*S** _{k,1}* =

*{*1,2

*}*for all

*k∈ {*1,2,3

*}*,

**5.3** **Description of real tropical cones by directed graphs**

We now describe how semilinear, monotone, homogeneous operators can be encoded by directed
graphs. To this end we take a directed graph *G⃗* := (V, E), where the set of vertices is divided
into Max vertices, Min vertices, and Random vertices, i.e., *V* := *V*Min *⊎V*Rand*⊎V*Max, where
the symbol*⊎* denotes the disjoint union of sets. We suppose that the sets of Max vertices and
Min vertices are nonempty. If*v∈V* is a vertex of *G⃗*, then by In(v) :=*{*(w, v) : (w, v)*∈E}*we
denote the set of its incoming edges, and by Out(v) :=*{*(v, w) : (v, w) *∈E}* we denote the set
of its outgoing edges. We suppose that the every vertex has at least one outgoing edge. If*v*is a
Min vertex or a Max vertex and*e∈*Out(v)is its outgoing edge, then we equip this edge with a
real number *r**e*. Furthermore, if*v* is a Random vertex, then we equip its set of outgoing edges
with a rational probability distribution. More precisely, every edge*e∈*Out(v)is equipped with
a strictly positive rational number *q*_{e}*∈*Q,*q*_{e}*>*0, and we suppose that^{∑}_{e∈Out(v)}*q** _{e}* = 1. We
also make the following assumption.

**Assumption C.** (i) Every path between any two Min vertices contains at least one Max
vertex;

(ii) Every path between any two Max vertices contains at least one Min vertex;

(iii) From every Random vertex, there is a path to a Min or a Max vertex.

We now construct a semilinear, monotone, homogeneous operator from such a graph. We
construct a stochastic matrix *P* *∈* [0,1]^{V}^{2} with transition probabilities *p**vv* := 1 for all *v* *∈*
*V*_{Max}*⊎V*_{Min}, *p** _{vw}* :=

*q*

_{(v,w)}if

*v*

*∈*

*V*

_{Rand}and (v, w)

*∈*Out(v), and

*p*

*:= 0 otherwise. In other words, if we consider a Markov chain with*

_{vw}*P*as its transition matrix, then every state of

*V*Max

*⊎V*Min is absorbing, and a trajectory of the Markov chain visits the states of

*V*Rand by picking at random, for each vertex

*v∈V*

_{Rand}, one edge in Out(v) according to the probability law given by

*q*

_{(v,}

_{·}_{)}, until it reaches a state of

*V*

_{Max}

*⊎V*

_{Min}. In this way, after leaving a Min vertex,

92 Chapter 5. Tropical analogue of the Helton–Nie conjecture
the trajectory reaches a Max vertex, and vice versa. If *e* is an edge and *v* is a Max or Min
vertex, then we denote by*p*^{e}* _{v}* the conditional probability to reach the absorbing state

*v*starting from the head of

*e. For the sake of simplicity, we assume that*

*V*

_{Min}= [n]and

*V*

_{Max}= [m].

**Definition 5.36.** The *operator encoded by* *G⃗* is the function *F*:R^{n}*→*R* ^{n}* defined as

*∀v∈*[n],(F(x))*v* := min

*e**∈*Out(v)

(

*r**e*+ ^{∑}

*w**∈*[m]

*p*^{e}* _{w}* max

*e*^{′}*∈*Out(w)

(*r**e** ^{′}*+

^{∑}

*u**∈*[n]

*p*^{e}_{u}^{′}*x**u*

))*.* (5.2)

**Lemma 5.37.** *The operator encoded by* *G⃗* *is semilinear, monotone, and homogeneous.*

*Proof.* Let*F* be the operator encoded by*G⃗*. Note that every*p*^{e}* _{v}*is rational since we have assumed
that the

*q*

*e*are in Q (see Remark 2.138). This means that the graph of

*F*is definable in

*L*og

and hence*F* is semilinear (by Lemma 2.103). Moreover, *F* is obviously monotone. We already
observed that the Max and Min vertices are absorbing states in the Markov chain constructed
from*G. Besides, Assumption C (iii) still holds in the subgraph obtained by removing the edges⃗*
going out of the Max and Min vertices. As a consequence, the Max and Min vertices are the
only recurrent classes in the Markov chain. Let*v* be a Min vertex, and*e∈*Out(v). We claim
that if *u* *∈* [n] is a Min state, then *p*^{e}* _{u}* = 0. Indeed, by Assumption C (i), any path from the
head of

*e*to

*u*in

*G⃗*contains a Max vertex. As a consequence, there is no path from the head of

*e*to

*u*in the subgraph in which we have removed the edges going out of the Max and Min vertices. We deduce that for every Min vertex

*v*and edge

*e∈*Out(v), we have

^{∑}

_{w}

_{∈}_{[m]}

*p*

^{e}*= 1.*

_{w}Analogously, we can show that for all Max vertices*w*and edge*e*^{′}*∈*Out(w),^{∑}_{u∈[n]}*p*^{e}_{u}* ^{′}* = 1. We
deduce that the operator

*F*is homogeneous.

In the following lemma, we show that any semilinear, monotone, homogeneous operator is encoded by some directed graph:

**Lemma 5.38.** *Let* *F*: R^{n}*→* R^{n}*be a semilinear, monotone, homogeneous operator. Then,*
*there exists a directed graph* *G⃗* *satisfying Assumption C such that* *F* *is encoded by* *G⃗.*

*Proof.* The idea is to identify the representation (5.1) to a special case of (5.2), in which the
probabilities *p*^{e}* _{w}* with

*e*

*∈*Out(v) and

*v*

*∈*

*V*

_{Min}take only the values 0 and 1. Formally, let

*A*

^{(1)}

*, . . . , A*

^{(p)}

*∈*Q

^{n}

^{×}*and*

^{n}*b*

^{(1)}

*, . . . , b*

^{(p)}

*∈*R

*such that Corollary 5.33 holds. We build*

^{n}*G⃗*as the graph in which the set of Min vertices is [n], the set of Max vertices is

*⊎*

*k*[M

*], and the set of Random vertices is*

_{k}*⊎*

*k*

*∈*[n], i

*∈*[M

*k*]

*S*

*. Let*

_{ki}*k*be a Min vertex. We add an edge (k, i) for every

*i∈*[M

*k*], with

*r*

_{(k,i)}:= 0. Moreover, for every

*i∈*[M

*k*], we add an edge(i, s) for each

*s∈S*

*ki*, with

*r*

_{(i,s)}:=

*b*

^{(s)}

*. Finally, if*

_{k}*i*

*∈*[M

*] and*

_{k}*s*

*∈*

*S*

*, we add an edge (s, l) with*

_{ki}*q*

_{(s,l)}:=

*A*

^{(s)}

*for every*

_{kl}*l∈*[n]such that

*A*

^{(s)}

_{kl}*>*0. The requirements of Assumption C are straightforwardly satisfied.

*Example* 5.39. The graph presented in Fig. 5.3 encodes the operator from Example 5.35.

The following proposition characterizes the semilinear, monotone, homogeneous operators associated with tropical Metzler spectrahedral cones.

**Proposition 5.40.** *Suppose that the graph* *G⃗* *fulfills Assumption C and has the following *
*prop-erties:*

*•* *every Random vertex has exactly two outgoing edges, the heads of these edges are diﬀerent,*
*and the probability distribution associated with these edges is equal to* (1/2,1/2);

5.3. Description of real tropical cones by directed graphs 93

3

1 2 1

2

*−*2 3

*−*2

5/2 1

1/4 4 3/4

14/3 1/3 2/3

Figure 5.3: Graph that encodes the operator from Example 5.35. Min vertices are depicted by
circles, Max vertices are depicted by squares, Random vertices are depicted by diamonds. We
put*r**e* = 0for every edge *e∈E* that has no label.

*•* *every edge outgoing from a Random vertex has a Max vertex as its head.*

*Let* *F*:R^{n}*→* R^{n}*denote the semilinear monotone homogeneous operator encoded by* *G⃗. We*
*extend* *F* *to a function* *F*:T^{n}*→* T^{n}*by applying* (5.2) *to all points of* T^{n}*.*^{2} *Then, the set*
*{x∈*T* ^{n}*:

*x*⩽

*F*(x)}

*is a tropical Metzler spectrahedral cone.*

*Proof.* Consider the Markov chain introduced before Definition 5.36, take a Min vertex*v∈*[n]

and an outgoing edge *e* *∈* Out(v). Under the assumptions over the graph *G⃗*, the absorbing
states reachable from the head of *e*form a set*{w*_{e}*, w*^{′}_{e}*} ⊂*[m]of cardinality at most 2(we use
the convention *w** _{e}* =

*w*

^{′}*if there is only one such absorbing state). Moreover, if*

_{e}*w*

_{e}*̸*=

*w*

^{′}*, then*

_{e}*p*

^{e}

_{w}*=*

_{e}*p*

^{e}

_{w}*′*

*e* = 1/2. Furthermore, observe that if *w∈*[m]is a Max vertex and *e*^{′}*∈*Out(w) is an
outgoing edge, then our assumptions imply that the head of *e** ^{′}* is a Min vertex. We denote it
by

*u*

_{e}*′*. With this notation, we have:

(F(x))* _{v}* = min

*e**∈*Out(v)

(
*r** _{e}*+1

2

( max

*e*^{′}*∈*Out(w*e*)(r_{e}*′* +*x*_{u}* _{e′}*) + max

*e*^{′}*∈*Out(w^{′}* _{e}*)(r

_{e}*′*+

*x*

_{u}*)*

_{e′}^{))}(5.3) for all

*v*

*∈*[n]. Out of this data we construct symmetric Metzler matrices

*Q*

^{(1)}

*, . . . , Q*

^{(n)}

*∈*T

^{m}

_{±}

^{×}*. If a Min vertex*

^{m}*v∈*[n]has an outgoing edge

*e∈*Out(v) such that

*w*

*e*

*̸=w*

^{′}*, then we define*

_{e}*Q*^{(v)}_{w}

*e**w*^{′}* _{e}* =

*Q*

^{(v)}

_{w}

_{′}*e**w**e* :=*⊖*max^{{}*−r*_{e}*′*:*e*^{′}*∈*Out(v) *∧ {w*_{e}*′**, w*_{e}^{′}*′**}*=*{w*_{e}*, w*^{′}_{e}*}*^{}}*.*
The diagonal coeﬃcients *Q*^{(v)}*ww* of the matrices*Q*^{(v)} are defined as follows:

• if*e*= (v, w)is an edge, but(w, v) is not, then we set*Q*^{(v)}*ww*:=*⊖(−r**e*);

• if*e*= (w, v)is an edge, but(v, w) is not, then we set*Q*^{(v)}*ww*:=*r**e*;

• if both*e*= (v, w) and *e** ^{′}* = (w, v)are edges, then we set

*Q*

^{(v)}

*ww*to

*⊖*(

*−r*

*e*) if

*−r*

*e*

*> r*

_{e}*′*, and to

*r*

_{e}*′*if

*r*

_{e}*′*⩾

*−r*

*.*

_{e}2We use the convention that*−∞ ·**x*=*−∞*for all*x >*0and*−∞ ·*0 = 0.

94 Chapter 5. Tropical analogue of the Helton–Nie conjecture
Finally, all the other entries of the matrices *Q*^{(v)} are set to *−∞*. Let *S* = *S*(Q^{(1)}*, . . . , Q*^{(n)})
denote the associated tropical Metzler spectrahedral cone. We want to show that *S* = *{x* *∈*
T* ^{n}*:

*x*⩽

*F*(x)

*}*. Note that we have the equivalence The last set of inequalities is equivalent to

*∀{w, w*^{′}*},* max

We want to show that the last set of constraints describes*S*. To do this, recall that an inequality
of the form max(x, α+*y)*⩾max(x^{′}*, β*+*y)* is equivalent tomax(x, α+*y)*⩾*x** ^{′}* if

*α*⩾

*β, and to*Moreover, note that if

*x∈*T

*verifies (5.4), then we have the equality*

^{n}max

Furthermore, observe that for any*w̸*=*w** ^{′}* we have the equality

*{{**v,e**}*:*e**∈*Out(v),max*{**w,w*^{′}*}*=*{**w**e**,w*^{′}_{e}*}}*(x_{v}*−r** _{e}*) = max