The Tangled Web We Weave

First a request for help.  I can’t stop myself, somebody please take my Mathematica license away.  Every time I hit the “Evaluate” button on this virality model, Mathematica disappears into shock for several minutes.  Most of the time I still get a result, but it is melting down on me with increasing frequency.  I’m afraid that one day I will get a call from Stephen Wolfram telling me to stop abusing his program.  (Naturally I will ask for another autograph)

Second an apology.  I have no idea if this new set of equations is meaningful in any way, I just had to try this to satisfy an intellectual itch.  Unfortunately the abstract virality properties added to this model make it very difficult to test unless you are Facebook.  My first EUREKA model was parameterized in a way that was intended to make it testable with real analytic data.  I believe that it is a very “practical” model.  The “problem” with it is that it doesn’t really attempt to model the “interior virality” of an application as it affects player behavior.  In other words I didn’t try to model how players interacting with one another from within the various EUREKA compartments altered their collective play behavior.  I just jacked in parameters that correspond to real analytic hooks that most games have access to and trust that the results of the games interior viral properties are implicitly captured within those parameters.  The property of “Virality” in media is fragile and changes rapidly, the damping nature of the logistic functions mediates the tendency of small differences in virality to “explode” or “collapse”.  In theory even if the basic EUREKA model is “wrong” it’s not likely to be “very wrong” because  in most real-world situations the model will tend to self-correct… or so I hope…

The itch, I just had to scratch, however was this;  The hardcoded rate parameters I built into the model, although practical for real analytics use, don’t try to model the fact that those parameters are actually emergent properties of the content itself… AND in the case of a highly viral social game, interior viral dynamics may be the dominant property of the game determining the rate of audience flow between compartments.  For example I could model average life span of an active user of a game in several ways;

1)       Just plug in the measured average life span as a parameter

2)      Model retention as a function of the rate of users flowing in to a compartment minus the users flowing out over time, as the first EUREKA model does

3)      Track every players interaction with every other player and model average life span as a function of how socially occupied they are over time… as the new EUREKA model supports

Theoretically all of the hard coded rate parameters in the first EUREKA model should be emergent properties of the content combined with the social interactions between players.  In practice tracking and correlating this kind of data is a major PITA.   Academically speaking I wanted to try to build a model that could, in theory, reproduce the same results from a graph analysis approach.  At the very least, having such a model might help me validate the first EUREKA model or at least determine the situations in which it may fail.

I have produced a new EUREKA model that is a complete superset of its predecessor which can model virality using either virality paradigm or mix and match them fluidly.  If you want to ignore the new internal virality parameters  (V1… V7), just set them all to ZERO and the model will be identical to my earlier SVD3 posting.   I presently have no idea if this is a meaningful achievement beyond being a pretty remarkably convoluted piece of mathematics.

That said… it does appear to have some very interesting properties.

The evidence that I’m not hopelessly lost in the weeds with this approach is as follows;

1)      The equations all appear to balance consistently in all tested permutations.  Since there are six equations with 37 parameters., this is fairly reassuring evidence that any possible “errors” are the result of false assumptions not incorrect math.

2)      By setting the rate equations of the first EUREKA model to ZERO and playing with just the new internal virality (V1… V7) parameters it appears that I can reproduce the same domain of results using this entirely different emergent virality paradigm.  *If I really felt motivated I would try to prove that the two different sets of equations can produce the same results.

3)      When I mix and match both types of parameters together the whole things stays balanced and does not do anything apparently inexplicable.  This may also be important if the constant rate parameters represent the intrinsic consumption behavior of the content itself independent of any viral social dynamics the game may have.

4)      I built the equations by laying out a state diagram of all of all the possible interactions between compartments that I needed to model and then converted that to a matrix that I filled out.   The equations all balanced on my first implementation with no “hackery”,  so EUREKA is a provably complete state machine.

I left the equations in their modular long form to make it easy for a knowable third party to analyze them for flaws.  I also included a simplified diagram of the state machine to help folks understand how it all fits together.  If you are such a knowledgeable party, please have at it, I’d appreciate some independent scrutiny.  Regrettably I no longer run a large social network, so I can’t access real-world data to test these models on.  The ideas for how to build these models were inspired by my experience analyzing Hi5.com’s 60 million user social network using a 64 server Hadoop cluster.

Links to a graphic presentation on the structure of the model and supporting equations

Now that my excuses and apologies are out of the way, let’s take a look at the new parameters added to the EUREKA model.  The basic idea is to model every combination of viral interactions that could occur between players in every compartment of the model except the “Immune” compartment.  For each combination of viral interactions that can happen between audiences in each compartment there is a new parameter that determines how strongly those interactions influence the flow of audience from one compartment to another.  For example if an Engaged Player interacts virally with an Exposed new player, there is a new V paramater to capture the increased probability that this Exposed Player will become Engaged themselves as opposed to how they may have behaved during the trial if they had encountered no friends already playing the game.

Link to live Mathematica EUREKA model

The EUREKA model “assumes” that social interactions always aggregately improve audience attraction and retention.  (Not a hardcoded assumption, just the way I chose to configure the default model) Thus in the new version of the model the basic traffic flow parameters are still present but they have a new meaning when used in combination with the internal virality parameters (V1… V7).  The existing parameters governing the flow of traffic between compartments represent the natural non-social properties of the content itself, in other worlds, how people play the game alone.  When the internal virality parameters are engaged they ADD additional engagement, retention and re-engagement to the games core non-social content.  In other words a highly social game becomes like a balloon whose volume is proportional to the HEAT or pressure of the gas molecules crashing around inside the balloon.  If the molecules slow down, the balloon deflates.  Contrast this to a classic content based single player console game which is more like a bowling ball composed entirely of high production value content that has a very dense core game behavior as a result and a known limited potential lifespan because once the content is consumed the game is over.

As a result of this work I believe that the EUREKA model is now a very promising model for describing virality for a broad spectrum of games from highly social Facebook games to highly content dense console games.

EUREKA! 6 Differential Equations, 4 Viral Compartments, 37 Parameters and every bulb lights up!

*Simple “proof” that the parameterized EUREKA model and viraly parametrized EUREKA model describe equivalent viral dynamics.  Both models describe the complete flow of traffic from all of the interacting compartments.  One model describes the flow with linear rates, the other with exponential viral functions.  Since both models describe 0%-100% traffic flow between compartments the linear model will include all of the rates that the exponential model produces as a non-linear distribution of linear parameter settings.  This means that IF internal virality is driving the games play behavior the linear model will do a “poor” job of describing the traffic behavior… BUT if the linear model is driven by live data the non-linear distribution of viral properties of content will be captured within the parameters and the linear model will produce a “reasonable” first order approximation of the games future behavior.  If the game is highly viral then this model may be a “poor” approximation of the games real-world behavior and engaging the new viral (V1… V7) parameters may produce a better result.

Comments

comments