# Intrinativeness: From Games to Random Walks +
## 4 Applications  
The victory graph of the Penney Game (Figure 1) shows a cycle at the core of its structure. In Ref. [3], we analyzed the transitive properties of random walkers on cycles, considering the influence of biased choice, partially absorbing traps, and the influence of the initial distribution. Simulations show that, for uniform initial distributions (all starting locations are possible), fully absorbing traps and unbiased choices (left-right displacements equally probable), \(\sigma=0\), but as soon as the process is biased it becomes intransitive, i.e., it is possible to choose the position of a trap that \"obscures\" the other. If the initial distribution is more localized (uncertainty is reduced), the transitivity region of the system widens, i.e., one needs higher biases to recover the possibility of obscuring the other trap. This result is quite obvious: if one knows the starting position of the walker and that they tend to go in one direction, the first player can place his/her trap in the appropriate neighbouring position and win. On the contrary, if all positions are equivalent and there is a drift, once that the first player has chosen a position, the opponent can place his/her trap upstream. The role of partial absorbance of traps is also that of weakening the intransitiveness, in a circle: if a walker can pass several times over a trap, the shading effect is lowered and the parameter \(\sigma\) diminishes towards zero. However, in more complex networks, more interesting phenomena may appear. While for cycles, increasing the value of the absorbancy of traps (\(\epsilon\)) always produces an increasing of \(\sigma\) (that is always positive), for networks, there may be an intermediate value of \(\epsilon\) for which \(\sigma>0\). This effect was shown for instance in Ref. [3], studying a small scale-free network (100 nodes). In this case, by varying the absorbancy \(\epsilon\) of traps, one observes an intermediate intransitive regions. In order to clarify the origin of this effect, which is due to the influence of \"leaves\" on the core cycle, let us consider a small network with two loops, see Figure 4b,c), and a uniform initial distribution. For very high absorbancy, it is clear that node 1, which has a larger \"basin\", dominates over all other position of traps. On the contrary, for very small absorbancy, it is node 2 that dominates, since all walkers from node 1 go first to node 2 and the to node 3, from where half go back to node 2 and half to node 2. For exactly vanishing absorbancy \(\epsilon\), the invariant distribution is given by \(x_{2}=x_{3}=2/5\) and \(x_{1}=1/5\), so for small but not null \(\epsilon\), nodes 2 and 3 dominates over node 1, which however still has the larger basin, so most of walkers pass through node 1 and end in the \"core\" of nodes 1, 2, 3, passing first from node 2, which, therefore, has an advantage over node 3. Increasing \(\epsilon\), we have a transition of dominance from node 2 to node 1, passing through a region in which the system is intransitive, with nodes 1, 2 and 3 that may capture most of walkers, depending on which is the second player. The next subject concerns hierarchical matrices, which are good models for communities. The network is defined by blocks, as indicated in Figure 5a [18]. In each element of a block \(k\), of size \(l_{k}\), there is a one (connection) with probability \(p_{k}\). As shown in Figure 5b, if communities are well isolated (\(p_{3}\simeq 0\)) and for enough absorbing traps, there is a region of intransitiveness. Since these matrices are in average symmetric, one does not expect intransitiveness. However, when the connection among communities is very low, \(\sigma\) becomes positive. This is due to the fact that for such low values of \(p_{3}\) (and absorbance high enough) there are very few connection among communities, so that the effective network is almost always asymmetrical.