Contagion in general life means the communication of disease from one person or organism to another by close contact. This definition can be extended by replacing the disease with a product or an ideology. Competitive Contagion is a type of contagion which deals with conflict and race of multiple firms who want to influence or infect more people than others. There are a lot of situations that can be described in such a way, for example: Two political parties trying to influence the citizens by giving incentives to some influential people in the country and directing them to advertise their ideology, Two mobile phone manufactures competing to advertise their mobile phones of same segment by hiring celebrities or tech reviewers and giving them incentives. So, it is important to simulate such an environment and provide algorithms and properties for optimal seed selection for the competitive contagion process. While doing such competition this work can be used by firms to select the initial spreaders or advertisers by analyzing their network topological properties.

Diffusion on networks is a fundamental process which involves spreading of an ideology (or infection) in a population, e.g., epidemic disease contagion, spread of innovation by word-of-mouth. Considering a network diffusion model, the influence maximization problem consists of finding a set of initial seed nodes so that the expected size of the resulting cascade is maximized. Supposing that there is a limit k on the number of nodes to target (e.g., due to advertising budgets), the goal is to efficiently find an appropriate set of k nodes with which to “seed” a diffusion process. Classical works by Kempe et al. (2003, 2005), on this subject are competitive unaware. They focused on designing models of spreading of a single influence (or idea) and algorithms to find out the optimal seed nodes for maximal adoption of a product of a single firm only. However in real life scenarios, several firms compete in the same market and multiple infections can occur simultaneously in the same network. This has led to the increase in effort toward finding seeds for more realistic settings. Recent works by Bharathi et al. (2007) and Goyal et al. (2014) focused on the modeling the competitive contagion of multiple firms using game theory. They proposed an algorithm to select the seed nodes in a network and discussed the Nash equilibrium when multiple firms compete against each other. Despite the considerable progress made toward finding the seeds in the social network in competitive settings, some very basic questions remain unanswered. Indeed, one approach used to make the influence maximization is to reduce the problem into the ranking of the nodes according to the centrality measures. In other words, the seed nodes are selected by mentioning its ranks using various centrality measure. This raises an important question as to which of the centrality measures should firms use to rank the nodes while selecting them as seed nodes. To answer this question, we compare various centrality methods for finding the rank of nodes. In order to do so, for all the centralities under investigation, we assign a seed strategy based on a centrality to one firm and another to the other one and we compare their spreading efficiency. We also address the question whether a firm should select a small number of highly influential nodes or a larger number of less influential nodes. To answer this question we assign each node a price tag using the best centrality measure found during our analysis and give a fixed and same amount of budget to both firms. Then one firm stakes his funds in buying large number of cheap seed nodes and other in buying small number of expensive seed nodes and then we compare their influence or number of nodes infected when stability is achieved.

The rest of paper is organized as follows: In section 2 basic terminologies and definitions mentioned in the paper are recalled. In section 3, the diffusion model is presented. Section 4 deals with comparing the efficiency of classical centrality measures in the competitive contagion in order to choose the seed nodes. Section 5 compares the strategies of using few highly influential seeds rather than a higher number of less influential seeds with the same budget. Section 6 concludes the paper.

We can classify the contagion processes (Dassios and Zhao, 2011) into three categories based on the dependency of one disease A to another disease B:

We need to calculate the importance of a node in the network to assign its price. Higher rank nodes will be considered costlier in comparison with lower rank nodes. Centrality is a measure for calculating the importance of a node based on its topological properties in the network. There are numerous centrality measures based on various topological properties of nodes which are used in order to assign a score of importance to every node (Gupta et al., 2015, 2016). In this work we restrain our attention to the most influential measures. Their definitions are given below:

There will be multiple eigenvalues λ for which non-zero solution exists. However, (by the Perron, 1907; Frobenius, 1912 theorem) only the greatest eigenvalue results in desired centrality measure.

i.e., the PageRank value for a page u is dependent on the PageRank values for each page v contained in the set B_u (the set containing all pages linking to page u), divided by the number L(v) of links from page v. The algorithm involves a damping factor for the calculation of the pagerank.

We study a competitive process of adoption of multiple products made by multiple firms who use their respective monetary resources for advertisement of their product to the consumers located in a network. Each firm has a fixed budget to advertise their products to the users in a social network. Therefore, each firm needs to optimally choose a set of seed nodes using the assigned budget for maximum adoption of their product. We use the generic game theoretic model (Osborne and Rubinstein, 1994) for the study of competition between firms. In view of game theoretic scenario in competitive market, we propose a diffusion algorithm for the spreading of any information about a product.

The proposed game theoretic model may be represented as:

Multiple firms may try to spread information about their products in the underlying social network. Here, in this work two firms are considered for spreading information about their two products, respectively. As the only action provided to firms is to choose the seed nodes at the beginning, so the result of the entire game depends on the strategy to choose the set of starting spreader (or seed) nodes. Each of the firms (A and B) comes into the open market to advertise their product with limited budget C_A and C_B. They have their node ranking algorithm using which they rank the nodes present in the network and then select the nodes whose price is less than their remaining budget, starting from highest rank (numerically lowest: Rank 1) till the funds remained are not enough to hire any node. The proposed diffusion Algorithm 1 used to simulate the dynamics is an extension of a previous work on simulating epidemic and rumor spreading (Kumar et al. 2018) in which we proposed a simple cascade algorithm for diffusion, discussed various characteristics of epidemic and rumor spreading and the relation among various attributes of epidemic and rumor spreading. This algorithm is a cascade based algorithm in which at each timestamp all the infected nodes try to transmit their disease (or ideology) to their direct connections and the probability that infection will transmit depends upon λ_A, λ_B, c_iA, and c_iB. Figure 1 shows the conversion model of nodes based on Algorithm 1.

λ_A and λ_B are the infection rates of infection A and infection B which are constant. c_iA and c_iB are the competitive measures of node i. It is 1 at time t = 0 for every node and changes when the node is infected by any infection. When ith node is infected by A, c_iA gets multiplied by α_A and if jth node is infected by B, c_jB gets multiplied by α_B. α_A and α_B are the competitive constants, larger the α_X more is the resistance of a node infected by X for another infection.

The population is divided into two compartments: Susceptible and Infected and infected is further divided into two compartments: Infected by A and Infected by B. Let N be the total population, S be the number of Susceptible nodes, X_A and X_B be the number of seed nodes of A and B, I_A and I_B be the number of nodes infected by A and B respectively. At the start of simulation I_A = X_A and I_B = X_B.

The agents present in the network take a fixed amount to advertise or spread the product of the firms and that value is decided in accordance to centrality value of the agents. To find out which centrality measure is more effective for finding the most influential nodes in a competitive contagion scenario, we compare the following five centrality measures: Page Rank, Degree, Betweenness, Closeness, EigenVector. To do so, we consider each method as a node ranking algorithm of a firm trying to advertise its product. Therefore, there are total (52) matches (Match of every centrality against every other centrality. Each firm ranks the top 10 nodes according to their node ranking algorithm and put them in their seed nodes set. As there are cases in which both competing centralities have common nodes in their top 10 list, we assign only unique nodes to each. An example of this distribution is given in Table S2. Table S2 contains the list of seed nodes for competitions of various centrality measures. After making the set of seeds for each match, we run the simulation for dynamics of infection using the Algorithm 1.

A general confusion among the firms is whether to choose small number of highly influential advertisers or large number of less or average influential advertisers. To solve this problem, we simulate a competition between a large group of less (or average) influential nodes and a small group of highly influential nodes both needing nearly same amount of budget.

For ranking the nodes while simulating the competition between group of small number of highly influential nodes and group of large number of less influential nodes we will use the most optimal centrality method found during the simulation discussed in section 4. We select the two sets such that both of them cost nearly same.

To investigate the high-less (highly influential nodes in small numbers) vs. low-more (low influential nodes in large number) competition. We took a set of less influential nodes mostly from different clusters in the low-more set and most influential nodes in the high-less set such that the cost of both is same.

We use three empirical network data-sets to perform the experiments [the wikipedia vote network which is available on: SNAP Stanford¹, the chess interaction network which is available on KONECT²], and the human contact network available on KONECT³. We have used the Wikipedia Vote Network as our primary data-set and others for verification. The results for networks other than chess interaction is added in the Supplementary Material. Details of the basic topological properties of all networks is given below in Table 1.

In this paper, we investigate empirically two linked issue in a competitive contagion setting. First, we simulate a set of competitions between strategies based on ranking according to various centrality measures. The goal is to choose the best centrality method to rank the nodes for initial adoption with a motive to maximize the adoption of the product or ideology. Results show that no centrality is universally best but it depends on the network properties of the network dataset used. The second part deals with solving the general dilemma of whether to choose group of small number of highly influential nodes or a group of large number of less influential nodes. We conclude that it is better to select a small number of highly influential nodes than a higher number of less influential nodes. We can extend this work by taking variable rate of spreading, cooperativity and competitive constant of the diffusion model. Future works could also be done by considering more sophisticated alternative network properties for selecting the seed nodes.

The datasets analyzed for this study can be found http://snap.stanford.edu/data/ego-Facebook.html.

PK and PV discussed the idea with AS. PK, PV, AS, and HC designed the methodology for the experiments. PK did the simulation and data pre-processing. PV did the data representation and image editing. AS and HC guided and helped with bugs and theoretical understanding. All authors wrote the article.