions where it is positive.
In these networks, the optimal grouping is often √(m) / 2 and m / 2.
4. Then, given the distribution of scores for each edge,
arbitrary. For example, in the network that begins with
we find a threshold such that all edges whose score is
all negative edges, the optimal solution changes half
below it become negative, and all others positive.
of them to positive edges. These solutions might have
difficulty deciding which nodes should make up each Results Figure 8. A network
half. separated into groups
This algorithm runs quickly (in polynomial time) and
after the Add/Remove
Symmetry poses the most problems in the algebraic terminates in a balanced solution. This solution, however,
algorithm has balanced it
aggregation approach; none at all, however, in the can be as bad as changing all the edges (m).
Figure 5. An unbalanced network divided into two cliques.
add/remove algorithm described in the next section.
This division would require changing 11 edges.
Resources Acknowledgements
[1] I. Giotis and V. Guruswami. Correlation Clustering with a Fixed Number of Clusters. In Theory of Computing. Volume 2, 2006, pp 249-266.
This project was funded by the Booth Ferris Foundation. Facilities provided by Oberlin College.
[2] F. Heider. (1958). The psychology of interpersonal relations. New York: John Wiley & Sons.