A Probabilistic Approach for Road Network Analysis

Abstract

Complex networks are networks whose structure is irregular, complex and which evolves over time and are used in various branches of science and technology, such as in biochemistry, in the study of interactions in quantum field theory, in the study of IT processes, topologies in geographical databases and also on the web, in social networks such as Facebook and Linkedin and in the Google model. The term “complex” itself derives from the Latin cum (together)—plexus (intertwined), “intertwined together”: it highlights that a complex network system is composed of a set of parts connected and “intertwined” in such a way that the result (the effect produced) is different from the sum of the constituent parts. Therefore, the behavior of a complex system cannot be inferred by a simple analysis of the elements that compose it, but it is necessary to carry out a systematic examination of the interactions that are generated between them and the constraints that determine their operation. In this chapter we show how the probability of intersections for the Road Network Analysis (RNA) can be useful. We use a geometric probabilities approach for transportation planning operations. We show the utility of the probability of intersections in the determination of a classification rule for raster conversions in Geographical Information System (GIS) and GRASS GIS.

1. Introduction

The realization of a public investment in mobility and logistic networks can be tackled by solving a problem of maximizing the “global utilities” associated with the planned intervention, through special algorithms that allow the systematic evaluation of the set of variability characteristics and constraints, expressed by equations and inequalities.

Constraints can also express conditions of nonnegativity of variables, such as when they represent physical quantities, or upper and lower limits on the value they can assume in the range of variation (budget, resources, etc.) [1, 2, 3, 4, 5, 6]. These constraints can be algebraically represented by the conditions to which the action variables are subject and can be expressed by means of equations or inequalities, classifiable as:

• Sign constraints used to indicate that the variables can assume positive value or equal to 0

  $$\mathrm{x}\ge 0\text{and}\mathrm{y}\ge 0$$E1

• Operational constraints used to indicate the maximum availability of resources, also expressed by equations or inequalities.

This approach must also consider the many probabilistic variables obtained from risk analysis in the various phases of operation and management, up to the decommissioning of the work, for the entire reference time scenario, in the interval [0, T], with T = useful life years of investment plans (such as the more complex ones in the PNRR).

The variables of the mathematical model used to optimize the results of the problem represent the unknowns on which the decision must be made and are characterized as “decision variables”, represented by specific indicator.

$$r_{\mathit{ij}},\mathit{con}\left\{\begin{array}{l}i=1,\dots ,n;\\ j=1,\dots ,m\end{array}\right.$$E2

The objective function representative of the problem can be minimized, as in the case of the cost flow analysis $C_i$ relating to the network infrastructure in question and the negative “impacts” on the ecosystem, or maximized if it is revenue flows or material and/or intangible benefits (including the so-called positive “externalities”, i.e. positive changes in utility) $B_j$, which must then be “discounted” in the economic calculation to outline the decision-making framework from an LCCA perspective—Life Cycle Cost Analysis (Figure 1) [7, 8].

Figure 1.

LCCA—Life Cycle Cost Analysis.

Therefore, having evaluated the flows of global benefits and those of global costs, both discounted, the calculation of the cost-benefit ratio can be expressed with the relationship:

$${\mathcal{B}}_{\mathrm{c}}=\frac{\sum b_{g_i}^{-w_i}\left(\frac{1}{{\left(1+\mathrm{r}\right)}^{\mathrm{t}}}\right)}{\sum c_{g_j}^{-w_j}\left(\frac{1}{{\left(1+\mathrm{r}\right)}^{\mathrm{t}}}\right)}\pm {\mathrm{V}}_{\mathrm{r}}\left(\frac{1}{{\left(1+\mathrm{r}\right)}^{\mathrm{N}}}\right),$$E3

With

$$\sum {\mathrm{w}}_{\mathrm{i}}=1\text{and}\sum {\mathrm{w}}_{\mathrm{j}}=1.$$E4

The term ${\mathrm{V}}_{\mathrm{r}}\left(\frac{1}{{\left(1+\mathrm{r}\right)}^{\mathrm{N}}}\right)$ of the expression indicates the residual value of the network infrastructure that is the subject of the investment, at the end of its useful life, estimated equal to “N” years. This can take on positive values, as in the case of the sale and/or reuse of the asset in a production cycle, or negative, when a cost must be borne for disposal, possible rehabilitation of the area where the public work is located, etc.

Indicating, then, with ${\mathrm{C}}_0$ the discounted initial investment cost if any $\left({\mathcal{B}}_{\mathrm{c}}-{\mathrm{C}}_0\right)>1$ the investment will be economically convenient and can be profitably implemented [8, 9]. For an n-variable mathematical model to maximize the objective function $\overrightarrow{\mathit{f}}$ it will be necessary to solve the typical maximum search problem

$$\mathit{max}\left[\overrightarrow{\mathit{f}}\left(x_{\mathit{1}},x_{\mathit{2}},\dots ,x_j,\dots ,x_n\right)\right].$$E5

In the systematic analysis to be conducted for investments in mobility and logistics networks, since the security of the “network system” is paramount, among the characteristic variables it is necessary to accurately evaluate all the probabilistic indicators of security.

$$P_X^s\left(x\right)$$E6

which can be defined algebraically by means of the general mathematical relation:

$$P_X^s\left(x\right)=f_X\left(C_{\mathit{gr}},L_{\mathit{tr}},G_C^{i=1,\dots ,n},A_r^{i=1,\dots ,n},G_S^{i=1,\dots ,k},T_r^{i=1,\dots ,n}\right).$$E7

This formula expresses the relationship between global risk management costs $C_{\mathit{gr}}$, the level of tolerability of the same $L_{\mathit{tr}}$, the degree of connection of the branches of the mobility network under study $G_C^{i=1,\dots ,k}$ and the relative accessibility of said branches $A_r^{i=1,\dots ,n}$, as well as the degree of intrinsic safety $G_S^{i=1,\dots ,k}$ of the k constituent elements of the infrastructure in question and the contribution made to safety by the technological devices operating in the network n itineraries that make up the transport system in question, which are indicated by $T_r^{i=1,\dots ,n}$.

Furthermore, the indicator $A_r^{i=1,\dots ,n}$ used above must be mathematically “weighted”, by introducing into the expression described a special factor “ε” associated with the specific intrinsic conditions of grid reliability, for the management of emergencies in critical situations (significant weather-climatic events, earthquakes, landslides, structural failure of works, etc.).

Finally, in the case of non-homogeneity of the level of risk tolerability $L_{\mathit{tr}}$ in the road axes and nodes of the network, the variable $L_{\mathit{tr}}$ will in turn be a function of the different specific levels of risk tolerability, according to the following relationship which also considers the degree of interconnection of the nodes of the graph in which the network is geometrically represented (Figure 2).

Figure 2.

Graph of EU freight corridors established by regulation EU/913/2010.

Denoting with ${\beta }_{r_i}$ the characteristic variable for the generic path “network path” which connects node $n_i$ by axes $r_i$ considered in the mobility system.

$${\beta }_{r_i}=\sum {\left({\beta }_{i\to j}^{i=1,\dots ,n}\right)}^{-w_{{\beta }_i}}$$E8

According to this approach, the risk tolerance level defined above $L_{\mathit{tr}}$ is expressed by the following relationship:

$$L_{\mathit{tr}}=\left[g\left(L_{{\mathit{tr}}_s}^{r,n}\right)\right]{\beta }_{\mathit{ri}}.$$E9

2. Graph network geometric model

On the basis of the mathematical formulation of the problem defined above, for the best representation of the problem, it is useful to carry out an appropriate modeling of the infrastructural network through a special “graph” characterizing the various its design components, such as the O-D paths (origin-destination), network (arcs of the graph) and “centroids” generating and/or attracting traffic (nodes of the graph), as in Figure 2 [10].

Indeed, this procedure makes it possible to carry out a systematic examination of the many variables characterizing the problem under study, through a relational structure composed of a finite set of objects (on a more strictly mathematical level, a finite set of points) and of a set relationship between pairs of them.

In particular, if the pair of nodes is ordered, i.e. if the arcs have an arrival node and a departure node, the graph is said to be “directed” and an ordered sequence of arcs identifies the “paths”.

If a quantitative characteristic is also associated with the latter, the graph gives rise to a particular mobility or logistics “network”. By operating, for example, a specific mathematical characterization for the railway network, it is possible to carry out a study of accessibility, even in an emergency (following relevant events), using two specific indicators:

• Single Accessibility Index (IAS)_i;

• Combined Accessibility Index (IAC)_i.

The variable ${\left(\mathrm{IAS}\right)}_{\mathrm{i}}$ represents the accessibility value of each arc “i” of the railway network, without considering the interactions between contiguous arcs (the accessibility value of neighboring arcs is not considered).

It may happen that an arc has a limited or zero accessibility value but is interposed between two highly accessible arcs: it will then be characterized by an accessibility value (even null), regardless of whether the two neighboring arcs are accessible and allow therefore to also access the arc in question.

The combined accessibility index ${\left(\mathrm{IAC}\right)}_{\mathrm{i}}$, on the other hand, considers the mutual interactions between contiguous arcs.

In this way, within the model, the possibility is considered that an arc with limited accessibility can still be reached using the neighboring (contiguous) arcs with greater accessibility, evaluated through the coefficient k.

The representation of the network with special evaluated graphs, integrated by particular mathematical models of stochastic geometry and geometric probabilities [11, 12, 13, 14, 15], such as those that will be described below, is an important decision aid tool for investment analysis or for solving design, management and maintenance problems, and above all for the optimization of operating safety, even in critical conditions, due to accidental obstacles along the network path, due to adverse meteorological events (landslides, instability, falling rocks, etc.), or seismic events, etc.

Methodologically, the problem can be analytically tackled through an appropriate interpretation of the “graph” constructed with “attributes” and thought of as formed by a whole union of elementary regular and/or irregular geometric figures, constituting special lattices R, in the geometric space $S^m$.

By means of specific test bodies (mathematical models) representative of the vehicles (passenger and/or freight trains for railway networks, or cars, busses, articulated lorries or other types of vehicles for road networks, etc.), it is possible to study the relative motion given speed V in the lattice R and study any “interferences” produced by “obstacles” (of given shape and size) that may be found along the generic arc h–k of the geometric lattice associated with the mobility network under examination.

These test bodies can be assumed as segments of suitable length l (to schematize a convoy with a large number of carriages, as for example occurs in the composition of a freight train, etc., or of an articulated lorry in the case of road transport), or made up of rectangles with sides $l_1$ and $l_2$ (as in the case of a high-speed train, a regional train, etc., or even a car along a street).

To arrive at a mathematical resolution of the problem, we will assume in the following developments that each side of the considered lattice offers the same resistance capacity to the advancement of the test-body (model).

Furthermore, to operationally address the problems in professional practice, reference can also be made in the schematization of the “lattices R union of elementary forms” to a “catalog” illustrated below, built on the results of scientific studies carried out on the different forms of the regular and irregular geometric figures constituting a generic network, such as the rail network in Figure 3, referring to East Asian high-speed trains.

Figure 3.

The high-speed train network in East Asia.

To then define the accidental obstacles on site (railway or road) and interfering with the movement of vehicles, a different generic shape is assumed (for example, triangular or circular like the section of a tree trunk accidentally fallen across a road), or a multi-sided polygon, representative of other possible cases¹ (Figures 4 and 5).

Figure 4.

Tree on the roadway.

Figure 5.

Accidental obstacles on a railway site.

The mathematical criterion described above makes use of a regular lattice formed by a union of equal regular polygons obtained starting from a figure called the “fundamental cell of the lattice”, with only segments of the geometric boundaries in common.

If in the vertices of the fundamental cell we consider some regular equal polygons, called “obstacles”, we then obtain a “regular lattice with obstacles”.

As a basic example for the interpretation of the method we can consider the fundamental cell formed by a square with side A and, as an obstacle, a square with side a with 2a < A (Figure 6)

Figure 6.

Lattice with square obstacles.

If we randomly throw a segment s of constant length l onto the lattice, the problem consists in determining the probability that such a segment s intersects one side of the lattice. This probability is equal to the probability that a segment s intersects the boundary of the fundamental cell $C_0$ (Figure 7).

Figure 7.

Lattice with rhombus obstacles.

We have that $\mathit{\text{area}}{C}_0=\frac{c^2}{2}\text{and}$ we consider the limit positions of s, with $\widehat{C_0\left(\rho \right)}$ being the figure determined by such positions (Figure 8).

Figure 8.

Lattice of limit positions.

Computing the areas, we obtain

$$\mathit{\text{area}}\widehat{C_0\left(\rho \right)}=\mathit{ab}-\frac{c^2}{2}-\left[\left(a-c\right)\mathit{\text{lcosρ}}+\left(b-\frac{c}{2}\right)\mathit{\text{lsinρ}}+\frac{l^2}{2}\mathit{sin}2\rho \right].$$E10

Denoting with M₁ the set of segments s which have their middle point in C₀ and with N₁ the set of segments s which have their middle point in C₀ but do not intersect the frontiers of C₀. Then, the probability is

$$P_{\mathit{int}}=1-\frac{\mu \left(N_1\right)}{\mu \left(M_1\right)},$$E11

where μ is the Lebesgue Measure in Euclidean plane [16].

The measures $\mu \left(M_1\right)$ and $\mu \left(N_1\right)$ are computed by the Poincaré Kinematic measure [17].

$$\mathit{dk}=\mathit{dx}\wedge \mathit{dy}\wedge \mathit{d\rho },$$E12

where x, y are the coordinates of P and $\phi$ the defined angle.

Since $\phi \in \left[0,\frac{\pi }{2}\right]$, by previous formula we have that:

$$P_{\mathit{int}}=\frac{\left(a+b-\frac{3c}{2}\right)l-l^2}{\pi \left(\mathit{ab}-\frac{c^2}{2}\right)}$$E13

In place of the fundamental cell and the obstacles (mathematical models) considered above, we can consider others representative of the numerous typical cases that it is possible to examine in the case of complex networks, both road and railway, or other material and/or immaterial networks (such as web networks, social networks, economic-financial networks, etc.).

For the practical applications of the method, the following “catalog” allows us to be able to consider even irregular grids, given that the conformation of the grid can be formed by a whole union of various geometric figures of all shapes and sizes.

The segment s considered (“test body”) in the problem of geometric probabilities can represent—as mentioned—a railway freight train, or a wheeled articulated lorry, but in other cases other test bodies can be considered, for example rectangles, such as the one that represents a passenger train (including those for HS/HC) or, in the road case, a simple car (two-dimensional figure, since one dimension is not negligible compared to the other) (Table 1).

Table 1.

Geometric probabilities applied to network security for different means of transport.

3. Extension model to immaterial networks and social network analysis

The mathematical model described above for material road and railway networks can be extended to applications for immaterial networks, such as Social Network Analysis (SNA) and to the study of the so-called “mental map”, for problem analysis and as a tool decision aid [18, 19, 20, 21, 22].

Indeed, any graphical representation of the thought and theories of the Social Sciences. Always places the accent on a particular dimension of social reality, making use of a reticular structure formed by nodes and arcs that connect them, giving rise to a specific graph of the immaterial network.

Social networks, as a representation of a reticular society or resulting from constraints and opportunities emerging from the relationships between subjects, give rise to graphs of complex networks,² such as the one shown in the following figure, where the individual subjects from which the various input/output solicitations originate/arrive are represented at the nodes, while the interrelationships are expressed by the various “branches”, following a mathematical approach scheme like that of an “O-D” (origin-destination) matrix of a transport network.

The content generated by the users of social networks has given birth to a new era of Business Intelligence & Analytics (BI & A) centered on textual analysis and unstructured content. SNA today increasingly represents a great opportunity for companies to interact bilaterally with the market, much more effective than traditional “one-way” marketings and “business-to-market” type techniques.

Furthermore, both in the economic and sociological fields and for applied sciences in general, the typological structuring of so-called “mental maps”, consisting of the graphic translation of the paths taken by the human brain for the elaboration of a thought, i.e. “conceptual maps”, in the exposition of a problem or a complex and articulated concept with interconnected logical frameworks, also lends itself well to the modeling interpretation provided here for making decisions.

Through the mind mapping technique, it is possible to describe the complex process in which the mind associate’s ideas in a non-linear way, processes information in a creative, intuitive, and figurative way, giving rise to a typical map with a radial structure, in the form of a particular oriented graph [23].

Some arcs are directed towards the most important node, represented by the thought processing center (the man who “travels” with his mind): the logical reference scheme is of the “input-output” type, with complex mechanisms of analytical reiteration (Figure 9).

Figure 9.

Social network analysis.

In the network there may also be associative relationships between the various nodes situated on the same level, but the previous and subsequent ones are given by precise structured hierarchical links (Figure 10).

Figure 10.

Lattice scheme for “mind maps”.

These tools allow you to put meta-reasoning into practice (allowing you to reflect on thought in an “introspective” way) and have practical applications that can be used not only to study, to memorize, or to plan (project management, etc.), but are of scientific and professional interest in various fields of application, such as forensic technology, engineering, economic and environmental sciences, artificial intelligence, medicine (treatment of dyslexia, dysgraphia, learning disabilities) and today they also find profitable use in the world of the web.

For example, in web marketing mind maps are exploited (using special software). To create website structures, to organize contents and strategies, while in the field of blogging they can be useful for identifying tags and categories but also for defining the contents to be addressed on a blog.

Furthermore, the contribution of the mathematical criterion proposed here to Social Network Analysis covers a vast range of operational applications and could even be employed to determine strategic and tactical actions in crisis areas or to counter terrorist actions (by collecting various sets of information and appropriately analyzing and aggregating them): in general, therefore, this methodology can constitute a valid support for the taking of complex decisions, when it is necessary that the information collected is examined by a human controller.

However, for immaterial networks, unlike those transported by road or rail, to mathematically define the probability of interference between network flows and the “obstacle” along the path which generates the connection defect in the graph, it is necessary to refer to a test-body consisting of a segment and look for a flat geometric figure able to characterize the seriousness of the functional impediment produced.

To this end, a specific degree of impedance of the network was identified (SNA, Mental Maps, etc.), in analogy to the physical quantity that represents the opposition force of an electric circuit to the passage of a variable current, in order to be able to apply the resolution algorithm corresponding to the pre-selected obstacle also to this type of network, according to what is reported in the aforementioned “catalog”, where—case by case—the probability of occurrence of interference is explained.

Based on the above, the catalog already reported for the material networks examined was completed with the subsequent supplementary abacus whereby, for each geometric form of “obstacle” considered therein (triangle, semicircle, square/rhombus, or circle), a precise “degree of impedance” is associated, synthetically defined according to an ordinal scale, formed by three characteristic levels:

1. Elevated (triangle)

2. Middle (semicircle)

3. Low (square or other geometric shape of the “catalog”)

Among intangible networks, one of fundamental importance is the economic-financial network aimed at defining the various logical links between “operators in the sector” for analysis of the type-flows to be studied (securities, etc.). However, the WEB network is also of great importance, considering that, at an international level, even the United Nations Human Rights Council (resolution A/HCR/20/L.13), has expressly considered the Internet a fundamental human right, included in art. 19 of the Universal Declaration of Human and Citizen Rights.

In this context, a performance element of the reticular system is the speed of the network (or of the links) in transmitting and receiving data securely.

In the case of a packet-switched network (finite and distinct sequence of data), the problem generally affects the operation of several networks (Figure 11) and the metrics that determine their performance are measured in terms of bandwidth, bit rate, throughput, latency (delays or delays) and packet loss.

Figure 11.

Description of the path of a data stream along multiple networks.

In any case, network congestion phenomena must be prevented by monitoring any traffic excesses that could block data paths along the graph representing the network structure.

4. Road network analysis with GIS and GRASS-GIS

The probability can be useful of intersections in the determination of a classification rule for raster conversions in Geographical Information System (GIS) and GRASS GIS for the Road Network Analysis (RNA) [15, 24, 25, 26, 27].

To store the referenced information in some regular lattice covering a projected portion of the Earth’s surface. These lattices might be quadratics in an ecological study region, square cells on a raster map, rectangular sheets in a map series, spatial partitions of a large geographical database, or zones in a georeferencing system.

The power of such solutions is to characterize the computational complexity of GIS algorithms and thereby estimate their efficiency [10, 28, 29].

Several papers have employed Buffon’s Needle framework to models of migration and travel distances. Rogerson [13] developed intraregional migration distance models; the probability that a move of a given distance will not cross a regional boundary was estimated under several scenarios. Kirby considered trip surveys and the probability of undercounting trips that did not cross survey boundaries.

Geometric probability is a branch of mathematics that is concerned with the probabilities associated with geometric configurations of objects. Among the most famous of these applications is Buffon’s Needle problem. The eighteenth-century French naturalist Buffon conceived of the problem while considering a popular game of chance in which a stick was thrown upon a tile floor. The problem continues to appeal to students of mathematics for its variety of elegant solutions [30, 31, 32].

In a previous paper Caristi et al. [33] considering the model introduced in the second paragraph, the variations of the classic Buffon’s Needle problem are of particular interest. Indeed, if we consider a tile $\mathcal{R}\left(a,b,\alpha \right)$ composed of irregular fundamental cells C₀ represented as in Figure 12.

Figure 12.

Lattice of limit positions with segment obstacles.

where $\alpha \in ]0,\pi /4]$ is an angle and $a>\mathit{\text{btgα}}$.

Denoting with M₁ and with M₂ the set of all segments s that have their centre in C₀₁ and C₀₃. Denoting, likewise, with N₁ and N₂ the set of all segment s completely contained within C₀₁ and C₀₃. We get:

$$P=\frac{\mu \left(N_1\right)+\mu \left(N_3\right)}{\mu \left(M_1\right)+\mu \left(M_3\right)},$$E14

where μ denotes the Lebesgue measure in the Euclidean plane [14].

To compute the above measures, we use the Poincaré kinematic measure [26]:

$$\mathit{dK}=\mathit{dx}\wedge \mathit{dy}\wedge \mathit{d\phi },$$E15

where x, y are the coordinates of the point 0 and $\phi$ the angle of s.

By Figure 1, $0\le \phi \le \pi /2-\alpha ,$ and we have

$$\mu \left(M_1\right)={\int }_0^{\frac{\pi }{2}-\alpha }\mathit{d\phi }{\iint }_{\left\{\left(x,y\right)\in C_{01}\right\}}\mathit{\text{dxdy}}={\int }_0^{\frac{\pi }{2}-\alpha }\left({\mathit{\text{areaC}}}_{01}\right)\mathit{d\phi }=\left(\frac{\pi }{2}-\alpha \right){\mathit{\text{areaC}}}_{01},$$

$$\mu \left(M_3\right)={\int }_0^{\frac{\pi }{2}-\alpha }\mathit{d\phi }{\iint }_{\left\{\left(x,y\right)\in C_{03}\right\}}\mathit{\text{dxdy}}={\int }_0^{\frac{\pi }{2}-\alpha }\left({\mathit{\text{areaC}}}_{03}\right)\mathit{d\phi }=\left(\frac{\pi }{2}-\alpha \right){\mathit{\text{areaC}}}_{03},$$

$$\mu \left(M_1\right)+\mu \left(M_3\right)=\left(\frac{\pi }{2}-\alpha \right)\frac{\mathit{ab}}{2},$$E16

and

$$\mu \left(N_1\right)={\int }_0^{\frac{\pi }{2}-\alpha }\mathit{d\phi }{\iint }_{\left\{\left(x,y\right)\in {\widehat{C}}_{01}\right\}}\mathit{\text{dxdy}}={\int }_0^{\frac{\pi }{2}-\alpha }\left[\mathit{\text{area}}{\widehat{C}}_{01}\left(\phi \right)\right]\mathit{d\phi },$$E17

$$\mu \left(N_3\right)={\int }_0^{\frac{\pi }{2}-\alpha }\mathit{d\phi }{\iint }_{\left\{\left(x,y\right)\in {\widehat{C}}_{03}\right\}}\mathit{\text{dxdy}}={\int }_0^{\frac{\pi }{2}-\alpha }\left[\mathit{\text{area}}{\widehat{C}}_{03}\left(\phi \right)\right]\mathit{d\phi }.$$E18

Integrating relations (17) and (18) and substituting in (14) we obtain

$$\begin{array}{ll}P=& 1-\frac{4}{\left(\pi -2\alpha \right)\mathit{ab}}\{\mathit{al}\left(1-\mathit{sin}\alpha \right)-\frac{\mathit{bl}}{2}\left(1-\mathit{sin}\alpha +3\mathit{cos}\alpha \right)\\ & -\frac{l^2}{2}\left[\mathit{cos}2\alpha +\left(\frac{\pi }{2}-\alpha \right)\left(1+\mathit{\text{ctgα}}\right)\right]\}.\end{array}$$E19

Denoting with

$$\left(\alpha \right)=\frac{2\mathit{al}\left(1-\mathit{sin}\alpha \right)-\mathit{bl}\left(1-\mathit{sin}\alpha +3\mathit{cos}\alpha \right)-l^2\left[\mathit{cos}2\alpha +\left(\frac{\pi }{2}-\alpha \right)\left(1+\mathit{\text{ctgα}}\right)\right]}{\pi -2\alpha }$$E20

we can write

$$P=1-\frac{2}{\mathit{ab}}f\left(\alpha \right).$$E21

We prove that there exists a system value for $\alpha ,a,b,l$ for which the probability P is maximum. Indeed, for $\alpha =\frac{\pi }{4}$ it is easy to verify that $f^{\prime }\left(\alpha \right)=0$ and ${f^{\prime }}^{\prime }\left(\alpha \right)>0$ then the probability P is maximum.

For our considered lattice we have: Example 1. What is the probability that the body test is missed during a 2D conversion to raster?

Theorem 1. The probability $P\_\mathit{int}$ that a random segment s of fixed length l, fulfilling the relation $l<a/2$, uniformly distributed in a bounded region of the plane, intersects a side of the lattice $\mathcal{R}\left(a\right)$ is:

$$P=\frac{4}{\left(\pi -2\alpha \right)\mathit{ab}}\left\{\mathit{al}\left(1-\mathit{sin}\alpha \right)-\frac{\mathit{bl}}{2}\left(1-\mathit{sin}\alpha +3\mathit{cos}\alpha \right)-\right.\left.\frac{l^2}{2}\left[\mathit{cos}2\alpha +\left(\frac{\pi }{2}-\alpha \right)\left(1+\mathit{\text{ctgα}}\right)\right]\right\}.$$E22

Corollary 1. For $\alpha =\frac{\pi }{4}$, by (21) we have that $0\le P_{\mathit{int}}\le \frac{2}{\mathit{ab}}f\left(\alpha \right).$.

We consider a tile $\mathcal{R}\left(a,\alpha ,\beta \right)$ composed of an irregular fundamental cell $C_0=C_{01}\cup C_{02}$ represented in Figure 13 where $\alpha \text{and}\beta$ are angles with $\frac{\pi }{4}<\alpha \le \frac{\pi }{3}$ and $\beta \le \alpha$ (Figure 13):

Figure 13.

Lattice of limit positions with rectangle obstacles.

In the same way, we have: Example 2. What is the probability that the body test rectangle is missed during a 2D conversion to raster?

Theorem 2. The probability $P_{\mathit{int}}$ that a random rectangle r of side a and b, uniformly distributed in a bounded region of the plane, intersects a side of the lattice $\mathcal{R}\left(a,\alpha ,\beta \right)$ is:

$$\begin{array}{ll}{P}_{\mathit{int}}=& \frac{1}{3a^2\left(\alpha \mathit{tan}\alpha +\beta \mathit{tan}\beta \right)}\left\{\mathit{al}\left.\left[6-4\mathit{cos}\alpha -4\mathit{cos}\beta +\frac{1}{\mathit{cos}\alpha }+\frac{1}{\mathit{cos}\beta }\right)\right]\right.\\ & -\left(2\mathit{sin}\alpha +2\mathit{sin}\beta -\frac{l^2}{2}\right.\left[\mathit{sin}\alpha \right.\left(\mathit{sin}\alpha +\mathit{cos}\alpha \right)+\mathit{sin}\beta \left(\mathit{sin}\beta +\mathit{cos}\beta \right)-\alpha \mathit{cot}\alpha -\beta \mathit{cot}\beta \left]\right)\\ & -\frac{m^2}{2}\left({\mathit{sin}}^2\alpha +{\mathit{sin}}^2\beta \right)-\alpha \mathit{cot}\alpha -\beta \mathit{cot}\beta \}.\end{array}$$E23

For m = 0 the rectangle r became a segment of length l and we find the probability determined in [30].

5. Conclusions

In this chapter we highlight the relevance of geometric probabilities for Road Network Analysis applications. In particular, we show how the probability of intersections and geometric probabilities approach for transportation planning operations can be useful.

To begin with we represent the network with special evaluated graphs, integrated by mathematical models of stochastic geometry and analytically tackled through an appropriate interpretation of the “graph” constructed with “attributes” and thought of as formed by a whole union of regular and/or irregular elementary geometric figures, constituting special lattices R, in the geometric space S^m. Subsequently, we extend the same representation for the Social Network Analysis and in conclusion we show the utility of the probability of intersections in the determination of a classification rule for raster conversions in Geographical Information System (GIS) and GRASS GIS.