HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-991102-A

Distance model

We initiate distances in the following manner (our graphs are directed):

a)

edge Let us denote the distance from condition $i$ to condition $j$ with $d_{i\rightarrow j}$. In the simplest case all of the distances are $1$, but usually $0 < d_{i\rightarrow j} < \infty$.

b)

way Going on subsequent edges the distances are summarized, so the distance is additive.

c)

between two nodes In this case we have to consider all of the ways connecting these two nodes. Thus, according to real life for the resultant distance $d_{i,j}$ we have $d_{i,j} \le$min $(d_{i\rightarrow j})$. The equality can holds only in degenerated cases, if one of the $d_{i\rightarrow j}$-s is $0$ or if all of the $d_{i\rightarrow j}$-s are $\infty$. Of course, usually $d_{i,j} \ne d_{j,i}$.

d)

between a node and a set (of nodes) Similarly, as in point c).

EXAMPLE Let us assume, that in a whole-type distance model from condition $c$ we can reach 2 dangerous conditions, $a_1$ and $a_2$ with distances $d_1:=d_{c,a_1}$ and $d_2:=d_{c,a_2}$, respectively.

System: $\quad$ a$_1$ $\longleftarrow$ c $\longrightarrow$ a$_2$

In this case obviously $d_{c,a} \le$min$(d_1,d_2)$, where $a$ symbolizes the resultant danger condition, and $d_{c,a}$ depends on $d_1$ and $d_2$. If e.g. $d_1 = \infty$, then $d_{c,a} = d_2$. A possible solution for this problem is the use of the harmonic average, so we get

$$d_{c,a} = \frac{1}{\frac{1}{d_1}+\frac{1}{d_2}} \quad \left(=\frac{d_1 \cdot d_2}{d_1 + d_2} \right).$$

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-991102-A