User consumption behaviour is a subject worthy of study. Because consumers’ consumption behaviours are dynamic and with individual differences, various factors need to be considered when establishing a fractional differential equation model of users’ online consumption behaviour. The two elements, namely advertising and price are more evident in influencing consumer behaviour. Therefore, the paper establishes a product diffusion fractional differential equation model of price and advertising presence or absence to study the impact of these two factors on consumer behaviour. It turns out that ignoring the advertisement and the cost of the product is related to the characteristics of the consumer network. When there are advertising and price factors, product awareness is related to price constraints.

#### Keywords

- complex network
- fractional differential equation
- user network consumption
- product diffusion
- marketing strategy

#### MSC 2010

- 35A24

Partnerships in social networks influence users’ decision-making behaviours. Nowadays, consumers increasingly rely on social networks to buy some products, especially when purchasing some high-tech or experiential products. There are two possible reasons: First, the advancement of modern network communication technology has reduced the difficulty and cost for consumers to obtain product/service-related information. The second is that some products have high prices, complex technical parameters, or limited information that consumers can get from advertisements and other mass media. Still, consumers’ purchase decisions require a large amount of high-quality product/service information.

Most classic product diffusion studies assume that the network between consumers is a regular network of interconnections [1]. However, the actual consumer social network is dynamically changing, and if the negative impact is significant, the product may not be able to spread entirely in the consumer social network. On the one hand, the continuous influx of new consumers with purchase needs in the consumer social system has supplemented the number of potential consumers. On the other hand, in some social networks such as forums or e-commerce sites, positive or negative word-of-mouth products can affect consumers’ purchasing decisions, especially when negative word-of-mouth has a more significant impact than positive word-of-mouth As a result, consumers may lose their desire to buy and withdraw from consumer social interaction – the internet.

This article also considers the dynamics of consumer social networks and the heterogeneity of consumer network structures and studies the proliferation patterns of monopolistic firms’ products. The dynamic nature of consumer social networks mainly refers to the continuous entry or exit of potential consumers and the withdrawal of purchasers from the web [2]. The heterogeneity of consumer social network structure refers to the difference in the number of consumers communicating, the difference in node degree. This article first constructs a product diffusion model without considering advertising or price strategies and analyses the threshold conditions for continuous product diffusion. Then we extend the research question to the situation where the company adopts only the advertising strategy, and to the situation where the company adopts both advertising and pricing strategies. Finally, we use numerical simulation to verify the correctness of the theoretical analysis.

Assuming that the size of the nodes in the network is N, undirected connections between nodes represent the interaction and exchange of information between individuals. Some nodes in the network are occupied by potential consumers and consumers who have purchased products [3]. Each consumer occupies only one node, and the remaining nodes are empty. We assign a state value to each node, represented by 0, 1 and 2. They correspond to three states, respectively, where 0 means an empty node, 1 indicates a potential consumer, and 2 represents a purchaser.

At time t, each new potential consumer entering the network will occupy an empty node. The pointless node changes to a node occupied by potential consumers with probability c, and the node state changes from state 0 to state 1. Some potential consumers lose interest in buying due to negative word-of-mouth or reduced expected utility of the product and exit the consumer social network with probability

This paper uses the node degree distribution of consumer social networks as the key feature. The number of consumers who choose to communicate information indicates the node degree of consumers [5]. The node degree distribution function is

Potential consumers are affected by any of the existing purchasers to purchase the product with probability a. _{k}_{k}_{k}_{k}_{k}_{k}_{k}

When the company does not implement advertising or price strategies, the change in the proportion of potential consumers with a node degree of k per unit time comes from three parts: The first is the new proportion _{k}_{k}_{k}_{k}

Available when the degree of the network is irrelevant

Among them

The classic diffusion theory defines consumers only affected by advertising and other mass media to purchase products as innovators [7]. Assuming that potential consumers buy products with probability b being influenced by advertising, the change is that the number of potential consumers with node degree _{k}_{k}

We assume that the price response function of the product is _{i}_{i}/ω_{0})^{−η}. Where _{H}_{L}_{0} represents the benchmark price of the product

When companies implement skimming pricing, high prices affect the market potential of consumer social networks. The market potential, the number of new potential consumers, and the number of existing potential consumers are all affected by the skimming pricing strategy. Therefore, the market potential of node degree k becomes _{H}_{k}_{k}_{H}_{H}_{k}_{k}_{H}x_{k}_{k}

Two equilibrium solutions
_{k}

Therefore, the necessary and sufficient condition for the existence of a unique positive solution of formula (1) is

_{1}_{1}_{2}_{2}_{Z},y_{Z}^{T} ∈ R^{2Z}|_{k}_{k} ≥_{k}_{k} ≤

Therefore, when
_{0} > 1 is a positive number. Therefore, from Lemma 2, we know that Proposition 2 is established.

Proposition 1 and Proposition 2 show that when potential consumers’ decision to purchase products is only affected by the purchasers. Therefore, the probability of potential consumers’ purchases of products affected by the internal influence of the consumer’s social network must be at least higher than a critical value to ensure the monopoly firm’s Products continue to increase in dynamic consumer social networks [10]. On the other hand, when the adoption probability is lower than the critical value, the monopolistic firm’s products cannot continue to spread, and the spread fails. This critical value is related to dynamic consumer social networks (node degree distribution and network scale). Therefore, the dynamics of consumer social networks (the proportion of new potential consumers, the probability of potential consumers losing their desire to buy, and the non-purchasing probability of publishing information or publishing invalid input) are related.

Assuming _{L}_{L}_{k}_{L} >_{L}_{L}

Proposition 3s shows that when companies implement advertising and penetration pricing strategies simultaneously, products continue to spread in dynamic consumer social networks. However, penetration pricing amplifies the effect of communication between advertising and consumers and makes product diffusion faster.

_{H} < ω_{1}

We substitute _{k}

_{H} >

The degree distribution of the network obeys the power-law distribution of dynamic consumer social networks, which can be obtained from Eq. (12)

When

Proposition 5 shows that when companies implement advertising and skimming pricing strategies simultaneously, they must ensure that the company’s products continue to spread in the dynamic consumer social network where the network’s degree distribution obeys the power-law distribution. There are constraints on product prices [13]. This price constraint has nothing to do with the network characteristics of the dynamic consumer social network. Still, it is only related to the proportion of new potential consumers in the dynamic consumer social network and the probability that the consumers who have purchased the product will not publish information or publish invalid information.

This section first compares and analyses the diffusion patterns of products in dynamic consumer social networks under different corporate strategies. Then it verifies the existence of critical conditions for continuous diffusion or diffusion failure when the enterprise does not adopt any approach. Finally, it demonstrates the impact of changes in the characteristics of dynamic consumer social networks on product proliferation [14]. The number of nodes in the network is N = 200, and the node degree of consumer social networks obeys the power-law distribution ^{−r}_{1} = 3_{2} = 2 has a new potential consumer proportion coefficient c = 0.01, internal influence coefficient a = 0.01, advertising influence coefficient b = 0.005, and the proportion of potential consumers who lose their desire to buy and withdraw from the consumer social system _{L}/ω_{0} = 0.75 is used for penetration pricing, _{H}/ω_{0} = 4/3 is used for skimming pricing, and the price influence coefficient is

First of all, the simulation diagram without considering advertising or price strategy is shown in Figure 2a. Next, a simulation diagram that only considers advertising strategies is shown in Figure 2b. Third, Figure 2c shows a simulation diagram that considers both advertising and penetration pricing strategies. Finally, the simulation diagram of the diffusion model considering both advertising and skimming pricing strategies is shown in Figure 2d.

The proportion of potential consumers in dynamic consumer social networks has continued to decline. Although the balance of the number of buyers has continued to rise, the trend of change gradually slowed down and finally reached a stable state. Unlike the classic Bass model, all potential consumers in the Bass model will eventually adopt the product ultimately. In the end, the proportion of potential consumers is equal to 0.

The comparison between Figure 2a and b shows that the time required for diffusion to reach a steady-state under the latter strategy is shortened. The proportion of the number of purchasers in the constant state is also relatively high. Comparing the slope changes of the curves in Figure 2c and 2d, it can be found that the company only implements the advertising strategy. Implements the advertising and penetration pricing strategy simultaneously, the product diffusion rate per unit time is faster under the latter approach.

This article also considers the dynamics of consumer social networks and the heterogeneity of the structure of consumer social networks. We have constructed a differential equation model for the product diffusion of monopolistic firms. The study found that the threshold conditions for continuous product diffusion without considering advertising or price strategies are related to the network characteristics of consumer social networks and the dynamics of consumer social networks. If innovators are in the network, the product will continue to spread faster. When companies implement advertising and skimming pricing simultaneously, the continued proliferation of products is related to price constraints.

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