Hertling’s Equation

I’m a huge fan of William Hertling. His newest book, The Turing Exception, is dynamite. It’s the fourth book in the Singularity Series, so you really need to read them from the beginning to totally get it, but they are worth every minute you’ll spend on them.

William occasionally sends me some thoughts for a guest post. I always find what he’s chewing on to be interesting and in this case he’s playing around with doing a Drake’s Equation equivalent for social networks. Enjoy!

Drake’s Equation is used to estimate the number of planets with currently communicating life, which helps us predict the odds of finding intelligent life in the universe. You can read the Wikipedia article for more information, but the basic idea is to multiply together a number of functions: the number of stars in our galaxy, the fraction of those that have planets, the average percent of planets that could support life, etc.

I’m currently writing a novel about social networks, and one of the areas that’s interesting to me is what I think of as the empty network problem: a new social network has little benefit unless my friend are there. If I’m an early adopter, I might give it a few days, and then leave. If my friends show up later, and I’ve already given up on it, then they don’t get any benefit either.

Robert Metcalfe, inventor of ethernet, coined Metcalfe’s Law, which says “the value of a telecommunications network is proportional to the square of the number of connected users of the system (n^2).”

Social networks actually have a more rigorous form of that law: “the value of a social network is proportional to the square of the number of connected friends.” That is, I don’t care about the number of strangers using a network, I care about the number of friends. (Friends being used loosely here to include friends, family, coworkers, business associates, etc.)

Drake’s Equation helps predict the success of finding life in the universe because it takes into account the rise and fall of civilizations: two civilizations must exist at the same time and within observable distance of each other in other to “find intelligent life”.

So there must be a similar type of equation that can help predict a person’s adoption of a new social network and takes into account that we’re only willing to try a network for so long before giving up.

Here’s my first shot at this equation:

P = (nN * fEA * fAv * fBE * tT) / (tB * nF) * B

P = probability of long-term adoption

nN = Size of my network (number of friends)
fEA = Fraction of my friends who are Early Adopters
fAv = fraction of those who have available time to try a new network
fBE = fraction that overcome the Barrier to Entry
tT = Average length of time people Try the network
nB = Average length of time it takes to see Benefit of the new network
nF = Number of Friends needed to see benefit
B = The unique benefit or desirability of the network

In plain English: The probability of a given person becoming a long term user of a new social network is a function of the number of their friends who also adopt and how long they remain there divided by the length of time and number of friends it takes to see the benefit multiplied by the size of the unique benefit offered by the social network.

Some ideas that fall out from the equation:

  • A network that targets teens, who may tend to have more time and may be more likely to be early adopters, will have an easier time gaining adoption than a network that targets busy executives, all other things being equal.
  • A network that has a benefit when even two friends connect will see easier initial adoption than one that requires a dozen friends to connect.
  • A network whose benefit applies to distant connections is at an advantage compared to one that only offers a benefit to close friends (because N is larger)
  • A sufficiently large unique benefit can overcome nearly any disadvantage.
  • Social gaming is interesting, because it provides benefit even when connected to no one, the benefit increases when connected to strangers, and then increases even more when connected to friends.

What do you think?

  • Typo for “nB” / “tB” in equation above

    Nice go – at 1st glance, one thing that jumps out is that nF should exponentially impact the equation instead of linearly. So lesser the number of friends who see the benefit, faster is the probability of success in a non-linear fashion.

    It also seems counter-intuitive for case nF= 1; not from an equation perspective but if viewed from a real world perspective. If nF=1 for 1 person, then 2 people come on the network. But considering nF changes for different people, theoretically there could be only two people in the network for a person for whom we calculated P where nF=1 (the highest probability). So we end up with a case of highest possible P for person 1 but only two people in the network.

    Another similar counter-intuitive case is where nF/tB = 0 but I dont think I am being coherent already so I will stop here!

  • Well will your proposed equation count for the “purpose” of the social network aside from just being social?. I’m thinking examples such as LinkedIn where the extrinsic benefits of being there are not proportional to close or distant friends but the “purpose” of why you’re joining / maintaining connections.

    Equally true with social platforms which seemingly broadcast information and allow for filtered readership based on ‘follows’ or ‘likes’. Keeping in mind the networks themselves have algorithms which need to be accounted for as they dictate heavily the use of the social network and hence the P = probability of long term adoption.

    At least maybe you should add a cSi = Complexity (maturity) of Social Interaction to add towards big P.

    • williamhertling

      LinkedIn in a very interesting case. It definitely has value. I’ve used it to make connections to people I didn’t have before. The value derives from the connections that it maps, which is the definition of a social network. In fact, in a 2003 conference I attended in San Francisco, I remember augmented social networks being the big hot topic, and LinkedIn was the example.

      But LinkedIn is not very social by any reasonable definition of the word “social”. I don’t spend time on there connecting with friends, or seeing what people are up to. I go in, I find a connection, or learn about a person, and I get out.

      • Actually I felt the same at the beginning, but now I do find it social to hear about new job positions people have, and LinkedIn added a type of blog posting, that has driven people to use the platform to communicate and share ideas / thoughts.

  • Hi Brad – your post is timely to me as we are getting ready to officially launch The Fitness Insider Social Network. We have combined educational content about fitness technology, integrated fitness challenges utilizing the latest fitness tech, such as Fitbit, and we require that new users share on their social networks before entering. Do you think this is enough to satisfy Hertling’s Equation? Can you think of something else we could to to improve the effectiveness of our social network?

  • We also need to consider that some Culture’s globally are more “wired” and openly embrace social networking– i.e. South Korea, Brazil….

  • WOW how cool. its not everyday that a VC churns out an equation! i’ll take a look into Drake’s equation and then check yours with more attention!


  • Very interesting, especially when you layer this kind of thinking with ChicagoBooth professor Ron Burt’s research on internal corporate networks.

  • conorop

    Are ‘unique benefit’ and ‘desirability of the network’ separate? With Instagram, I had a truly unique benefit of easily posting great looking photos long before I had or cared about a network. (My timehops showing photos with 1 like remind me of this every day)

    Whereas with Facebook, the desirability to connect with my college friends is what caused me to signup.

    This post hit home because I’m working on a new network now. We’re trying to bake in some unique benefit, so it works well for a single user and only gets better when friends join.

    • williamhertling

      I think that is a powerful effect, when the unique benefit presents itself even before the network is there.

    • “I’m working on a new network now. We’re trying to bake in some unique benefit…”

      You might better serve your market by coming up with a compelling benefit and then building a network around it.

  • small typo for “nB” / “tB” in equation/legend

  • Rupen

    I’ve spent a lot of time slicing and dicing financial data. Looking for formulas that can model trends and prices. I wanted to build trading systems in my last startup. I can show the optimization report (saved from Tradestation) for trading system that on paper produced over 1000 trades, no losses. I would be nervous to try it with real money. It only worked in a particular kind of market. How would you verify that system mathematically? That’s when I started getting into the mathematics of portfolio optimization. I decided Tradestation was not giving me what I needed so I would build my own. That startup failed because I took on too much, ran out of money, should have got investors with my prototype. Instead went back to bootstrapping and eventually decided on a different direction.

    One problem with a formula like this the inherent confirmation bias. You try with a small sample and seems to work. You pick another sample, seems to work. Life is never this clean. When you try with larger amount of data you will see more and more edge cases, more variables for your formula with until it no longer works. The best you will get is a rough approximation. The more variables you have, the more likely it will be wrong. You will only ever fit small portion data. Metcalfe’s law seems like such a rough approximation. I would need to test it before accepting it. This is why non-linear machine learning approaches are popular. They help deal with the noise of life. But often you need a rough approximation (model) to use as input for the machine learning algorithms. Often the output of one machine learning algorithm produces a model for input to another.

    • williamhertling

      I think the real value of positing a formula such as this is in enabling discussions. I agree, I’m sure there’s a big confirmation bias. But hopefully getting more input from more people helps you explore those edge cases. In the end, you learn from the dialogue.

      • Rupen

        I should make a correction. I cannot find the optimization report with 1000 winning trades. I seem to remember that. It was 2009. My memory could be playing tricks on me. Seems suspicious. I did find one with 149 winning trades, 1 loss, nice equity curve and draw down characteristics.

        People are human. There are always a lot of biases at play. Another would be hindsight bias. One of the things i want to do with my current startup is help mitigate the effect of bias.

        This formula is a model of the world. How well does it fit the real world? In this case I would aim for algorithms more than a formulas. Algorithms can and often are based on multiple models of the real world (often formulaic). That’s what I did with my trading systems.

  • Bingo in many ways with one caveat!

    “the value of a social network is proportional to the square of the number of connected friends.” I tend to agree in the case of social networks 1.0, when they are just about friends connecting. But as you stated later:

    “A sufficiently large unique benefit can overcome nearly any disadvantage.”

    And in our way of thinking, social networks 2.0 offer unique benefits or
    some core value beyond your friends and acquaintances – Instagram being
    the perfect example in allowing people to enjoy the photography of
    strangers and do cool things with their own photos in an easy way.

    And I couldn’t agree more that: “Social gaming is interesting, because it provides benefit even when connected to no one, the benefit increases when connected to strangers, and then increases even more when connected to friends.”

    I built Hookist, which is a collaborative songwriting platform where fans, music
    lovers and aspiring artists get to write an original song with a beloved and respected artist. We are really a proof-of-concept-that-won’t-quit at the moment, but our users did not start out as friends. They are mostly strangers from all over the world yet it has become a very tight community, and I have seen many friendships form thru — and because of — the process of creating art together – our unique benefit.

    A line from our pitch deck: “Hookist harnesses the power of a loyal fan base with the reach of social media and the compulsion of social games” to create a very
    sticky environment. Users submit one or two lines at a time based on a
    theme and the guest artist chooses a winning lyric each day or so, and
    together we build a song together, line by line.

    This combination of competition, collaboration and potential kudos from both
    the community and a “famous” artist is a very powerful daily draw. And
    even at this very early stage in our evolution and with limited
    functionality, users spend an average of 9.5-14 minutes per visit and
    pay an average of $26 per song (in our last collaboration – $15 overall)

    So I essentially agree with all your points, but don’t know if friends
    being there is always the key component. Definitely agree about unique
    benefits and the power of social games.

    • williamhertling

      “social networks 2.0 offer unique benefits or some core value beyond your friends and acquaintances”

      I definitely agree with this, and I think it’s absolutely necessary for any new network that arrives.

      Instagram is an interesting example. Compared to other networks, I tend to follow more people I don’t know on Instagram, just because they share pretty pictures. But I also don’t feel that I have much of a connection to them. If their photos stopped showing up in my feed, I might not notice. I suspect that has some interesting implications, but I’m not sure what.

      Hookist sounds interesting, because what’s important in getting value out of that network is not necessary connection with peers as much as it is connection with artists, who have a special, elevated role in your system, and who presumably start out with a relatively large set of followers external to Hookist, which they can then transfer over.

      • Thanks so much, William! I totally agree about Instagram. I, too,
        follow lots of strangers and I love their work, but there is so little
        interaction that I don’t feel a strong connection to them either.

        With Hookist, the initial draw for many is, indeed, their devotion to a particular artist and, for sure, if Tom Petty were to choose my lyric or give me some kind of kudos, it would be far more meaningful to me than the comments and votes of the community – natch! But because users are actually CRAFTING lyrics from their hearts – as corny as that may sound- they are deeply and emotionally invested, so the comments and votes from the community also offer a significant feeling of reward and satisfaction. Similar to the satisfaction you get on FB when people like your posts, only magnified substantially because you’ve put more of yourself into it.. And the fact that everyone’s lines end up melded together in the song and we watch it unfold and take shape together, day after day, there is a strong feeling of connection and a tight community formed very quickly!

        Our hope was to build a vibrant creative community devoted to creating, sharing and discovering new music, and it has been a real thrill seeing people have very meaningful experiences and actually caring about the community. We are competing in the Women Who Tech Start Up Challenge and our users from
        around the world came out to support us, even donating big chunks of money! Really moving, thrilling experience!

        We have a new song starting next week with the amazing Sasha Dobson from Puss N Boots (her band with Norah Jones) if you want to come along for the ride! It’s still very much a work in progress but I’d love any comments or feedback you or
        anyone from this community care to offer! You get free credits when you sign up, but let me know if you run out. Thanks so much for connecting!

    • Meredith, you literally took every single one of my points and it’s almost amusing how fundamentally similar our projects are. Great minds think alike I suppose.

      I’m about to launch a social network that brings people together on the basis of a shared sense of mission and engages them with game dynamics. Our theory, which is quickly proving correct, is that, now that most of the world is on Facebook and connected with “friends”, the value is in socializing with people you don’t know, but taking it a step beyond Twitter by engaging them on the basis of shared values and doing something fun and of mutual interest together.

      We are incentivizing friend invites through Facebook, Twitter and Gmail auth but specifically as a way to grow our user base organically with like-minded people rather than facilitating a familiar environment for our users.

      • Would like to know more / name of the project?

      • Oh funny! Great minds, for sure! All sounds great. Let me know if I can help in any way. We have a strong charitable bent (we try to have a charity involved for every song) and always looking for partners doing cool things! Best of luck with it!

  • Scott27

    Always great thoughts, Brad.

    I think your equation misses an additional factor – that the length of time tT that people try the network will be overlap with other early adopter friends on the network. In other words, there may be another factor which explains why some of the big networks were successful in launching at SXSW where you had not only a large population of early adopters, but that same population was trying the same new networks at the same time.

    • williamhertling

      Great point. Getting that high density of early adopters on at the same time though an event helps demonstrate the network effect earlier than it would otherwise be achieved.

  • What do you think about social networks that connect people by something other than a direct one-to-one connection? For instance, ribl (http://ribl.co) connects users with content created by other users at their current location. Sorry for the shameless plug, but it’s relevant!

    • williamhertling

      I think it’s interesting, and probably runs into the same variation of the problem that other location-based social services like dating sites have: until a certain density of usage in a given location is achieved, then I’m on what amounts to an empty network. So how do you keep early adopters interested in using the network until more people show up?

      • Yeah, the cold start problem, always an issue!

  • The equation can work when you are talking about a social network based on social connections. But many social networks are based on other types of connections. Twitter is based on following people whose news you value; Pinterest is based on following people whose images you value and so on. Facebook’s main advantage is that it is the strongest tool for social connections, however it is currently conflicted between its social role and its role as a news provider. You may be interested in Ben Thompson’s article, which touches on this a little: https://stratechery.com/2015/facebook-and-the-feed/

    • williamhertling

      I would interpret the word “friends” very loosely to mean people of interest, whomever they might be be. I may follow people of interest on Twitter, but if I’m on Twitter before those people of interest, then I can’t follow them, and don’t get that benefit. People usually aren’t fungible: I can’t substitute a a random stranger for someone I’m interested in.

      • To clarify I should have mentioned interest-based social networks, where you interact with people based on their particular interest. In those cases the interest is more important than the social relationship. An example of that is right here: we are interested in Brad Feld, but we are not (yet) interested in each other socially, other than having a chat. Is the network about the people you know, or about what you have in common with other people? I believe a mapping of all the different kinds of relationships people have will be part of the next generation of social networks — it’s something I am working on on my news-based social network.

  • Diego Torres

    A simple formula such as this one is a compact way to present several ideas. However, if the objective is specific predictions on adoption of new social networks or data analysis of existing ones, there are social network theories from academic studies that are better suited to modeling the probabilities, such as the SIS model (“susceptible-infected-susceptible”) that is frequently used in a different context (epidemiology). The models might be quite complicated but the predictions are pretty simple, making them easy to implement, and should fit the data in a much better way.

  • It matters that your peers nN * nEA start to try the network in a synchronised manner – to build an “accessible” Beachhead

    And that they are proximal (ie if this were skype (synchronous use) you need them simultaneously on-line – so possibly same time zone, or somehow co-ordinated but for (linkedIn asynchronous use) this is not so.

    In the synchronous case it matters that they are observable (peer discovery) when accessible (skype does a lousy job of propagating when contacts are online or not)

    These considerations suggest in some networks local (geographic, or field of interest, or common timing) launches should be co-ordinated to achieve a threshold interaction density – think facebook campus roll-out. Noting that the ones that matter are those most marginal.

    It also suggests that despite perhaps one marketing channel being most effective, that at launch-time local multi-channel (some less efficient alone) communication could have network benefits. Ie run co-ordinated awareness campaigns across media in launch areas, but fall back on high converting / $ media in established sites.

  • Rupen

    How do you objectively measure “benefit”, “need” or “desirability”? Probability is some portion of something. Benefit means different things to different people.

  • Rupen

    Does not account for the probability your friends will ever join or that they will join as “your friend” on the network. They may join but for whatever reason not connect with you. Not that good a friend, don’t know you are there, forgot about you, or such a good friend no need to connect online. So many people, so many variations. How do you quantify the noisiness of life? Usually done through simplification. Over a very large set, probability your friend will be on X network is Y. But will vary between networks. Eventually, have to ask. With so many variables and assumptions, what does the number that comes out really mean?

    • Rupen

      Philosophical value is still value.

    • williamhertling

      I think it’s more valuable as a framework for discussion than specifically to try to get a value to come out out the formula.

  • Robert Ellis

    I was late to twitter and I haven’t tried to connect with friends unless they find me. I use twitter for an entirely different reason. I follow outspoken industry leaders and people who have a tendency to tweet pertinent or interesting content. I haven’t found many of them, but those I’ve found keep me pretty well occupied. Honestly, I’ve found twitter to be a great way to stay on top of what’s going on in your profession and/or general business & economic news. Especially breaking news. It’s the best source for this stuff I’ve ever found. Most of my tweets are retweets. Several of my clients are startups or execs with startups, and I regularly email them stuff I got off of twitter that’s new & pertinent to them.

  • I like your equation because its quantitative elements are comprehensive, but there’s a bit of rear-view mirroring in it, if you try to reverse engineer an explanation to what actually happened.

    I think there are emotional and psychological factors that enter this equation and can boost the whole thing. It’s kind of, a key cause for adoption. We tend to justify later with quants, but the spark is the emotion and appealing to our motivations. Without that, nothing happens.

  • Beautiful analogy Brad. Makes me wonder about the equation for a two-sided business…

  • Randall Makin

    Great Post Brad. I am thinking that what you are describing may also be useful as a guide in change management for business adopting new technology.

  • kamera sistemleri

    thank you very much . People are human. There are always a lot of biases at play. Another would be hindsight bias. One of the things i want to do with my current startup is help mitigate the effect of bias. http://www.intechguvenlik.com

  • Max Rehkopf

    As a 22 year old, content is a key missing component in the equation above. For myself and my peers, social media has become much more about content consumption than it is about networking and social connections. I have zero friends on vine, yet I get a ton of value from the platform(and check it daily) because of the awesome content. Two factors ought to be considered: the frequency of posting(twitter: high and instagram: low), and the quality of posts(instagram: high and twitter: low). Platforms will succeed with either high content frequency or high content quality. Otherwise, good work!

    • williamhertling

      I would say my use of reddit is similar. I’m on reddit, it is definitely social, and yet I don’t really maintain any connections there.

      Social networks (like facebook) derive their power from the relationships. Take away my friends from Facebook, and there’s nothing there for me. Take away content, and we’re left with friends chatting, which is just fine.

      Social media derive their power from the content. Take away friends, and the content is still there, as-is ad-hoc social interactions. Take away content, and we’re left with strangers chatting…which can work. Lots of BBSes were like that in the 80s. But it’s probably not ideal.

      I think it’s interesting to think about. Thanks for bringing it up!

  • Kevin Stanley

    With most social networks there’s also the “fan” connect that can extend the time of the grace period while you’re giving the social network a chance. With Twitter, Instagram, and Facebook you have the opportunity to follow celebrities, businesses, and anyone who can inspire.

    While it may not be the focal point of most social networks, it creates an incentive to continue with the platform while your network is still adapting.

  • Scholars Learning

    NCERT Solutions for class 5

    Scholars learning provide online education for CBSE students
    as well as helps in the searching of Educational Institutions and study
    material in CBSE NCERT Solution across India. We provide animated videos, NCERT
    solutions, Study notes, Online Test Practice papers and solved papers for the complete and easy solution understanding
    for their children.


    For further information or help, contact at [email protected]

    Visit our web page at http://www.scholarslearning.com

    Telephone- 0120-4540415/ 6

    Solutions for class 5 e v s

    solutions for class 5 english

    solutions for class 5 mathematics