In viral marketing advertisements (usually entertaining video clips, images, text messages, apps, etc.)

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1 On modeling and optimization of viral marketing revenues Jiří Mazurek (Silesian University in Opava, the Czech Republic) In viral marketing advertisements (usually entertaining video clips, images, text messages, apps, etc.) are spread through social networks by network users themselves in the process resembling spreading of a virus, hence the name viral. This type of marketing was enabled by a dynamic development of information and communication technologies and Internet in the last two decades. By viral marketing both specific target groups of consumers or broad audience of millions of Internet users can be addressed. Today, large international companies such as Volvo, Burger King, Old Spice or Anheuser- Busch take advantage of this new advertising tool. The article is divided into three parts. In the first part a novel mathematical model for viral marketing is presented. The model consists of a messenger, a number of messages sent to a set of target groups, time period, target groups forming a network, a number of retransmitted messages among target groups, a utility function expressing a total revenue of viral marketing at a given time, etc. In the second part several optimization problems within the model framework are formulated, and the last part of the paper deals with problem s solution. Presented optimization model of viral marketing might contribute to a successfull advertising campaign as well as it enables simulating marketing s outcomes under different conditions. The use of the model is illustrated by examples. Key words: mathematical modeling, model, optimization, viral marketing. Introduction In viral marketing advertisements (usually entertaining video clips, images, text messages, apps, etc.) are spread through social networks by network users themselves in the process resembling spreading of a virus, hence the name viral. For an introduction to viral marketing see e.g. the following recent publications: Penenberg (2009), Good et al (2010), Adams (2013), Berger (2013) or Scott (2013). This type of marketing was enabled by a dynamic development of information and communication technologies and Internet in the last two decades. By viral marketing both specific target groups of consumers or broad audience of millions of Internet users can be addressed. Recently and today, large international companies such as Hotmail, Volvo, Burger King, Old Spice or Anheuser-Busch take advantage of this new advertising tool. Usually, the aim of viral marketing is to address as many recipients of some network as possible with the lowest costs, as a production of each advertisement is connected with some (considerable) expenses. Generally, models of viral marketing consist of elements such as messenger, message, cost of a message, network of target groups with their importance (expressed by weights), expected revenue, etc. Target groups might represent groups, individuals, enterprises, etc. The article is divided into three parts. In the first part of the paper a novel mathematical model for viral marketing is presented. The model consists of a messenger, a number of messages sent to a set of target groups, time period, target groups forming a network, a number of retransmitted messages among target groups, a utility function expressing a total revenue of viral marketing at a given time, etc. In the second part several optimization problems within the model framework are formulated, and the last part of the paper deals with problem s solution. Presented optimization model of viral marketing might contribute to a successfull advertising campaign as well as it enables simulating marketing s outcomes under different conditions. The use of the model is illustrated by examples. 1. Motivation example Before we get into general formulation of the problem of viral marketing optimization, we will start with a simple illustrative example. Consider the following (social) network R of three target groups

2 with retransmission coefficients r 1 a group i to a group by a period of time τ:, where r expresses the rate of messages retransmitted by R For example r means that a group 1 sends to a group 3 exactly 0.6 (60 %) of received messages on average during a given period of time (for example a day or a week). Furthermore, consider a period of time given by 2 time units (two days, two weeks, etc.), and the initial vector v0 (1,0). This vector expresses that in the beginning (at zero time), a messenger sends one message to a group 1, an no message to groups 2 and 3. But because all groups are connected and sent to each other a fraction of received messages given by coefficients r, the message reaches all groups eventually. The number of received messages by all groups in time τ = 1 (v 1 ) is computed as the product of the (row) initial vector v 0 and the matrix R: v1 v0 R 1, , 0.3, As can be seen, a part of a message was reached by groups 2 and 3. Now we repeat the process of multiplication to obtain the vector v 2 for time τ = 2: v2 v1 R 0.2, 0.3, , 0.27, Though the process resembles Markov chains, it is a different process, because the matrix R is not stochastic in general. After two time units we obtain the overall impact on all three target groups: 2 v v v v (1.51, 0.57, 0.90) i0 i Now, let the worth (weight) of a message addressed to a target group be w, 3 1 w 1, for example let w (0.2, 0.3, 0.5). The weight reflects importance of a target group. One group can be considered more important, when it consists of more individuals, or these individuals have higher purchasing power, or ust these individuals are considered more suitable for an advertising campaign by a messenger. Then the utility function U (which can be considered the equivalent of the total revenue) of viral marketing is given as a dot product of the overall impact vector and the weight vector: 3 2 U vi w (1.51, 0.57, 0.90) 0.2, 0.3, i0 =0.902 The numerical value of U is not direct pay-off in euro or some other currency, it enables a comparison with other feasible configurations of an initial vector, matrix R, time period, etc. In our example one might ask if it is not better to send the message to a group no. 3, as this group has the largest weight. So now consider the same network R and the same time interval, but a different initial vector : v0 ( 1). By the same procedure we obtain:

3 v1 v0 R , 0.3, 0.1, v2 v1 R 0.2, 0.3, , 0.12, vi v0 v1 v2 (0.41, 0.42,1.35), i0 3 2 and finally U vi w (0.41, 0.42,1.35) 0.2, 0.3, i0 Surprisingly, the utility function is smaller than in the first case! The reason is that although the group no. 3 is the most important, it has low coefficients r, so it retransmits a smaller amount of received messages, which in turn causes the smaller value of the utility function in overall. For completeness, if the vector v0 (1, 0) ceteris paribus, then U = So if a messenger has only one message and can send it to only one group, than the best choice is to sent it to a group no. 2. Of course, one might ask what does mean that for example 0.6 of a message was delivered. One possible answer is that this value can be understand as a probability (60 %) that a message was delivered, so there is some uncertainty in the setting, which is quite a common situation in decision making. On the other hand, it is possible to define the floor function of a utility function U, and in this case only 100 % delivered messages matter. In the following sections both approaches are to be followed parallely. Now, from this motivation example we will turn to general cases in the following Section., 2. The model of viral marketing 2.1. Notation and variables In this section notation and variables of the model are provided as well as some explanations concerning the model: n...the number of target groups, g g,..., 1 gn...the vector of target groups g i, τ...time period (one day, week, etc.), r 1... the rate of messages retransmitted by a group i to a group by a period of time τ, R... n n retransmission matrix with elements 1 r, v ( v,..., v )...the initial (row) vector, v 0i is the number of messages sent to a target group i by n an messenger at time τ = v ( v,..., v )... the state (row) vector, v 1 n i is the number of messages retransmitted from all other target groups at time τ =, w...the vector of message worth (weights) associated with a vector of target groups g, w 0, n w 1, 1 U ( r, w, )...the utility function of viral marketing revenue,

4 U ( r, w, )...a floor utility function of viral marketing revenue. The state vectors are computed from preceding vectors with a recurrent formula (1): v v R (1) k k1 The utility function U ( r, w, ) is defined as follows: n U R, w, 0, i w 1 i0 The floor utility function U ( r, w, ) is defined as follows: n U R, w, 0, i w 1 i0 - Model s input variables include: n, g,, r, w. - Variables, k, and U/U can be both input or output variables depending on the formulation of 0 a problem, see section 3. - The model iteratively computes state vectors v k from a preceding state vector by a formula (1). After all iterations are done, the value of a utility function is established by relations (2) or (3). - The floor utility function U is introduced because one may ask what does for example vk (0.88,1.65, 0.57) mean: how many messages are received, when the values of v k are not integers? Non-integer values can be understood as probabilities (value 1.65 means that 1 message was surely received and the second message was received with 65% probability by the second target group). But if one takes into account only messages that were surely received, then he/she can use the floor utility function. - The utility function U ( r, w, ) is assumed to be directly proportional to the viral marketing revenue, so the higher is U ( r, w, ) / U ( r, w, ), the higher is the revenue. By the value of U ( r, w, ) / U ( r, w, ) all feasible initial states (v 0 vectors) can be compared and ordered from the best to the worst. - Utility function (2) is linear in v, as U v U U and U kv ku the problem belongs to (multiobective) linear modeling, and techniques such as simplex method can be used for the solution. - Square retransmission matrix R is neither stochastic nor symmetric generally. But if it is stochastic, then state vectors are Markov chains. If some r 0, then it can be interpreted as there is no communication from a group i to a group, rii 0 Also, it is assumed that elements r are constant in time. To show how the model works consider the following example. Example 1. Consider three target groups (n = 3), five messages (k = 5), initial vector v0 (2,1, 2) 1, w (0.5,0.3, 0.2) and the matrix Solution: According to (1): v1 v0 R (0.8, 0.4,1). From (2) we obtain the utility function: i R Find the utility value (2) U ( R, w, v ) v i w (2.8,1.4,3) (0.5,0.3,0.2) (2) (3), so

5 One can easily verify that if all 5 messages were send to the group 1, so v0 (5,0), then U would be higher (U = 3.05), so it makes good sense to find the initial vector maximizing the utility function under given conditions. 2.2 Formulation of selected optimization problems In the previous section 2 it was demonstrated that the utility function U depends on the initial vector v 0 ceteris paribus. In this section this and other viral marketing optimization problems are formulated: Problem no. 1: Find the initial vector v 0 so that the (floor) utility function is maximal for a given number of k messages (for a given cost) and given time τ. On the other hand, when a messenger wants to spread his advertisement among a given number of recipients (for example 10 million Internet users in 100 groups), he may ask how many messages have to be sent to a given social network (in other words how many messages have to be produced at a given cost). This results in formulation of the second problem of viral marketing: Problem no. 2: Find the minimal initial vector v 0 so that the (floor) utility function attains (exceeds) a given value K at given time τ. A minimal initial vector is a vector with a minimal length given by l 1 metric. Problem no. 3: Find the minimal initial vector v 0 so that all target groups are impacted at least by L messages at given time τ. Problem no. 4: Find the minimal initial vector v 0 so that all target groups are impacted at least by L messages at given time τ, while the (floor) utility function attains (exceeds) a given value K at given time τ. In problem 3 it is required to deliver at least a given number L of messages to all groups, so all groups are impacted by a viral marketing to some (desired) degree. As viral marketing might be multiobective, in problem 4 both conditions from Problems 2 and 3 are combined into one problem. Of course, also other interesting problems can be formulated within the model, which might be important for a messenger. As for problems solutions, the easiest is Problem 1. Due to the linear nature of the model it suffices to send one message to each target group, evaluate the utility function in all cases, and then select a target group with the highest value of the utility function (the most important target group). In the next step all messages are sent to this target group until the desired value of the utility function is achieved. Problem 2 can be solved analogically. A solution to Problem 3 and 4 (and other feasible more complex problems) can be searched among all possible redistributions of messages to all target groups in general. We start with one message, distribute it to each and every target group successively and evaluate the utility function. Then the process is repeated with two messages redistributed over all target groups, etc., until conditions of problem s formulations are satisfied with a minimal number of M messages. From combinatorics it is known that the number of possible redistributions of k obects (messages) into n boxes (target groups) is given as: n k 1 C( k, n) n 1 The formula (4) is a well-known formula for combinations with repetition. (4) Now, with the use of formula (4) we obtain the number of cases (NoC) which must be searched through generally in Problems 3 and 4:

6 M n k 1 NoC k1 n 1 (5) In (5) k is the number of messages sent by an messenger, M is the lowest number of messages which satisfy problem s formulations and n is the number of target groups. If there is no heuristic or simplification, then finding of the solution of the problem requires to compute the utility function of all cases given by (5), which might be a difficult task without some software tool. For these purposes free software tool called VIRAMARK is currently developed, and can be downloaded from the following address: 3. The illustrative example In this Section the solution of Problem 1 is illustrated. Consider 4 target groups with the network given by the matrix R (see below), 2 w 0.4, 0.2, 0.1, 0.3 and k = 8. The goal is to maximize, U ( R, w, v0, ). Hence, a messenger wants to maximize the utility function for 8 messages under given circumstances R In the first step we evaluate the utility function with the use of relations (1) and (2) for 1 message sent to each group separately: v0 1, 0 U ( R, w, v0, ) 0.706, v0 1,0 U ( R, w, v0, ) 0.523, v0 1,0 U ( R, w, v0, ) 0.469, v0 1 U ( R, w, v0, ) And then we choose the group with the largest value of the utility function, which is group no. 1. Hence, all 8 messages should be sent to this group so the impact of the viral marketing would be maximal. Conclusion The aim of the paper was to present a new simple mathematical model for modeling viral marketing revenues, which might be useful in preparing or assessing an advertisement campaign in all fields of economics. The model can be considered the first step towards more realistic and complex models of viral marketing followed in near future. To facilitate computations the MS Excel add-in VIRAMARK is soon to be developed, which is free and can be downloaded from Acknowledgment The work was supported by the Grant Agency of the Czech Republic no

7 Bibliography Adams, R. L. (2013). Viral: How to Spread your Ideas like a Virus. CreateSpace Independent Publishing Platform. Berger, J. (2013). Contagious: Why Things Catch On. Simon & Schuster. Good, R., Bazzano, D. and Lombardi, E. (2010). What Is Viral Marketing: Key Principles And Strategies. (accessed on August 15, 2013). Penenberg, A. L. (2009). Viral Loop: From Facebook to Twitter, How Today's Smartest Businesses Grow Themselves. Hyperion. Scott, D. M. (2013). The New Rules of Marketing & PR: How to Use Social Media, Online Video, Mobile Applications, Blogs, News Releases, and Viral Marketing to Reach Buyers Directly. Wiley.