Proceedings of the International Conference on Social Modeling and Simulation, plus Econophysics Colloquium 2014 by Hideki Takayasu Nobuyasu Ito Itsuki Noda & Misako Takayasu

Author:Hideki Takayasu, Nobuyasu Ito, Itsuki Noda & Misako Takayasu
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

16.2 Dynamical Graph Model

In order to investigate the lifetime distributions of mutually interacting systems, we propose a simple dynamically evolving model which was originally introduced for biological community assembly [22]. A system is represented by a weighted and directed network, which self-organizes by successive migrations and eliminations (extinctions) of nodes. Each node i has a state variable called “fitness” f i , which is defined as the sum of the weights of incoming links, i.e., , where a ij is the weight of a link from node j to i. Node i can survive if its fitness is larger than or equal to zero, otherwise it is eliminated from the system.

At each time step, a new node is added to the system. New links between existing nodes and the new node are randomly assigned with probability c, whose weights are randomly drawn from the Gaussian distribution with mean 0 and variance 1. After a migration, the species with minimum fitness is identified and is eliminated from the system if the minimum fitness is negative. Since the extinction of a node affects the fitness of other species, successive extinctions can happen. This process is repeated until all the fitness values in the system become non-negative for each time step. (See Fig. 16.1.)

Fig. 16.1An example of the model dynamics. Nodes and arrows denote species and interactions, respectively. (a) Before the migration of species D, species A, B, and C coexist (b) When species D immigrates into the community, species B goes extinct (c) Then, another resident species A goes extinct due to the extinction of species B. After extinctions of species A and B, all remaining species (C and D) have positive fitness values. The figure is taken from [22]

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