Social Structure And Network (A Mathematical Model For Social Behaviour)

Analogy and metaphor are often used by social scientists to explain a social phenomenon because certain social concepts are otherwise very difficult to comprehend. For example, a physical structure like ‘building’ or a biological structure like ‘organism’ is compared to define the concept ‘social structure’. Actually, social structure is not a physical structure. An abstract concept which can’t be seen is explained in a simplified way by using an analogy which can be seen easily by everyone. Physical scientists use a model to test the predictions. If the predictions are correct when the model is tested every time then the model constructed is perfect. Otherwise, the model is suitably modified and then the predictions are tested again. This process is continued until the model becomes perfect. Do we have a grand model of social structure that can be used to test social predictions? In this article, an attempt is made to understand how far network theory is useful in explaining social structure and whether social predictions can be made using the network.

Radcliffe-Brown was one of the earliest to recognise that the analysis of social structure would ultimately take a mathematical form. Radcliffe-Brown defines social structure as a ‘set of actually existing relations at a given moment of time, which link together certain human beings’. According to Oxford dictionary, ‘relations’ means the way in which two persons, groups, or countries behave towards each other or deal with each other. The phrase, ‘link together certain human beings’ can be compared with a ‘net work’ of connections.

Network is defined as a closely connected group of people who exchange information. Each point (person or agent) in the network is called a ‘node’ and the link between two nodes is connected by a line called an ‘edge’. When two nodes have a direct social relation then they are connected with an edge. So when a node is connected with all possible nodes with which the node has social relations, it produces a graph. The resulting graph is a social network. The number of edges in a network is given by a formula nc2, where ‘n’ is the number of nodes. For example, if there are 3 people in a party then the number of handshakes will be 3. If there are 4 people then the number of handshakes will be 6. If there are 5 people then it will be 10. If there are 10 people then the number of handshakes will be 45. If there are 1000 people then the number of handshakes will be 499,500. When the number of people has increased 100 folds from 10 to 1000, the number of handshakes has increased 10,000 folds. So the number of relationships increases significantly as ‘n’ increases. The network theory was developed by the Hungarian mathematicians, Paul Erdos and Alfred Renyi, in the mid twentieth-century. Networks of nodes that can be in a state of 0 or 1 are called Boolean networks. It was invented by the mathematician George Boole. In Boolean networks, the 0 or 1 state of the nodes is determined by a set of rules.

If two nodes are connected then the network of the two nodes assumes four states (00, 01, 10, and 11). The number of states of network grows exponentially as the number of nodes increases which is obtained by a formula 2n, where ‘n’ is the number of nodes. When n is greater than 100, it is quite difficult to explore all the possible states of the network even for the world’s fastest computer. In a Boolean network we can fix the number of states as 0 and 1. In a Boolean network, if there are three nodes A, B, and C which are connected directly by edges then the state of C can be determined by fixing the states of A and B. It means the state of C depends upon the states of A and B in some combination. Further it implies that if we know the state of C then we will know the combinational behaviour of A and B. But in a social network of persons, we do not know how a person’s behaviour is deterministic. Further, in a Boolean network, the behaviour of the nodes can be studied in controlled experiments as nodes here are objects. But in a social network, nodes which are individual persons can’t be treated as objects. In a social network how do we define the states of a person? How many states does a person have? What is the nature of a state? If the expected behaviour of a person is reduced to two states like ‘yes’ or ‘no’, then the number of states of a network will be 2n. Out of this, only one state will show up at a given moment of time. How do we predict that one particular state?

Family is a micro network within the network. The family members are closely connected with each other. Most of the members are also connected to other networks external to the family. Interactions take place within the family among the members who also have interactions outside the family. So there are several edges proceed from one node of a family towards nodes within the family and nodes outside the family. The edges within a family show intimate relationship, whereas the edges connecting nodes outside the family do not necessarily show intimate relationship. This intimate relationship is a very important assumption that we have to consider so as to reduce the number of states of the social network. For example, the likelihood of a family member to conform to the family norms will be higher. Similarly, the likelihood of a person to side with a close friend will be higher. Also, the likelihood of a member of a particular group to conform to group norms will be higher. These assumptions are necessary to measure the probability of how the whole network behaves in a certain way.

Interaction takes place along the nodes. The connection of one node to the other is either direct or indirect. For example, a person’s friend is connected to the person directly; the person’s friend’s friend is connected to the person indirectly, separated by one friend or technically by one degree. Research (Stanley Milgram, 1967) shows that every person in the world is separated only by six degrees to any other person. This implies that every person is connected directly or indirectly with other persons in the network except for an isolated community whose members do not have any contact with outside world. The six degrees of separation is only an approximation. For example, if you know the targeted person then the degrees of separation is zero. If your friend knows the targeted person then the degrees of separation is one and so on. Milgram’s conclusion was if you have selected a person to be targeted at random, then the maximum degrees of separation would have been six. However, the number of degrees of separation depends upon the number of critical nodes in the network in question. We will discuss about critical nodes later. So, connectivity is more or less a social reality. The question is whether this connectivity can be used as a tool to study social phenomena? If the answer is affirmative, then where can we apply this tool?

If we analyse social structure in terms of a network system, then it may be useful to understand the nature of ‘dynamism’. The state of a system at the current moment is a function of the state of the system at the previous moment and some change between the two moments. Therefore, ‘a set of actually existing relations at a given moment’ depends upon the actually existed relations at the previous moment. It implies the importance of time interval, whatever the interval may be. That means if we want to know why a particular type of social structure prevails over a society at a given point in time, then we should necessarily bring ‘historical perspective’ to the study. Change is an important ingredient of dynamic system. A change at the micro level sometimes doesn’t affect the system. But, in other occasions the system becomes chaotic. It depends upon the nature of change in time and space. What is to be noted here is, a person’s behaviour is shaped by the person’s past experiences and the present situation.

Moreover, a person in a social network is connected to different smaller networks which are dispersed widely. After all, a social network is networks within networks. But we should note that the system behaves differently with respect to a particular behaviour of different persons; it depends upon who the person is and how the person is placed in the hierarchy of the network. The network landscape is not even; it contains persons with different status and position. A person moves vertically and horizontally as well as deletes and adds connections. This brings change frequently at the micro level of the network. A person who is in power can easily influence others to follow an idea which need not be correct and a person who is not in power may not be able to influence others though the idea may be correct and good for the society. An idea doesn’t arise in a vacuum; it comes from the mind of a person. Even if an idea is correct, sometimes our society takes a lot of time to accept it. For example, it took a lot of time for our people to accept the fact that the earth is revolving around the sun and not the other way.

In a social network, (1) each node is unique as two individuals can’t be treated as two similar objects; (2) a node may have a large number of edges connected to it directly or indirectly though it may not influence the behaviour of other nodes; (3) a node may not have a large number of edges connected to it directly or indirectly, yet it may influence the behaviour of other nodes in its network; (4) a node may have both larger connectivity and the power of influence over other nodes. So it is necessary that each node is to be studied and graded according to its connectivity and power of influence. Once this is done, we will be able to predict, to some extent, how a particular network would behave. A critical node is a node that has a larger connectivity as well as the power of influence. Why people took a lot of time to accept that the earth is revolving around the sun and not the other way: It was because the critical nodes might not have been immediately ready to accept the fact for certain reasons; secondly, each node is required to be connected with at least one critical node in order to get influenced quickly; finally, a node was in confusion because it might have been connected to two critical nodes which had opposite views.

Though network is a good analogy to explain the concept of social structure, it has certain limitations: (1) The states of a network increases exponentially as the number of nodes increases; (2) The number of states of each node and its dependency on other nodes can’t be fixed as it can be done in Boolean network; (3) The number of edges (social relationships) increases as the number of nodes increases by a formula nc2; (4) Edges do not have uniform relationship; (5) Each node is unique and continues to change; (6) Information of opposing values continues to flow in the edges on both directions.

Though the number of relationships increases significantly as the number of nodes increases in a social network, it does not increase the complexity of the network. Society has certain norms. People are expected to follow these norms. These norms regulate the behaviour of people. Social regulations tend to reduce the noise in the network.

Though the behaviour of a node in the social network is difficult to determine, we can measure it by applying the theory of probability. For example, a family may hold a particular value. As the family is a closely knit group, all the members are expected to hold the same value. If we attribute a colour to this particular value, then the nodes of the family network will have the same colour and will look distinct. When the information pertaining to this value flows out from the family network to other networks through the edges, the information will have this colour. Therefore, the other nodes which receive and value this information will be influenced by this colour. Similarly, the nodes of a family will also be influenced by other colours as different information flow into the family network. The colour of a node depends upon how strong the node holds a particular value. Suppose, a certain node is surrounded by several nodes of a distinct colour, then the probability that this particular node will have a strong influence of that particular colour is higher. This is what happens when a person joins a group; the person will be strongly influenced by the values of that group. And when this person interacts with other nodes, those group values are transmitted. Therefore, if we know (a) the network of a particular node, (b) the colour of other nodes in the network, and (c) the colours of the critical nodes in the network, then we will be able to determine the probable behaviour of the particular node by giving weighted measure to each node of the network according to its location, distance, and colour. Though the nodes in a social network are not objects, the nodes can be studied objectively in this manner with a probabilistic determination.

Suppose a node is a drug addict and living in the neighbourhood of other nodes who are drug users and sellers, then we have reasons to believe that the node’s addiction is due to its location and easy availability. But we can’t attribute the same reason to a node’s addiction to drugs if the node doesn’t live in the neighbourhood of drug users and sellers. The circumstances under which the two nodes have got addicted to drugs would be different. There may be many causes for a node to become a drug addict. However, network analysis with probabilistic determination will be useful to find out the significant cause. In the former case, the node should be treated leniently because the probability to become a drug addict is higher due to its location and easy availability. The node is prone to be a victim of circumstances. The circumstances could be due to retreatism, a concept developed by the sociologist Robert Merton (1968).

According to Merton, retreatism is a response to inability to succeed; it is the rejection of both cultural goals and means, so that, in effect, one drops out. The sale of illegal drugs itself is another kind of deviant behaviour which Merton defined as innovation. Innovation involves accepting the cultural goals but rejecting conventional means. This excessive deviance results from particular social arrangements. Whereas in the latter case, the node’s probability to get addicted to drugs is lower and the circumstances are not obvious. It could be a personal choice or the drug sellers’ spread to new locations. If it was a personal choice, then in addition to social arrangements, biological and psychological factors would also be considered to find the causes. In this case, the node’s deviant behaviour needs a different treatment.

The above example illustrates how a social phenomenon can be studied using a network analysis with probabilistic determination. The probability of a node’s behaviour will guide us to make social predictions such as how a particular neighbourhood will behave in a certain situation at a given moment of time. One problem which would arise in this model is how the nodes are coloured. Suitable research method is to be employed to arrive at the probable nature of a node. The probable nature of a node depends upon the probable nature of other nodes in the network. The researcher should proceed from the established and known nature of certain nodes. For example if a group’s values are overtly known to everyone then the group will be coloured accordingly. However, the researcher should note that if the nodes are wrongly coloured, the measures will be wrong and so our predictions.

Another problem is the dynamic nature of the system. The behaviour of a node is constantly changing. However, the change in a particular node does not bring about a change in the system immediately in most of the cases. The change in the system is felt only after reaching a tipping point. Social change does not take place every second. After all, a period of 1,000 years is just a blink of an eye in the biological evolutionary time scale. Hence, social predictions can be made at a given moment of time. A third problem is, there can be individual differences within a family or group. This fact is to be considered in the research method before colouring a node.

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