Math Breaking Magic: The Network Effect in Knowledge Work

Josh Redd

April 11, 2023
Unlock the power of Metcalfe’s Law and revolutionize your team’s productivity by creating a unified digital network for knowledge sharing and collaboration.

Let’s talk about a guy you’ve never heard about and his law that governs your world: Robert Metcalfe. He co-invented ethernet (thank you Mr. Metcalfe 🙏) and presented his namesake’s law in 1980. In short, Metcalfe’s law states that the value of a network exponentially increases as more users connect. That sounds boring as crap but stick around and you’ll see why understanding this simple concept can supercharge your life and productivity. Gain a new perspective with me.

Imagine a land line telephone – are you old enough (I mean wise enough) to remember those? – but you have the only one in existence. One of those by itself is absolutely worthless. You pick the thing up and you hear dial tone. You dial any number but since yours is the only phone in existence, you get no one. They had no screens or functionality besides simply connecting to another user and opening the line of communication. You could talk to anyone anywhere that was also connected to the network so long as you had their number.

The phone allowed you as the user to be connected to the network. Even if you weren’t actively using it, the phone was still connected and anyone could call you anytime of day or night. The door was open. The phone line was an intermediary for communication between two people, specifically using voice.

So imagine one other person in the world has a phone just like you and you have each other’s numbers. Amazing. You have a friend. Representing yourselves as dots, you can now draw one line between your two dots. You’re connected. This is very valuable. Rather than meeting in person which is expensive and time consuming, or sending mail which is time consuming but much cheaper, now you can communicate cheap AND fast with your one friend.

You convince another friend to get one of these bad boys called a home phone. Now there are three dots. This new dot has the ability to connect to both of the pre-existing dots and so you can draw 2 more lines. Now you have a triangle – three dots, three lines. Did you catch the effect already?

By adding the first phone to our fledgling network, we went from 1 to 2 dots and we added our first line. By adding one more dot, we were able to add not 1, but 2 lines. The increase in connection is outpacing the increase in connected users from the very beginning. With 5 users, you have 10 connections. Your connections are now double the number of users.

With 8 phones, we have 28 connections. Our connection are now more than triple the amount of users. This is growing fast!

NERD ALERT – Skip this section if you don’t care about math. No one will judge you.


Metcalfe’s law shows us that the value of our network increases exponentially as we add more and more users. In fact, it’s simple math: $n( n – 1 ) /2$. Take 8 phones and subtract 1 to get 7. Multiply that by the original 8 to get 56. Divide by 2 to get 28. While “28” may not seem like a meaningful qualifier for the value of a network, it comes to life when compared against other network sizes. Think about it this way. Every time you add one more user to the network, you add the total number of existing users as new connections. The larger the network gets, the more valuable each additional user becomes.

If 10 connected users has a network “value” of 45 and 20 connections has a value of 190 – doubling the size of network users more than quadruples it’s value, 4.2x to be exact. (is 42 the answer to the universe??) The increase in connections is exponential as you add more and more users.

By simply dividing our connection number by our user number, we can get a “power” number. We’re answering the question, how many more connections are there than users? That number is linear. It increases by 0.5 for every additional user added to the network. We’re basically re-engineering our original equation. Divide your user number by 2 and subtract 0.5 to get the power. You can then multiply your user number by that power number to get your connection number.

And again this is based on a pure logical model, where every connection is perfect and clear. In reality, we don’t communicate clearly, we forget, and we run up against things like Dunbar’s number. Whether that number is 150 or more likely much lower, the theory is what’s most important. We can only cognitively keep up with a limited number of connections. At scale, we need help.


In the case of our phone example – the phone company does its job to provide a clear transfer of sound across potentially thousands of miles within milliseconds. The phone hardware also has to be up to snuff and built to standard to take sound waveforms and transform them into a signal that can be turned back into waveforms on the other end. Over time, we’ve nailed this concept, we take it for granted. technical standards and infrastructure have been in place for decades, if not over a century.

But we’ve come a lot further than land lines these days. I don’t need to tell you about the power in your pocket, but Metcalfe’s law still applies and sites like Facebook, Twitter, Instagram, Linkedin and TikTok have all leveraged this effect to their prosperity. Our computers and devices can perform the same function as those old land lines but now we’re using a visual medium. Social media sites created a medium for people to share bits of information in a standardized and friction-free way. I like to think of this as a Common Denominator of Thought (CDT) – A tweet, a status update, an instagram post, etc. There is a widely understood standard and the world is now hyper-connected because of them. Just like the phone company, these sites, leveraging already established infrastructure (the internet), allow people to connect their dots and the value of the network is incredible because of the vast number of people connected. There is a reason they are called “Social Networks.”

If you’re reading this, you understand this intuitively and your personal life is connected on many networks to create your unique network. You share life this way. You create a legacy of thought and connection when you interact on these networks.

But what about your work life? I don’t just mean social connections with your coworkers either. I mean what about the knowledge you bring to the table at work? A company is full of talent, people with knowledge who do things according to that knowledge. That knowledge is shared ad hoc with apps like Slack, Salesforce, Email, etc. When you need the info, you ask for it from the person who knows, and you get it. If your company is on top of it, they’ve got documents of established knowledge. Maybe you’ve got things in Google Docs or actual paper files. You might project manage in Asana, Todoist, Trello, or Jira.

Some of these are starting to create a good network while some of them are inefficient or just plain bad and frustrating to use. But every tool is essentially a different network that you have to synthesize individually into your own network of knowledge. You have to manage each network with different connections. You act as your own router. It’s not unthinkable that you might even have to connect with a higher up who then has to network with someone else higher up to get you the information you really need to continue your work. At this point, this sounds like insanity. It’s frustrating at the least and it’s inefficient to the core. If companies fail to adapt and focus on integrating and uniting their communication and information networks – they’re going to lose.

The thing about knowledge work is that knowledge is the key (wow). The talent on the roster of the company brings a set of knowledge that should have free flowing channels to make the network more valuable. It SHOULD be an exponential value add with each new talent hire. Instead, there is no streamlined network, its a smattering of sub-networks with broken links and gatekeepers to knowledge. **The average worker spends 30% of their time just looking for the information they need** just to do the job that they want to do and that they’re good at. Besides killing morale and destroying the ability for talent to thrive in a fulfilling career, it’s an incredible waste of resources when the solution is in sight.

What knowledge workers need is a Second Brain. “Second Brain” is the hot new keyword for anyone in the productivity space since the release Tiago Forte’s book “Building a Second Brain.” To give you the one sentence synopsis – human brains are great at creating ideas, not storing them, so we need a place to store our thoughts and ideas in a structured way for later use. The Second Brain. For individuals, any number of apps including a simple notebook and pen will suffice. Where it becomes really interesting is when we apply this idea to teams. We’re looking to create a team brain – a place where a team can collectively share and store their own ideas – but also be inspired by the work of their colleagues. It’s digital serendipity.

If teams had a Second Brain and established CDT’s (Common Denominator of Thought) – if the infrastructure was established and the network lines were open and clear – team potential becomes explosive. I’m convinced that this is what makes super teams. This is what unicorn companies are made of. This is how scaling fast becomes realistic. People can hack it at small scale. They can individually own their area if they have the personality and drive to go get their knowledge from others. But at some critical point, this becomes unstable, untenable. And without a network in place, more dots keep getting added to the picture but not every line gets connected. People just can’t manage that many connections efficiently. A lot of noise starts to get introduced and workers become relays for knowledge rather than knowledge bearers, gatherers and executors.

Teams need established infrastructure and a clear network with CDT’s (standards of information packets). This is their rocket fuel. This wasn’t a realistic expectation for the average company even 10 years ago. But with the proliferation of tech and options out there like Notion, this is not only possible, it’s responsible.

Enter the Network Effect. Metcalfe’s Law. Metcalfe’s law explains how the value of a network increases exponentially as more users are connected. It’s a great theoretical model and it proves this one point very well. But it does not speak at all to the efficacy of the network itself. It assumes a perfectly clear signal path between all connected users. In reality, there’s a lot of noise on the network. If you were to model the average team today and include annotations to show signal strength – you would have a tangled mess. Some lines would be strong representing clear communication channels, but most would be faded, broken, or require multiple stops to get to the final destination. They’d probably take a different route on the way back too.

We want the true network effect in a team. Not just the theoretical connections but a real network of communication. A medium where ideas translate and transfer with low noise and high signal. This is the secret sauce of a truly connected team. Unicorns in the making. This is what we call digital math-breaking magic because it’s where 1+1=3. The team on this kind of juice is greater than the sum of its parts. Innovation and inspiration are born in this kind of environment.

To be clear, adding team members to a roster does NOT assume that they join in this network effect. They become a dot on the map but the network effect takes place only when they become connected with every single other person in some way. But at some point, one person cannot maintain 1:1 connections with every other person in the organization. We’ve covered this and this is why we need information to meet in the middle. We need common denominators of thought. We need a standard of ideas and information where they can be packaged into atomic bits and sent through the network without any delay. We need a digital interface for this to happen in. We need a tool that lets us build the system to suit our needs. It’s time for Notion to shine.

This is typically where teams fail to scale. Small teams work really well when they can manage individual connections with all the other talent in the company. But as the company grows – this direct connection is unsustainable. The strict network effect – if its 1:1/person:person – does not scale. Thats where Notion comes into play. It becomes a common interface that all people connect to. Its the router, and for what? The most important thing being shared between people – ideas/knowledge/thought.

Team members translate their thoughts into written knowledge and add it to the common brain, Notion. By communicating to the central brain of the organization – they’ve just made it possible for anyone else to jump in and receive that information at their leisure or as needed. No more signal-to-noise problems, no more all hands meetings that only relate 10% to each person – just clean communication of information as needed. It’s the knowledge workers version ofJIT (Just In Time) supply. In classic supply chain, you’re dealing with physical goods that require serious logistics to manage the movement of those goods for manufacturing. Luckily for the knowledge workers, we work with digital goods that can be transferred instantaneously and stored basically for free. But by no means should we not take our logistics seriously. A little love goes a long way here.

Companies who take this objective seriously will reap the greater reward. As every individual is unique, so is every company – thus the team brain will also be. This is getting to the core of why we, at Notion State, are in business. We believe in the unique identity of every company, and we believe in the power of a cohesive platform to effectively think together. There is now a proliferation of SaaS tools in the marketplace for teams to think together. But these tools have largely stuck to a single lane. Salesforce handles customers, Slack handles instant messaging, Jira handles product development, Asana handles project management, and so on. These have increased the speed and efficiency of thought within teams, but they are still sub-networks. They exist in silos to create the larger virtual network of a team. We believe Notion is the game-changer. It is a genre-creating software in a class of its own. No more silos – Notion is a wrapper for all of your “applications” because it allows you to create them for yourself using the same lego blocks for Each one communicates perfectly with the others.

Notion is not just another shiny toy in the arsenal – it’s a game changer for those willing to go all in. It’s about creating an intermediary for thought – a second brain that accepts common denominator’s of thought between individuals. It’s the closest thing we have to reading someone’s mind because we now share a mind. If you made it this whole way and haven’t tried Notion yet – make the jump.