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**Complex systems and emergent property- An Introduction to social networks**

Let us understand social networks with the help of an example. Suppose people are asked to clap in a group, all clap in a different order. However, after sometime, they start clapping in a rhythm and synchronization even if no one decides what to follow. Everyone starts to follow the same frequency.

In starting, everyone frequency is different. Let ‘i ‘denotes person.

So, i is from 1, …… to N where N is the total number of people in the group. Let fi be the frequency of clap for a time and time is the constrain.

So, at time 0,

fi(1) (0) ≠ fi(2) (0)

After some time, everyone claps at similar time.

fi(1) (∆t) ≈ fi(2) (∆t) ≈ F

where F is the common frequency.

All this is done without anyone guiding.

**How did it happen?????**

Following can be the reasons: -

1. Frequency is distributed in the interval [0,100] where 0 means extremely fast and 100 means extremely slow.

2. There is a common pattern or rhythm.

Let the common rhythm is f^-i that everyone is following except you so you also adapt to it.

So,

Where Fi is the average frequency.

N-1 is included to normalize the equation.

Suppose, after some time ( ⸹) you don’t change your rhythm.

And if you adapt to the new rhythm,

Where a is the factor between 0 and 1, because you don’t adapt completely usually.

0 if you did not adapt at all and 1 if you adapt fully.

Error is introduced as you can’t always replicate the same.

This is the microscopic description if what is happening. This same can be interpreted for networks.

**Neighbors**

Neighbors are the local connection of points. Each point is called Node and each connection is called Link.

‘Ola’ occurs when there is synchronization one after the other. For example, fake news spread from you to your friends, from which it is spread to their friends and so on. So this is spread from small network and we cannot pin-point a single person who spread the news. This is called emergent property.

In linear regression, emergent property is there as there is always cluster or groups of points.

**Complex systems**

Complex system is a system that exhibits emergent property.

Example: — social networks, online communities.

How is system is said complex?

Following factors are true for complex systems: -

1. Numerosity: There should always be lot of elements in network not one or two. This is called numerosity.

2. Unpredictable: We know the network now but we cannot predict it in the future, it is unpredictability.

3. Chaotic: — When a network starts with very less variation, but later it ends with much variability, it is called chaotic.

4. Opacity: — It is very hard to completely describe what is going in the network, it is called opacity. For example: — In a car engine, there are bits and pieces which cannot work at alone but can work only when put together. This is opacity.

**Challenges in complex systems**

The system is quite opaque, since we can’t keep everything, so it is important to find that what is important factor and what is not so important. Discovering this is a great challenge.

**Network and Friendship paradox**

For example, there are 7 people in a room.

Let consider them as nodes.

Nodes = {A, B, C, D, E, F, G}

If we ask color preference,

A like yellow, B likes red and C like blue. D also likes red, E also likes red and F likes yellow and G likes blue. When asked about the number preference, B and C likes number 4.

If we make loops for common connections, we get,

Now let’s make a network by observing these links.

Lets calculate the links for each nodes.

A: 1

B: 3

C: 2

D: 2

E: 2

F: 1

G: 1

If we observe, node B has 3 links and node C has 2 links and node D has 2 links.

Average link of B and C = 2+3 / 2 = 2.5

It is greater than links D has which is also a friend of B. This is called friendship paradox.

A single node sometimes gets more popular and has lot many links.