Topological Distance Research Articles

Leak Localization in Water Distribution Networks Using Pressure and Data-Driven Classifier Approach

Leaks in water distribution networks (WDNs) are one of the main reasons for water loss during fluid transportation. Considering the worldwide problem of water scarcity, added to the challenges that a growing population brings, minimizing water losses through leak detection and localization, timely and efficiently using advanced techniques is an urgent humanitarian need. There are numerous methods being used to localize water leaks in WDNs through constructing hydraulic models or analyzing flow/pressure deviations between the observed data and the estimated values. However, from the application perspective, it is very practical to implement an approach which does not rely too much on measurements and complex models with reasonable computation demand. Under this context, this paper presents a novel method for leak localization which uses a data-driven approach based on limit pressure measurements in WDNs with two stages included: (1) Two different machine learning classifiers based on linear discriminant analysis (LDA) and neural networks (NNET) are developed to determine the probabilities of each node having a leak inside a WDN; (2) Bayesian temporal reasoning is applied afterwards to rescale the probabilities of each possible leak location at each time step after a leak is detected, with the aim of improving the localization accuracy. As an initial illustration, the hypothetical benchmark Hanoi district metered area (DMA) is used as the case study to test the performance of the proposed approach. Using the fitting accuracy and average topological distance (ATD) as performance indicators, the preliminary results reaches more than 80% accuracy in the best cases.

Simulation of the multiphase configuration and phase transitions with quantum walks utilizing a step-dependent coin

Quantum walks are versatile simulators of topological phases and phase transitions as observed in condensed matter physics. Here, we utilize a step dependent coin in quantum walks and investigate what topological phases we can simulate with it, their topological invariants, bound states and possibility of phase transitions. These quantum walks simulate non-trivial phases characterized by topological invariants (winding number) $\pm 1$ which are similar to the ones observed in topological insulators and polyacetylene. We confirm that the number of phases and their corresponding bound states increase step dependently. In contrast, the size of topological phase and distance between two bound states are decreasing functions of steps resulting into formation of multiple phases as quantum walks proceed (multiphase configuration). We show that, in the bound states, the winding number and group velocity are ill-defined, and the second moment of the probability density distribution in position space undergoes an abrupt change. Therefore, there are phase transitions taking place over the bound states and between two topological phases with different winding numbers.

Central nervous system compartmentalization of HIV, an interesting model of viral evolution: from theory to clinical implications

The human immunodeficiency virus (HIV) infects the entire human body. Genetic diversity within the viral population of an infected individual is considerable and heterogeneous. Viral subpopulations develop during infection, which can lead to independent evolutions according to anatomical compartments -organs, tissues or cells- i.e. the phenomenon of viral compartmentalization. The identification of compartmentalization relies on different tests based on phylogenetic analysis of sequences of viral populations, tree topology and genetic distances. Compartmentalized viral populations are found in the central nervous system in nearly half of HIV-infected individuals. This is particularly the case when severe neurocognitive disorders, including dementia, are present. Different genetic and phenotypic properties are associated with compartmentalized HIV variants of the central nervous system, in particular the enhanced ability to infect macrophages. This is an interesting model for studying viral compartmentalization given the tightness of the blood-brain barrier, the peculiar cellular environment for this virus, and the clinical implications for the pathophysiology of HIV infection.

Synthesis, Characterization, Antibacterial and Antioxidant Potency of NSubstituted- 2-Sulfanylidene-1,3-Thiazolidin-4-one Derivatives and QSAR Study.

Rhodanine is known for its potential and important role in the medicinal chemistry since its derivatives exhibit a wide range of pharmacological activities such as antibacterial, antifungal, antidiabetic, antitubercular, anti-HIV, antiparasitic, antioxidant, anticancer, antiproliferative and anthelmintic agents. Since N-substituted rhodanine synthons are rarely commercially available, it is desirable to develop a straightforward synthetic approach for the synthesis of these key building blocks. The objective was to synthesize a series of rhodanine derivatives and to investigate their antimicrobial and antioxidant activity. Also, in order to obtain an insight into their structure-activity relationship, QSAR studies on the antioxidant activity were performed. 1H and 13C FTNMR spectra were recorded on Bruker Avance 600 MHz NMR Spectrometer, mass analysis was carried out on ESI+ mode by LC-MS/MS API 2000. 2,2-Diphenyl-1- picrylhydrazyl radical scavenging activity (% DPPH) was determined in dimethylsulfoxide (DMSO) as a solvent. The antibacterial activity was assessed against Bacillus subtilis, Staphylococcus aureus (Gram positive) and Escherichia coli, Pseudomonas aeruginosa (Gram negative) bacteria in terms of the minimum inhibitory concentrations (MICs) by a modified broth microdilution method. A series of N-substituted-2-sulfanylidene-1,3-thiazolidin-4-ones were synthesized and characterized by 1H NMR, 13C NMR, FTIR, GC MS, LCMS/MS and C,H,N,S elemental analysis. Most of the synthesized compounds showed moderate to excellent antibacterial activity (MIC values from 125 μg/ml to 15.62 μg/mL) and DPPH scavenging activity (from 3.60% to 94.40%). Compound 2-thioxo-3- (4-(trifluoromethyl)-phenyl)thiazolidin-4-one showed the most potent activity against Escherichia coli (3.125 μg/mL), equivalent to antibiotic Amikacin sulphate and against Staphylococcus aureus (0.097 μg/ml), 100 times superior then antibiotic Amikacin sulphate. It has also shown a potent antioxidant activity (95% DPPH scavenging). Two best QSAR models, obtained by GETAWAY descriptor R7p+, Balabans molecular connectivity topological index and Narumi harmonic topological index (HNar), suggest that the enhanced antioxidant activity is related to the presence of pairs of atoms higher polarizability at the topological distance 7, substituted benzene ring and longer saturated aliphatic chain in N-substituents. A series of novel N-substituted-2-thioxothiazolidin-4-one derivatives were designed, synthesized, characterized and evaluated for their antibacterial and antioxidant activity in vitro. Majority of the compounds showed excellent antibacterial activity compared to ampicillin and few of them have an excellent activity as compared to Chloramphenicol standard antibacterial drug. The QSAR study has clarified the importance of presenting a pairs of atoms higher polarizability, such as Cl and S at the specific distance, as well as the substituted benzene ring and a long saturated aliphatic chain in N-substituents for the enhanced antioxidant activity of 2-sulfanylidene-1,3- thiazolidin-4-one derivatives.

Reversible Spatiotemporal Control of Induced Protein Degradation by Bistable PhotoPROTACs.

Off-tissue effects are persistent issues of modern inhibition-based therapies. By merging the strategies of photopharmacology and small-molecule degraders, we introduce a novel concept for persistent spatiotemporal control of induced protein degradation that potentially prevents off-tissue toxicity. Building on the successful principle of bifunctional all-small-molecule Proteolysis Targeting Chimeras (PROTACs), we designed photoswitchable PROTACs (photoPROTACs) by including ortho-F4-azobenzene linkers between both warhead ligands. This highly bistable yet photoswitchable structural component leads to reversible control over the topological distance between both ligands. The azo-cis-isomer is observed to be inactive because the distance defined by the linker is prohibitively short to permit complex formation between the protein binding partners. By contrast, the azo-trans-isomer is active since it can engage both protein partners to form the necessary and productive ternary complex. Importantly, due to the bistable nature of the ortho-F4-azobenzene moiety employed, the photostationary state of the photoPROTAC is persistent, with no need for continuous irradiation. This technique offers reversible on/off switching of protein degradation that is compatible with an intracellular environment and, therefore, could be useful in experimental exploration of biological signaling pathways—such as those crucial for oncogenic signal transduction. Additionally, this strategy may be suitable for therapeutic intervention to address a variety of diseases. By enabling reversible activation and deactivation of protein degradation, photoPROTACs offer advantages over conventional photocaging strategies that irreversibly release active agents.

Incremental Updating Information Extraction and Topology Conflict Detection Method for Updating Road Network

Abstract. In recent years, Located Based Service (LBS) has become one of the hotspots on the application of geographic information to the government and the public. Providing a good performance map to the users is an important part of the location service system. Compared with the traditional web spatial information service system, service system of map information also put forward higher requirements of real time and veracity. The problem of updating electronic map in real-time cannot be effectively solved, hindering the updating of multi-scale electronic map and restricting the development of location service market. As a very important branch for the updating data field of geographic information science, incremental update has become a hot issue in the current international GIS research field. The extraction of incremental information includes two key technologies, namely road network matching and road network selection. After incremental updating roads were inserted into the database, coordinating topological conflicts caused by these inserted roads was the mainly task. However, the accuracy and algorithm efficiency of each technology need further improvement to provide the users with the latest and the most accurate basic geographic data. As the "skeleton" element of a map, the road network rapidly changes, carrying out more urgent requires for the incremental road update. Aiming at key technology problems of the updated road network, the paper focused on the following three parts:(1) To address the problems that the traditional probabilistic relaxation method only adopted geometric constraints as one of road matching criterions and could not respond to M: N matching pattern, an improved probabilistic relaxation method was proposed from the combined views of local optimization and global one, integrating geometric indicators with topology ones to achieve an effect with local optimization, as well as identifying M: N matching pattern by inserting virtual nodes to achieve a globally optimal effect. Then the matching strategies and corresponding implement algorisms were designed for different matching patterns. The case test showed that the overall matching accuracy of each evaluation indictor reached over 90%, increasing by 6%–12%; the evaluation indicators on both spatial and attribute properties increased by 4%–6%; the proper buffer threshold could be defined as twice the average value of the closest distances from all nodes in the candidate matching dataset.(2) Aiming at the low accuracy and irrational structural selection caused by only using linear pattern or areal pattern, an improved method was proposed combining linear method as well as areal ones. The proposed method improves the Stroke generation algorithm of linear pattern using OLS model taking the overall information from the roads to be connected. Meanwhile, it partitioned road network by weighted Voronoi diagrams. The roads were selected under the process of calculating the importance of Stroke and its road density threshold. The test demonstrated that the improved method added the accuracy of Stroke generation results, and the results of road selection could maintain the overall characteristics of the road network as well as its spatial distribution information. Compared with method only using one pattern, the combined method had a better accuracy of road selection.(3) According to the characteristics of detection topology for an updated road network, this paper proposed a topology conflict detection algorithm considering the incremental update of multi-scale network. The algorithm designed K-level topological neighbourhood to identify incremental neighbourhood road segments, built a topological refinement model based on geometric metrics, proposed checking rules based on comprehensive operation operator, and detected the conflict topology by using improved topological distance. The experimental results showed that 1) the accuracy and recall of the proposed method were more than 90%; 2) considering conflict topology caused by the generalization, the accuracy increased by 29.2%; 3) the average path length of a road network could be used as the reference of the best K-value in the K order incremental neighbourhood method.

Post-Traumatic Cerebral Microhemorrhages and their Effects Upon White Matter Connectivity in the Aging Human Brain.

Cerebral microbleeds (CMBs), a common manifestation of mild traumatic brain injury (mTBI), have been sporadically implicated in the neurocognitive deficits of mTBI victims but their clinical significance has not been established adequately. Here we investigate the longitudinal effects of post-mTBI CMBs upon the fractional anisotropy (FA) of white matter (WM) in 21 older mTBI patients across the first ~6 months post-injury. CMBs were segmented automatically from susceptibility-weighted imaging (SWI) by leveraging the intensity gradient properties of SWI to identify CMB-related hypointensities using gradient-based edge detection. A detailed diffusion magnetic resonance imaging (dMRI) atlas of WM was used to segment and cluster tractography streamlines whose prototypes were then identified. The correlation coefficient was calculated between (A) FA values at vertices along streamline prototypes and (B) topological (along-streamline) distances between these vertices and the nearest CMB. Across subjects, the CMB identification approach achieved a sensitivity of 97.1% ± 4.7% and a precision of 72.4% ± 11.0% across subjects. The correlation coefficient was found to be negative and, additionally, statistically significant for 12.3% ± 3.5% of WM clusters (p <; 0.05, corrected), whose FA was found to decrease, on average, by 11.8% ± 5.3% across the first 6 months post-injury. These results suggest that CMBs can be associated with deleterious effects upon peri-lesional WM and highlight the vulnerability of older mTBI patients to neurovascular injury.

Towards a Phenomenology of the Digitalization of Consciousness. The Virtualization of the Social Space

The digitalization of the everyday experience is a process which achieves virtual reality through a process of replacing the original and authentic experience with digital artefacts: cybersex, augmented reality, revolutions 2.0. We are even proposed a full digitalization of the contents of consciousness, and the development of our own existence in virtual spaces, as a means of prolonging life indefinitely. The objective of the paper is to identify and analyze certain philosophical perspectives on the mutations at the level of the cultural model called transmodern society , following the process of “virtualization of the social space”. The virtual space is non-tridimensional. The characteristic of topological distance , specific to any human cohabitation is being replaced with that of intersubjective distance, regarded as the subjective intensity of the communication . The understanding of the virtualization of social space may open new horizons of investigation in the philosophy of mentalities, cultural anthropology, communication sciences and tehnoethics.

Development of R-Group Fingerprints Based on the Local Landscape from an Attachment Point of a Molecular Structure.

Molecular fingerprints are indispensable in medicinal chemistry for quantifying chemical structures. Fingerprints can be calculated for substructures with attachment points, which are positions where a substructure and a corresponding core structure connect. Because structures with attachment points can be crucial for understanding structure-activity relationships, fingerprints specialized for representing this structural feature are required. R-group fingerprints and R-group descriptors were proposed previously for this purpose; however, these molecular representations have limitations. Current R-group fingerprints do not emphasize information about attachment points, and R-group descriptors are too sensitive to changes in the topological path length from an attachment point. In the present work, we developed novel R-group fingerprints, termed R-path fingerprints, which contain substituent information from an attachment point without being sensitive to small differences in topological distances. The concept of the R-path fingerprints is to describe a chemical substructure from the viewpoint of an attachment point, to distinguish atomistic information around the attachment point and other parts of the substructure. This was achieved by considering all the paths on the shortest path between the attachment point and each atom in a substituent. Benchmark testing was conducted, including comparisons of similarity distributions and potency prediction for R-group substituents. The results showed that R-path fingerprints should be useful for classifying and comparing substructures with attachment points.

QUANTITATIVE STRUCTURE ACTIVITY RELATIONSHIP STUDY ON THE INHIBITORY ACTIVITY OF COPPER SCHIFF BASE COMPLEXES AGAINST CANDIDA ALBICAN

Quantitative Structure Activity Relationship (QSAR) study was performed on Copper Schiff base complexes. Multiple Linear Regression analysis and genetic function algorithms was employed to derive QSAR model for better activity. The derived QSAR model having Squared Correlation Coefficient R 2 = 0.8345, Cross Validation Squared Correlation Coefficient Q 2 = 0.6681 and predicted R squared (R 2 pred ) = 0.5980. The predictive ability of the derived model was also confirmed by internal and external cross validation techniques. The QSAR model indicate that the descriptors (MATS4p) Moron autocorrelation of lag 4 weighed by polarizability,(RCI) Ring Complexity index, (G2m) 2 nd component symmetry directional WHIM index/weighted by mass, BI0 [N-O] Presence/absence of N-O at topological distance 10 and (nF) Number of Fluorine atoms plays an important role in predicting the activities against anti-candida albican. The result obtained in this study was used in designing more potent Copper Complexes as anti-candida albican agents.

The E. coli transcriptional regulatory network and its spatial embedding.

Usually complex networks are studied as graphs consisting of nodes whose spatial arrangement is of no significance. Several real biological networks are, however, embedded in space. In this paper we study the transcription regulatory network (TRN) of E. coli as a spatially embedded network. The embedding space of this network is the circular E. coli chromosome, i.e. it is practically one dimensional. However, the TRN itself is a high-dimensional network due to the existence of an adequate number of long-range connections. We find that nodes in short topological distance l = 1, 2 tend, on average, to be in shorter spatial distances r indicating an abundance of short-range connections as well. Community analysis of the TRN reveals the interesting fact that highly interconnected subnets consist of nodes that tend to be in spatial proximity on the circular chromosome. We also find indications that for certain transcriptional aspects of the E. coli it is advantageous to treat the circular genome as two line segments starting from the OriC and ending to Ter.

Orientational versus esthetical urban street morphology parameterization in Space Syntax

ABSTRACTCurrent topological and geometrical distances in Space Syntax are based on the premise that each change of direction along a path is a mental cost because one loses orientation. This paper proposes to extend the analysis to the case in which esthetic and variety, rather than orientation, are the key elements of street selection. It is widely recognized that most people are attracted to curvy paths rather than straight ones; therefore, when one is not worried about losing orientation in her walk, or when the preferred path is well recognizable even when it requires changes of direction, we should adopt another criterion to weigh distances: morphoahestetic and networkvariety criteria are shown as potential parameterization for Space Syntax distances.

Statistical force-field for structural modeling using chemical cross-linking/mass spectrometry distance constraints.

Chemical cross-linking/mass spectrometry (XLMS) is an experimental method to obtain distance constraints between amino acid residues which can be applied to structural modeling of tertiary and quaternary biomolecular structures. These constraints provide, in principle, only upper limits to the distance between amino acid residues along the surface of the biomolecule. In practice, attempts to use of XLMS constraints for tertiary protein structure determination have not been widely successful. This indicates the need of specifically designed strategies for the representation of these constraints within modeling algorithms. A force-field designed to represent XLMS-derived constraints is proposed. The potential energy functions are obtained by computing, in the database of known protein structures, the probability of satisfaction of a topological cross-linking distance as a function of the Euclidean distance between amino acid residues. First, the strategy suggests that XL constraints should be set to shorter distances than usually assumed. Second, the complete statistical force-field improves the models obtained and can be easily incorporated into current modeling methods and software. The force-field was implemented and is distributed to be used within the Rosetta ab initio relax protocol. Force-field parameters and usage instructions are freely available online (http://m3g.iqm.unicamp.br/topolink/xlff). Supplementary data are available at Bioinformatics online.

Enhanced Connection Adaption Strategy With Partition Approach

The control of dynamical networks is a topic of great value with applications across multiple disciplines. An efficient control scheme based on the connection adaption strategy (CAS) has been proposed by Zhou et al. who revealed that the controllability is restricted by the topological distance between the units and the controller. In this paper, based on the framework of the CAS, we propose a distributed control scheme by partitioning the network space into grids. With this partition approach, the distance between units and controller could be much reduced, resulting in a significant enhancement of the controllability of the dynamical networks. In addition, we introduce an effective length to measure the time-varying topological distance between the mobile units and the controller. The introduced effective length could well reflect the controllability of the dynamical networks with mobile units.

Facility Location with Tree Topology and Radial Distance Constraints

Let Gd = (V, Ed) be an input disk graph with a set of facility nodes V and a set of edges Ed connecting facilities in V. In this paper, we minimize the total connection cost distances between a set of customers and a subset of facility nodes S⊆V and among facilities in S, subject to the condition that nodes in S simultaneously form a spanning tree and an independent set according to graphs and Gd, respectively, where is the complement of Gd. Four compact polynomial formulations are proposed based on classical and set covering p‐Median formulations. However, the tree to be formed with S is modelled with Miller–Tucker–Zemlin (MTZ) and path orienteering constraints. Example domains where the proposed models can be applied include complex wireless and wired network communications, warehouse facility location, electrical power systems, water supply networks, and transportation networks, to name a few. The proposed models are further strengthened with clique valid inequalities which can be obtained in polynomial time for disk graphs. Finally, we propose Kruskal‐based heuristics and metaheuristics based on guided local search and simulated annealing strategies. Our numerical results indicate that only the MTZ constrained models allow obtaining optimal solutions for instances with up to 200 nodes and 1000 users. In particular, tight lower bounds are obtained with all linear relaxations, e.g., less than 6% for most of the instances compared to the optimal solutions. In general, the MTZ constrained models outperform path orienteering ones. However, the proposed heuristics and metaheuristics allow obtaining near‐optimal solutions in significantly short CPU time and tight feasible solutions for large instances of the problem.

Breaking Permutation-Based Mesh Steganography and Security Improvement

Permutation-based steganography in polygonal meshes can provide considerably large embedding capacities for hiding secret messages. However, corresponding steganalysis techniques against such methods have never been studied. This paper identifies the essential differences between naturally generated meshes and meshes produced by permutation-based steganography methods. It is found that the two types of meshes differ significantly in the distribution of topological distances between consecutive mesh elements. Therefore, by measuring the orderliness of the vertex list and the face list of meshes, we develop solutions for several mesh steganalysis problems. These solutions are effective, leading to high detection accuracy; and they are also universal, requiring no knowledge such as which steganography method is used and what data embedding rate is adopted for the detection mechanism to work. Moreover, this paper also presents a security-improved permutation-based mesh steganography approach, by taking advantage of the connectivity information of polygonal meshes and establishing a good trade-off between embedding capacity and undetectability. Without bringing global changes, our approach embeds secret messages into local neighborhoods on meshes. As a result, meshes generated by the proposed steganography approach tend to have natural structures that are unlikely to draw suspicions to steganalyzers.

Combinatorial Design of Molecule using Activity-Linked Substructural Topological Information as Applied to Antitubercular Compounds.

Generating a large number of compounds using combinatorial methods increases the possibility of finding novel bioactive compounds. Although some combinatorial structure generation algorithms are available, any method for generating structures from activity-linked substructural topological information is not yet reported. To develop a method using graph-theoretical techniques for generating structures of antitubercular compounds combinatorially from activity-linked substructural topological information, predict activity and prioritize and screen potential drug candidates. Activity related vertices are identified from datasets composed of both active and inactive or, differently active compounds and structures are generated combinatorially using the topological distance distribution associated with those vertices. Biological activities are predicted using topological distance based vertex indices and a rule based method. Generated structures are prioritized using a newly defined Molecular Priority Score (MPS). Studies considering a series of Acid Alkyl Ester (AAE) compounds and three known antitubercular drugs show that active compounds can be generated from substructural information of other active compounds for all these classes of compounds. Activity predictions show high level of success rate and a number of highly active AAE compounds produced high MPS score indicating that MPS score may help prioritize and screen potential drug molecules. A possible relation of this work with scaffold hopping and inverse Quantitative Structure-Activity Relationship (iQSAR) problem has also been discussed. The proposed method seems to hold promise for discovering novel therapeutic candidates for combating Tuberculosis and may be useful for discovering novel drug molecules for the treatment of other diseases as well.

Computational studies on α-aminoacetamide derivatives with anticonvulsant activities

Computational methods were used to study the structural parameters that influence that anticonvulsant activity of some α-aminoacetamides which were active in maximal electroshock seizure test. Their molecular structures were optimized with B3LYP/631G** density function theory method using Spartan 14 software. Modified-K-mediod clustering algorithm was used for data division, genetic function algorithm was used for variable selection and multiple linear regressions method was used for model construction. Developed model was statistically significant with coefficient of determination (R2) of 0.957, cross-validated R2 i.e. Q2 of 0.927, variance ratio (F4,15) of 82.94, Y-randomization R2 i.e. cR2P of 0.840 and predicted R2 (R2Pred) of 0.812. The molecular descriptors contained in the model were GATS8c (Geary autocorrelation of lag-8/weighted by atomic charges); VCH-7 (valence chain of order 7); VE3_D (Logarithmic coefficient sum of the last eigenvector from topological distance matrix) and RDF100p (radial distribution function − 100/weighted by polarizability). Molecular docking result showed that studied compounds had high binding affinity for neuronal sodium channel (PDB: 2KaV). Their binding affinity compared favorably with that of phenytoin, a validated sodium channel blocker. In addition, a linear relationship existed between anticonvulsant activity of studied compounds and their binding affinity for neuronal sodium channel.

Implementation and efficiency analysis of composite DNS-metric for dynamic server selection

User requirements for a high availability and fast response time of network services require placing more than one server for accessing particular network service, often using multiple communication links and locations. Dynamic Server Selection (DSS) is a new DNS method for the optimal server selection of a multiple available network service that allows dynamic selection of a server on the client side based on the information of the server load and its network topological distance from the client. The server selection is based on the calculations of a composite DNS-metric in which servers, whose IP addresses are sent in a DNS response, are ranked from the optimal to the least suitable. Calculation parameters are server response time, which the client measures for each server independently, and the server load, which is specified by the server administrator and is forwarded to the client together with the rules for calculating the composite DNS-metric. The DSS method has the lowest overall network service response time in comparison with the other four observed methods (Geographical, Hops, Random and RTT) which, in measurements done in a real time environment for two servers accessible by two independent internet links each, have a longer response time from 8.5 to 26.8% compared to DSS. Results of the proposed analytical model for calculating the efficiency index of the DSS method are compared with results of the practical measurements confirming the relevance of the analytical model. In measurements, the DSS method achieved a high average efficiency index of 1.23.

Estimation of geometric route distance from its topological distance: application to narrow road networks in Tokyo

The structure of road networks has been investigated in accordance with the development of GIScience. By classifying road networks into wide and narrow ones, we can define the route as the path from the route’s origin (also called the root) on a wide road network to a narrow road segment which consists of the sequence of narrow road segments arranged by ascending order of the number of steps of adjacency to its root. The length of the route can be defined with the following geometric and topological terms: the route distance, measuring the length along the route and the depth, counting the number of road segments on the route. The depth plays the important role of being a substitute for the route distance in modelling road networks as a planar graph. Since road networks clearly exhibit irregular patterns and road segment lengths are non-uniform, it is considered appropriate to adopt a stochastic approach rather than a deterministic one to analyse the route distance. However, the relationship between the route distance and its depth has not been sufficiently investigated stochastically. Therefore, the research question is how can we estimate the route distance from its depth? Based on an empirical study in the Tokyo metropolitan region, it was found that (1) the statistical distribution of the route distance can be formulated as an Erlang distribution whose parameters are its depth and the inverse of the mean length of narrow road segments, and (2) this length is constant and close to 40 m. Therefore, we can estimate the route distance from only one parameter, the depth. Also, as a practical application, accessibility to the kth depth link in terms of firefighting was evaluated because the maximum length of the extension of fire hoses is approximately 200 m. It was found that (1) even if k ≤ 5, the probability that the route distance to the kth depth link is equal to or longer than 200 m ranges from 0 to 0.45; and (2) if k ≥ 8, the probability is approximately 1. These indicate the limitation of the deterministic approach because, on the basis of complete grid patterns (with intervals of 40 m between intersections), k = 5 corresponds to a distance of 200 m from wide road networks and the route to the 5th depth link can be covered with fire hoses. Moreover, it was found that the connectivity of wide road networks is higher than that of narrow ones in terms of the smaller ratio of cul-de-sacs and the larger ratio of four-way intersections. These answers contribute substantially not only to constructing a science of cities that provides a simple model and specifies the most important parameter, but also to our understanding of the structure of narrow road networks within several hundred metres of wide road networks.