Product marketing and competition between different companies are common in the commercial world. At the information age today, competition of market shares for one company largely depends on that of its product information spreading. And that is why we can see various kinds of advertisement everywhere in the world. Generally speaking, product information spreading is roughly divided into commercial advertising and public opinion spreading. The public opinion for one product are the experience and comments told by the consumers who used the product and are spread from one person to another. As the spreading behaviors are not controlled by anyone and non-commercial, these comments look much closer to the truth and more convincible for consumers, compared with the other ways of promotion. That is the reason why it will take an important role in product competition campaign today.
Public opinion spreading works in social networks. Recently, the researches concerning complex networks and its development provide us with new methods and ideas to understand and study social networks and also various types of spreading behaviors in these networks. The earliest study of complex networks is traced back to the research of mathematical graph theory, which is one of the bases of discrete mathematics. The fact that Euler’s celebrated 1735 solution of the Königsberg bridge problem is often cited as the first true proof in the theory of networks. In 1960s, two Hungarian mathematicians Erdös and Rényi [
When competing information of two products flow freely, different cognition about two products from consumers will interact with each other. Consumers’ knowledge and attitude to a product influences that on their acquaintances’ attitude to products via mouth that of their acquaintances by words of mouth, and then influence the market shares of these two products. In this paper, consumers are divided into three types in the assumption that there are two competing products A and B in market. Type A are supporters for product A; type B are supporters for product B and type C are neutrals. Neutrals have no knowledge of any products, so they have no preference first. Their attitude is likely to be influenced by other consumers’ information but doesn’t affect the others’. Type A can be classified into firm proponents and ordinary supporters. Firm proponents can be regarded as loyal customers of product A, who are good spreaders of product A and aren’t affected by other customers. Ordinary supporters of two products can affect other people and be affected by others. Denoting the subclass of firm proponents of type A as type A and that of type B as type B, we build the spreading model based on competing public opinion information of two products on a BA network, and simulate its spreading pattern through a computer simulation as follows.
We use variable 
to express attitude of node i (ith consumer) towards two products at time t,
. The closer the value of 
approaches 1, the stronger the consumer support product A. On the contrary, the closer the value of 
approaches −1, the stronger the consumer support product B. Time t takes discrete value and its step size is 1.
Firstly, we build a BA network as follows:
1) Growth: Starting with a small number (m0) of nodes, at every timestep we add a new node with 
edges that link the new node to m different nodes already present in the system.
2) Preferential attachment: When choosing the nodes to which the new node connects, we assume that the probability Π that a new node will be connected to node i depends on the degree 
of node i, such that
Secondly, we investigate the evolution rules of public opinion on BA networks:
1) Initialization of node state: Every node is assigned to a value which comes from 
randomly, and take the value as the corresponding node’s initial state. Namely, initial state of node i is denoted by
, which represents node i’s first cognition about two products.
2) Classification rules of nodes: we classify nodes according to their state and the rules are showed as follows.
Type A set
; they are proponents for product A.
Type B set
; they are proponents for product B.
Type C set
; they are neutrals and maybe effected but can’t affect other nodes.
Type A set
; it’s a subset of
, and nodes in it are firm supporters of product A. They can affect other nodes but no one can affect them.
Type B set
; it’s a subset of
, and nodes in it are firm supporters of product B. They can affect other nodes but no one can affect them.
3) Evolution mechanism: every time step, a node is selected randomly. The node will be affected by its neighbor nodes and its new state is produced. That means a customer have a new understanding of products because he or she is affected by their neighbor’s information. For example, at the eleventh time step, node No. 3 is selected, and its state changes for its neighbor node No. 8 affects it. Then its next state can be expressed by a state transition equation:
where 
shows state of node i in time t; 
shows affect strength of node No. 8. If 
is bigger, that means the state of No. 3 closer to 
after affected; if 
is smaller, No. 3 will be closer to its previous state after affected. In real life, the influence of type A and B nodes to other nodes may depends on how strong their promotion is. In the model, 
and 
are taken as IF
(impact factor) of each node in type A and type B respectively. Then we give the state transition equation showing how node i is affected by its neighbor node j (1):
. The number of evolution times is 50000, and sampling interval is 500 when drawing.
and
. The vertical coordinate shows proportion of three types by total numbers of people. We see from 
, and value of 
changes from 0.6 to 0.9. We calculate the difference of the proportion of type A by total numbers of individuals and the proportion of type B: 
and graph it in 
in 
is positive in 
from 0.5 to 0.9 at interval of 0.05, and the same to
. And there are 81 ways of combination consisting of different 
and
, and we do 81 experiments, starting from the largest to the smallest of the value
. Then 
: absolute square of
. 
in experiments with smaller serial numbers (
is bigger) are growing faster, namely the competition costs shorter time, while the values of 
in experiments with bigger serial number (
is smaller) are growing slower, namely the competition costs longer time. This experiment indicates that the bigger difference between two IF is, the faster determine a winner is determined.