How to reasonably measure the natural growth of users?

Although according to my personality, I like to narrate this matter in a purely logical order, but because this is a brand new theory that was recently produced by independent thinking, and there is no reference object, the purpose of this article is to discuss the feasibility of this theory with more people. Therefore, I will raise a few questions to make the audience a little confused, so that a few viewers are willing to read it and discuss with me:

1. The more money I invest, the more users I can bring in?
2. Every time I invest a similar amount of money, can I bring in a similar number of users?
3. Is there an optimal value between capital investment and user growth ?
4. Is the widely-publicized “exponential growth” really achieved exponentially? What does exponential growth mean for the product? What is the crisis of non-exponential growth products?

At this point, those who are not interested should have left, and I will discuss the issue of growth with those who choose to stay.

Personal subjective interpretation of the concept:

In this article, I will break down growth into three parts:

User-generated natural growth: growth generated by spontaneous user dissemination without any resource investment.

Investing resources to promote natural user growth: Invest some resources, but the resources are not used to directly increase the number of users. Instead, resources are invested to reach users, introduce a part of the product to users, arouse their interest, and users enter spontaneously; growth methods that can attract users with benefits and will not directly leave the product also fall into this category.

Violent promotion brings direct growth: directly giving users real income or benefits. The condition for income and benefits is downloading/registering the product, and it does not care whether the user will leave the product immediately in a short time.

In this article, I will classify the first two categories as natural growth.

Why did this question suddenly occur to me?

Before I talk about natural growth, I want to talk to you about two phenomena I observed, which prompted me to start thinking about this issue:

1. Growth indicators

In many companies, I have found such a phenomenon: especially for To C products, when everyone sets a growth goal for a certain period, there seems to be a formula that always holds true:

Indicators = experience + guesswork

Many people make a spur-of-the-moment decision based on their years of experience in the industry. As you can see, they are basically integer multiples of 1000, 10000, 100000… This makes me think, for a naturally growing product, isn’t there a simple and easy way to predict how many users we will have in the next stage?

2. Is the equation “resource quantity = growth rate” correct?

Of course, everyone will think the above equation is stupid, so where is the limit to growth?

To clarify the problem, I assume the simplest scenario

(Concept: fission means that when users are in a state of natural growth, one user can bring in several new users. At this time, I think the original user has undergone fission; this kind of growth is called fission growth, and you can use cell division as an analogy to it)

If we were to describe user growth in its entirety at the individual level, it would look like this:

In XXX time, X% of users can generate fission, and each of these users invites X new users:
Among these users, X are brought by natural dissemination; X are brought by investing XX resources.

We can extract four important independent variables from it:

• What percentage of users?
• A while?
• How many fissions?
• How many resources should be invested?

But if I consider so many variables at the same time, I need a multivariate function solution, and I'm sure you won't believe the answer that the computer fits. So, I assume the simplest case:

If there is a developing invitation-based community product, users must have an invitation code to register, and each person can only issue one invitation code per month; according to the current user growth rate, the number of users will double every month (that is, all users who enter each invite one person);

Then, one month later, the user volume limit of this community will be e (natural constant, ≈2.71828…) times the current value; (This is based on the assumption that every new user who enters will immediately invite the next user after an infinitesimal period of time, which you can assume is 1ns)

The above conclusion is obtained from a very simple formula:

We bring it into the scenario and get:

The following part is the derivation: (If you have a good mathematical foundation and know how this conclusion comes from, you can skip it directly):

Suppose there is a single-celled organism that divides once every 24 hours.

Then it is obvious that the number of this creature will double every day. Today is 1, tomorrow will be 2, and the day after tomorrow will be 4. We can write a formula for this growing quantity:

The x in the above formula represents the number of days. The total number of this kind of creatures in x days is 2 to the power of x. This formula can be changed to the following:

Among them, 1 represents the original quantity, and 100% represents the growth rate per unit time.

We continue to assume that every 12 hours, halfway through the division, the newly produced half of the cell is ready to divide again.

Therefore, the 24 hours in a day can be divided into two stages, each stage increasing by 50% based on the previous stage.

At the end of the day, we had 2.25 cells in total. Among them, 1 is the original one, 1 is the new one, and the other 0.25 are the new cells that have divided halfway.

If we continue to modify the hypothesis, this cell will have the ability to divide independently every 8 hours, which means dividing 1 day into 3 stages.

So, in the end we can get about 2.37 cells.

Naturally, if we further imagine that this division is continuous and that new cells have the ability to continue dividing every minute and every second, then how many cells can we get at most in a day?

When n tends to infinity, the extreme value of this formula is equal to 2.718281828….

Therefore, when the growth rate remains constant at 100%, we can only get a maximum of 2.71828 cells per unit time. Mathematicians call this number e, which means the maximum value that can be reached by continuous doubling growth per unit time.

This value is the limit of natural growth, so the logarithm with base e is called the natural logarithm.

So, I have obtained a growth in the simplest case, and I further generalize it: (The following are all extreme cases, that is, the fastest/maximum that can be achieved, which is actually difficult to achieve)

If my base is 100 users, then I will have 100e users after one month

If users are not limited to inviting only one person, but can invite 2 people (or even no limit, according to recent data observations, the average number of people invited by one person is 5); or if not all users invite new users, according to recent data observations, the average number of users invited new users is 15%.

Definition: rate = "% of users" * "number of people invited by each user"

The limit is:

What if we want to find the growth after two months?

In the case of time t, the general formula is:

Similarly, we can calculate the time required to grow to a certain scale:

For example, if 5% of users send invitations, and one user invites one person, the fastest time for 100 users to become 200 is

The result is 13.86 months: (So don’t fail to accelerate natural growth for the sake of stable growth, fearing that it will damage user growth. Of course, violent growth has always been discouraged.)

Above, we have discussed all the limiting cases and obtained the growth function in the limiting case:

The scale of users after a period of time is predicted as follows: (m is the current user scale, rate = "time t" "% of users participating in invitations" "number of invitations by each user")

This limit formula is not only applicable to the limit of user growth, but also to predicting indicators such as economic scale, sales growth, product sales, etc. that grow in a relatively stable environment. In fact, this value does not grow as fast as we often do in mathematics (i.e. exponential growth). In actual scenarios, growth is often the front end of the limit:

What's the point of it?

We can see that the curve of this limit formula is not that steep before it rises to a huge scale. In other words, even if I invest more resources, the limit I can reach is this curve. Therefore, it is impossible to achieve any growth indicators that exceed the data points on this limit curve by relying on natural growth (fission growth).

In addition: In actual business cases, the initial users are often not what we start from 1 and let it grow naturally. We will accurately place them in a group of places where we think our target customers gather, which actually does not conform to this curve. Therefore, our classic product curve is as follows:

It’s like a push notification about a new Xiaomi feature. Forwarding it to our Moments is the same as sending it to the Xiaomi Forum. Sending it to the Xiaomi Forum is not in line with natural growth, but when the feature starts to go beyond the Xiaomi Forum, it starts to spread, so our actual curve often starts from here:

Now that I have reached the limit, how can I predict the indicator data for my product in the next stage?

We continue to deduce along the final conclusion of the limit form:

• When the unit of measurement is not month, this formula can also be applied. The e-fold growth rate is based on one unit of measurement as one unit of time, such as one year, one day, or one second. We only need to define how long one stage is. For example, we can define "1 second = 1", "one year = 1", or "one year = 365". x changes with the definition of this stage time. "30/x", "365/x" or "1/x" can all be defined by ourselves.
• We do the processing to remove the limit, that is, we can get the number of users in the next stage as: (x means that the user starts to fission after x days)

• If after all users enter, the user base does not grow at 100% (not one user pulls in another), and the growth rate after x days is a% (the rate mentioned above), and you want to know what the user base will be like after 30 days:

If it is not 30 days, and the time promotion conclusion of the previous item is also included, then the number of users in the next stage will be:

So, we have a limit value - an exponential function about e (determines the upper limit of growth), and we have a calculation formula within the normal range (predicts the amount that will be reached in the next stage). What can we do:

1. More accurately predict the growth of a certain indicator (business volume, number of users, etc.), breaking the original unreasonable prediction method ("indicator = experience + guesswork").

2. The growth limit is exponential growth. Therefore, no matter how much resources are invested, it can only make the process of "1→e". Even many products themselves can achieve "exponential" growth. Therefore, investing more does not necessarily lead to faster growth. There is a limit.

3. The base of exponential growth is e, which is the limit. However, most companies will invest resources in promotion and marketing departments. Therefore, not all traffic is natural growth. There are also situations where artificial promotion promotes users’ understanding and willingness to invest in growth, and promotion forces user indicators to grow in a short period of time. We can judge through the growth observed in the growth curve over a period of time whether the current investment is reasonable, whether users really recognize the product, or whether they enter the market only because of market investment. That is, if the growth does not meet the conventional growth quantity forecast value (each stage is too low), we can say that this product has no user recognition. Although it is still growing, if it continues, the product will decline and collapse (before the 16% population determined by theories such as the "innovation diffusion curve" and before other conditions limit the limit, it is established in stages, which will be explained in this example).

Let’s use these two models to analyze a specific case.

Growth Case Study

Perfect example of explosive growth: Slack

Slack is a cloud-based team collaboration software. It doesn’t even have a marketing team and only relies on word-of-mouth marketing and brand influence to acquire users (Slack has only recently established a marketing team. Why is it necessary to do so? The reasons will be explained after the analysis), which means it is purely organic growth.

The above figure is a curve showing the number of slack users. Let’s try to take a data point from the curve and analyze it ourselves:

Because the graph basically conforms to the exponential growth of e, that is, the number of Slack users is growing at a nearly limit growth rate, so we have to use the limit state formula to analyze its growth:

By comparison we can get the following information:

1. The growth rate of Slack is 46.74%, that is, "percentage of fission users" * "number of fission users per person" = 46.74%. Therefore, we can easily get the number of Slack users in the next period of time. For example, we can calculate that in February 2015 (that is, the independent variable is 10), the number of users will be 532k. From the figure, the number of Slack users in February 2015 is about 520k, and the error rate is 2%. For prediction, this error is completely acceptable.

2.Why is Slack growing at an almost exponential rate?

Because in the analysis of the above figure, the reference dimension n we chose (the horizontal axis, which is also the "stage" defined earlier) is months. After each Slack user enters, they are ready to invite the next user to join. This time period x is very small compared to a month. Therefore, Slack eventually grows almost exponentially in this dimension.

3.Slack is growing at a rate close to its limit. What does this mean?

(2) Slack’s product is good enough that users recognize it and are willing to promote it spontaneously (on the contrary, if it is not an exponential growth, your product may not be recognized by many users. If your strategy does not change, your product crisis is coming, but it has not started yet, just like the US real estate securities in 2008).

I need to explain this point. The exponential growth that is often talked about by many media is often the result of artificially driven growth with a lot of money invested (even the result of violent promotion. In this case, I will believe it even if you exceed the exponential growth), because the limit of growth is an exponential function. In fact, if the funds invested in each user decrease, it will no longer grow exponentially. It is not like Slack, which has no investment and is driven purely by product power. It can always maintain exponential growth.

Of course, in actual products, making steady investments in each user, inducing users to trust the product, and then achieving natural growth (that is, the second type of natural growth that I defined at the beginning) is also a good way to accelerate growth, and it is sustainable.

(3) If we want to increase the user growth rate, the money we invest is actually to increase the user fission rate and fission amount, that is, to encourage more recommendations from old users, and encourage old users to recommend more people; because the limit of the growth curve has been reached, measures such as expanding new users will not actually make the growth rate faster, but will only increase the growth base (the growth rate may slow down due to marketing aversion, marketing targets not meeting the target user profile, etc.).

4. If I already have exponential growth, and I want to invest resources to accelerate this natural growth (still natural growth, not brute force promotion), and ultimately achieve the changes shown in the following figure, how do I evaluate my input-output ratio? Is there an optimal investment? We will discuss this issue in the second stage, because the author is still exploring this part, and I hope you can discuss it with me.

5. Will exponential growth continue?

Of course not. As can be seen from the Slack data, the table only gives the curve of Slack to a user scale of 500,000. It does not show other trends simply because the current growth has not reached the time point for change. So, where is the turning point of the curve for mass products?

The turning point of product growth - Taking NetEase Cloud Music as an example

In this part, I would like to explain what the future trends of general growth star products will be. (Due to limited data, I can only use NetEase Cloud Music as an example. If you have suitable data and want to discuss it with me, that would be great.)

Generally speaking, from the press releases we have seen, we summarize the growth of NetEase Cloud Music as follows:

It looks like a product with exponential growth, right? But in fact, the product will not grow in the above trend. According to the "NetEase Cloud Music 2016 First Half User Behavior Big Data " released by NetEase itself, the number of users of NetEase Cloud Music is as follows:

Although the authenticity of this trend has not been verified, we assume that this trend is correct and analyze on this basis why NetEase Cloud did not grow exponentially all the way to 200 million, but instead reached a turning point at 55 million users, and why it resumed its growth trend after 100 million. (Below I will only analyze the analysis methods that are common to any industry. I believe that readers are more familiar with the different special factors in different industries that affect this inflection point.)

After synthesizing more than 508 diffusion studies, communication professor Alfred Rogers proposed the famous "Innovation Diffusion Curve" theory in his book "Duffision of Innovation". Rogers used this theory to describe the process by which individuals and organizations adopt innovations. You may have heard this theory before in many famous speeches by various celebrities, but I will just show it to you again:

The blue line indicates that among the entire population, 2.5% are called innovators. They are the people who would queue up all day and all night in front of Apple stores to buy a new phone as soon as it is released (of course, they don’t have to do that now); 13.5% are called early adopters. They may have seen the new iPhone posted by the first group of people in the circle of friends, heard that it was good, and then bought it; the next 68% are people who followed the crowd in because of the gradual influence of the first two groups and market changes; the last 16% are laggards. They are the people who still insist on using keypad phones in an era when touch-screen phones are widely popular. (If you are engaged in an industry that is strongly related to sales, you can also pay attention to the yellow line here, which is the market share curve. I will not go into details in this article.)

Let’s use this theory to look at the music industry. According to the National Bureau of Statistics’ 2010 census data, the number of people aged 16-44 is 548 million (of course, you may think that this data has changed now. This is just an example estimate. In the actual industry, we all know how big the scale of our target user base is).

• I assume that 548 million is the number of users that will eventually be reached;
• According to the urban and rural population census, the urban population accounts for 57.35%. I think that for a new music player product, the urban population is the first to receive it (assuming this is the so-called industry influencing factor). I get that the urban population is about 314 million.
• Multiply the urban population by 2.5% and you get 7.85 million, which is the basic number of users that the player needs to reach in order to become a mass product. Multiply it by 16% (2.5% + 13.5%) and you get 50 million, which is the total number of innovators and early adopters, corresponding to the turning point on the NetEase Cloud Music graph, because at this time the product is about to enter a transition period of slower growth, transitioning from attracting people who "like new products" to attracting "conformists."

(4) How did the growth from 100 million to 200 million occur? I assume that this period is also exponential as shown in the figure. In fact, this mainly corresponds to the second part of natural growth. Each user invests a certain amount of money on average to promote natural growth of users. It can be seen that NetEase Cloud has made many moves during this period. The result is that under a stable growth-promoting strategy, after users have passed the transition period, the followers also have natural growth due to guidance.

However, if we are more realistic, in every small stage of the product life cycle , we will inevitably have some big actions, such as the launch of functions that affect the number of target user groups, exposure in well-known media, successful product marketing, etc. Is there any way for us to predict the number of users in this case?

Phased exponential growth - Taking Didi as an example

If we are just an ordinary product, we cannot rely purely on the first type of natural growth, and the speed of change is extremely fast. There will be many major actions that affect user growth and even the number of target user groups. At this time, it is often inaccurate to use a function to predict indicators for the entire life cycle. The following is an example of Didi Taxi : (This is actually the data of Didi downloads, not the number of users, so it is not accurate, but it is enough to explain the problem to you)

Similarly, we analyzed a piece of data from Didi. Since the observed growth trend roughly conforms to the exponential growth limit, I still use the extreme method to try to get close to the data of Didi Taxi:

The dotted line is the function I tried to fit. You can see that the error is very large, and even the trend is different. I re-observed the original data graph and found that in the data segment I took, Didi had a major version iteration, and there was no change node for version iteration. I checked and found that Didi announced a new financing of US$700 million on December 7, 2014. So I made an attempt first, and I fitted the data between the change nodes separately:

At this time, we discovered that after one big event happened and before another big event happened, the number of users still showed a stable exponential growth trend, but every time a big event happened, this growth trend would be replaced by another new exponential growth trend.

That is to say, in actual situations, if the environment changes, before the next change, we believe that the growth is stable (for healthy products recognized by users), and within this range it can still be predicted by the previous limit formula and a normal growth formula.

When does this theory fail?

This theory will fail for any product that is about to collapse (users do not recognize it, growth is not exponential but driven by marketing and promotion investment) or is already collapsing (there is a net loss of users at each stage). Of course, we can judge from the growth trend whether the product is growing but will collapse if it continues. This has been explained in the previous example discussion, so I will not repeat it here.

The most reasonable value of input-output ratio

We know that if we invest resources steadily, we can reach the growth limit by promoting natural growth; if we continue to invest resources, we can control the other three parameters, namely shortening the time of a stage, increasing the ratio of users who undergo fission, or increasing the number of users who can fission from each user. (As shown in the following formula, n and a% will increase)

So, because the changes caused by changing the exponent between exponential functions are not linear, there is a most reasonable input-output ratio in a mathematical sense. Next, I will discuss this unfinished part with you.

Where is the limit in mathematical sense?

Let’s look at the initial growth breakdown:

Here I assume the total number of users of the first category who grow spontaneously and naturally is A, and the sum of the first and second categories of users who promote natural growth is B, assuming that we do not have growth users brought about by violent promotion (the author of the article also strongly recommends not introducing such users). At this time, I think "BA" is the user growth part brought by investment. I convert all resource investment into funds, which is C.

Input-output ratio = C/(BA)

So if we bring in B and A, we get the following result:

Obviously, this value is too large. No economy in the world can reach the investment scale of C/(e-1), so this mathematical limit is actually not applicable.

What about starting from reality?

Although the above inference is wrong, the idea is correct. At the same time, I found that I could use my existing "user scale (x)" * "average investment per user (c)" to replace the investment, but user growth is also related to user scale, so I can get the following formula:

Assume that the scenario here is that forwarding in circle of friends brings natural growth, here f(x)=a b c*x (a is the average number of users that each user can convert , b is the forwarding rate, and c is the retention rate ), and then the maximum value can be obtained. However, this formula is limited to solving the optimal input-output of the forwarding channel , and its applicability is also questionable. Therefore, I look forward to further discussion with you on the input-output ratio.

Summarize

Because the purpose of this article is to discuss theories with others, the entire derivation process is described. For the convenience of application, two formulas of prediction indicators that are actually applied in this article are extracted here (of course, you need to find the understanding of trend curves in the article):

Growth formula under the limit state: (right side of the equation)

Under normal circumstances, the formula for predicting the number of users (indicator scale/economy) in the next stage is: (Note that this formula is only used to calculate the value of each point and does not represent a trend function of the entire growth)

Of course, any trend described in this article is only logical. No matter at what stage the actual situation is, as long as capital is invested to promote violent growth or cause the product to lose (collapse), the trend can be changed arbitrarily.

However, I just want to tell you what the trend of natural growth is. Only by investing resources in line with the natural growth stage can you get the biggest return and the highest input-output ratio. If the same growth indicators are set at different stages, the result will be that growth after reaching 16% becomes extremely difficult, and more and more funds must be invested to maintain growth, but retention is not very good.

Because in fact, forcing a group of "late followers" to the status of "innovators" requires this person to do something contrary to his or her character, which is bound to be very difficult. Even if exponential growth is maintained in the short term, it is actually not worth the cost.

The author of this article @MrMa compiled and published by (Qinggua Media). Please indicate the author information and source when reprinting!

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