Statistical Analysis of Bus Networks in India
SStatistical Analysis of Bus Networks in India
Atanu Chatterjee ∗ , Manju Manohar † , and Gitakrishnan Ramadurai ‡ Department of PhysicsWorcester Polytechnic Institute100 Institute Road, Worcester, MA 01609, USADepartment of Civil EngineeringIndian Institute of Technology MadrasChennai-600036, India
Abstract
In this paper, we model the bus networks of six major Indian cities as graphsin L -space, and evaluate their various statistical properties. While airline andrailway networks have been extensively studied, a comprehensive study on thestructure and growth of bus networks is lacking. In India, where bus trans-port plays an important role in day-to-day commutation, it is of significantinterest to analyze its topological structure and answer basic questions on itsevolution, growth, robustness and resiliency. Although the common feature ofsmall-world property is observed, our analysis reveals a wide spectrum of net-work topologies arising due to significant variation in the degree-distributionpatterns in the networks. We also observe that these networks although,robust and resilient to random attacks are particularly degree-sensitive. Un-like real-world networks, like Internet, WWW and airline, which are virtual,bus networks are physically constrained. The presence of various geographi-cal and economic constraints allow these networks to evolve over time. Ourfindings therefore, throw light on the evolution of such geographically andsocio-economically constrained networks which will help us in designing moreefficient networks in the future. From the neural architecture of the brain to the patterns of social interactions,many physical systems and real-world phenomena are being formulated as networkmodels [1, 2, 3, 4, 5, 6, 7, 8]. These models are complex because of their sizeand the various emergent properties that arise due to their inter-nodal connections.Any physical, chemical, biological or social system can be visualized as a complex ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ phy s i c s . s o c - ph ] M a y etwork; the constituting elements are known as nodes, and the interactions betweenthem identified as links. Based on the nature of the links, these networks can bebroadly classified into virtual and spatial networks. In the former category, the linksare physically absent, e.g., social networks or collaboration networks, whereas, inthe latter case, the links are physically present, i.e., geographically embedded roador railway networks [9, 10, 11, 12]. In between these two broad classes there existnetworks in which the links although physically absent are, however, geographicallyconstrained. The structure of the real-world networks such as bus or electric powergrid are dependent upon the structure of the physically constrained, geographicallyembedded networks on which they grow and evolve.In the field of transportation science, the use of networks to understand the flowof entities including vehicles, cargo and pedestrians, has a long history. This tra-ditional network flow formulation has answered many interesting engineering ques-tions related to optimality of cost, maximality of flows and the classical, shortestpath determination [13, 14]. But there exist questions that deal with the topolog-ical structure of the network, which are primarily concerned with the inter-nodalconnectivity and evolution of the network, which the traditional formulation failsto address. In order to answer interesting questions, such as estimating the impor-tance of a particular node in a network, identifying existence of hubs, analyzingthe pattern of variation in shortest paths with the network size, or the robustnessand resiliency of the network, we need to look at the statistical and topologicalproperties of the network.Mathematically, a network is a graph, G , characterized by the presence of nodes, N , and links, L , connecting the nodes, such that G = ( N, L ) where the set ofnodes belong to the Euclidean space of two or three dimensions. Specific to publictransit, networks are often modeled either in L -space or in P -space [15, 16]. Inboth the configurations, the nodes remain the same, for example, bus stops, metroor railway stations, whereas the pattern of the link connectivity changes. In L -space formulation, each pair of consecutive neighboring nodes lying along a routeis considered to be connected by a link, whereas in P -space formulation, everypossible pair of nodes belonging to a route are connected by a link. Thus, L -spaceconfiguration helps in understanding the relationship between the stops or nodes ingeneral, and P -space helps in studying the transfers between different routes in thenetwork.For each node in the set of nodes, N = { n , n , n ...n i | i ∈ I , ∀ n i ∈ R n } , weidentify the degree of a node, k i as the number of links to which that particularnode is connected to. The pattern of the inter-nodal connectivity, specifically thedegree-distribution, P ( k ), of the nodes, leads to the emergence of several interestingproperties of the network. Based on the degree-distribution of nodes, two promi-nent network models have been identified: a ) the random network model and b ) thescale-free network model. The random network model was first studied by Erd¨os andR´enyi, and they provided two generative models where either the number of nodesand edges are fixed or each node is associated with some probability [17]. Althoughthe Erd¨os-R´enyi random graph is an important model for comparison purposes, itfails to capture the essence of real-world networks, such as presence of clusters, com-munities, and the small-world phenomena. A more interesting model was proposed2y Watts and Strogatz (WS) in order to understand real-world networks in greaterdepth, which is commonly known as the small-world network [6, 18]. However, ithas been observed that most of the real-world networks show a heavy-tailed degreedistribution where the degree of few of the nodes significantly exceeds the averagedegree of the nodes in the network. This inhomogeneity in degree-distribution oftengives rise to striking properties in the network, that has been extensively studied byBarab´asi and Albert (BA) [1, 2, 3]. Both BA and WS models advocate the small-world phenomenon, which is a characteristic feature of real-world networks, such aselectric power grids, WWW, Internet, social-networks, protein-yeast (metabolite)interaction networks, citation networks, and movie-actors collaboration networks[1, 2, 3, 5, 6, 19, 20, 21, 22, 23].Interestingly, the above mentioned properties have been reported in various pub-lic transit networks as well [11, 15, 16, 24, 25, 26, 27, 28, 29]. The small-world phe-nomenon in transportation networks is expected since transportation facilities in acity are planned to provide maximum convenience by allowing travel between placesin minimum possible time. Most transportation networks are pre-planned networkswhere the initial design of the network decides the presence of hubs. Transporta-tion networks are not as large as social-networks or the Internet, and are subjectedto geographical as well as socio-economical constraints. Studies on public tran-sit networks for different cities around the world (inclusive of all modes: buses,trams, metros and monorails) have been shown to exhibit scale-free behaviour withvarying values of the power-law exponent, γ [15, 27, 28, 29]. Airline and metro-networks show scale-free degree distribution patterns whereas degree-distributionin bus and rail networks tend more towards exponential patterns. The reason forthis contrasting behaviour could be attributed to the two following observations:( i ) airline-networks are not bounded by geographical constraints and ( ii ) metro-networks are local often catering to a part of the city whereas, bus and railway-networks are global as they are spread throughout the entire state and sometimesacross the entire country. Specific to Indian scenarios exhaustive studies on publictransit networks as a whole are yet to be conducted. Previous work have shown thatthe pattern of nodal connectivity of the Indian Railway Network (IRN) drasticallydiffers from that of the Airport Network of India (ANI) [11, 24]. The nature ofIndian bus networks still remains understudied.Bus transport networks have been studied elsewhere. Analysis of the statisticalproperties of bus transport networks (BTNs) in China revealed their scale-free de-gree distribution and small-world properties. The presence of nontrivial clusteringindicated a hierarchical and modular structure in the BTN. Weighted analysis of thenetwork was done considering routes as nodes and weights as the number of commonstations between the routes. The weight distribution followed a heavy tailed powerlaw, and the strength and degree were linearly dependent [30]. In another study,an empirical investigation was conducted on the bus transport networks (BTNs)of four major cities of China. When analyzed using P -space topology, the degreedistribution had exponential distribution, indicating a tendency for random attach-ment of the nodes. The authors also evaluated two statistical properties of BTNs,viz., the distribution of number of stops in a bus route ( S ) and the number of busroutes a stop joins ( R ). While the former had an exponential functional form, the3atter had asymmetric unimodal functional forms [31]. The statistical analysis ofthe urban public bus networks of two Chinese cities, Beijing and Chengdu revealedscale free topology and small world characteristics. Presence of more hubs in theBeijing network led to a comparatively smaller exponent of degree distribution andlarger clustering coefficient. Similar location of bus stops in the two cities has ledto a hierarchical structure, denoted by power law behaviour (with nearly same ex-ponents) of the weights characterizing the passenger flows [32]. The rail (RTS) andbus transportation systems (BUS) in Singapore were studied with respect to theirtopological as well as dynamic perspectives. The stations in RTS had high aver-age degree indicating high connectivity amongst them, while the BUS had a smallaverage degree. Both networks had an exponential degree distribution indicativeof randomly evolved connectivity. Strength of nodes defined as the sum of weightof incident edges, appeared scale free for both networks indicating the existence ofhigh traffic hubs. The BUS network exhibited small world characteristics and had ahierarchical star like topology. RTS had slightly negative topological assortativity,while the weighted BUS displayed disassortative nature [33]. An extended space(ES) model with information on geographical location of bus stations and routeswas used to analyze the spatial characteristics of bus transport networks (BTNs)in China [34]. The ES model consisted of directed weighted variations of the L -and P -space networks designated as ESL and ESP networks respectively, and thesymmetry-weighted ESW network that stored information of the short-distance sta-tion pairs (SSPs). Often, two bus stations which are geographically close to eachother may not have any direct bus route link between them. Such stations whichare at walkable distances from each other, are defined as SSPs. The SSPs greatlyinfluence the BTNs by reducing the transfer times as well as the number of busroutes. The average clustering coefficient of the ESW networks was considerablylarge, denoting a nearly circular location of the SSPs around a station. Majority ofthe route sections in the bus routes were short, while a few route sections connect-ing cities downtowns and satellite towns or special purpose BRT routes were long,leading to a power law edge length distribution of the ESL networks.Majority of the above studies have looked into the structural properties of thebus networks in both L - and P -spaces. The ESW network is one such networkwhich has looked into the aspect of network redundancy due to geographical place-ment of the nodes. In this paper, we do a comparative study of the bus networksof some of the major Indian cities, namely Ahmedabad (ABN), Chennai (CBN),Delhi (DBN), Hyderabad (HBN), Kolkata (KBN) and Mumbai (MBN). In orderto understand the structure of bus networks in India we calculate various metrics,such as clustering coefficients, characteristic path lengths, degree-distribution andassortativity. We also simulate network robustness and resiliency by first remov-ing nodes at random, followed by targeted removal based on degree, closeness, andbetweenness. This provides us with interesting results on network (nodal) redun-dancy, as well as structural invariance. It may seem at first that the complexity ofa bus transportation network is much lesser than that of other large-scale networks,however it is the nature of the growth and the penetrative effect of these networksthat makes them not only complex but interesting and worthwhile to investigate.4 Methodology
We obtain the route data for all the bus networks from the respective state gov-ernment websites. Every stop is considered a node, and the routes joining thestops form the set of links. We define a graph, G = ( N, L ) where the set N =( n , n , n , ... ) with each n i as a bus-stop, and the set L = ( l , l , l , ... ) where each l i connects the node pair ( n i , n j ). The set of nodes belong to the n-dimensionalEuclidean space, R n , and the set of links form the Cartesian product over R n .We define the set of routes as the set R such that ∪ i l i ∈ R for some i . In orderto analyze the networks, we generate the graph adjacency matrix, A ij such thatany matrix element a ij of A ij is either equal to one or zero depending upon theexistence of a connecting link between node-pair ( i, j ). The degree of any node isgiven as k i = Σ i a ij . The above formulation generates a L -space network withoutweights. In order to assign weights, we calculate the route overlaps between a pairof nodes which we call edge-weights, w ij . The degree strength matrix is given by s ij = a ij × w ij and the weighted degree or node-strength as s i = Σ i s ij . Since theflow of transport is along both the directions, we consider the network links to beundirected. The local clustering coefficient is given by C ( i ) = | a ij :( n i ,n j ) ∈ N,a ij ∈ A ij | k i ( k i − where a ij is the link connecting node pair ( i, j ), and k i are the neighbours of thenode n i . The neighbourhood, n i , for a node, i is defined as the set of its immediatelyconnected neighbours, as n i = { n j : l i ∈ L ∧ l j ∈ L } . For the complete network,Watts and Strogatz defined a global clustering coefficient [5, 6], C = Σ i C i /n . Theweighted clustering coefficient is given as [35] C w ( i ) = s i ( k i − Σ j,h w ij + w ih a ij a ih a jh .Another important measure is the characteristic path length, l ij which is definedas the average number of nodes crossed along the shortest paths for all possible pairsof network nodes. The average distance from a certain vertex to every other vertexis given by d i = Σ i (cid:54) = j d ij | N ( G ) |− . Then, l ij is calculated by taking the median of allthe calculated d i ∀ i ∈ R n . In order to check the small-world property, we generaterandom graphs of same size, i.e., keeping network size N constant. However, thenetwork topology of a random graph is governed by a wiring probability, p w whichdetermines the connectedness of the network (or the number of edges of the net-work). In order to generate random networks of comparable sizes (similar numberof nodes and edges), we calculate the wiring probability as p w N ∼ N . The cen-tralities, betweenness and closeness tell us the relative importance of nodes in thenetwork. Betweenness centrality of any node is calculated as, C B ( i ) = Σ s (cid:54) = i (cid:54) = t σ s,t ( i ) σ s,t ,where σ s,t is the number of shortest paths connecting s to t and σ s,t ( i ) number ofshortest paths connecting s to t but passing through i . Likewise, closeness centralityfor any node is calculated by C C ( i ) = Σ i a ij . The average closeness is the harmonicmean of the shortest paths from any node to every other node. In weighted net-works, usually the edge weights are considered as cost functions; therefore, largerthe edge weight, lesser is the node’s closeness, as the cost of travel would be large.However, in our case the edge weights play an altogether different role signifying the‘ease’ of travel. Hence, we take the inverse of edge weights during the calculationof weighted C C as in collaboration networks given by C wC ( i ) = min Σ i ( w ij ).The degree-assortativity or the Pearson correlation coefficient of degree between5airs of linked nodes is given by Σ jk jk ( e jk − q j q k ) σ q , where e jk is the joint probabilitydistribution of the remaining degrees of the two vertices at either end of a randomlychosen edge with Σ j,k e jk = 1 and Σ j e jk = q k . Here, q k is the normalized degree-distribution of the remaining degrees, and σ q is the variance of the distribution q k given by[36] σ q = Σ k k q k − [Σ k kq k ] . The degree-distribution P ( k ) gives the prob-ability of finding a node with a degree k in the network, which basically representsthe ratio of all the nodes with degree equal to k to the size of the network, N . Thedegree-distribution is observed to follow a heavy-tailed function. The equation forthe power-law or exponential fits (in Table 1 and Figure 2) are calculated usingMaximum Likelihood Estimation (MLE) and the Kolmogorov-Smirnov test is em-ployed to check for goodness of fit [37]. The degree-strength correlation is evaluatedusing linear-regression model, and the least-square error is calculated. The datasets were obtained from the government websites of Ahmedabad BRTS(ABN), MTC (CBN), DTC (DBN), APSRTC (HBN), CSTC (KBN) and BEST(MBN). In Figure 1, we plot the network structure using force directed algorithms.The figure compares the structural construct of the networks. We can clearly ob-serve the nature of connectivity between the nodes in the different networks. WhileDBN is densely packed, CBN, HBN and KBN are sparse. The network structure ofMBN is particularly striking. The long branches with multiple intermediate nodesas seen from the figure cause the characteristic path-length, l ij of MBN to increaseabnormally (see Table 1). We also calculate the modularity of the networks to iden-tify community structure. Networks with high modularity have dense connectionsbetween the nodes within the same modularity class but weak connections betweennodes in different modularity class. In order to identify communities we colour-codethe nodes based upon the modularity classes. Community detection in bus networkshelp us in identifying the different zones of operation. As large as six communitieswere identified for CBN and MBN whereas fewer (four or less) communities wereidentified for ABN, DBN, HBN and KBN.In Table 1, we present the statistical analysis for the various networks in atabular form. It can be seen from the table that the network sizes of all the citiesare comparable to each other, except that of KBN because CSTC is localized andoperates as a subdivision of West Bengal Surface Transport Corporation (WBSTC)that operates buses in the entire state. The network density, ρ , which is the ratio ofthe number of edges in a given network to the corresponding completely connectedgraph varies from 0 .
001 to 0 . l ij from as low as 3 .
87 to as high as 10 .
02. In order toget a deeper insight into the structure of these networks, we carried out a weightedanalysis by assigning a weight corresponding to the overlap of routes connectinga particular pair of nodes that helps us understand the potential flow of trafficbetween that nodal pair. The weighted degree of a node or its strength is observedto follow a heavy-tailed distribution on a double logarithmic scale, and the nodestrength and node degree are found to be related non-linearly. This implies that the6 us routes Nodes Edges l ij C av γ Assortativity < k >
ABN
CBN
DBN
HBN
KBN
518 884 5.72 0.08 4.96 -0.01 6.72
MBN l ij = characteristic path length, C av = average weighted clusteringcoefficient, γ = power-law exponent, and < k > = average node degree).potential traffic at a node due to route overlaps increases exponentially as comparedto the actual number of routes it is connected to [35].We observe that the average clustering coefficient, C av also shows a remark-able variation from 0 .
07 to as high as 0 .
26. We check the presence of small-worldphenomenon in the above networks by generating random graphs with the samenumber of nodes and comparable number of edges, and calculate the characteristicpath length, l randij and average clustering coefficient, C randav in each case. Upon com-paring with the data in Table 1, we find that C av >> C randav each time, whereas l ij is either comparable to l randij or l ij < l randij . Based upon the above comparisons, wecan state that the bus networks show small-world phenomenon. As we discussedearlier, L -space formulation merely gives the relationship between bus stops andbus routes, whereas it is the P -space formulation which helps in determining thenumber of transfers, or in this case, number of bus changes. We can estimate thenumber of bus changes required by looking at the average number of bus stopspresent in each of the routes. CBN and MBN typically show the largest magni-tudes of characteristic path-lengths in L -space. A P -space analysis for both CBNand MBN, reveals the number of transfers as low as 2 −
3. Thus, all the networksstudied in this paper show small-world behaviour in P -space topology [38].As discussed earlier, node-degree distribution plays an important role in un-derstanding the structure and evolution of complex networks. In Figure 2 (a), weplot the degree distribution for all the networks on a double logarithmic scale. Thedegree-distribution patterns show mostly heavy-tailed characteristics, with MBNshowing a slight deviation from the power-law behaviour. In Figure 2 (b), we plotthe centrality distributions (closeness and betweenness), P ( C C ) and P ( C B ) in thefirst two rows for ABN, HBN (scale-free) and MBN (non scale-free) on a doublelogarithmic scale to contrast the differences between scale-free networks and nonscale-free ones. We find that the distribution function follows an exponential decaygiven by P ( C C ) ∼ exp( − λC C ) (similarly for C B ) where the value of the exponent λ is shown in each of the plots. In the last row, we plot the variation of betweennesscentrality with the degree of a node which follows a power-law relationship, givenas C B ∼ k α with the magnitude of the exponent α also shown in the plots.In Figure 3, we plot the response of the network’s characteristic path length, l ij to random and systematic perturbation. We simulate the robustness and resiliency7f the networks by modeling perturbations as node removals. Due to their strongassortative nature, MBN and CBN disintegrate into separate entities very quickly,whereas the other networks remain connected upto atleast 4% of node removals.It is observed that in all the cases the targeted node removals are crucial for thenetwork to remain connected. In the regime of p i ≤ l ij does not change much (at most it increases by one ‘hop’). Finally,in Figure 4, we plot the degree-distribution for ABN and MBN after removing20% , In this paper, we analyzed the statistical properties of the bus routes of the six In-dian cities, namely Ahmedabad, Chennai, Delhi, Hyderabad, Kolkata, and Mumbai.Our analysis suggests that the bus networks show a wide spectrum of topologicalstructure from power-law to exponential with varying magnitude of the power-lawexponent γ . Ahmedabad (ABN) is particularly interesting in this regard becauseit has a BRTS (Bus Rapid Transit System) with dedicated lanes - a type of pub-lic transit system that is yet to be introduced at a large scale in India. ABN’sBRTS, thus, holds a structural advantage by the presence of many hubs to whichextreme routes are connected, a structure similar to WWW or the airline networks(WAN and ANI) [24, 35]. As we saw in the earlier sections, CBN and MBN do notshow the small-world property in L -space. They, however, do show the small-worldproperty in terms of transfers ( P -space topology), as majority of the places can bevisited by making as little as 2 to 3 bus changes [38]. The structural relationshipbetween bus stops as observed from the degree-distribution plots in Figure 2 is ofparticular interest. In Figure 2, we plot the weighted degree-distribution of thenetworks which capture the strength of the nodes with respect to the traffic han-dled in terms of the number of routes. In order to check for correlations betweennode degree, k and node weighted-degree, s we plot them on a double-logarithmicscale. Interestingly, ABN shows a strong correlation as, s ∼ k β with β = 1 . R = 0 .
91, whereas the other networks fail to show such strong relationships(CBN, KBN and HBN show similar relationships with β ∼ . − .
08, however,with lower correlation coefficients, R ∼ . − . γ as 2 .
47, whereas the degree-strengthexponent, β is found to be 1 .
27. This implies that the strength of a node increasesfaster as compared to its degree indicating a sense of order in ABN where higherdegree nodes, for example, large or important bus stops, handle heavy traffic asmajority of the routes pass through them. This is definitely missing in the othernetworks where the edge weights or routes seem to be more randomly distributed.Also the topological structure of the road networks in the city of Ahmedabad showa scale-free degree distribution with γ = 2 . l ij = 5 .
20, which is very similar toABN [10] (see Table 1).In Figure 2 (b), we plot the centrality distribution for betweenness ( C B ) and8loseness ( C C ). We consider betweenness and closeness because they play a crucialrole from a transportation perspective. C C is a measure of a node’s relative im-portance in the network due to the existence of shortest paths from that particularnode to every other node in the entire network. C B on the other hand acts as abridging node connecting different parts of the network together. When travelingfrom one node to the other, it is often beneficial to get to the node with the high-est value of C C first if a direct path does not exist between the origin-destinationpair. Often transportation network of a city is planned in a way such that the hubsallow maximum number of routes to pass through them, and all other nodes in thenetwork to be easily reachable from them. Since, centrality is positively correlatedto node degree, the hubs in a network also tend to have the largest degrees. Wefound this pattern in all the networks, ( C B ∼ k α ); however, in DBN and MBN therelationship between degree and centrality is not that strong perhaps due to thepresence of noise in the network due to random attachment of nodes (see Figure 2(b) last row). The noise or the presence of redundant nodes (links) due to randomattachment of the nodes in the network causes the degree-distribution patterns toshift from a purely power-law decay to truncated power-law and exponential de-cays. The presence of these redundant nodes increase the degree of non-centralnodes which is observed in the degree-centrality plots (see Figure 2). These nodesdue to their random placement tend to appear at random places in the networkcausing hindrance in the direct connectivity of the hubs. The networks (exceptCBN and MBN) therefore show disassortative or weakly assortative behaviours.We also observe that the centrality-distribution functions follow exponential decay,as P ( C C ) ∼ exp( − λC C ) (similarly, for C B ) which shows that nodes in a networkare different, i.e., some nodes are more ‘central’ as compared to other nodes. An in-teresting observation is that nodes in the networks tend to connect to existing highdegree nodes preferentially whereas such a preferential attachment rule is missingwhen, for example, node-betweenness is considered as the metric. A close observa-tion in Figure 2 (b) reveals that nodes with high betweenness certainly have highdegrees however, the reverse is not true.Some nodes do not play any significant role in the network’s overall functionality,i.e., they are redundant. In Figure 3, we evaluate the network’s response to externalperturbations by random and directed removal of nodes. We fix an importantmeasure l ij and check its variation upon percentage removal of nodes (bus stops).As we saw earlier, CBN and MBN due to their strong assortative behaviour, seem tobe very sensitive to node removals as they quickly disintegrate, whereas ABN, DBN,HBN and KBN do not show any significant change in l ij upto 4% of node removal.This basically amounts to roughly 40 −
70 nodal redundancy (in numbers), that ifremoved can reduce cost of construction, operation, and maintenance significantlyin the network. However accessibility for all users has to be carefully studied beforeremoving any node. We also observe that the clustering coefficient C varies inverselywith the node degree which implies that the nodes with low clustering coefficientstend to have higher degrees and vice-versa. This is because nodes (bus stops) havinghigher degree will be a part of multiple bus routes whereas, those bus stops throughwhich fewer bus routes pass will have lower degree. Thus, it is more likely for thenodes in the later case to form clusters as compared to the ones which are connected9o multiple bus routes.Finally in Figure 4, we observe that the topological structure of the networksare preserved (Figure 4 (a) and 4 (b)) when the networks are subjected to largenumber of node removals. It can also be clearly seen that the networks are degree-sensitive. Degree-biased node removal causes the heavy tails in the degree-topologyto disappear thus signifying gradual decrease in the number of hubs. Interestingly,a similar effect is also observed when the nodes are removed based upon theirbetweenness centralities. Although, the effect is relatively less significant, it is morewhen compared to closeness biased node removals and random node removals. InFigure 4 (b), we plot the degree-distributions for MBN with respect to percentagenode removal. In case of MBN it is particularly interesting to note that the degree-distribution plots, which originally showed a better fit for exponential distribution(Figure 2 (a)), evolves into a scale-free topology (as can be observed from straightline slope in the double-logarithmic scale) with varying power-law exponent, γ ,when nodes are removed. At 20% node-removal, MBN starts showing heavy-taileddegree topology. The above phenomenon could be attributed to the reduction ofnoise (randomness of connectivity and nodal redundancy) due to removal of nodes.Also, from Table 1, we observe that the bus networks, like all other surfacetransport networks are assortative in nature with HBN and KBN showing weakdisassortative behaviour. The strong assortativity observed in these networks resultin increased characteristic path-lengths. Since, the nodes (bus stops) are spatiallydistributed throughout the city, the tendency of similar nodes to attach to nodeswith similar statistical properties causes the characteristic path-lengths to increasesignificantly. From a transportation perspective, assortative mixing is beneficialas this will allow direct connectivity between hubs. However, it will also increasethe number of hops in traversing from any given source to a destination withinthe network. In terms of transfers, the small-world property is retained, yet thetraveling time between any random origin-destination pair will increase, due todelays associated with numerous intermediate stops.As noted earlier, bus networks form a specific class of complex networks thatgrow and evolve over physically constrained spatial networks. Road intersectionsare usually separated by a distance which is geographically much smaller as com-pared to the distance between bus stops; therefore, our results emphasize thattransportation undoubtedly brings the world closer. What we observed from ourpaper is that bus networks show scale-free topology and small-world property inthe number of transfers. Also, from the above analysis we observe that the bus net-works although structurally different, show similar as well as self-similar topologicalstructures. With the exception of MBN, all the networks show scale-free topologywith MBN showing slight deviation towards an exponential distribution. The pres-ence of heavy-tails in the degree-distribution plots imply a preferential attachmentrule, the tendency of high degree nodes to cluster with low degree nodes reveal a hi-erarchical organization, and the stability of characteristic path length with gradualremoval of nodes reveal the presence of nodal redundancy in the network.The present study opens before us new horizons for efficient transportation net-work designing and planning. Questions such as: what are the statistical propertiesof the network that will ensure efficiency or how network topology is related to the10tatistical properties and vice-versa would be both challenging and worthwhile toanswer. It would be exciting to come up with innovative models to capture thegrowth and evolution of real-world large scale public transit networks. Developinggenerative methods to reduce noise (in the network) due to random node attachmentby including geographic and socio-economic constraints such as demand, flow, andcost, to maximize certain network parameter(s) or node-utility function(s) basedon the above constraints is another promising area of future work. References [1] Barab´asi AL, Albert R. Emergence of scaling in random networks. Science.1999;286(5439):509–512.[2] Albert R, Barab´asi AL. Statistical mechanics of complex networks. Reviewsof Modern Physics. 2002;74(1):47.[3] Albert R, Jeong H, Barab´asi AL. Internet: Diameter of the world-wide web.Nature. 1999;401(6749):130–131.[4] Dorogovtsev SN, Goltsev A, Mendes JFF. Pseudofractal scale-free web. Phys-ical Review E. 2002;65(6):066122.[5] Newman ME. The structure and function of complex networks. SIAM Review.2003;45(2):167–256.[6] Watts DJ, Strogatz SH. Collective dynamics of ‘small-world’ networks. Nature.1998;393(6684):440–442.[7] Ravasz E, Barab´asi AL. Hierarchical organization in complex networks. Phys-ical Review E. 2003;67(2):026112.[8] Niwa HS. Power-law versus exponential distributions of animal group sizes.Journal of Theoretical Biology. 2003;224(4):451–457.[9] Jiang J, Calvao M, Magalhases A, Vittaz D, Mirouse R, Kouramavel F, et al.Study of the Urban Road Networks of Le Mans. arXiv preprint arXiv:10020151.2010;.[10] Porta S, Crucitti P, Latora V. The network analysis of urban streets:a dual approach. Physica A: Statistical Mechanics and its Applications.2006;369(2):853–866.[11] Sen P, Dasgupta S, Chatterjee A, Sreeram P, Mukherjee G, Manna S.Small-world properties of the Indian railway network. Physical Review E.2003;67(3):036106.[12] Chatterjee A, Ramadurai G. Scaling Laws in Chennai Bus Network. In:4th International Conference on Complex Systems and Applications, France.https://halshs.archives-ouvertes.fr/halshs-01060875/document; 2014. p. 137–141. 1113] Ahuja RK, Magnanti TL, Orlin JB. Network flows. DTIC Document; 1988.[14] Bertsimas D, Sim M. Robust discrete optimization and network flows. Math-ematical programming. 2003;98(1):49–71.[15] Derrible S, Kennedy C. Network analysis of world subway systems using up-dated graph theory. Transportation Research Record: Journal of the Trans-portation Research Board. 2009;(2112):17–25.[16] Zhang Y, Zhang Q, Qiao J. Analysis of Guangzhou metro network based onL-space and P-space using complex network. In: Geoinformatics (GeoInfor-matics), 2014 22nd International Conference on; 2014. p. 1–6.[17] Erd¨os P, R´enyi A. On the evolution of random graphs. Publ Math Inst HungarAcad Sci. 1960;5:17–61.[18] Strogatz SH. Exploring complex networks. Nature. 2001;410(6825):268–276.[19] Albert R, Albert I, Nakarado GL. Structural vulnerability of the North Amer-ican power grid. Physical Review E. 2004;69(2):025103.[20] Bork P, Jensen LJ, von Mering C, Ramani AK, Lee I, Marcotte EM. Pro-tein interaction networks from yeast to human. Current opinion in structuralbiology. 2004;14(3):292–299.[21] Jeong H, Mason SP, Barab´asi AL, Oltvai ZN. Lethality and centrality inprotein networks. Nature. 2001;411(6833):41–42.[22] Easley D, Kleinberg J. Networks, crowds, and markets: Reasoning about ahighly connected world. Cambridge University Press; 2010.[23] i Cancho RF, Sol´e RV. Least effort and the origins of scaling in human language.Proceedings of the National Academy of Sciences. 2003;100(3):788–791.[24] Bagler G. Analysis of the airport network of India as a complexweighted network. Physica A: Statistical Mechanics and its Applications.2008;387(12):2972–2980.[25] Von Ferber C, Holovatch T, Holovatch Y, Palchykov V. Public transport net-works: empirical analysis and modeling. The European Physical Journal B.2009;68(2):261–275.[26] Woolley-Meza O, Thiemann C, Grady D, Lee J, Seebens H, Blasius B, et al.Complexity in human transportation networks: a comparative analysis ofworldwide air transportation and global cargo-ship movements. The EuropeanPhysical Journal B. 2011;84(4):589–600.[27] Guimera R, Mossa S, Turtschi A, Amaral LN. The worldwide air transportationnetwork: Anomalous centrality, community structure, and cities’ global roles.Proceedings of the National Academy of Sciences. 2005;102(22):7794–7799.1228] Sienkiewicz J, Ho(cid:32)lyst JA. Statistical analysis of 22 public transport networksin Poland. Physical Review E. 2005;72(4):046127.[29] Angeloudis P, Fisk D. Large subway systems as complex networks. Physica A:Statistical Mechanics and its Applications. 2006;367:553–558.[30] Xu X, Hu J, Liu F, Liu L. Scaling and correlations in three bus-transportnetworks of China. Physica A: Statistical Mechanics and its Applications.2007;374(1):441–448.[31] Chen YZ, Li N, He DR. A study on some urban bus transport networks.Physica A: Statistical Mechanics and its Applications. 2007;376:747–754.[32] Ma K, Wang Z, Jiang J, Zhu G, Li W. Power law and small world prop-erties in a comparison of traffic city networks. Chinese Science Bulletin.2011;56(34):3731–3735.[33] Soh H, Lim S, Zhang T, Fu X, Lee GKK, Hung TGG, et al. Weighted complexnetwork analysis of travel routes on the Singapore public transportation sys-tem. Physica A: Statistical Mechanics and its Applications. 2010;389(24):5852–5863.[34] Yang XH, Chen G, Chen SY, Wang WL, Wang L. Study on some bus trans-port networks in China with considering spatial characteristics. TransportationResearch Part A: Policy and Practice. 2014;69:1–10.[35] Barrat A, Barthelemy M, Pastor-Satorras R, Vespignani A. The architecture ofcomplex weighted networks. Proceedings of the National Academy of Sciencesof the United States of America. 2004;101(11):3747–3752.[36] Newman ME. Assortative mixing in networks. Physical review letters.2002;89(20):208701.[37] Clauset A, Shalizi CR, Newman ME. Power-law distributions in empiricaldata. SIAM review. 2009;51(4):661–703.[38] Chatterjee A. Studies on the Structure and Dynamics of Urban Bus Networksin Indian Cities. arXiv preprint arXiv:151205909. 2015;.
The authors acknowledge the support from Center of Excellence in Urban Trans-port at the Indian Institute of Technology, Madras, sponsored by the Ministry ofUrban Development, Government of India and the Information Technology Re-search Academy, a Division of Media Labs Asia, a non-profit organization of theDepartment of Electronics and Information Technology, funded by the Ministry ofCommunications and Information Technology, Government of India.13
Author contributions statement
A.C. and G.R. conceived the idea, A.C. and M.M. collected the data and wrote themanuscript, A.C. wrote the codes and ran the simulations, A.C. and G.R. analyzedthe results. All authors reviewed the manuscript.
Competing financial interests
The authors declare no competing financial in-terests. 14 i g u r e : F i g u r e s h o w s t h e n e t w o r k s tr u c t u r e o f t h e d i ff e r e n t bu s r o u t e s w h e r ee a c hn o d e r e p r e s e n t s a bu ss t o p . T h e p l o t s a r e g e n e r a t e du s i n g f o r ce d i r ec t e d a l go r i t h m s a nd t h ec o l o u r o f n o d e s p a rt i t i o n t h e n e t w o r k s i n t o d i ff e r e n t c o mm un i t i e s . W h e r e A B N , D B N a nd H B N s h o w t y p i c a l s c a l e - f r ee s tr u c t u r e o b s e r v e t h e l o n g r o u t e s p r e s e n t i n C B N a nd M B N . P ( k ) on a double logarithmicscale; (b) Figure shows centrality distribution for betweenness ( C B ) and closenesscentralities ( C C ) with the decay exponent λ (inset). The plots in the last row showdegree-betweenness dependency with exponent α (inset).16 i g u r e : F i g u r e s h o w s t h e v a r i a t i o n i n l i j w i t hn e t w o r k s i ze up o n r a nd o m a nd t a r g e t e dn o d e r e m o v a l s . T h e d a r k li n e r e p r e s e n t s d e g r ee - b a s e dn o d e r e m o v a l s , a nd t h e li g h t li n e r e p r e s e n t s r a nd o m n o d e r e m o v a l s . T h e X - a x i s r e p r e s e n t s t h e f r a c t i o n o f n o d e s r e m o v e d . i g u r e : F i g u r e s h o w s d e g r ee - d i s tr i bu t i o np l o t s f o r( a ) A B N a nd ( b ) M B N s ub j ec t e d t o r a nd o m a ndd i r ec t e d a tt a c k s f o r p e r ce n t ag e o f n o d e r e m o v a l s , p = % , % a nd % ..