Before discussing second-order characteristics of network intensity functions in detail, some notation and terminology are introduced. For an in-depth treatment of graph theory, we refer the interested reader to the monographs of Bondy and Murty (2008) and also Diestel (2010).
Notation and terminology
We consider a graph \({\mathcal {G}}\) as a pair of two finite sets: vertices \({\mathcal {V}}\) and edges \({\mathcal {E}}\). The terms network and graph are used interchangeably. The shape of \({\mathcal {G}}\) could be undirected, directed or partially directed such that pairs of vertices in \({\mathcal {V}}\) are linked by at most one edge, namely a line or an arc. In general, elements of \({\mathcal {V}}\) and \({\mathcal {E}}\) will be expressed in lower cases.
For a given network, certain sets are of interest and are intensively used in the remainder of this paper. Any pair of distinct vertices which is linked by an edge is called adjacent. In this case, the vertices are termed the endpoints of an edge and the edge is incident to its endpoints. The set of all vertices which are joined by an undirected edge to node \(v_i\) is the neighborhood \({{\,\mathrm{ne}\,}}(v_i)\). The degree of \(v_i\) (\({{\,\mathrm{deg}\,}}(v_i))\) is the number of distinct vertices in \({{\,\mathrm{ne}\,}}(v_i)\). Similarly, for any directed graph, we define the parents \({{\,\mathrm{pa}\,}}(v_i)\), resp. children \({{\,\mathrm{ch}\,}}(v_i)\) of \(v_i\), as the set of nodes pointing to \(v_i\), resp. with root \(v_i\). Taking the union over both sets results in the family \({{\,\mathrm{fam}\,}}(v_i)\). Analogously to \({{\,\mathrm{deg}\,}}(v_i)\), we express the number of distinct parents of \(v_i\) by \({{\,\mathrm{deg}\,}}^-(v_i)\) and the number of distinct children of \(v_i\) by \({{\,\mathrm{deg}\,}}^+(v_i)\).
A path is any sequence of distinct nodes and edges, and any nodes \(v_i\) and \(v_j\) which are joined by a path \(\pi _{ij}\) are called connected. If all edges along a path are directed, the path is called directed path where we assume that the path is direction preserving. That is, we do not consider sequences of directed edges in which a head-to-head or tail-to-tail configuration exists. A directed path from \(v_i\) to \(v_j\) will be indicated by \(\vec {\pi }_{ij}\). In addition, we call any vertex \(v_i\) pointing to \(v_j\) an ancestor of \(v_j\) and write \({{\,\mathrm{an}\,}}(v_j)=\lbrace v_i\in \vec {\pi }_{ij-1}\rbrace \) to denote the set of ancestors of \(v_j\). Similarly, we say that \(v_j\) is a descendant of \(\lbrace v_i\rbrace \) if \({{\,\mathrm{an}\,}}(v_j)=\lbrace v_i\rbrace \). The set of descendants of \(v_i\) is indicated by \({{\,\mathrm{de}\,}}(v_i)\).
Motivation
For motivation, we consider an arbitrarily shaped spatial network with vertices \(v_1\) to \(v_{11}\) and a set of edges joining some, but not all pairs of vertices, as depicted in Fig. 1. For simplicity, assume Fig. 1 displays a traffic network such that edges correspond to roads and vertices correspond to segmenting entities such as crossings or ends, whence each road has at least two ends which need not necessarily be interconnected to any alternative segment in the network. However, to obtain further parsimony, auxiliary vertices might be included into the network structure. We also note that a real-world road is a continuous one-dimensional structure, and our network-based construction is a good (perhaps not perfect) approximation of the road itself. Typically, certain roads are unidirectional by nature such that traffic can only flow in one direction, while other roads remain bidirected. That is, our spatial network contains directed as well as undirected edges and movements along the network appearing as a sequence of either directed or undirected edges. However, alternative sequences might also be present in real-world spatial networks and corresponding sequences could easily be defined. Despite such heterogeneity, some roads might also be affected by speed limits such that movements along such network sections are decelerated.
Given a collection of random point locations (a spatial point pattern) over a traffic network where the locations themselves are assumed to be governed by an underlying stochastic mechanism (a spatial point process), one could be interested in the description of egdewise, nodewise or pathwise characteristics such as the number of events that felt onto a specific road segment or took place within a certain neighborhood structure or along a path. Such characteristics have been addressed by Eckardt and Mateu (2017) by means of edgewise, nodewise and pathwise counting measures, first-order intensity functions as well as various K-functions for directed, undirected and mixed networks. Unlike alternative adaptations of Ripley’s K-function to the network domain, these K-functions are related to the expected number of events that fall into a certain distance d subject to an integer-valued threshold \(\xi \) which, in turn, is computed as the number of vertices from i to j such that \(d(i,j)\le \xi \) holds (see Eckardt and Mateu (2017) for a detailed treatment and specification of alternative second-order point process characteristics).
We note that different from the linear network formalism presented in Ang et al. (2012), Baddeley et al. (2014), and Baddeley et al. (2017) among others, where the point pattern over the network is initially given independent on the structure network itself, and both configurations are only joint a posteriori, the above specification of the traffic network in the form of a graph through interconnected edges and sets of vertices is the base used to compute a wide range of different point process characteristics. Note that in our context, the specification of the particular subgraph structures determines the calculation of, for example, the nodewise first-order intensity function. That is, unlike the linear network formalism, the computation of network intensity functions is initially linked to the structural specificity of the network and thus explicitly controls for the structural constraints of the network on the point locations. In consequence, the specification and thus the characteristics of the graph strongly determine the characteristics of the point pattern such that different specifications of one particular network might lead to slightly varying locally computed point pattern characteristics, for example edgewise intensity functions. At the same time, the global information on the point pattern over the complete graph will yield similar results, which also holds for more globally computed characteristics such as pathwise intensity function.
Despite first-order characteristics, one might also be interested in the variation or association between pairs of edges, neighborhood structures or paths. However, considering two disjoint edges, neighborhood structures or paths, multiple second-order characteristics can be defined addressing either similar or diverse shapes. That is, the second-order edgewise intensity function could either refer to pairs of directed edges, pairs of undirected edges or, alternatively, consider pairs of one directed and of one undirected edge. In addition, pairs of directed edges might also have a diametrical orientation in the network. Similarly, for second-order neighborhood characteristics, one could be interested in the characterization of events that fell into two neighborhoods in case of undirected or mixed networks, or consider either two sets of parents, two sets of children or only one set of parents and one set of children. In addition, for higher-order neighborhood structures, one could also consider pairs of ancestors or descendants.
An illustration of three different pairs of paths is shown in Fig. 1. Figure 1a highlights two undirected paths joining \(v_3\) to \(v_6\) and \(v_4\) to \(v_{11}\). In contrast, two diametrically shifted paths are shown in Fig. 1b. Finally, Fig. 1c contains one undirected path (\(v_9\) to \(v_5\)) and one directed path (\(v_6\) to \(v_3\)). For any of these paired paths, one might be interested in the expected number of points, the variation in number of points or the correlation between the number of points that felt onto both paths. In addition, as for classical spatial point patterns, one could also be interested in the probability of an event in path a given an event in path b.
Recapitulating first-order network intensity functions
Before we discuss the second-order statistics for spatial networks, we briefly present the basic ideas of counting measures and statistics with respect to points contained in \({\mathcal {S}}_{E(G)}\) for different types of networks and recapitulate different first-order network intensity functions. Here, we first treat undirected graphs and discuss directed and partially directed graphs consecutively. Extension to higher-order characteristics is straightforward and follows naturally as generalizations of well-known point pattern characteristics. In general, three different types of network intensity functions can be addressed referring to different levels of network resolutions. These are the edgewise, the nodewise and the pathwise intensity functions.
Following the ideas and notation of Eckardt and Mateu (2017), we address the set of nodes at fixed locations \({\mathbf {s}}_{v}=({\mathbf {x}}_{v},{\mathbf {y}}_{v})\) contained in a spatial network \({\mathcal {G}}\) by \({\mathcal {V}}_s({\mathcal {G}})\) and refer to the set of edge intervals connecting pairs of fixed locations in \({\mathcal {G}}\) by \({\mathcal {S}}_{E_s({\mathcal {G}})}=\lbrace s_{e_1},\ldots ,s_{e_k}\rbrace \). In addition, we express the locations of a point process \(X({\tilde{\mathbf {s}}})\) over \({\mathcal {S}}_{E_s({\mathcal {G}})}\) by \(\tilde{{\mathbf {s}}}=(\tilde{{\mathbf {x}}}, \tilde{{\mathbf {y}}})\). The location of node \(v_i\) is \(s_{v_i}=(x_{v_i},y_{v_i})\). Clearly, under this definition, point patterns are only allowed to occur within a given edge interval contained in \({\mathcal {G}}\). That is, the locations \(\tilde{{\mathbf {s}}}\) are said to occur randomly within edge intervals spanned between any two fixed locations \(s_{v_i}\) and \(s_{v_j}\) of \({\mathbf {s}}_{v}\), for example on road segments. By this, we understand a path as a sequence of consecutive edge intervals and the distance \(d_{\mathcal {G}}(v_i,v_j)\) between any two nodes in \({\mathcal {V}}_s({\mathcal {G}})\) is the number of consecutive edges joining \(v_i\) and \(v_j\), that is, the length of a path. Hence, the shortest path distance is the minimum number of consecutive edges needed to move from \(v_i\) to \(v_j\) along a network.
We explicitly note that these definitions lead to fundamentally different concepts of length as considered in Ang et al. (2012), Baddeley et al. (2014) and Baddeley et al. (2017) who defined the length of a path as the sum over Euclidean distances between consecutive nodes contained in a path, and in Rakshit et al. (2017) who also considered alternative metric distances. By this, the shortest path distance is the minimum of metric distance totals of all paths joining two locations and it is not defined as the minimum number of traversed edges along a path. Unlike the above metrics, the present definition related to the number of edges along a path allows for the specification of polynomial non-circular areas of the network. Also our definition allows for the formulation of alternative point pattern characteristics such as the network pair correlation or network K-function defined over the set of points along different edge intervals whose ends are reachable along the network in \(\xi \) or less steps. While the linear network framework characteristics are defined through discs with their relative versions yielding circular-type relations, the proposed graph theoretic formulations related to the number of edge intervals traversed along a path do not yield necessarily circular relations, which implicitly control for the general non-unique edge–vertex distribution and specification of the network. That is, for example, the number of traffic accidents might be closely related to external factors as speed limitations due to overcrowded streets such that high numbers of accidents are more likely to happen along high-speed areas such as motorways which usually consist of less crossings compared to dense traffic areas in the city center.
First-order network intensity functions for undirected networks
Let \(N(s_{e_i})\) be the number of points that fall into the undirected edge interval \(s_{e_i}\) and \(ds_{e_i}\) denote an infinitesimal interval containing \(s_{e_i}\) such that \(N(ds_{e_i})=N(s_{e_i}+ds_{e_i})-N(s_{e_i})\). Then, we have for the first-order edgewise intensity function
$$\begin{aligned} \lambda (s_{e_i})=\lim _{|ds_{e_i}|\rightarrow 0}\lbrace \mathbb {E}\left[ N(ds_{e_i})\right] /|ds_{e_i}|\rbrace \end{aligned}$$
(1)
Using this expression, we obtain the nodewise mean intensity function \(\lambda (v_i)\) for any given node \(v_i\) contained in \({\mathcal {G}}\) by averaging (1) over the set of adjacent nodes.
Besides, apart from any such average intensities of points per neighborhood, one can define neighborhood intensity functions using the set of incident edges. To this end, let \(\flat (v_i)\) denote the set of edge intervals with endpoint \(v_i\), \(N(\flat (v_i))\) be the number of points in \(\flat (v_i)\) and \(d\flat (v_i)\) denote an infinitesimal area covering \(\flat (v_i)\). By this, we define the non-averaged neighborhood intensity function \(\lambda ({{\,\mathrm{ne}\,}}(v_i))\) as
$$\begin{aligned} \lambda ({{\,\mathrm{ne}\,}}(v_i))=\lim _{|d\flat (v_i)|\rightarrow 0}\lbrace \mathbb {E}\left[ N(d\flat (v_i))\right] /|d\flat (v_i)|\rbrace . \end{aligned}$$
(2)
Using the same ideas as for \(\lambda (v_i)\) and \(\lambda ({{\,\mathrm{ne}\,}}(v_i))\), we can define an averaged and a non-averaged version of pathwise intensity functions for a path \(\pi _{ij}\) joining \(v_i\) to \(v_j\) (cf. Eckardt and Mateu (2017)). As for (2), writing \(\wp _{ij}\) to denote the set of edge intervals traversed once along \(\pi _{ij}\), we define the non-average pathwise intensity function as
$$\begin{aligned} \lambda (\pi ^*_{ij})=\lim _{|d\wp _{ij}|\rightarrow 0}\lbrace \mathbb {E}\left[ N(d\wp _{ij})\right] |d\wp _{ij}|\rbrace \end{aligned}$$
(3)
where \(N(d\wp _{ij})=N(\wp _{ij}+d\wp _{ij})-N(\wp _{ij})\) and \(|d\wp _{ij}|\) is the area contained in \(d\wp _{ij}\).
We note that in the classical formulation of an intensity function, we only take the number of neighbors within a particular distance from an event of interest. But here, as we go deeper into the geometry of the network, we are able to provide more specific intensity function that highlights many other possible configurations of the points depending on whether they are in the same path or neighborhood.
First-order network intensity functions for directed networks
To cover directed graphs, slight modifications of the previous notations are required. To this end, let \(N(s_{e_i}^\mathrm{in})\) express the number of events on an edge leading to and \(N(s_{e_i}^\mathrm{out})\) be the number of events on an edge departing from a vertex of interest, and \(ds_{e_i}^\mathrm{in}\) and \(ds_{e_i}^\mathrm{out}\) denote infinitesimal intervals containing \(s_{e_i}^\mathrm{in}\) and \(s_{e_i}^\mathrm{out}\).
Substitution of \(N(s_{e_i}^\mathrm{in})\) or \(N(s_{e_i}^\mathrm{out})\) for \(N(s_{e_i})\) in (1) yields the directed first-order edgewise intensity functions whose average, in turn, leads to the parentwise mean intensity function \( \lambda ^\mathrm{in}(v_i)\) and the childrenwise mean intensity function \(\lambda ^\mathrm{out}(v_i)\). As in the undirected case, one can define non-averaging versions of \(\lambda ^\mathrm{in}(v_i)\) and \(\lambda ^\mathrm{out}(v_i)\) with respect to the sets of incident edge intervals with head or tail \(v_i\), namely incident edge intervals pointing to \(v_i\) (\(\lambda ({{\,\mathrm{pa}\,}}(v_i))\)) and incident edge intervals departing from \(v_i\) (\(\lambda ({{\,\mathrm{ch}\,}}(v_i))\)). Defining \(\flat ^{\mathrm{in}}(v_i)\) (resp. \(\flat ^{\mathrm{out}}(v_i))\) as the set of edge intervals pointing to (resp. departing from) \(v_i\) and using the same terminology as before, we obtain the non-averaging parentwise (resp. childrenwise) intensity function by substituting \(N(d\flat ^{\mathrm{in}}(v_i))\) (resp. \(N(d\flat ^{\mathrm{out}}(v_i))\)) for \(N(d\flat (v_i))\) and \(d\flat ^{\mathrm{in}}(v_i)\) (resp. \(d\flat ^{\mathrm{out}}(v_i)\)) for \(d\flat (v_i)\) in (2).
Extensions of pathwise intensity functions to directed networks follow naturally as a generalization of \(\lambda (\pi _{ij})\). For the directed path \(\vec {\pi }_{ij}\) pointing to \(v_j\), we have
$$\begin{aligned} \lambda (\vec {\pi }_{ij})=(|{\mathcal {N}}_{\vec {\pi }}|)^{-1}\sum _{v_i\in \vec {\pi }_{ij}}\lambda (s_{e_i}) \end{aligned}$$
(4)
where \({\mathcal {N}}_{\vec {\pi }}\) is the cardinality of consecutive edge intervals along \(\vec {\pi }_{ij}\). We note that in general, different from nodewise calculations, (4) is defined for an ordered pair of endpoints of a directed path such that \(\lambda (\vec {\pi }_{ij})\) and \(\lambda (\vec {\pi }_{ji})\) refer to different sequences of edge intervals contained in \({\mathcal {G}}\). However, \(\vec {\pi }_{ij}\) and \(\vec {\pi }_{jk}\) are allowed to have a common endpoint, for example \(v_j\). Apart from (4), we define the directed non-average pathwise intensity function \(\lambda (\vec {\pi }^*_{ij})\) as
$$\begin{aligned} \lambda (\vec {\pi }^*_{ij})=\lim _{|d\vec {\wp }_{ij}|\rightarrow 0}\lbrace \mathbb {E}\left[ N(d\vec {\wp }_{ij})\right] /|d\vec {\wp }_{ij}|\rbrace \end{aligned}$$
(5)
where \(\vec {\wp }_{ij}\) is the set of edges intervals traversed once along a directed path with root \(v_i\) and head \(v_j\), \(d\vec {\wp }_{ij}\) is an infinitesimal interval contained in \(\vec {\wp }_{ij}\) and \(|d\vec {\wp }_{ij}|\) is the area covered by \(d\vec {\wp }_{ij}\).
Apart from directed pathwise intensity functions, one could also consider the information contained in the ancestors or descendants of a distinct node. For example, writing \(\wp ^{-i}_{ij}\) for the set of edge intervals contained in \({{\,\mathrm{de}\,}}(v_i)\), a modification of (5) yields to \(\lambda ({{\,\mathrm{an}\,}}(v_j))=\lim _{|d\wp ^{-j}_{ij}|\rightarrow 0}\lbrace \mathbb {E}\left[ N(d\wp ^{-j}_{ij})\right] /|d\wp ^{-j}_{ij}|\rbrace . \)
First-order network intensity functions for partially directed networks
As partially directed networks are defined as hybrids of directed and undirected networks, we obtain various types of network intensity function as union over the directed and the undirected intensity functions. Using the results of Sects. 2.3.1 and 2.3.2, we obtain the nodewise mean intensity function for partial networks \(\lambda ^{cg}(v_i)\) by
$$\begin{aligned} \lambda ^{cg}(v_i)=(|{{\,\mathrm{deg}\,}}^{cg}(v_i))|)^{-1} \lambda ^\mathrm{out}(v_i)\cup \lambda ^\mathrm{in}(v_i)\cup \lambda (v_i). \end{aligned}$$
(6)
Alternative versions of (6) follow naturally by modification of the union sets. For example, the union \({{\,\mathrm{pa}\,}}(\cdot )\cup {{\,\mathrm{ch}\,}}(\cdot )\) would only consider directed adjacent edges, whereas the union \({{\,\mathrm{ne}\,}}(\cdot )\cup {{\,\mathrm{ch}\,}}(\cdot )\) will exclude any edge pointing to a node of interest. Using the previous results for directed and undirected networks, we can define non-average versions of (6) as unions over \(\lambda ({{\,\mathrm{ne}\,}}(v_i)),\lambda ({{\,\mathrm{pa}\,}}(v_i))\) and \(\lambda ({{\,\mathrm{ch}\,}}(v_i))\) such as the familywise intensity function \(\lambda ^{cg}({{\,\mathrm{fam}\,}}(v_i))=\lambda ({{\,\mathrm{pa}\,}}(v_i))\cup \lambda ({{\,\mathrm{ch}\,}}(v_i)) \) which expresses the expected number of counts along all directed edge intervals which are incident to node \(v_i\).
Second-order intensity and covariance density functions for planar networks
Having point patterns over spatial networks under study, one could be interested in the variation of intensity functions among two different graph entities, e.g., the pairs of distinct edges, neighborhoods or paths contained in the graph. For classical point pattern statistics, such variations are usually expressed by means of second-order properties of the point pattern such as the second-order intensity or the auto- and cross-covariance density functions. This section covers extensions of both functions to pairs of distinct edge intervals, pairs of distinct nodewise sets of edge intervals or pairs of sequences of edge intervals contained in spatial networks. These functions can then be used to characterize the locations of events over the spatial network, which in turn could exhibit randomness, clustering or regularity.
Edgewise second-order intensity and covariance density functions
Consider \(s_{e_i}\) and \(s_{e_j}\) denote two distinct edge intervals of possibly different shape or length contained in \({\mathcal {G}}\). Then, for any distinct edge intervals contained in any such pair, we can define either directed or undirected counting measures. First, assume that \({\mathcal {G}}\) is undirected. Then, using the same notation as before, we obtain the second-order edgewise intensity function \(\lambda (s_{e_i}, s_{e_j})\) as
$$\begin{aligned} \lambda (s_{e_i}, s_{e_j})=\lim _{|ds_{e_i}, ds_{e_j}|\rightarrow 0}\lbrace \mathbb {E}\left[ N(ds_{e_i})N(ds_{e_j})\right] /|ds_{e_i}\times ds_{e_j}|\rbrace \end{aligned}$$
(7)
where \(s_{e_i}\ne s_{e_j}\). Less formally, \(\lambda (s_{e_i}, s_{e_j})\) is the expected number of counts for pairs of distinct undirected edge intervals. However, although (7) can be used to define edgewise versions of Ripleys’ K-function (Ripley 1976), it does not provide a suitable characterization of the theoretical properties of the spatial point pattern usually expressed by the location and scale under different spatial model specifications, for example the first two moments of a particular theoretical point process distribution. The counterpart version of the K-function based on (7) would reflect the number of edgewise neighbors and provides different information from the one obtained using a linear K-function which does not consider edgewise, nodewise or pathwise structures.
An alternative second-order characteristic which better describes these theoretical properties of the spatial point pattern subject to these two distributional parameters is the edgewise covariance density function \(\gamma (s_{e_i}, s_{e_j})\):
$$\begin{aligned} \gamma (s_{e_i}, s_{e_j})=\lambda (s_{e_i}, s_{e_j})-\lambda (s_{e_i})\lambda (s_{e_j}). \end{aligned}$$
As discussed in Sect. 2.3, several different second-order edgewise intensity and covariance functions can be defined. An overview of second-order edgewise intensity functions and edgewise auto-covariance functions which can be defined for directed, undirected and partially directed networks is given in Table 1.
Table 1 Edgewise second-order intensity functions Nodewise second-order intensity and covariance density functions
Similar to the edgewise second-order intensity functions, we could also be interested in the characterization of variations among distinct subsets of edge intervals contained in a spatial network. For this, one could address either the pairwise variation with respect to distinct nodes such as the second-order or covariance density functions for pairs of neighbors, or the pairwise variation with respect to an identical vertex, for example the variation of intensities between the parents and children of a specific node.
Given two sets of distinct neighborhoods \({{\,\mathrm{ne}\,}}(v_i)\) and \({{\,\mathrm{ne}\,}}(v_j)\) where \(v_i\ne v_j\), the nodewise second-order intensity function \(\lambda ({{\,\mathrm{ne}\,}}(v_i),{{\,\mathrm{ne}\,}}(v_j))\) results directly from generalization of (7). Using the same arguments as for the edgewise second-order intensity function, the auto-covariance density function \(\gamma ({{\,\mathrm{ne}\,}}(v_i),{{\,\mathrm{ne}\,}}(v_j))\) can also be computed from \(\lambda ({{\,\mathrm{ne}\,}}(v_i))\) and \(\lambda ({{\,\mathrm{ne}\,}}(v_j))\), the non-averaged nodewise intensity functions of \(v_i\) and \(v_j\) as defined in (2). An overview of different nodewise second-order intensity and auto-covariance functions is given in Table 2.
Table 2 Nodewise second-order intensity functions where \(\flat ^{fam}(v_j)=\flat ^\mathrm{in}(v_j))\cup N(d\flat ^\mathrm{out}(v_j))\) Pathwise second-order intensity and covariance density functions
Lastly, we can also consider the variations among distinct pairs of paths contained in a network. In general, any such variation can be defined for pairs of paths with either common or different endpoints such as V-structures in the form of \(\pi _{ij}\) and \(\pi _{ik}\), inverse V-structures in the form of \(\pi _{ij}\) and \(\pi _{hj}\), elliptic O-structures in the form of \(\pi ^{(1)}_{ij}\) and \(\pi ^{(2)}_{ij}\) where any edge interval is only allowed to traverse once in either \(\pi ^{(1)}_{ij}\) or in \(\pi ^{(2)}_{ij}\), or in the form of two distinct paths \(\pi _{ij}\) and \(\pi _{kl}\).
In general, for \(\pi ^*_{ij}\) and \(\pi ^*_{kl}\) and adopting the same ideas as before, we have
$$\begin{aligned} \lambda (\pi ^*_{ij},\pi ^*_{kl})=\lim _{|d\wp _{ij},d\wp _{kl}|\rightarrow 0}\lbrace \mathbb {E}\left[ N(d\wp _{ij})N(d\wp _{kl})\right] /|d\wp _{ij}\times d\wp _{kl}|\rbrace \end{aligned}$$
and \(\gamma (\pi ^*_{ij},\pi ^*_{kl})=\lambda (\pi ^*_{ij},\pi ^*_{kl})-\lambda (\pi ^*_{ij})\lambda (\pi ^*_{kl})\) where \(\lambda (\pi ^*_{ij})\) and \(\lambda (\pi ^*_{kl})\) are non-averaged pathwise first-order intensity functions as introduced in (3).
As for the edgewise and nodewise second-order characteristics, various types of pathwise second-order intensity and auto-covariance functions can easily be introduced, see Table 3 for a detailed list. We remark that differently from edgewise or nodewise calculations, the second-order pathwise properties either include or exclude the endpoint of a directed path such that \(\lambda (\pi ^*_{ij},\pi ^*_{kj})\ne \lambda ({{\,\mathrm{de}\,}}_{ji},{{\,\mathrm{an}\,}}_{ij})\). That is, while the edge interval \(s_{e_j}=(v_{j-1},v_j)\) is included by \(\vec {\pi }^*_{ij}\), it is excluded by \({{\,\mathrm{an}\,}}(v_j)\) as \({{\,\mathrm{an}\,}}(v_j)\) only considers all edge interval along the path \(\vec {\pi }^*_{ij-1}\).
Table 3 Pathwise second-order intensity functions Contrasting edge-, node- and pathwise second-order intensity and covariance density functions
The above definitions have introduced three subfamilies of second-order point characteristics for spatial network point process data which in sum provide a detailed picture on the observed network pattern, each focusing on a different scale from a more locally to a more globally structural description of the observed events. In general, while edgewise intensity functions are the base underpinning both the node- and pathwise characteristics, both alternative subfamilies of network intensity functions focus on different aspects of the point distribution over the network. Being defined through sets of either undirected, directed or mixed sets of edge intervals which are connected to a particular node, the proposed nodewise intensity functions reflect the spread of points within (pairs of) polynomial (sub)areas centered at pairs of distinct fixed nodes and are whence suitable tools for either hot- or cold-spot detection; they are also useful to decide on regularity or clustering subject to the number of points over different graph theoretic subsets such as the neighborhood or the parents. In particular, for directed and partially directed graphs, these characteristics could be used to analyze the flow of events over consecutively interrelated subsets such as the parents and children.
Different from the nodewise characteristics, the subfamily of pathwise first- and second-order intensity functions provides helpful characteristics which describe the intensity of points over sets of distinct interconnected edge intervals over the complete network and, thus, allow, for example, to evaluate and compare different routes along the network from i to j.