Using synchronous Boolean networks to model several phenomena of collective behavior

In this paper, we propose an approach for modeling and analysis of a number of phenomena of collective behavior. By collectives we mean multi-agent systems that transition from one state to another at discrete moments of time. The behavior of a member of a collective (agent) is called conforming if the opinion of this agent at current time moment conforms to the opinion of some other agents at the previous time moment. We presume that at each moment of time every agent makes a decision by choosing from the set {0,1} (where 1-decision corresponds to action and 0-decision corresponds to inaction). In our approach we model collective behavior with synchronous Boolean networks. We presume that in a network there can be agents that act at every moment of time. Such agents are called instigators. Also there can be agents that never act. Such agents are called loyalists. Agents that are neither instigators nor loyalists are called simple agents. We study two combinatorial problems. The first problem is to find a disposition of instigators that in several time moments transforms a network from a state where a majority of simple agents are inactive to a state with a majority of active agents. The second problem is to find a disposition of loyalists that returns the network to a state with a majority of inactive agents. Similar problems are studied for networks in which simple agents demonstrate the contrary to conforming behavior that we call anticonforming. We obtained several theoretical results regarding the behavior of collectives of agents with conforming or anticonforming behavior. In computational experiments we solved the described problems for randomly generated networks with several hundred vertices. We reduced corresponding combinatorial problems to the Boolean satisfiability problem (SAT) and used modern SAT solvers to solve the instances obtained

The cycle of length 4 for the network with both conformists and anticonformists.

<p>The agents-conformists are marked with "C" and agents-anticonformists are marked with "A". The network contains 7 instigators (crimson vertices). At the initial time moment all simple agents are inactive.</p

The behavior of the Watts-Strogatz network with 30 vertices under the influence of instigators and loyalists.

<p>In the upper part of the figure the functioning of the network under the influence of 1 instigator is shown. In the lower part of the figure the functioning of the network under the influence of 1 instigator and 9 loyalists is shown. Dispositions of instigators and loyalists were found as solutions of <b>Problem 1</b> and <b>Problem 2</b>.</p

The behavioral dynamics of the network under the influence of two different dispositions of instigators.

<p>In the initial state all simple agents are inactive. In the first case (left part of the figure), 5 instigators after 5 steps activate 17 simple agents. In the second case (right part of the figure) 3 instigators after 5 steps activate 26 simple agents.</p

Results of the computational experiments for Erdos-Renyi networks with 500 vertices.

<p>Results of the computational experiments for Erdos-Renyi networks, averaged for 10 tests (for each value of parameter ). <b>Pr1</b> and <b>Pr2</b> stand for <b>Problems 1</b> and <b>2</b> of finding dispositions of at most 50 instigators and at most 100 loyalists, respectively.</p><p>Results of the computational experiments for Erdos-Renyi networks with 500 vertices.</p