Existing social networks
A social network is a website or app where people connect, share and talk. There are several kinds. Messaging apps (WhatsApp, Telegram) are for chats and groups. Photo and video apps (Instagram, YouTube, Snapchat) are for sharing pictures and clips. Short-post and discussion sites (X, Reddit) are for opinions and news. Professional networks (LinkedIn) are for jobs. Wikipedia and online games have communities too.
Every account is a profile. You can follow people, make friends, post, like, comment and share. Some links are two-way (friends) and some are one-way (you follow a singer, the singer does not follow you).
Graphs and their measures
Mathematicians draw a network as a graph. A person is a node (dot). A friendship is an edge (line). If you follow someone and they do not follow back, the edge is an arrow: a directed graph.
Degree = how many edges touch a node. A person with 6 friends has degree 6. The node with the biggest degree is a hub (an influencer, a popular teacher). The sum of all degrees = 2 × number of edges, because each edge is counted at both ends.
A path is a chain of edges from one node to another. Its length is the number of edges. The distance between two people is the shortest path. A graph is connected if there is a path between every pair. A cluster is a group whose members are mostly friends with each other, like a class.
Small world and the Milgram experiment
In 1967 the psychologist Stanley Milgram asked people in one part of the USA to send a letter to a stranger far away. They could only hand it to someone they knew by first name, who passed it on. The letters that arrived took about 5 to 6 hand-overs. This gave the idea of six degrees of separation: almost any two people are linked by a short chain of acquaintances.
Why so short? A few hubs and a few long-distance links join separate clusters. In our 3D, the purple hub cuts the distance from A to B a lot. Large online networks have been measured with an average distance of only about 4 to 5. The result is a small world. Note: many letters never arrived, so the experiment is a hint, not a proof.
Business model: why is it free?
Running servers, engineers and storage costs a lot of money, yet most apps are free. The money comes from advertising. The app learns what you watch, like and search (your data), and also keeps you looking (your attention). Advertisers pay to show their ad to the right group, for example "students aged 15 to 18 who like cricket". Other income: premium subscriptions, selling gifts or stickers, and a fee on shopping in the app.
So the real customer of a free app is the advertiser, and you are the audience. This is why settings like privacy, ad choices and screen-time limits matter. Apps are designed to be hard to put down, so decide your time limit before you start.
Cyberviolence and cyberbullying
Cyberviolence is harm done through phones and the internet. It includes cyberbullying (repeated insults, threats, rude memes, leaving someone out of a group), harassment and stalking, sharing someone's private photos or details without permission (doxxing), spreading lies (fake news and rumours), pretending to be someone else (impersonation) and cheating people for money (scams).
It hurts more than we expect: it can reach the victim at any hour, many people can see it, it can be copied and it is hard to delete. A post spreads along edges like a wave (see the 3D).
If it happens to you: 1) Do not reply or fight. 2) Take screenshots as proof. 3) Block the person. 4) Report the post to the app. 5) Tell a parent, teacher or helpline. In India you can call the Childline number 1098 or report cyber crime at cybercrime.gov.in; other countries have similar helplines.
If you see it happen to someone: do not share or laugh. Support the person and report it. Before you post, think: is it true, is it kind, would I say it face to face?
Try it
Draw your own graph. Pick 6 friends. Draw a dot for each, and a line between two who also know each other. Count the degree of each dot. Who is the hub? Pick two dots that are not joined and find the shortest chain between them.
In the 3D, step 4, press "New pair" and guess the number of steps before you read it.
Key formulas and definitions
- Node = person, edge = friendship
- Degree of a node = number of edges at it
- Sum of all degrees = 2 × number of edges
- Distance = number of edges on the shortest path
- Milgram: about 6 steps (six degrees of separation)
- Free app: the advertiser pays; your attention and data are the product
Worked examples
1. A friend group has 4 people: Asha, Ben, Chen and Dev. Asha is friends with all three. Ben is also friends with Chen. Draw the graph in words and find each degree.
Edges: Asha-Ben, Asha-Chen, Asha-Dev, Ben-Chen. That is 4 edges. Degrees: Asha 3, Ben 2, Chen 2, Dev 1. Check: 3 + 2 + 2 + 1 = 8 = 2 × 4. Asha is the hub.
2. In the same group, how many steps from Dev to Ben?
Dev is only linked to Asha. Asha is linked to Ben. So the path is Dev - Asha - Ben: 2 steps. There is no 1-step path, so the distance is 2.
3. A network has 100 people and each has exactly 6 friends. How many friendships (edges) are there?
Sum of degrees = 100 × 6 = 600. Each edge is counted twice, so edges = 600 / 2 = 300.
4. A free video app shows 3 ads per video. Each ad pays 0.50 rupee. One user watches 40 videos a day. How much does the app earn from that user per day?
Ads per day = 3 × 40 = 120. Money = 120 × 0.50 = 60 rupees. The user paid nothing, but the advertisers paid 60 rupees for this user's attention.
5. You get a rude message from a classmate in a group chat. Put the best actions in order.
First do not reply. Then take a screenshot. Then block the person and report the message. Finally tell a parent or teacher. Do not forward the message to others.
Common mistakes
- Thinking "free" means no cost. The cost is your time, attention and data.
- Mixing up node and edge. The person is the node (dot); the friendship is the edge (line).
- Reading "six degrees" as exactly six. It is a short chain on average, often 4 to 6.
- Sharing a hurtful post "just for fun". Every share is a new edge it spreads along, and it is part of the bullying.