#### Presentation Title

#### Advisor Information

Dora Matache

#### Location

Criss Library Room 249

#### Presentation Type

Oral Presentation

#### Start Date

1-3-2019 2:15 PM

#### End Date

1-3-2019 3:15 PM

#### Abstract

Boolean Networks are being used to analyze models in biology, economics, social sciences, and many other areas. These models simplify the reality by assuming that each element in the network can take on only two possible values, such as ON and OFF. The node evolution is governed by its interaction with other nodes in its neighborhood, which is described mathematically by a Boolean function or rule. For simplicity reasons, many models assume that all nodes follow the same Boolean rule. However, real networks often use more than one Boolean rule and therefore are heterogeneous networks. Heterogeneous networks have not yet been studied in-depth, only specific cases have been looked at. This research looks at sixty-four generalized totalistic rules and analyzes the behavior of heterogeneous Boolean networks in which a certain percentage of the nodes in the network update based on one totalistic rule and the rest of the nodes updated based on a second totalistic rule. Totalistic rules consider only how many nodes in a neighborhood are ON or OFF and not the placement of those nodes within the neighborhood. Bifurcation diagrams are used to visualize the behavior of these networks using mathematical tools such as mean-field approximations. The findings show that it can be predicted which of these networks will have a stable long-term evolution and which will not. The findings of this analysis provide insights into the dynamics of these networks, letting us predict the likely behavior of some real networks that can be modeled this way.

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Heterogeneous Boolean Networks with Two Totalistic Rules

Criss Library Room 249

Boolean Networks are being used to analyze models in biology, economics, social sciences, and many other areas. These models simplify the reality by assuming that each element in the network can take on only two possible values, such as ON and OFF. The node evolution is governed by its interaction with other nodes in its neighborhood, which is described mathematically by a Boolean function or rule. For simplicity reasons, many models assume that all nodes follow the same Boolean rule. However, real networks often use more than one Boolean rule and therefore are heterogeneous networks. Heterogeneous networks have not yet been studied in-depth, only specific cases have been looked at. This research looks at sixty-four generalized totalistic rules and analyzes the behavior of heterogeneous Boolean networks in which a certain percentage of the nodes in the network update based on one totalistic rule and the rest of the nodes updated based on a second totalistic rule. Totalistic rules consider only how many nodes in a neighborhood are ON or OFF and not the placement of those nodes within the neighborhood. Bifurcation diagrams are used to visualize the behavior of these networks using mathematical tools such as mean-field approximations. The findings show that it can be predicted which of these networks will have a stable long-term evolution and which will not. The findings of this analysis provide insights into the dynamics of these networks, letting us predict the likely behavior of some real networks that can be modeled this way.