1、Introduction

Social networks have become integral to our daily lives, enabling the rapid spread of information, opinions, and behaviors among individuals. From the propagation of ideas in online communities to the transmission of diseases within close-knit social circles, understanding the dynamics of diffusion in social networks is crucial for various fields, including public health, marketing, and social sciences. One base framework for studying epidemic spread is the Susceptible-Infectious(SI) model, which has been widely adopted and studied in epidemiology.

In this study, we will utilize computational simulations to explore the diffusion process in social networks，we simulate the diffusion process in social networks using the SI model

2、SI Model

The SI model is a simplified mathematical framework used to study the spread of infectious diseases within a population. It focuses on two main compartments: susceptible (S) and infectious (I).

• 1. Susceptible (S): This compartment represents individuals who are susceptible to contracting the disease. These individuals have not yet been infected and have no immunity or resistance to the disease.
• 2. Infectious (I): This compartment represents individuals who are currently infected and capable of transmitting the disease to susceptible individuals. Infected individuals can spread the disease through direct contact, airborne transmission, or other means depending on the specific disease being modeled.

The SI model describes the transition process of an individual from S (susceptible) to I (infected) state. Assuming that in a closed group, there is no birth, death and migration, and assuming that individuals are evenly mixed in the group, that is, the probability of being infected is the same, the parameter $p$ can be used to represent the infection rate.

3、R implementation

Different from traditional diffusion research, network diffusion considers the influence of network structure on the diffusion process. In network diffusion, the interconnections and relationships between individuals play a key role and can influence the spread of information, ideas, or diseases.

The basic algorithm for simulating network diffusion using R language is very simple:

• Generate a network: We create a graph containing nodes and edges, where nodes represent individuals and edges represent connections between individuals. This network can simulate real social network data.
• Select seed nodes: We randomly select one or several nodes as seed nodes (seeds), and these nodes are initially considered to be infected. Seed nodes are the starting point for diffusion, they will start spreading information or disease.
• Propagation process: Each infected person affects the nodes connected to it with probability p. This means that there is a certain probability of transmission between the infected person and its neighbor nodes. The higher the probability, the faster the transmission speed.
• The diffusion process is continuous: by repeating the spreading steps, the infected person will continue to spread the virus, information or ideas to its neighbor nodes. This process can continue until no new nodes are infected or a predetermined stopping condition is reached.

3.1、Example of CDF

Take 100 nodes and a propagation probability of 0.4 as an example，the initial structure of the network is as follows:

Randomly select a node as the propagation source, as follows: