Sk,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 := VMin ⊎VRand⊎VMax, where the symbol⊎ denotes the disjoint union of sets. We suppose that the sets of Max vertices and Min vertices are nonempty. Ifv∈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. Ifvis a Min vertex or a Max vertex ande∈Out(v)is its outgoing edge, then we equip this edge with a real number re. Furthermore, ifv is a Random vertex, then we equip its set of outgoing edges with a rational probability distribution. More precisely, every edgee∈Out(v)is equipped with a strictly positive rational number qe ∈Q,qe >0, and we suppose that∑e∈Out(v)qe = 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]V2 with transition probabilities pvv := 1 for all v ∈ VMax⊎VMin, pvw := q(v,w) if v ∈ VRand and (v, w) ∈ Out(v), and pvw := 0 otherwise. In other words, if we consider a Markov chain with P as its transition matrix, then every state of VMax⊎VMin is absorbing, and a trajectory of the Markov chain visits the states of VRand by picking at random, for each vertex v∈VRand, one edge in Out(v) according to the probability law given byq(v,·), until it reaches a state ofVMax⊎VMin. 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 bypev the conditional probability to reach the absorbing statevstarting from the head of e. For the sake of simplicity, we assume that VMin = [n]and VMax= [m].
Definition 5.36. The operator encoded by G⃗ is the function F:Rn→Rn defined as
∀v∈[n],(F(x))v := min
e∈Out(v)
(
re+ ∑
w∈[m]
pew max
e′∈Out(w)
(re′+ ∑
u∈[n]
peu′xu
)). (5.2)
Lemma 5.37. The operator encoded by G⃗ is semilinear, monotone, and homogeneous.
Proof. LetF be the operator encoded byG⃗. Note that everypevis rational since we have assumed that the qe are in Q (see Remark 2.138). This means that the graph of F is definable in Log
and henceF 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 fromG. 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. Letv be a Min vertex, ande∈Out(v). We claim that if u ∈ [n] is a Min state, then peu = 0. Indeed, by Assumption C (i), any path from the head of eto u in G⃗ contains a Max vertex. As a consequence, there is no path from the head of e tou in the subgraph in which we have removed the edges going out of the Max and Min vertices. We deduce that for every Min vertexv and edgee∈Out(v), we have ∑w∈[m]pew = 1.
Analogously, we can show that for all Max verticeswand edgee′ ∈Out(w),∑u∈[n]peu′ = 1. We deduce that the operatorF 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: Rn → Rn 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 pew with e ∈ Out(v) and v ∈ VMin take only the values 0 and 1. Formally, let A(1), . . . , A(p)∈Qn×n and b(1), . . . , b(p) ∈Rn such that Corollary 5.33 holds. We buildG⃗ as the graph in which the set of Min vertices is [n], the set of Max vertices is ⊎k[Mk], and the set of Random vertices is ⊎k∈[n], i∈[Mk]Ski. Let k be a Min vertex. We add an edge (k, i) for every i∈[Mk], with r(k,i) := 0. Moreover, for everyi∈[Mk], we add an edge(i, s) for each s∈Ski, with r(i,s) := b(s)k . Finally, if i ∈ [Mk] and s ∈ Ski, we add an edge (s, l) with q(s,l) := A(s)kl for every 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 putre = 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:Rn → Rn denote the semilinear monotone homogeneous operator encoded by G⃗. We extend F to a function F:Tn → Tn by applying (5.2) to all points of Tn.2 Then, the set {x∈Tn:x⩽F(x)} is a tropical Metzler spectrahedral cone.
Proof. Consider the Markov chain introduced before Definition 5.36, take a Min vertexv∈[n]
and an outgoing edge e ∈ Out(v). Under the assumptions over the graph G⃗, the absorbing states reachable from the head of eform a set{we, w′e} ⊂[m]of cardinality at most 2(we use the convention we =w′e if there is only one such absorbing state). Moreover, if we ̸=w′e, then pewe =pew′
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 ue′. With this notation, we have:
(F(x))v = min
e∈Out(v)
( re+1
2
( max
e′∈Out(we)(re′ +xue′) + max
e′∈Out(w′e)(re′+xue′))) (5.3) for allv ∈[n]. Out of this data we construct symmetric Metzler matricesQ(1), . . . , Q(n)∈Tm±×m. If a Min vertex v∈[n]has an outgoing edgee∈Out(v) such thatwe̸=w′e, then we define
Q(v)w
ew′e =Q(v)w′
ewe :=⊖max{−re′:e′∈Out(v) ∧ {we′, we′′}={we, w′e}}. The diagonal coefficients Q(v)ww of the matricesQ(v) are defined as follows:
• ife= (v, w)is an edge, but(w, v) is not, then we setQ(v)ww:=⊖(−re);
• ife= (w, v)is an edge, but(v, w) is not, then we setQ(v)ww:=re;
• if bothe= (v, w) and e′ = (w, v)are edges, then we set Q(v)ww to⊖(−re) if−re> re′, and tore′ ifre′ ⩾−re.
2We use the convention that−∞ ·x=−∞for allx >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 ∈ Tn: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 describesS. 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′ ifα⩾β, and to Moreover, note that ifx∈Tnverifies (5.4), then we have the equality
max
Furthermore, observe that for anyw̸=w′ we have the equality
{{v,e}:e∈Out(v),max{w,w′}={we,w′e}}(xv−re) = max