Reinforced random walks

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Reinforced random walks

Term interpretations for different systems. To introduce reinforcement, we update the conductivity Dij in response to the current after the moving of the particle as 2. These rules define our current-reinforced random walk. At first, the particles spread over the whole network, and then the shortest path is formed and reinforced.

Finally, at equilibrium, the flow converges to the shortest path. In general, simulation of this system on mazes converges to the shortest path. The simulation also responds effectively in a dynamic environment. Electronic supplementary material, movie S1 is an example of the current-reinforced random walk in which the network structure is changed in the same way as the experiment by Reid et al.

Importantly, this mean field updating of conductivity is used in the Physarum solver model, first described by Tero et al. In the Physarum solver, the body of slime mould is assumed to be a network of pipes.

In terms of Physarum biology, the flow rate of protoplasm can be thought of as and the thickness of the pipe as Dij. In such a situation, Ni and Dij are, respectively, fast and slow variables allowing time separation between them.

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Ni can thus be treated as in equilibrium, i. By solving this equation instantaneously every time after Dij is updated, one can obtain the corresponding Ni and in advance get the flow rate.

This is the simulation scheme used by Tero et al. The scheme of Tero et al. This point is important both regard to the biological realism of the model and if the model were to be implemented efficiently on a parallel computer.

Pemantle : Phase Transition in Reinforced Random Walk and RWRE on Trees

Density reinforcement We now contrast the current-reinforced model with models in which reinforcement is based on density. In their model, each ant at node i walks down an edge ij with probability 2. The update rule for Dij is then the same as in equation 2. Under this scheme, the particle flow in both directions between two nodes are taken into account, i.

The mean flow rate between node i and j is given by 2. However, there are two important differences between the ant system and the Physarum solver.

Reinforced random walks

In the sense of electric network, each node now can be considered as a capacitor, and Ci is thus the capacitance. The particles are trapped in their own feedback loop and cannot establish the shortest path between the source and the sink.

In their simulation, they let only one ant at a time go through the maze. This ant does not deposit pheromone until it gets to the sink.

Once it has arrived at the sink, all loops in its path are removed, and pheromone is laid on the paths not involving loops.

The Rip-offs & Making Our Original Game

These loop-erased random walks are ultimately equivalent to the random walks based on current reinforcement. This can be seen by noting that loops are equally likely in either direction and thus cancel out, leading back to reinforcement based on potential differences.

The advantage of our current-reinforced random walk over loop erasion is that it is based entirely on local information. In biological terms, this is equivalent to ants placing themselves at all positions in the foraging arena, but moving only on the path which is the shortest!

We can improve the biological realism by thinking about the decisions by an individual particle or ant about how to move through the network when responding to the local pheromone density.

We first assume that is the probability that an ant on node i will chose to move to node j. As a result, edges linked to nodes with large number of ants can still have flow if the two nodes have different total pheromone density. Larger Ci decreases the potential of the node and the ants will move to a lower potential node and create a flow.

Electronic supplementary material, movie S1 provides an example of the current-reinforced random walk in which the network structure is changed in the same way as the experiment by Reid et al.

When the network is changed, both the ants and the flow focus on the shortest path. Non-symmetric Physarum solver In the earlier-mentioned numerical examples, our current-reinforced random walks converge to the shortest path between a source and a sink. In order to prove the convergence of these mechanisms in a general sense, Johansson and co-workers have proposed a slight modification to the Physarum solver in which conductivity is updated in both directions between two linked nodes [ 2324 ].

In this modification, between node i and j, there are edges in both directions. Because the conductivity will be reinforced on edge ij, but at the same time so the conductivity decreases on edge ji, thus the conductivity will be updated non-symmetrically.At school, Houseago was a recalcitrant student, struggling to read and occupying himself mainly with drawing, daydreaming, drinking, and brawling.

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