We have considered our model to be completely random, where all cops and robbers perform a random walk in the network. The movement of all the cops and robbers in the network is completely random. Combination of both the cop's and the robber's moves are considered to be one round of the game.
Variants of Cops and Robber Problem
Both the cops and the robber can only move to its neighboring nodes and cannot jump over the nodes. At any moment either only the robber moves or the police move to another location. The robber in their game knows about the police's previous moves, but not the upcoming ones.
They studied the effect of reducing the visibility of the police officer on the graph, i.e. the robber can only be seen by the policeman if the distance between them is no more than a certain threshold. In this problem, a fire starts at some node of the graph and continuously spreads to all neighboring nodes in the graph. Another version of the cops-robbers game, motivated by the angel-devil theory, was studied in [17].
In this game, the police have a special power to build a wall over a corner they move to and the robber cannot move to a corner with a wall. To demonstrate the behavior of the discrete system, a concept of transition system is used in the field of calculus.
Temporal Logic
Linear Temporal Logic (LTL)
LTL was first introduced by Amir Pnueli in [20] to verify programs based on time reasoning. LTL is essentially modal time logic that can be used to determine the paths generated by any time-stepping system. It determines the infinite sequence of events in any system based on the property of linear time.
Xφ1 : This is the next operator that evaluates to true if φ1 holds in the next condition. Gφ1 : G is the global operator that says that the formula φ1 is true in all the future states. A trace of any transition system is the sequence of states visited in the system in discrete time steps.
Probabilistic BLTL
Model Checking
Probabilistic Model Checking
There are many probabilistic systems in the real world scenario that exhibit non-deterministic and probabilistic behavior. To model such systems, probabilistic models such as discrete time Markov chains (DTMCs) are used. Probabilistic Model Checking (PMC) is an adaptation of the actual model checking technique used to verify the models that have the stochastic behavior.
This is one of the well-known techniques for modeling and verifying real-world applications. A standard model checking technique also takes a model of the system in question and a property specification expressed in some probabilistic time logic such as PCTL (Probabilistic Computation Tree Logic) as input and checks the properties of the system. Transitions in the system model are based on probabilities expressed in the transition matrix of the model.
For further reading on model checking, readers may refer to Chapters 1 and 10 of the book [22].
Statistical Model Checking
Statistical model checking is based on simulating a finite number of executions of the system and using various statistical techniques to infer whether the property is satisfied or not. In addition, these execution paths or traces generated by simulations can be used to estimate the probability that a property is satisfied in a system. In contrast, estimation methods estimate the probability that a given property is true.
These parameters determine the strength of the test and limit the error probability, the probability of accepting K when H is true is bounded by α and is called Type-I error. Similarly, the probability of accepting H when K holds is bounded by β and is called Type II error. Single Sampling Plan (SSP) : In this method, for n simulations of the model, the value is accumulated and it crosses a constant c then H0 is accepted, otherwise.
But to estimate the probability that the property in the model is satisfied, estimation methods are used that estimate the probability based on the simulations from the model. Once we have the value of N, the number of samples required for the system, we keep track of the number of samples in which the property is satisfied and the ratio of samples in which the property is satisfied to the total number of samples gives the probability of property is true in the system within the approximation limits.
Random Graph Models
- k-Regular Graphs
- Erd¨ os-R´ enyi (ER) Model
- Barab´ asi Albert (BA) Model
- Street Network
- Problem Statement
- SMCA Tool
This model is used to generate random graphs named after Paul Erd¨os and Alfr´ed R´enyi. To generate the network, a priority snapping mechanism is used, which is the process of distributing things between entities based on what they already have. Agent-based modeling of real-world systems has become increasingly popular over the past few years.
System model: This part of the tool receives input from the user in the form of the JAVA code that tells the model how the different agents move in the network at each time step. This is given as input by the user in the pre-described format of the tool and can be changed as per the user's requirements based on his system. Property Specification: It specifies the property that the user wants to check the modeling of the system and is given as input by the user in the predefined format.
This specification should essentially define which parameter should be checked across all simulations of the model for all agents in the network. Statistical Techniques: This component provides options for the statistical techniques that can be applied to estimate the probability of satisfying the property in the model with the given constraints.
Implementation Details
It simulates the system for the required number of samples based on bounded errors and returns the result in the form of estimated probability. This component considers all simulations and produces the result in the form of estimated probability. We describe the results based on different graph models and throughout all experiments the cops and robbers are performing a random walk on the graph.
We have performed two different sets of experiments for k-regular graphs, in one set we take 5 robbers performing a random walk in the network and in the other set the number of robbers is 10. The number of nodes in all experiments is set to 500, the number of police and in the network it is set to 300. In this experiment, we considered 5 robbers moving in the network and trying to escape from the police indefinitely, and estimated the probability of catching all the robbers in a certain time frame.
In this experiment, it is assumed that 10 robbers perform the random walk in the graph. The idea of this experiment is to verify the results of previous experiments with five robbers and also to see how the probability values for the monitored property change as the number of robbers in the network increases.
ER Model & BA Model
- Varying the number of cops
- Varying the time bound
- Varying the number of edges in the network
- Multiple cops & multiple robbers
The number of nodes in the network is 500 and the number of edges in the network is 61,915 for both graph models. The number of agents in the network varies from 60 to 420 in this experiment and the time limit is set to 150 in all runs of this experiment. As the results show, the probability of catching the robber increases with the increase in the number of agents , as with more agents in the network it becomes difficult for the robber to evade arrest.
Also as observed in the experiment with varying number of policemen, in this experiment also the probability values for BA model are significantly higher than the probability estimated for ER model, thus again proving that the capture of the robber definitely depends on the topology of the network. In this analysis, we therefore change the number of edges in the network and estimate the probability of catching the robber within the specified time. The number of police officers in this experiment is fixed at 300 which is divided equally in 3-color police setup while all 300 are of the same color in 1-color police setup.
Counterintuitively, it can be inferred from the results that changing the size of the network does not have a major effect on the probability of apprehending the robber in the network generated by the ER model. In this experiment, we considered more than one robber in the graph for the ER and BA models and looked at the probability estimate values for them.
Street Network
This thus again shows that the BA model gives better chances of catching the robber in police and robber scenarios. In this work, we have studied a variation of the police and robber problem, where multicolors take random walks in the network and aim to apprehend the robber in the network. In our work, we have studied our model on different random graph models and analyzed the effect of changing different parameters in the network.
We used statistical model checking methods to estimate the probability that all the robbers will be caught within the given specified time limit. Also in random k-Regular graphs, the chances of catching the robber do not really depend on the K-value of the graphs. We believe that it would be interesting to investigate the police and robber problem with more complex police and robber behavior using statistical model checking.
The technique will prove particularly useful when it is difficult to obtain closed solutions and complex queries must be made to the system. To that end, we are building a statistical model checker optimized for generic versions of the cops and robbers problem.
