Outward Accessibility in Urban Street Networks: Characterization and Improvements

Out w ard Accessibilit y in Urban Street Net w orks: Characterization and Impro v emen ts Bruno Augusto Nassif T rav en¸ colo and Luciano da F ontoura Costa Institute of Physics at S˜ ao Carlos, Uni versity of S˜ ao Paulo, PO Box 369, S˜ ao Carlos, S˜ ao Paulo, 1356 0-970 B r azil (Dated: October 23, 2018) The dynamics of tra nsp ortation through tow ns and cities is strongl y affected by the topology of the connections and routes. The curren t w ork describ es an approac h com bining complex netw orks and self-a voi ding random w alk dynamics in order to quantify in ob jective and accurate mann er, alo ng a range o f spatial scales , the accessibilit y of places in to wns and cities. The tra nsition probabilities are esti mated for sev eral lengths of the walks and used to calculate the outw ard accessibilit y of eac h node. The potential of the meth odology is illustrated with respect to the characteriza tion and impro vemen ts of the accessibilit y of the tow n of S˜ ao Carlos. The intrinsic rela tio nship b etw een s tructure and dy- namics seems to sca ffold ma n y dynamica l pro cesses in nature, from the fl ight of birds to the binding of pro teins . Because of its inherent a bilit y to r epresent and mo del the most diverse types o f discrete structures, complex ne t- works ha ve r eceived growing atten tion. Having init ially fo cused attention on the characteriza tion and mo deling of the top olog y of in terconnectivities (e.g. [1, 2, 3]), complex net work res earch progr essed steadily to enco mpass the r e- lationship b etw een s tr ucture and dy na mics in the most diverse systems (e.g . [2, 4]). Though the connectivity do es not completely define dynamics, it s trongly affects it. This ha s b ecome clear through in vestigations of rela- tionships be tw een struc tur e and sev er al types of dynam- ics, including diffusio n (e.g. [5]), sy nc hroniza tion (e.g. [6]) and neurona l netw or ks [7 , 8]. Particularly , when the dy- namics is mo deled in ter ms of ra ndom walks (e.g. [5, 9]), the displacements of the resp ective moving a gents are strongly influenced b y sev eral topo lo gical facto rs suc h a s the num b er of connectio ns at ea ch no de and the s ho rt- est path le ngths b etw een no des. This t yp e o f sto chastic dynamics pr esents a n intrinsic p otential for mo deling the displacement of p eople within towns o r cities . How ever, bec ause traditiona l linear random walks allow a moving agent to visit edges and no des more than once, implying nu ll av erage displacement in the long term, it b ecomes impo rtant to cons ider more purp osive types of displa ce- men ts. Self-avoiding walks (e.g. [1 0, 11, 12, 13]) represent a na tural simplified choice for mo deling ur ba n displace - men ts, implying the ag ent s to mov e aw ay from their ini- tial p osition in a mor e effective wa y while no t rep eating edges or no des. Complex net works hav e b e e n used to characterize im- po rtant top olog ical, dyna mical and spa tial prop erties of cities (e.g., [14, 15, 1 6, 17, 18]). One imp orta n t pra c- tical applica tion of the structure-dyna mics relationship concerns the characterization, mo deling and pla nning o f urban displacements. In a previous study , Rosv all et al. [14] considered shortest pa th lengths in order to quan- tify a nd compar e the information needed to lo cate sp e- cific addresses in different cities. The curr ent work ap- plies the recently intro duced concept of ac c essibility [1 9] in orde r to q uantif y in a n o b jective and comprehens ive wa y the outw ar d a ccessibilities of each no de of a town (i.e. intersection or beg inning of routes). The metho dol- ogy is illustrated with resp ect to the sp ecific applica tio n to the Bra z ilian town of S˜ ao Carlo s. Imag e pro cessing and ana ly sis metho ds w ere used to trans fo rm the pla n of the town into a resp ective geog raphical planar net- work, where the node s r epresent the crossing s and the beg inning of routes, while the edges c o rresp ond to the streets. Figure 1 shows the netw ork derived fro m the central par t of the town. Then, by sim ulating a ser ies of self-av oiding walks starting at a ll no des, trans ition prob- abilities from one no de to ano ther were estima ted with resp ect to v arying lengths of the walks. The out war d ac c essibility o f ea ch no de — expr essing the diversity of routes b etw een thos e p oints as well as the p otential of the moving agent to visit a set o f no des in the s ho rt- est time — is quantified in ter ms of the entropy o f the obtained tr ansition probabilities, so that v alues close to 1 indicate ma ximum outw ar d acces sibilit y . One imp or- tant pro per t y of the outw ard a c cessibility measur e men t is that se lf-avoiding walks initiating fr om no des charac- terized by high a c cessibility for a given pa th length tend to visit all rea chable no des at that leng th in the short- est per io d of time. In addition, the outw ar d accessibility int rinsically considers the n umber o f a lter native r outes from the initial no de to the r eachable no des. No des with high outw ard accessibility therefor e hav e mor e balanced nu mber of ro utes leading to the rea chable no des [19, 2 0]. W e sta rt by pr esenting the ba sic concepts ab out net- work r epresentation. An un weighted a nd undirected complex netw or k c a n b e r e presented by a matrix K , called adjac ency matrix . The dimension of this matrix for a netw or k with N no des a nd E edges is N xN . If the no des i and j a r e connected thro ugh an edge, the ele- men ts K ( i, j ) a nd K ( j, i ) of the a djacency matrix are set to 1 ; otherwise K ( i, j ) = K ( j, i ) = 0. Two no des o f the net work are said to b e adjac e nt if they shar e one edg e. Two edges of the netw ork are said to b e adjacent if one extremity of ea ch edg e shar e the s ame no de. The de gr e e of a no de i is the num b er of its immedia te neighbors. A walk over the netw ork is co mpo sed by a sequence of adjacent no des, starting fro m an initial no de and pro- ceeding through succ e ssive steps h . A self-avoidi ng walk is a walk where the no des and edge s do not app ear more than once. 2 FIG. 1: N etw ork of urban streets of S ˜ ao Carlos, Brazil. The gray level of the no des indicates th eir resp ective av eraged outw ard accessibilit y accordingly to th e legend at the right-hand side. The dashed lines represent the hypothetical ad d itional edges aimed at improving the accessibilit y . After a real-world structur e has b een pr op erly r epre- sented as a complex net work, a diversit y of measur es can be obtained, r anging from simple featur es such as the no de degr ee and clustering co e fficie nt, to mor e sophisti- cated s uc h as shor test path lengths and b etw eeness cen- trality . Thes e measures hav e allowed the comprehensive description and characterization of several co mplex sys - tems [3, 4]. The tr ansition pr ob ability that a moving agent no de reaches a no de j after depar ting fro m a no de i , through a self-av oiding walk after h steps, is henceforth expres sed as P h ( i, j ). In order to estimate this pro babilit y , a total of M self-av oiding ra ndom walks, star ting from the no de i and pro ce e ding S steps, a re p erformed. Note that these walks stop when o ne of the following three conditions is met: (i) the walk r eaches the maximum pre-defined v alue of steps ( S + 1 no des ); (ii) the walk rea ches an extremit y no de, i.e., a no de with degree o ne; or (iii) the walk ca nnot pro ceed further b ecause all o f immediate neig hbors of the no de at step h were already visited. The probability P h ( i, j ) can then be estima ted in terms of the num b er of times that the walks depa rting from a no de i r each the no de j after h steps, divided by the n umber of walks, M . Note that P h ( i, j ) is typically different of P h ( j, i ). After the pro babilities a re estimated for each no de, it is po ssible to calculate the diversity en tr opy signatur e E h ( i ) of a no de i a fter h steps a s [20]: E h ( i ) = − N X j =1 ( 0 if P h ( i, j ) = 0 P h ( i, j ) l n ( P h ( i, j )) if P h ( i, j ) 6 = 0 (1) Although the diversity entrop y provides an in teresting 3 quantification of the a c cessibility of the no des, the out- war d ac c essibility has b een prop os e d as a no rmalization of the diversity entr opy , allowing direct compar ison with other measur es r egarding non-linear transient dynamics of the netw o r k (e.g. inwar d ac c essibility [1 9 ]). The out- ward acce s sibilit y OA h ( i ) of a no de i after h steps can be immediately calcula ted from the diversit y ent ropy as OA h ( i ) = exp ( E h ( i )) N − 1 (2) The outw ar d accessibility was estimated for the central part o f the ur ban streets of S˜ ao Carlos, which is r epre- sented b y a netw or k with N = 2 812 no des and E = 4713 edges (Fig. 1). The total length of the self-avoiding ran- dom walks pe rformed for each no de w as S = 60 , and 10 . 000 walks were per formed so as to obtain acc uracy in probability es tima tio ns. In Fig. 1 the gr ay le vels o f the no des co rresp onds to their r esp ective a ccessibility , av er- aged over all the steps. An int eresting result which is evident in this figure is that most part o f the highly ac- cessible no des cor r esp onds to the down town S˜ ao Carlo s, lo cated at the central region of the map. Another im- po rtant prop erty is the high spatial discr iminative p ower provided by the outw a rd acces sibilit y mea surement: it can b e clea rly seen from Fig. 1 that the no des situated at the b order of the netw ork hav e the smallest o utw ard v alues, while the inner no des hav e the highest outw ar d v alues. Interestingly , no des with low outw a rd access ibil- it y can b e found even do wnto wn. It is in teresting to recall that the ability of the acce s sibilit y approa ch to identify the most central (ag ainst the b or ders) parts of a netw ork is not restricted to geogr aphical netw orks, but can be im- mediately applied to a n y other t yp e o f complex netw o r k. In or der gain mor e insights rega r ding the accessibility of the different parts of the town, we also considered hy- po thetical new edges (i.e., new streets) connecting so me no des of the p eriphery a nd int ernal reg ions o f the town (represented as dashed lines in Fig. 1). This new a r- rangement allow ed a study of the p otential impact of the new edges in the a ccessibility of their neighborho o d. The mean ac c e ssibility was computed co nsidering the no des lo cated up to s even blo cks aw ay fro m the no des that re- ceived the new connections . Figure 2 shows these v al- ues for the original netw o rk a nd for the enhanced net- work. Note an increas e of 21% in the a ccessibility after approximately h = 15 steps for the place where the new edges were added. This r e sult s hows that ma jor improv e- men ts of accessibility ca n b e achiev ed by adding just a few stre e ts at stra tegic lo cations. The authors thank the S˜ ao Carlos town hall for provid- ing and gra n ting the per mission for using the city pla n and Matheus P . Viana fo r the desig n of the image pro- cessing ro utines. Br uno A. N. T r aven¸ colo is grateful to F APESP for financ ia l s uppor t (2007/ 02938 -5) and Lu- ciano da F ontoura Costa thanks to CNPq (30 1303/ 06-1) and F APESP (05/ 00587 -5) for financial supp ort. [1] R. Alb ert and A. Barab´ asi, Rev. Mo d. Phys. 74 , 47 (2002). [2] M. E. J. N ewman, SI AM R ev. 45 , 167 (2003). [3] L. da F. Costa, F. A . Ro drigues, G. T ravieso, and P . R. Villas Boas, Ad v. Phys. 56 , 167 (2007). [4] L. d a F. Costa, O. N. Oliveira Jr, G. T ravieso , F. A. Ro drigues, P . R. V. Boas, L. 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