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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Five Considerations for Realistic Network Synergy Savings

A primary reason for merger and acquisition or rival companies in the telecommunications industry is to achieve cost savings. One source for those savings comes from the network integration of both companies’ owned and leased network assets and expenses. These is a plethora of options to eliminate redundant network costs, but buyers of companies are often too optimistic in their expected synergy savings. As a result, they often failed to achieve their initial savings target. This article explains why many of these mergers failed to achieve their financial goals and what you can do to improve them.

There are many methods to identify network savings. Although there is no set rule for identifying network synergy savings, a general method is to group network savings by optimization types or by geographies after eliminating redundant savings.

Unfortunately, this popular process of identifying network synergy savings fails to address key issues. There are five additional considerations before investing capital to acquire a company.

Consideration 1: Coordination of Multiple Optimization Projects – Some optimization projects aim to eliminate different parts of network costs within the same geographic area. Extra attention is required when dealing with many projects. Although these projects may deal with different parts of network costs, they are often inter-dependent of each other. The network planner often fails to understand these key relationships between projects and his lack of understanding can result in under-estimation of the project timeline while over-estimating project savings.

For example, the planner initially finds two projects in one geographic area. The first opportunity is to eliminate the leased access expenses by grooming them to the seller’s metro rings. Another opportunity is to consolidate the collocation space where local traffic is aggregated. The planner would like to consolidate the collocation space first, but the buyer needs to acquire additional space to accommodate seller’s equipment as well as incurring a higher penalty for eliminating seller’s collocation space in the first year of the merger. As a result, the planner waits for initiating the collocation consolidation plan. Since the terms under leased circuits are expired and billed at month-to-month term, the planner would like to initiate the groom plan to the metro ring as quickly as possible. The main problem is that the planner has to establish an interconnection between two collocation sites to route these circuits back to the buyer’s backbone network since it is not possible to groom these circuits onto the seller’s backbone due to its current transport route. These additional expenses of establishing the interconnection between buyer and seller, the revised saving is now substantially reduced. Another option is to re-term these circuits until the space consolidation project is feasible, but this decision will lead to the lower cost savings.

It is a worthwhile exercise to evaluate the relationships across different network components and how each of these projects would affect rather than looking at each project as a separate entity. We need to ask questions such as “Does it make sense to initiate a project ‘A’ first, then to project ‘B’?” “If a project sequence is reversed, how would other project impacted?” “Does it make sense to renew a leased circuit or implement a short-term solution while waiting for another project to launch?”

The planner must appropriately adjust the timing of synergy savings, as other projects may need to wait until completion of the predecessor project. The planner may consider a short-term fix such as renewing a leased circuit while waiting to initiate other inter-dependent projects.

Consideration 2: Network Evolution – There are short- and long-term network integration strategies. Attempting to maximize short-term savings can create constraints to implementing a long-term network solution. The planner must strike a balance between short- and long-term network decisions to find the integration strategy leading to a best Net Present Value (NPV). Unfortunately, the planner is tasked to realize savings fast, the decisions he makes would not lead to an optimal NPV over a measured period.

For example, there is a limited capital funding to integrate the networks in the first year of the merger. Where there is no capital available, the planner decides to establish leased network hubs to consolidate both companies’ leased circuits instead of building a network to eliminate leased circuits. The planner is uncertain that the capital will be available in the following year to build a network; thus, he decides to establish the leased hubs with a five-year term commitment. Because the termination liability charges to eliminate the leased hubs will be substantial as well as the cost of re-grooming will be prohibitive, the network builds will not be approved in the second year of the merger as indicated by the lower NPV. The first-year decision to establish leased hubs constrained the combined company to initiate the better project in the second year.

Consideration 3: Implementation and Network Scaling – Most people are fundamentally optimistic. It is no different for estimating integration savings. Actual results often show lower than expected savings and longer than expected project timelines. Some issues that can temper optimism are:

1) An integration of network would require dealing with different processes, cultures, and systems. These issues lead to a communication break-down resulting in a higher cost of network integration.

2) A post-merger is a hectic time for companies. Additional resources brought in to help speed up the network integration efforts with people unfamiliar with the company’s process and procedures result in more errors and re-works.

3) Network builds may require obtaining a permit or private easement negotiation prior to starting constructions, which can add more time to complete projects.

4) A target company’s network may not be scaled to support the traffic demand. Just because it has network assets at the right place, it does not mean that there is enough capacity to support the project. If the network is not scaled to support the capacity demand, then the additional capital investments and operating expenses will be required to augment networks.

The planner should become familiar with possible “soft” issues associated with network integrations such as cultural and communication issues. He needs to ask key information from the seller such as the current network capacity to avoid any future surprises. In addition, the planner should collect data to help estimate the realistic timelines for completing different project types so that he can adjust savings appropriately.

Consideration 4: Risky Projects – Just as there are risky stocks with above average expected market returns, a risky optimization project tends to result in a higher potential savings. The planner must be careful when estimating synergy savings. Normalize network savings by risk levels when comparing several network optimization approaches in order to avoid disappointments later.

For example, establishing a metro ring to a customer’s premise is a risky project. Grooming to the metro ring eliminate 100% of leased network expenses. Another strategy is to place a node in an Incumbent Local Exchange Carrier’s End Office (ILEC EO) where partial savings can be achieved. Without risk adjustments, the network builds to the customer’s site show higher savings. Unfortunately, establishing the metro ring to the customer premise would require the customer to get involved in the circuit groom. Not all customers are willing to groom circuits under this scenario unless service providers share savings from groom projects. Although the optimization project at the ILEC EO would not deliver a high cost reduction, it does not require the customer involvement. Groom projects without customer’s involvement lead to higher project completion rate. As a result, the planner must appropriately adjust savings when he compares optimization projects with different risk characteristics.

Consideration 5: Circuit Life – A cash-flow improvement from network cost reduction initiatives over long-term is harder to predict for reasons.

1) Customers continue to groom, upgrade, and re-configure circuits. These activities often result in circuits being prematurely disconnected. If the planner must prepare a long-term view of synergy savings, it is likely that the majority of circuits will not be kept after few years. The planner must factor an attrition rate into estimating the practical long-term synergy savings.

2) A majority of customers request circuit renewals at a lower price point. Customers are aware of the expanded networks from the merger, and they will demand a price reduction. Any cost savings projects will be short-lived.

Although I discussed different risks for over-estimating the integration savings up to this point, there is a case where the planner fails to identify further opportunities due to his lack of a seller’s perspective. The planner must ensure the completeness of synergy savings analysis by including opportunities for both buyer and seller’s point of views.

In addition, the planner may be required to combine both companies’ savings opportunities in order to justify a project. Although it is not always easy to do, the planner should incorporate both sides’ network expenses when identifying synergy projects.

A penalty for incorrectly estimating synergy savings can be high; however, the opportunities for maximizing savings are also substantial. There are few tricks that the planner can use to his analysis to increase the accuracy of synergy savings. Although addressing all the issues described can be a daunting task, it is possible to adopt few changes to improve the accuracy of estimating synergy savings. Just be mindful of these issues when you lead an integration of networks.

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