1. F INDING B ALANCE IN S OCIAL N ETWORKS What is BALANCE ? These are BALANCED TRIADS. They are considered less stressful arrangements. The first is balanced because of the principle “the friend of my friend is my friend.” The second is balanced because “the enemy of my enemy is my friend.” These are UNBALANCED or FRUSTRATED TRIADS, and tend to morph into either of the balanced states. In the first example, this is justified by the idea that it is stressful to stay friends with two people who are enemies one another. In the second example, it is easier to ally with one of your enemies against the greater enemy. BALANCE THEORY, initially formalized by Fritz Heider in 1958 [4], described the interactions within a group of three people. In this model, relationships could either be positive (+) or negative (-), denoting friendship or enmity. Balance in the context of SOCIAL NETWORKS + - - - - - - - - - + + + + + SOCIAL NETWORKS are collections of nodes and edges representing people and the relationships between them. We assign each relationship either a positive value or a negative value, to denote a pair of people as either friends or enemies. An ASSIGNMENT is a function mapping the set of edges to their assigned (+) or (-). An assignment on a particular social network is considered balanced when every triangle of nodes is in a balanced state. A COMPLETE network is one where every person knows (has an edge to) every other person. An important property of complete social networks is that, given a balanced assignment of relationships, the nodes can be divided into two groups such that all relationships within each clique are friendly, and all relationships between the two cliques are unfriendly. Figure 1 shows a balanced network divided into its two groups. During the term of our research, we have focused on complete networks and the idea that, given an arbitrary network, dividing it into two groups and assigning edge values as specified above will necessarily lead to a balanced assignment of relationships on that network. Fig. 1 Michael BrooksKatie Kuksenok GOAL: GOAL: given a social network that is unbalanced and complete, find a balanced configuration while changing the fewest number of relationships. Since any balanced complete network can be split into two groups such that all links within each group are friendly, and all links between the two groups are unfriendly (see Fig.1), we can abstract the given problem to dividing the network into two groups with as few “bad” relationships as possible. “Bad” relationships are negative connections within groups or positive connections between groups. If we were to adopt the resulting assignment, we would be guaranteed to have a balanced assignment, but at the price of having to negate all “bad” relationships. A GGREGATION A PPROACHES First, we considered algorithms with the general strategy of allowing every node in a network to assert its opinion about the outcome with the goal of finding balance based on completely self-interested behavior. ADD-REMOVE ALGORITHM 1. PHYSICAL MODEL Here, the positive edges exert an attractive force on their two nodes and negative edges cause a repulsive force. The idea is that over time the nodes diverge into two groups that can be separated easily and that the division would require a low number of relationship changes. 2. ALGEBRAIC AGGREGATION This algorithm essentially translates the principle of the physical approach described above to a series of calculations. While the physical model allowed continuous, real-time interaction between the 'people' in a social network, it was unclear when a particular division can be considered the end result. The proposed algorithm has the property of being quickly terminating and more readily measurable. METHOD: Under this method, we say that 'people' only care about minimizing the number of changes to relationships they themselves are involved in. (1) Every point makes a proposed division into two groups, placing itself and its friends into one group and all others into the opposite group. (2) Each such proposal is given a weight based on the quality of the proposal—the inverse of the total number of edges that must be flipped in order for that partition to be adopted. (3) Then, each relationship between two people, a and b, is assigned a score – the sum of the weights of all proposed partitions where a and b appear in the same group. (4) Then, given the distribution of scores for each relationship, we find a value such that all relationships whose score is below it become negative, and others – positive. This value is found by trying values in the given range at certain intervals and taking the value that leads to least edges changed and most balance created. RESULTS: ISSUES: First, all symmetry is preserved. In some cases, it is better to set symmetry aside in favor of a more efficient balance; but, qualitatively, preservation of symmetry makes sense in the context of the nature of social networks. METHOD: In order to test the viability of this idea, we wrote a simulator that iteratively calculates the effects of the attractive and repulsive forces on the positions of the n nodes in an n-dimensional space. Points are pulled together and pushed apart until they tend toward some configuration. We allowed the nodes to move in n-dimensions so that it was possible to start the nodes in at equidistant locations. The figures show the positions projected into a plane. (Fig. 2)‏ RESULTS: In many cases, this method resulted in a good grouping. In graphs with little symmetry, the points would tend into two groups so that we could find an appropriate division between them. ISSUES: Problems arose in networks where all or many of the points are effectively interchangeable (symmetric). In these networks, there is no obvious way for the nodes to be grouped, so the simulation can fail to group the nodes optimally. Fig. 3 Supervisor: Alexa M. Sharp, Computer Science Department BACKGROUND THE GAME-THEORETIC APPROACH. We can use a a game-theoretic algorithm, defining the players to be the people in the social network and the strategies to be one of two possible groups. Then, we allow every player to successively choose strategies to minimize the number of edges that player must change the value of based on the other players' current strategies until no player wants to change his or her strategy. FIRST, we know this process terminates, and fairly quickly. SECOND, we know that the best possible solution is an equilibrium; unfortunately, there are many equilibria, and the worst is no better than SECOND changing half all relationships to positive- which is as inefficient as we get. THE ADD-REMOVE ALGORITHM is a very complicated algorithm that attempts to mimic the game-theoretic approach but with as clever a decision-making BALANCE AND CORRELATION CLUSTERING [1] J. Scott. Social Network Analysis....something [2] T. Antal, P. L. Krapivsky and S. Redner. Social Balance on Networks: The Dynamics of Friendship and Enmity [3] I. Giotis and V. Guruswami. Correlation Clustering with a Fixed Number of Clusters. In Theory of Computing. Volume 2, 2006, pp 249-266. [4] Heider, F. (1958). The psychology of interpersonal relations. New York: John Wiley & Sons. SOURCES  The problem of dividing a group of nodes with positive or negative associations into groups such that as few positive edges are between groups and as few negative edges are within groups has been studied by [3].  Finding the optimal division into k groups has been shown to be NP-Complete, or probably not solvable in a polynomial number of steps. There are approximation algorithms for this problem though.  Since the problem of dividing up nodes into k groups while minimizing “bad” edges is NP-complete, and any algorithm which solves that problem could be used to balance a network with minimum flips, we know that our problem is NP-complete. WHAT? THAT'S NOT WH^!!! RESULTS: ISSUES: First, all symmetry is preserved. In some cases, it is better to set symmetry aside in favor of a more efficient balance; but, qualitatively, preservation of symmetry makes sense in the context of the nature of social networks.

2. Balance Theory Initially formalized by Fritz Heider in 1958 [4], Balance Theory describes the dynamics of the attitudes of groups of three people. Relationships can either be positive (+) or negative (-), denoting friendship or enmity. These are BALANCED triads. Of the four possible arrangements of positive and negative values, these are considered less stressful. The first is balanced because of the principle “the friend of my friend is my friend.” The second is balanced because “the enemy of my enemy is my friend.” a cb - ++ a cb - -- a cb + ++ a cb - +- These are UNBALANCED or FRUSTRATED triads. They tend to morph into either of the balanced states. In the first example, this is justified by the idea that it is stressful to stay friends with two people who are enemies one another. In the second example, it is easier to ally with one of your enemies against the greater enemy. BACKGROUND Social Networks These are collections of nodes and edges representing people and the relationships between them. An ASSIGNMENT is a function mapping the set of edges to their assigned (+) or (-). An assignment on a particular social network is considered balanced when every triangle of nodes is in a balanced state, as described below. A COMPLETE NETWORK is one where every person has an edge to every other person. Figure 3 is an example of a complete network. An important property of complete social networks is that, given a balanced assignment of relationships, the nodes can be divided into two groups (called CLIQUES) with the following two properties: 1. all relationships within each clique are friendly 2. all relationships between the two cliques are unfriendly Figure 4 shows a balanced network divided into its two groups. Given an unbalanced complete network, balance can be achieved by forcing the nodes into two groups and flipping some edge values to satisfy the two balance properties. Properties of Complete Networks Figure 3. A complete network Figure 2. Unbalanced triads Figure 1. Balanced triads Figure 4. A balanced network divided into two cliques GOAL Finding Balance Quickly Our objective was to find a way to balance an unbalanced network while flipping as few edges as possible. We developed and tested several different algorithms that attempt to solve or approximate this problem. SOLUTIONS 1. The Physics-based Solution Given that balance theory is supposed to reflect changing degrees of attraction between people, it seemed natural to define the input as a physical simulation. The positive edges exert an attractive force on their two nodes and negative edges cause a repulsive force. The idea is that over time the nodes diverge into two groups that can be separated easily. Hopefully, making that division stable would require a minimum or at least low number of relationship changes. Method We wrote a simulator that iteratively calculates the effects of the attractive and repulsive forces on the positions of the n nodes in an n-dimensional space. Points are pulled together and pushed apart until they tend toward some configuration. We allowed the nodes to move in n dimensions so that it was possible to start the nodes at equidistant locations. The figures show the positions projected into a plane. (Fig. 2)‏ Results In many cases, this method resulted in a good grouping. In graphs with little symmetry, the points would tend into two groups so that we could find an appropriate division between them. Issues Problems sometimes arise in networks where all or many of the points are interchangeable, or symmetric. In these networks, the optimal grouping is often more arbitrary. Also, it can be unclear when the simulation should terminate, making the quality of this method difficult to analyze. Figure 5. An unbalanced network divided into two cliques. This division would require changing 11 edges. 2. Algebraic Aggregation This algorithm is intended to translate the principles of the physical approach described above to a single, non-iterative procedure. While the physical model uses repeated interaction between the 'people' in the network. Unlike the physical solution, this algorithm has a definite termination and result. Method Under this method, we say that 'people' care only about minimizing the number of edge changes in their incident edges. There are four steps: 1. Every point makes a proposed division into two groups, placing itself and its friends into one group and all others into the opposite group. 2. Each such proposal is given a weight based on the quality of the proposal—the inverse of the total number of edges that must be flipped in order for that partition to be adopted. 3. Each edge is assigned a score—the sum of the weights of all proposed partitions where its end- nodes appear in the same group. 4. Then, given the distribution of scores for each edge, we find a number such that all edges whose score is below it become negative, and all others positive. We find the value by searching the range of scores and taking the value that leads to the least edges changed. Issues As with the physical solution, symmetry can be a problem. If nodes are symmetric, there will be symmetry in the scores which can lead to a less than optimal division. 3. Add/Remove Method This algorithm is an attempt to improve on the following game-like procedure: Repeatedly allow every player to choose to join one clique or the other so as to minimize the number of edges he must flip. We proved that this process terminates, and in a polynomial number of steps. We also know that the optimal solution could be the final result of this process. Unfortunately, this process is not guaranteed to find the optimal solution, and could be as bad as changing half of the edges to positive. THE ADD-REMOVE ALGORITHM is a very complicated algorithm that attempts to mimic the game-theoretic approach but with as clever a decision- making Results Issues NP-Completeness Correlation Clustering A version of the Correlation Clustering problem, called MinDisagree[k], is defined as follows: Given a graph where the edges have positive or negative value, divide the nodes into k groups such that as few positive edges are between groups and as few negative edges are within groups. This problem has been analyzed by Giotis and Guruswami [3]. Finding the optimal division into k = 2 groups is known to be NP-Complete. This means that the problem is probably not solvable in a polynomial number of steps. However, there are fast algorithms to find approximate solutions. Since the problem of dividing up nodes into 2 groups while minimizing “bad” edges is NP-complete, and this is exactly our problem, we know that our problem is NP-complete. We developed and tested several different algorithms that attempt to solve or approximate this problem.