1. DYNAMICS OF CITY BIKE SHARING NETWORKS Kasia Samson & Claudio Durastanti

2. Intro: City bike sharing system Taking a bike in station x; Going from the station x around the city to the station y Leaving the bike in the station y

3. Station Distribution We can reasonably consider the stations as uniformly distributed around the city Our ‘ideal’ city can be described as a fully connected network

4. Two different approaches By Netlogo –Agent Based Model By Matlab: –Statistical Approach

5. MATLAB PROGRAM STRUCTURE Velov Create_transition_matrix2 Exp_factor Thr_bike Move_bike Create_transition_matrix Time_step Check_bikes Dislocation Perturbation Initialize Iteration

6. Assumptions & Distributions 1.BIKER WILL AXIOM 2.CITY CENTER AXIOM 3.CONSERVATION OF THE NUMBER OF BIKES LAW 4.NEIGHBOR SELECTION PRINCIPLE 5.NON STUPIDITY OF BIKER COROLLARY 6.FOOTBALL MATCH EFFECT LAW 7.WORLD ROTATION DEPENDENCE POSTULATE 8.BIKERS SLEEPING NECESSITY COROLLARY

7. BIKER WILL TRANSITION After few cycles, the system approaches the stable equilibrium point The bikes are almost uniformly distributed among all the station NIGHT TIME CONFIGURATION

8. GEOGRAPHY-BASED TRANSITION After few cycles, the system approaches the stable equilibrium point. The bikes are mainly concentrated in a centered portion of station. DAY TIME CONFIGURATION

9. Transition matrices Distance matrix D(i,j) is Gaussian-standardized and normalized D(i,j) is weighted with the exponential probability associated to the arrival station (and its distance from the center) and then normalized D(i,j) is weighted with the exponential probability associated to the arrival station (and its distance from the center) and then normalized

10. Why stability? Both the transitions are linear. Eigenvalues of both the associated matrices are smaller/equal to 1. System approaches equilibrium in less than two step for every initial condition It is possible to obtain a more realistic simulation by summing the two contributes

11. NEIGHBORS 1.The arrival station x is full. 2.The biker looks for a place in the nearest station y 3.If also this station is full, the biker look for a place in the station nearest to y. 4.If the station nearest to y is already visited, the biker look for the nearest unvisited station.

12. PERTURBATION Hypothetically, the behaviour of the bikers could depend on the ‘important events’ in the city By defining the geographic position of these events, the bikers find other attraction points

13. PERTURBATION For each cycle, a random generated value is compared with a fixed threasold (very high). If this value is higher than threasold, the transition is modified by summing another term

14. COMPARISON

15. THE BEHAVIOUR OF SOME STATIONS

16. TIME EVOLUTION OF THE SYSTEM: AN EXAMPLE Number of bikes The city

17. AGENT-BASED MODEL NetLogo Model: 1.Stations: fixed number random location 2.Bikes 3.Directed links

18. SETUP Initial Assumptions No city centre axiom Initial threshold on stations Conservation of the number of bikes law World rotation dependence postulate Non-stupidity of bikers corollary Bikes x Station

19. BIKE MOVEMENT Some no of bikes move at each step from location to random location Bikes and locations chosen upon the Matlab weight matrices (daytime vs night time)

20. EMERGENT DAY TIME PATTERN

21. CENTER FORMATION

22. CONNECTIONS Statistical simulations can set realistic parameters Agent based simulations are ‘lighter’ and systems could be studied for longer time periods Nature of simulations is ‘station-centered’ Nature of simulations is ‘bike-centered’ It is possible to obtain different and convergent measures of the system

23. POSSIBLE IMPROVEMENTS On Matlab Simulations – Comparing with real data; – Adding non linear terms to the equation – Refining the current equation On Netlogo – Adding more constraints

24. End