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- 1. Lake Como 2016 Spatial network Theory and applications Marc Barthelemy CEA, Institut de Physique Théorique, Saclay, France EHESS, Centre d’Analyse et de Mathématiques sociales, Paris, France marc.barthelemy@cea.fr http://www.quanturb.com
- 2. Lake Como 2016 Outline n I. Introduction: space and networks n II. Tools q Irrelevant tools q Interesting tools n Typology (street patterns) n Simplicity n Time evolution (Streets, subway) n III. Some models q “Standard” models n Random geometric graph, tessellations n Optimal networks q “Non-standard” n Road networks q Scaling theory n Subway and railways
- 3. Lake Como 2016 Models of spatial networks n Large classes of ‘standard’ models q 0. Tessellations q 1. Geometric graphs (i and j connected if distance < threshold) q 2. Spatial generalization of ER networks (hidden variables, Waxman) q 3. Spatial generalization of small-world (Watts-Strogatz) networks q 4. Spatial growing networks (Barabasi-Albert) q 6. Optimization (global and local)
- 4. Lake Como 2016 Some classical models Could be useful null models
- 5. Lake Como 2016 Importance of models n Choose the null model wisely n Needs to satisfy constraints and should be ‘reasonable’ n MST, Voronoi tessellation, etc. Planar Erdos-Renyi ? (Masucci et al, EPJ B 2009)
- 6. Lake Como 2016 Voronoi-Poisson tessellation n Take N points randomly distributed n Construct the Voronoi tessellation V (i) = {x|d(x, xi) < d(x, xj), 8j 6= j}
- 7. Lake Como 2016 Voronoi-Poisson tessellation n Spatial dominance (Okabe): local centers (1,20) (2,18) (3,15) (4,3) (5,7) (6,11) (7,1) (8,16) (9,3) (10,15) (11,5) (12,3) (13,6) (14,12) (15,2)
- 8. Lake Como 2016 Voronoi-Poisson tessellation n Spatial dominance (Okabe): local centers (1,20) (2,18) (8,16)
- 9. Lake Como 2016 Voronoi-Poisson tessellation n Spatial dominance (Okabe) 1 2 8
- 10. Lake Como 2016 Incidentally: census of planar graphs n BDG bijection between a rooted map and a tree (Bouttier, Di Francesco, Guitter, Electron J Combin, 2009) n Approximate tree representation of a weighted planar map (Mileyko et al, PLoS One, 2012 Katifori et al, PLoS One 2012)
- 11. Lake Como 2016 Random geometric graphs n i and j connected if d(i,j)<R n Large mathematical literature n Continuum percolation: existence of a threshold n Renewed interest: wireless ad hoc networks q Existence of a giant component ?
- 12. Lake Como 2016 Random geometric graphs Dall Christensen 2002
- 13. Lake Como 2016 Spatial generalization of Erdos-Renyi n Erdos-Renyi random graph (1959) n Spatial generalization n Example the ﬁtness model (Caldarelli et al, 2002) F(x, y) = ✓(x + y z) P(x) = e x ) P(k) ⇠ 1 k2
- 14. Lake Como 2016 Spatial small-worlds n Watts-Strogatz model (1998) n Spatial generalization
- 15. Lake Como 2016 Spatial small-worlds n Kleinberg’s result on navigability (Nature 2000)
- 16. Lake Como 2016 Growth models
- 17. Lake Como 2016 Models for growing scale-free graphs § Barabási and Albert, 1999: growth + preferential attachment § Generalizations and variations: Non-linear preferential attachment : Π(k) ~ kα Initial attractiveness : Π(k) ~ A+kα Fitness model: Π(k) ~ ηiki Inclusion of space Redner et al. 2000, Mendes et al. 2000, Albert et al. 2000, Dorogovtsev et al. 2001, Bianconi et al. 2001, MB 2003, etc...
- 18. Lake Como 2016 A model with spatial effects • Growing network: addition of nodes + distance with: Many other models possible, but essentially one parameter η=d0/L : Effect of space Interplay traffic-distance
- 19. Lake Como 2016 Optimal networks
- 20. Lake Como 2016 Optimal network design: hub-and-spoke n Point-to-point vs. Hub-and-Spoke …See paper by Morton O’Kelly
- 21. Lake Como 2016 Optimal network design: general theory n Optimal network design: minimize the total cost (usually for a fixed number of links) Cost per user on edge e Trafﬁc on e on a given network
- 22. Global optimization: simple cases Shortest path tree (SPT)
- 23. Global optimization: simple cases Euclidean minimum spanning tree (MST) Important null model: provides connection to all nodes at a minimal cost Average longest link (Penrose,97) M ⇠ s log ⇢ ⇢
- 24. “Xmas” tree Global optimization: simple cases
- 25. Optimal traffic tree (OTT) Network which minimizes the weighted shortest Path Global optimization: simple cases
- 26. Global optimization n Resilience to attacks to fluctuating load Minimize the total dissipation (total cost fixed): where Pe is the total power dissipation when Edge e is cut Corson, PRL 2010 Katifori, Szollosi, Magnasco, PRL 2010 R = X e Pe Pe = X e06=e C(e0 )(V (i) V (j))2 ( Istem = N Ii6=stem = 1 X e C(e) = 1
- 27. Lake Como 2016 Optimization and growth
- 28. Local optimization n Global optimization not very satisfying: limited time horizon of urban planners; growing, out-of-equilibrium, self-organizing cities n However, locally, it is reasonable to assume that cost minimization prevails
- 29. Lake Como 2016 § Growing Networks+optimization: Fabrikant model § A new node i is added to the network such that is minimum. - large: EMST - small: star network Optimization and growth Fabrikant et al, 2002
- 30. Lake Como 2016 § Growing Networks+optimization Optimization and growth Gastner and Newman, 2006
- 31. Lake Como 2016 n Centers (homes, businesses, …) need to be connected to the road network n When a new center appears: how does the road grow to connect to it ? A simple model for the road/streets network n We assume that the existing network creates a ‘potential’ V(x) n Two main parameters: “freedom” and “wealth” (number of connections) P(x) ⇠ e V (x)
- 32. Lake Como 2016 A (very) simple model n Algorithm q (0) Generate initial seed of a few centers connected by roads q (1) Generate a center in the plane with proba P(x) q (2) Grow the n (n depends on the wealth) roads from the center to the existing network q (3) back to (1)until N centers MB and Flammini 2008, 2009; Courtat et al, 2010
- 33. Lake Como 2016 A (very) simple model MB and Flammini 2008, 2009
- 34. Lake Como 2016 Illustration: presence of an obstacle MB and Flammini 2008, 2009
- 35. Lake Como 2016 A simple model: Problem of the area distribution q Empirically: the density decreases with the distance to the center (Clark 1951) => Generate centers with exponential distribution Surprisingly good agreement ! MB and Flammini 2008, 2009
- 36. Lake Como 2016 n The new centers are not uniformly distributed: economical factors Co-evolution of the network and centers q Choice of location (for a new home, business,…): depends on many factors. q We can focus on two factors: rent and transportation costs n Rent price increases with density n Centrality q Very simplified model, but gives some hints about possible more complex and realistic models P(x) ⇠ e V (x)V (x) = Y CR(x) CT (x) / ⇢(x) g(x)
- 37. Lake Como 2016 Co-evolution of the network and centers n Competition renting price- centrality Most important:rent Most important:centrality
- 38. Lake Como 2016 A simple model for the road/streets network A large variety of patterns (Courtat et al, 2010)
- 39. Lake Como 2016 A simple model n Local optimization seems to reproduce important features of the road network n Points to the possible existence of a common principle for transportation networks n Simple economical ingredients lead to interesting patterns
- 40. Lake Como 2016 Cost-benefit analysis of growth
- 41. Lake Como 2016 Railway growth model n Add a new link of length which maximizes where n Crossover from a ‘star-network’ ( small) to a minimun spanning tree ( large) n Emergence of hierarchical networks Tij = K PiPj da ij
- 42. Lake Como 2016 Railway growth model n Crossover from a ‘star-network’ ( small) to a minimun spanning tree ( large) n Emergence of hierarchical networks n Most empirical networks display: where is obtained for Benefit≈Cost ⇤
- 43. Lake Como 2016 Spatial network growth model n Most networks in developed countries are in the regime where the average detour index is minimum (due to the largest variety of link length) ⇠ ⇤
- 44. Lake Como 2016 Scaling for transportation systems How are network quantities related to socio-economical factors ?
- 45. Lake Como 2016 Scaling n Network properties - Total length L - Number Ns of stations - Ridership R (per year) n Socio-economical quantities - GDP G (or GMP for urban areas) - Population P - Area A n Difference subway-railway - subway: urban area scale - Railway: country scale
- 46. Lake Como 2016 Scaling – general framework n Iterative growth: add a link e such that is maximum n In the `steady-state’ regime: operating costs are balanced by benefits Z(e) = B(e) C(e) Z = X e Z(e) ⇡ 0
- 47. Lake Como 2016 Scaling – Subways n Benefit: R (total ridership per unit time); f ticket price n Costs: per unit length (and time) for lines; and per unit time and per station. n Estimate of R ? For a given station i, we have where the “coverage” is Zsub = Rf ✏LL ✏sNs Ri = ⇠iCi⇢ Ci ⇡ ⇡d2 0
- 48. Lake Como 2016 Scaling – Subways n We then obtain n Linear fit gives d0≈500meters R ⇡ ⇠⇡d2 0⇢Ns
- 49. Lake Como 2016 Scaling – Subways n Estimate of d0: where the average inter-station distance is n Interstation distance constant ! (138 cities) 2d0 ⇡ `1 `1 = L Ns `1 ⇡ 1.2km
- 50. Lake Como 2016 Scaling – Subways Ns / G ✏s n Relation with the economics of the city where G is the GMP (Gross Metropolitan Product) n Large fluctuations…
- 51. Lake Como 2016 n The railway connects cities distributed in the country n The intercity distance is where A is the area of the country. n The total length is Scaling – The railway case ` = r A Ns L = Ns` ⇠ p ANs
- 52. Lake Como 2016 Scaling – The railway case L = Ns` ⇠ p ANs n A power law fit gives an exponent ≈0.5
- 53. Lake Como 2016 n For railways we write n T is the total distance travelled is the relevant quantity (not R) n fL ticket price per unit distance n In the steady-state regime and assuming Scaling – The railway case Ztrain ' TfL ✏LL T ⇠ R` R ⇠ ✏LNs fL
- 54. Lake Como 2016 Scaling – The railway case R ⇠ ✏LNs fL n Large fluctuations…
- 55. Lake Como 2016 n Relation with the economics of the city where G is the GMP (for railways Cost(lines)>>Cost(stations)) n There is some dispersion. Importance of local specifics. Scaling – Railways L / G ✏L
- 56. Lake Como 2016 n A simple framework allows to relate the properties of the networks (R, L, Ns) to socio-economical quantities such as G, P, A. n These indicators allow to understand the main properties. Fluctuations are present and might be understood, elaborating on this simple framework n Fundamental difference subway-railway - The interstation distance is imposed by human constraints in the subway case - Railways: the network has to adapt to the spatial distribution of cities Scaling – Railways
- 57. Lake Como 2016 Discussion n Few models of (realistic) planar graphs n Even for the evolution of spatial networks n Interesting direction: socio-economical indicators and network properties…
- 58. Lake Como 2016 Thank you for your attention. (Former and current) Students and Postdocs: Giulia Carra (PhD student) Riccardo Gallotti (Postdoc) Thomas Louail (Postdoc) Remi Louf (PhD student) R. Morris (Postdoc) Collaborators: A. Arenas M. Batty A. Bazzani H. Berestycki G. Bianconi P. Bordin M. Breuillé S. Dobson M. Fosgerau M. Gribaudi J. Le Gallo J. Gleeson P. Jensen M. Kivela M. Lenormand Y. Moreno I. Mulalic JP. Nadal V. Nicosia V. Latora J. Perret S. Porta MA. Porter JJ. Ramasco S. Rambaldi C. Roth M. San Miguel S. Shay E. Strano MP. Viana Mathematicians, computer scientists (27%)! Geographers, urbanists, GIS experts, historian (27%)! Economists (13%)! Physicists (33%)! ! http://www.quanturb.com marc.barthelemy@cea.fr