Analyze networks and connections in a mathematical perspective

in steemstem •  last year 
Steemians friends must have heard or were familiar with the term network. Network words are often associated with computers, internet connections, or in Indonesian, we interpret them as networks. Not infrequently people interpret the network as a number of acquaintances.

Example, we just moved to a new place of residence, moved to a school, or met new friends, what did we do first? Maybe most are looking for a network of friendship networks. Well, apparently this network can be learned through mathematics, you. Curious, how do you do it? Let’s find it out!

Like the story above when we are in a new place, we will look for our connection. Now in mathematics, network connections that are often denoted by points or nodes can be shown as shown below.

Suppose there is a city called Regular City. Everyone in the Regular City will have only 4 acquaintances, or in other words, have 4-degree nodes. So, each point will be connected by 4 other points through the line, and so on.

By repeating a system like this on a network or network, every point at the end will soon be filled with 4 points and the network will continue to grow.

If we describe the percentage of all nodes to the degree of the node (the number of contacts of each person) we will get the results as shown below. The majority of people have 4 acquaintances, while those who have less than 4 acquaintances are those on the side or in the corner.

Of course the Regular City is not the right model to describe a network. We cannot limit that everyone must have 4 acquaintances. Maybe there are those who have more than 4, or who have less than 4 acquaintances. Let's study the other forms of networks in Lhokseumawe City.

In Lhokseumawe City, each individual can relate to as many people as possible. When the existing network is growing, the points contained in the network no longer only relate to the 4 existing points (square) as in the picture above, but are more complex.

We can also have fewer networks so we don't complete the 4 points. In conditions like this we will find that for each inter-point connection there is an angle formed. The smaller the angle of the node, the more connections. Conversely, the greater the angle of the node, the fewer connections.

In the case in Lhokseumawe City we can mention that everyone in the village has its own node so that each node has a different angle (picture below). Suppose there are 10 people in Lhokseumawe City and there is a possibility of one person to be friends with the other 9 people.

The number of friends that appear seems to be easily determined, which is 9 x 10 = 90 connections. But wait, we have calculated twice for the same friendship. Therefore, the possible number of friends is 90: 2 = 45.

If one person is randomly chosen, how likely is the chosen person to know us? The answer, we can count as 9/45 = 20%. In this case, 9 is the number of people who know us. The value of 20% means there is a 20% chance for each person to be known in one city of Lhokseumawe with a population of 10 people.

Now if the population increases, the probability of a person being known will change even if only a little. The distribution function will shift slightly left or right. However, the probability of someone having too little or too many friends is very small. When described, the distribution function will fulfill the binomial form as shown below.

Both examples in a regular City and Lhokseumawe City are simple models of a network. People with very many friends are not abnormal things like the binomial curve above. So, how do you form a more realistic network model?

A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as {\displaystyle P(k)\ \sim \ k^{\boldsymbol {-\gamma }}} P(k) \ \sim \ k^\boldsymbol{-\gamma} where {\displaystyle \gamma } \gamma is a parameter whose value is typically in the range 2 < {\displaystyle \gamma } \gamma < 3, although occasionally it may lie outside these bounds. Many networks have been reported to be scale-free, although statistical analysis has refuted many of these claims and seriously questioned others. Preferential attachment and the fitness model have been proposed as mechanisms to explain conjectured power law degree distributions in real networks. Wikipedia source

The more we form connections with friends or with friends from our friends, the structure of our friendship is more like a form of scale-free networks. This tissue structure is found in many food tissues, protein interactions, and the internet.

Scale-free structure meets the principle of "attachment interest" (preference attachment). This principle is similar to the rule of "the rich get richer". The more connections a person has, the faster the person's connection becomes more numerous.

Reference:

Wikipedia source

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