## 3 Numerical Examples
### Linear Link Cost Functions  
Table 1 shows the smallest (\(\varphi\) min) and largest (\(\varphi\) max) values of the fraction of contrarian travelers that ensure fixed point stability, computed by means of expression (54) for selected values of the other model parameters. Since \(0\leq\varphi\leq 1\), the lower limit of the stability region is set to zero whenever its calculated value turns out to be negative, and the upper limit is set to one whenever its calculated value exceeds unity. The results displayed in Table 1 provide numerical evidence in support of the conclusions ensuing from the preceding analytical study. In particular, when sensitivities of costs to flows and of route choices to costs are low (case \(\gamma\mu=1\)), the fixed point turns out to be stable for any combination of direct and contrarian behaviors. As \(\gamma\mu\) grows, the stability region tends to shrink progressively; however, the increase of the lower stability limit (\(\varphi\) min) is offset by appropriately reduced day-to-day flow and cost updating rates, and therefore becomes apparent only for the higher values of \(\alpha\) and \(\beta\). Intuitively, this happens because travelers' inertia and memory depth tend to act as damping forces against the destabilizing effects of high \(\gamma\) and \(\mu\) values. Examples of possible patterns of evolution of the network state over 100 time periods are presented in Figure 1 through 4 in order to illustrate the effects of the model parameters on the long-term outcome of the system's dynamics. In these figures, \(t\) denotes the generic period (day) of the dynamic process and \(Z(t)\) represents the difference between the mean  
Figure 4: Time evolution of dynamical system for \(\alpha=0.5;\beta=0.5;\gamma\mu=10;\varphi=0.8;Z(0)=0.01\) (case of linear link cost functions). Figure 3: Time evolution of dynamical system for \(\alpha=0.9;\beta=0.9;\gamma\mu=5;\varphi=0.15;Z(0)=-3\) (case of linear link cost functions). Figure 2: Time evolution of dynamical system for \(\alpha=0.75;\beta=0.75;\gamma\mu=10;\varphi=0.23;Z(0)=1\) (case of linear link cost functions). The implications of contrarian route choice behavior for the performance of the network as a whole are illustrated in Figures 5 and 6, which show the relationship between mean travel cost of all users and the fraction of contrarians for different values of \(\gamma\mu\) and for two pairs of values of \(\alpha\) and \(\beta\). The results reported in these figures represent long-term states attained by the system after 1000 time periods, and were derived assuming \(K^{0}=1\) in expressions (48) and (49). While Figure 5 refers to a population of users exhibiting a high level of inertia (\(\alpha=0.1\)) and a deep memory of past travel costs (\(\beta=0.1\)), Figure 6 addresses the case of highly reactive (\(\alpha=0.9\)) and nearly memoryless (\(\beta=0.9\)) travelers. Both graphs clearly show that mean travel cost is lowest and remains constant over the region of stability of the \(\widetilde{Z}=0\) fixed point, but increases steadily as \(\varphi\) moves outside this region. Comparison of the two figures further suggests that higher inertia and memory depth (Figure 5) tend to elongate this region leftwards by counteracting the instability which otherwise arises when contrarian subjects are underrepresented within the traveling population. Figures 7 and 8 provide evidence about the relative performance of the two types of travelers (as measured by the ratio of the respective mean travel costs) over the range of  
Figure 5: Long-term overall mean travel cost as a function of \(\varphi\) for \(\alpha=0.1,\beta=0.1\) and different values of \(\gamma\mu\) (case of linear link cost functions). Figure 6: Long-term overall mean travel cost as a function of \(\varphi\) for \(\alpha=0.9,\beta=0.9\) and different values of \(\gamma\mu\) (case of linear link cost functions). values of \(\varphi\), having assigned to the other parameters the same values as in Figures 5 and 6.