This is a natural generalization of the previous work by Dan, "Modeling and Simulation of Diffusion Phenomena on Social Networks," to appear in The proceedings of 2011 Third International Conference on Computer Modeling and Simulation. In this paper, we consider the diffusion phenomena of personal or secret information on the variety of networks, such as complete, random, stochastic and scale-free networks.
The development of information and network technology enables us to share information at anywhere and anytime. The social networks on the Internet, for example BBS, SNS, Twitter etc. in the Web, have become the personal network media of information. They are quite useful in information sharing each other.
On the other hand, we faced the difficulties in the confidentiality of sensitive content, which have the risk of diffusion of personal or secret information over the Internet. Once the personal and secret information diffuses on the Internet, people on the Internet can know the information anywhere and anytime even in future.
This paper presents mathematical models for diffusion phenomena of personal or secret information on the Internet in particular. We investigate the behavior of the models using analytical and computational methods with numerical Monte Carlo simulation. We also consider the structure and dynamics of diffusion on networks constructed by a large number of people with interaction or communication each other.
As is well known that social networks have grown rapidly on the Internet. The community on the Internet is in general visible from access logs in servers, rather than that in the real world, so that we can easily analyze the structure and dynamics of the social networks. Not only the element but also the link of a social network determines the behavior of social systems.
A network [8] is a set of points (also called vertices or nodes) connected by lines (also called edges or links). We may call complex networks, if the number of points and links is so large that only computational calculation can analyze them. Any system with coupled elements can be represented as a network, so that our world is full of networks [5]. This is a natural generalization of the previous work by Dan [3]. In this paper, we consider the diffusion phenomena of personal or secret information on the variety of networks, such as complete, random, stochastic and scale-free networks.
In this section, we provide each definition of the corresponding models of networks. The dynamics of diffusion or percolation depends on the structure of networks. We see the property of networks under the definition, and consider the characteristics of each network.
A complete network is the network all of whose two vertices have an edge. There is no pair that does not have edge in the network. When the number of vertices is n, the network has n(n -1)/2 edges.
As the previous work, Dan [3] investigaed the mathematical modeling and computer simulation of diffusion phenomena on social networks for complete networks.
A random network is the network whose vertices have edges at random. Randomness is assumed for not only uniform distribution, but also any possible function of distribution. In this paper, we assume uniform distribution of randomness.
A stochastic network is the network whose each edge has probability of the value between zero and one. Each edge mediates the information at the probability that depends on the edge. One can communicate on the edge at the probability p, on the other hand, one cannot communicate on the edge at the probability 1 -p. The possibility of communication depends on the probability p defined each on the edge.
A scale-free network is defined the power law for the number of edges. There are some vertices, which are called hubs, that have comperable large number of edges. On the other hand, almost all vertices have only a few edges. The graph of the number of edges indicates the law of power. Scale-free networks are first proposed as small-world networks by Watts and Strogatz [13].
It is known that scale-free networks have high cluster coefficients like regular lattices. However, these networks have small characteristic path lengths like random networks.
Table 1: Link matrix for random networks 1 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1
Let us begin with the setting of constructing the structure of the networks in the simulation.
As random networks, we provide edges between two vertices on the network at uniform probability of 1/2. We expect that there are n/2 edges at random.
Figure 1 indicates a link matrix for the random networks we constructed. The ( i, j ) element of the matrix is 1 if the vertex i and j are connected, 0 if not. Therefore, this matrix is symmetric.
As stochastic networks, we provide the probabilities to all of edges on the network. The value of the probabilities take zero to one uniformly. We expect that any edge has probability of 1/2 in average.
Figure 2 indicate a probability matrix for stochastic networks. This matrix is also symmetric. All of edges in the network has vakues between 0 and 1, although the value of diagonal elements have no sense.
Despite two matrices have different elements, the averages of all elements are expected same as 1/2. That is,
where r ij are the elements of the link matrix of random networks and p ij are the elements of the probability matrix
This content is AI-processed based on open access ArXiv data.
