Suppose we are given

(i) an undirected connected graph with vertex set $\mathcal V$ and a countable set $\mathcal E$ of edges, where each edge apart from specifying a relation between two vertices is viewed as a set $e$ which is in bijective correspondence with some non-empty open interval;

(ii) $\mathcal V$ and any edge $e\in\mathcal E$ are disjoint, and the edges are pairwise disjoint.

Then we call the triple of $\mathcal V$, $\mathcal E$, and the bijective mappings/edge coordinates for a graph with Euclidean edges, and we denote this triple by $\mathcal G$ and the whole graph set by
$L=\mathcal V\cup \bigcup_{e\in\mathcal E}e$. In the special case where each edge $e$ is just an open line segments whose endpoints agree with the adjacent vertices associated to $e$, then $L$ is a linear network as
considered in connection to for example road networks, dendrite networks of neurons, and brick walls. Now, for any points $u,v\in L$,

the edge coordinates lead naturally to a geodesic distance $d_{\mathcal G}(u,v)$ given by shortest path distance in $\mathcal G$. If the vertex set is contained in the Euclidean space $\mathbb R^k$ and the edges are smooth subsets $\mathbb R^k$, we may require that condition (i) and not necessarily (ii) is
satisfied: In fact there is then a natural one-to-one correspondence to a graph with Euclidean edges, and this naturally induces a geodesic distance $d_{\mathcal G}(u,v)$ but taking into consideration whether $u$ (or $v$) is a certain vertex or it belongs to a certain edge. We notice that
$d_{\mathcal G}(u,v)$ may then be different from the usual geodesic distance $d_L(u,v)$ on $L$ which is given by shortest path-connected curve distance.

Our main goal is to establish sufficient conditions on the existence of positive definite functions of the form $K(d_{\mathcal G}(u,v))$ for all $u,v\in L$. Then the Kolmogorov Extension Theorem establishes the existence of a separable (Gaussian) random field $Z={Z(u): u\in S }$ on $\mathcal G$ with covariance function
$$
\text{cov}(Z(u), Z(v)) = K\left(d_{\mathcal G}(u,v)\right)\quad\forall u,v\in L.
$$
(Since the covariance function depends on the graph with Euclidean edges, we prefer using the terminology "random filed on $\mathcal G$" rather than "random field on $L$".)
We say then that the covariance function is {\it pseudo-stationary} and that the random field $Z$ is *second-order pseudo-stationary*, noting that we do not require that the mean function $\mathrm EZ(u)$ is constant. Note that our setting is different from that in research on random fields on directed trees such as in a network of rivers or streams where water flows in one direction. Then special techniques are appropriate for constructing covariance functions of the form above. However, our techniques will be different, since we deal with undirected graphs.

One motivation for considering a second-order pseudo-stationary random field $Z$ is that for any geodesic path $p_{uv}\subseteq L$ connecting two points $u,v\in L$, the restriction of $Z$ to $p_{uv}$ has the same covariance structure as the random field $\tilde Z(t)$ defined on $t\in[0,t_0]\subset \mathbb R$ where $\text{cov}(\tilde Z(t), \tilde Z(s))= K(|t-s|)$ and $t_0 = d_{\mathcal G}(t,s)$. In brief, $Z$ restricted to a geodesic path is indistinguishable from a corresponding Gaussian random field on a closed interval.

Another motivation is that given a covariance function of the form above, we can construct {\it second-order intensity-reweighted pseudo-stationary (SOIRPS) point processes} on $\mathcal G$, meaning that the point process has a pair correlation function of the form $g(u,v)=g_0(d_{\mathcal G}(u,v))$ for all $u,v\in L$. A Poisson process on $L$ is SOIRPS but to the best of our knowledge, apart from the Poisson process, models for SOIRPS point processes on point processes with Euclidean edges have not yet been specified in the literature. We show that for a log Gaussian Cox process (LGCP) $X$, i.e.\ when $X$ conditional on a Gaussian random field $Z$ on $L$ is a Poisson process with intensity function $\exp(Z(u))$, $u\in L$, second-order pseudo-stationarity of $Z$ is equivalent to SOIRPS of $X$. We also specify moment and Palm measure theoretical results for LGCPs. Further examples of SOIRPS point processes on graphs with Euclidean edges will be discussed in the talk.

Joint work with Ethan Anderes (University of California at Davis), and Jakob G. Rasmussen (Aalborg University).