### Balance Theory: From Psychology to Cancer Dynamics

May 25, 2019

#### Social Balance Theory

Imagine Alice and Bob have just gotten married and they have a mutual friend called Chris. In this scenario, they form a triangle of husband-wife-friend relationship which every edge of this triangle is a friendship link. Obviously, in this triplet of relationships people like each other and they can spend a lot of time together without any conflicts! So this triplet is said to be balanced because everything is fine there and people enjoy their current states. But If something goes wrong between Alice and Bob and they decide to get divorced then the link between the husband and wife in the triangle turns into enmity and from now on, this change makes Chris to rethink about his relationship with this ex-couple because he cannot simultaneously be friend with these guys who are enemy of each other! Chris has to change his attitude towards Alice or Bob, at last he must decide to be in a friendship status only with one of them. The triplet of relationship is now imbalanced and Chris finally decides to turn his relationship with Alice or Bob into enmity (turning a positive link to a negative one) which leads to another balanced state for this triangle of relationship after experiencing an imbalance . The dynamics of this change is mimicked in the picture below.

So as you may guess, in a triplet of relationships if odd numbers of the links are positive (friendship) the triangle is said to be balanced and otherwise, it is imbalanced. A balanced triangle, therefore, fulfills the adage that: (i) a friend of my friend is my friend, (ii) an enemy of my friend is my enemy, (iii) a friend of my enemy is my enemy, and (iv) an enemy of my enemy is my friend. This notion of balance in social systems was firstly introduced by Heider (1946) an Austrian psychologist whose work was related to the Gestalt school.

#### The Energy of Social Networks

The key concept in Heider Balance Theory is that “We adjust our relationship based on reducing the psychological stresses”. On the other hand, an imbalanced triangle is analogous to a frustrated plaquette in a random magnet. Physicists know how to assign energy to a system based on the interactions of its constituents. Therefore, inspiring by a random magnet, we can assign energy to a triplet of relationships by just multiplying the strength of each link in the triangle of relationships. So in the case of husband-wife-friend triangle, the energy is:

$$E = -R_{hw}R_{wf}R_{fh}$$

which $R_{hw}$ is the strength of the relationship between husband and wife. For a friendly relationship $R_{hw}$ is a positive number and for an unfriendly relationship $R_{hw}$ is a negative number. The minus sign is to make the analogy between the concept of social balance and stability in physics. In physical terms, the imbalanced triangles must have higher energy because the dynamics is toward more stable states with lower energy levels. The same reason that an apple falls off from a tree and going from a higher energy level at a higher altitude to lower energy levels toward the ground.

Now with this mechanism, calculating the energy of triangles formed in a social network is handy, we can just simply find the energy of each triangle and by aggregating them come to the energy of that network! The higher energy of a network, the more tendency toward attitude change is expected in that network!

#### The Cancerous Cell Has Less Energy!

When we look at the social life of the genes in the cell, we see that genes make dialogue with each other. It means that genes form a social network in a cell that the relationships between each pair of the genes can be positive or negative, in a sense that genes can alter the state of other genes through the so-called regulatory effects. So, again if the strength of the relationship between each pair of the genes is known we can find the energy of the cell. For the case of Breast Cancer, Abbas Karimi et al. at Shahid Beheshti University in Tehran have inferred the strength of the relationship between each pair of the genes and showed that the energy level of the normal cell is higher than the cancerous one, meaning the normal cell has more tendency to change its attitude and therefore it can tackle more challenges it experiences through its life cycle.

This is how interdisciplinary research is done nowadays in the field of complex systems. The point is to be open to ideas from different disciplines and to be creative enough to construct a pathway to solve real-world problems. From a psychological theory from 1946, we have come to a question in 21st century systems biology with a resolution equipped with the toolbox of physics!