Long Memory Properties and Complex Systems
are weighted in order to assume one or another state are :
• The individual past belief state , given by S ( t – 1 )
• Overall average of the individuals past belief state ( condensed into “ prices ” between 0 and 1 — continuous variable ), given by P ( t – 1 )
• External Information , represented by a Bernoulli Random Variable , given by B ( t ), with a probability of q obtaining a 1 and ( 1 – q ) of obtaining a 0
The overall average of individuals beliefs are condensed into collective belief states ( which resemble a price ) according to the following mathematical expression :
Furthermore , a mathematical function updates the belief state according to the following expression : where
and
Moreover , z is the individual bias , generated randomly for each agent .
In the implementation of this code , for the purpose of this paper , in order to generate heterogeneity between the individuals , it was imposed that z ~ N ( 0.5,0.1 ), fixed at the first simulation step .
For simplicity , in this paper it was adopted a probability q = 0.5 , w 1
,
w 2
= 0.3 , and w 3
= 0.4 , in order to avoid any apparent bias in the generated time series and any very strong autocorrelation over individual past states .
Thus , basically , this set of rules represents a simple Boolean Network , where all agents are interconnected ( which simulates a situation of synchronous information and perfect information symmetry ), simplified by the existence of an external “ Market Maker ” agent , which condensates all agents ( nodes ) beliefs into something that resembles a price . On the other hand , the state of each agent does not depend on any spatial position , since they are all virtually connected and information flows instantaneously , resembling individuals interconnected by a mechanism such as Internet .
Mathematically speaking , it turns out that this network configuration implies on a linear feedback relationship between the agents behavior and their respective collective behavior , which can amplify the system oscillations or stabilize them , depending on the parameter value ( w 3
).
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