# Lossy Gossip and Composition of Metrics

- 685 Downloads
- 1 Citations

## Abstract

We study the monoid generated by \(n \times n\) distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \), and we compute the structure of this fan for \(n\) up to \(5\). The monoid captures gossip among \(n\) gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.

## Keywords

Min-plus matrix multiplication Finite metrics Gossip Tropical algebraic groups## 1 Introduction and Results

*car–bike metric*for the remaining ordered pairs leads to the picture on the right in Fig. 1. The corresponding matrix computation is

*tropical*or

*min-plus*matrix multiplication, obtained from usual matrix multiplication by changing plus into minimum and times into plus. Note that the resulting matrix is not symmetric (the transpose corresponds to the “first bike, then car” metric), and that it does not satisfy the triangle inequality either. The bike metric and the car metric were both picked from the \(3\)-dimensional cone of symmetric matrices satisfying all triangle inequalities. Hence one might think that such tropical products sweep out a \(3+3=6\)-dimensional set. However, if we perturb the travel times in the two metric matrices slightly, then their min-plus product moves only in a

*three*-dimensional space, where the entry at position \((1,3)\) remains the sum of the entries in positions \((1,2)\) and \((2,3)\), while the entry in position \((3,1)\) moves freely. This preservation of dimension when tropically multiplying cones of distance matrices is one of the key results of this paper.

*tropical identity matrix*, with zeroes along the diagonal and \(\infty \) outside the diagonal. A phone call between \(E\) and \(P\), for example, corresponds to tropically right-multiplying that tropical identity matrix with

*lossy gossip*, where each phone call between gossipers \(k\) and \(l\) comes with a parameter \(q \in [0,1]\) to be interpreted as the fraction of information that gets broadcast correctly through the phone line, and where each gossiper \(j\) knows a fraction \(p_{ij} \in [0,1]\) of \(i\)’s gossip. Assume the (admittedly simplistic) procedure where \(k\) updates his knowledge of gossip \(i\) to \(q \cdot p_{il}\) if this is larger than \(p_{ik}\) and retains his knowledge \(p_{ik}\) of gossip \(i\) otherwise, and similarly for gossiper \(l\). In this manner, the fractions \(p_{ij}\) are updated through a series of lossy phone-calls. Passing from \(p_{ij}\) to the

*uncertainty*\(u_{ij}:=-\log p_{ij} \in [0,\infty ]\) of gossiper \(j\) about gossip \(i\) and from \(q\) to the

*loss*\(a:=-\log q \in [0,\infty ]\) of the phone line in the call between \(k\) and \(l\), the update rule changes \(u_{ik}\) into the minimum of \(u_{ik}\) and \(u_{il}+a\), and similarly for \(u_{il}\). This is just tropical right-multiplication with the matrix \(C_{kl}(a)\) having \(0\)’s on the diagonal, \(\infty \)’s everywhere else, except an \(a\) on positions \((k,l)\) and \((l,k)\). So

*lossy gossip is tropical matrix multiplication*. Note that lossy gossip is different from

*gossip over faulty telephone lines*discussed in [2, 11], and also from gossip algorithms via multiplication of doubly stochastic matrices as in [5] (though the elementary matrices \(W_{kl}\) there are reminiscent of our matrices \(C_{kl}\)).

This paper concerns the entirety of such uncertainty matrices, or compositions of finite metrics. Our main result uses the following notation: fixing a number \(n\) (of gossipers or vertices); let \(D=D_n\) be the set of all *metric* \(n \times n\) matrices, i.e. matrices with entries \(a_{ij} \in {\mathbb {R}}_{\ge 0}\) satisfying \(a_{ii}=0\) and \(a_{ij}=a_{ji}\) and \(a_{ij}+a_{jk} \ge a_{ik}\). For standard notions in polyhedral geometry, we refer to [20].

Throughout the paper, we give \([0,\infty ]\) the topology of the one-point compactification of \([0,\infty )\), i.e. the topology of a compact, closed interval.

### **Theorem 1.1**

The set \(\{A_1 \odot \cdots \odot A_k \mid k \in {\mathbb {N}},\ A_1,\ldots ,A_k \in D_n\}\) is the support of a (finite) polyhedral fan of dimension \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \), whose topological closure in \([0,\infty ]^{n \times n}\) (with product topology) is the monoid generated by the matrices \(C_{kl}(a)\) with \(k,l \in [n]:=\{1,\ldots ,n\}\) and \(a \in [0,\infty ]\).

We will denote that monoid by \(G_n\), and call it the *lossy gossip monoid* with \(n\) gossipers. The most surprising part of this theorem is that the dimension of \(G_n\) is not larger than \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \). We will establish this in Sect. 7 by proving that \(G_n\) is contained in the tropicalisation of the orthogonal group \(\mathrm {O}_n\).

### **Theorem 1.2**

For \(n \le 5\) the fan in the previous theorem is pure and connected in codimension \(1\). Moreover, for \(n \le 4\), there is a unique coarsest such fan. This coarsest fan has \(D_2,D_3,D_4\) among the \(1,7,289\) full-dimensional cones; and in total it has \(1,2,16\) orbits of full-dimensional cones under the groups \({\text {Sym}}(2),{\text {Sym}}(3),{\text {Sym}}(4)\), respectively.

For some statistics for \(n=5\) we refer to Sect. 6. We conjecture that the pureness and connectedness in codimension \(1\) carry through to arbitrary \(n\).

About the length of products we can say the following.

### **Theorem 1.3**

For \(n \le 5\) every element of \(G_n\) is the tropical product of at most \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) lossy phone call matrices \(C_{kl}(a)\), but not every element is the tropical product of fewer factors.

We conjecture that the restriction \(n \le 5\) can be omitted.

Our next result concerns “pessimal” ordinary gossip (the least efficient way to spread information, keeping the gossipers entertained for as long as possible).

### **Theorem 1.4**

Any sequence of phone calls among \(n\) gossiping parties such that in each phone call both participants exchange all they know, and at least one of the parties learns something new, has length at most \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \), and this bound is attained.

This implies a bound on the length of *irredundant products* of matrices \(C_{kl}(0)\), i.e. tropical products where leaving out any factor changes the value of the product.

### **Corollary 1.5**

In the monoid generated by the matrices \(C_{kl}(0),\ k,l \in [n]\) every irredundant product of such matrices has at most \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) factors.

Our motivation for this paper is twofold. First, it establishes a connection between gossip networks and composition of metrics that seems worth pursuing further. Second, the lossy gossip monoid is a beautiful example of a submonoid of \(({\mathbb {R}}\cup \{\infty \})^{n \times n}\); a general theory of such submonoids also seems very worthwhile. Note that sub*groups* of this semigroup (but with identity element an arbitrary idempotent matrix) have been investigated in [14].

The remainder of this paper is organised as follows: Sections 2 and 3 contain observations that pave the way for the analysis for \(n=3,4\) in Sects. 4 and 5. In Sect. 6 we report on extensive computations for \(n=5\). In Sect. 7 we discuss tropicalisations of the special linear groups and the orthogonal groups, and use the latter to prove the first statement of Theorem 1.1. Interestingly, no polyhedral-combinatorial proof of Theorem 1.1 is known. In Sect. 8 we study the monoid generated by the ordinary gossip matrices \(C_{kl}(0),\ k,l \in [n]\): using the ordinary orthogonal group we prove Theorem 1.4, and for \(n \le 9\) we determine the order of this monoid. We conclude with a number of open questions in Sect. 9.

## 2 Preliminaries

Fixing a natural number \(n\), we define \(\overline{D_n}\) to be the topological closure of \(D_n\) in \([0,\infty ]^{n \times n}\), and we denote by \(G_n\) the monoid generated by \(\overline{D_n}\) under min-plus matrix multiplication. We call \(G_n\) the *lossy gossip monoid* with \(n\) gossipers. This terminology is justified by the following lemma.

### **Lemma 2.1**

The lossy gossip monoid \(G_n\) is generated by the *lossy phone call matrices* \(C_{kl}(a)\ (k,l \in [n], a \in [0,\infty ])\) having zeroes on the diagonal and \(\infty \) everywhere else except for values \(a\) on positions \((k,l)\) and \((l,k)\).

### *Proof*

Lossy phone call matrices lie in \(\overline{D_n}\), so the monoid that they generate is contained in \(G_n\). For the converse it suffices to show that every element \(A\) of \(\overline{D_n}\) is the product of lossy phone call matrices. We claim that, in fact, \(A=\prod _{k<l} C_{kl}(a_{kl})=:B\), where the \(a_{kl}\) are the entries of \(A\) and the product is taken in any order. Indeed, the \((i,j)\)-entry of \(B\) is the minimum of expressions of the form \(a_{i_0,i_1}+a_{i_1,i_2}+\cdots +a_{i_{s-1},i_s}\) where \(s \le \left( {\begin{array}{c}n\\ 2\end{array}}\right) \), \(i_0=i\), \(i_s=j\), and where the \(C_{i_0,i_1},\ldots ,C_{i_{s-1},i_s}\) \(\big ({\hbox {with}}\; s\le \left( {\begin{array}{c}n\\ 2\end{array}}\right) \big )\) appear in that order (though typically interspersed with other factors) in the product expression for \(B\). By the triangle inequalities among the entries of \(A\), the minimum of these expressions equals \(a_{i,j}\).\(\square \)

Although elements of \(G_n\) need not be symmetric, they have a symmetric core.

### **Lemma 2.2**

Each element \(A\) of \(G_n\) satisfies \(a_{ij} = a_{ji}\) for at least \(n-1\) pairs of distinct indices \(i,j\). The graph with vertex set \([n]\) and these pairs as edges is connected.

### *Proof*

We need to prove that for any partition of \([n]\) into two nonempty parts \(K\) and \(L\) there exist a \(k \in K\) and an \(l \in L\) such that \(a_{kl}=a_{lk}\). Write \(A=C_{i_1,j_1}(b_1) \odot \cdots \odot C_{i_s,j_s}(b_s)\) with \(b_1,\ldots ,b_s \in {\mathbb {R}}_{\ge 0}\). If there is no \(r\) such that \(i_r\) and \(j_r\) lie in different sides of the partition, then \(a_{kl}=a_{lk}=\infty \) for all \(k \in K\) and \(l \in L\). Otherwise, among all \(r\) for which \(i_r\) and \(j_r\) lie in different parts of the partition, choose one for which \(b_r\) is minimal. Then \(a_{i_r,j_r}=a_{j_r,i_r}=b_r\). \(\square \)

### **Lemma 2.3**

Every connected graph on \([n]\) occurs as symmetric core of some element of \(G_n\).

### *Proof*

Observe that \(C_{kl}(a) \odot C_{kl}(b) = C_{kl} (a \oplus b)\), where \(\oplus \) denotes tropical addition defined by \(a \oplus b = \min (a,b)\). Thus Lemma 2.1 exhibits \(G_n\) as a monoid generated by certain *one-parameter submonoids*, reminiscent of the generation of algebraic groups by one-parameter subgroups. This resemblance will be exploited in Sects. 7 and 8.

We define the *length* of an element \(X\) of \(G_n\) as the minimal number of factors in any expression of \(X\) as a tropical product of lossy phone call matrices \(C_{kl}(a)\). A rather crude but uniform upper bound on the length of elements of \(G_n\) is the maximal number of factors in a tropical product of lossy phone call matrices in which no factor can be left out without changing the result. We call such an expression *irredundant*, and we have the following bounds.

### **Lemma 2.4**

The number of factors in any irredundant tropical product of lossy phone call matrices in \(G_n\) is at most \(n^2(n-1)/2\). In particular, the length of every element of \(G_n\) is bounded by this number.

### *Proof*

### **Lemma 2.5**

There exists an expression that is an irredundant tropical product of \(\left( {\begin{array}{c}n+1\\ 3\end{array}}\right) \) lossy phone call matrices in \(G_n\).

### *Proof*

### **Proposition 2.6**

The closure of \(D_n\) under tropical matrix multiplication is the support of some finite polyhedral fan in \({\mathbb {R}}_{\ge 0}^{n \times n}\) and equals \(G_n \cap {\mathbb {R}}_{\ge 0}^{n \times n}\). Its topological closure in \([0,\infty ]^{n \times n}\) equals \(G_n\).

Note that this is Theorem 1.1 minus the claim that the dimension of that fan is (not more than) \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \); this claim will be proved in Sect. 7.

### *Proof*

By Lemma 2.1 and the proof of Lemma 2.4 the closure of \(D_n\) under tropical matrix multiplication is a finite union of images of orthants \({\mathbb {R}}_{\ge 0}^k\) with \(k \le n(n-1)^2\) under piecewise linear maps. Such an image is the support of some polyhedral fan. The remaining two statements are straightforward.

From now on, we will sometimes use the term “polyhedral fan” for the topological closure in \([0,\infty ]^N\) of a polyhedral fan in \({\mathbb {R}}_{\ge 0}^N\). Thus \(G_n\) itself is a polyhedral fan in \([0,\infty ]^{n \times n}\).

*Kleene star*of \(A \in [0,\infty ]^{n \times n}\) is defined as

### **Lemma 2.7**

The Kleene star maps \(G_n\) into its subset \(\overline{D_n}\).

### *Proof*

## 3 Graphs with Detours

In the next two sections we will visualise elements of the lossy gossip monoids \(G_3\) and \(G_4\), as well as the polyhedral structures on these monoids. We will do this through combinatorial gadgets that we dub *graphs with detours*. We first recall realisations of ordinary metrics, i.e. elements of \(D_n\) (see, e.g. [8, 13]).

Let \(\varGamma = (V,E)\) be a finite, undirected graph and \(w: E \rightarrow {\mathbb {R}}_{\ge 0}\) be a function assigning lengths to the edges of \(\varGamma \). The weight of a path in \((\varGamma , w)\) is the sum of the weights of the individual edges in the path. A map \(\ell : [n] \rightarrow V\) is called a *labelling*, or \([n]\)-*labelling*, if we need to be precise, and the pair \((\varGamma , \ell )\) is referred to as a labelled graph, or an \([n]\)-labelled graph.

A weighted \([n]\)-labelled graph gives rise to a matrix \(A(\varGamma , w, \ell )\) in \(D_n\) whose entry at position \((i, j)\) is the minimal weight of a path between \(\ell (i)\) and \(\ell (j)\). We say that the weighted labelled graph \((\varGamma , w, \ell )\) *realises* the matrix \(A(\varGamma , w, \ell )\). Any matrix \(X \in D_n\) has a realisation by some weighted, \([n]\)-labelled graph, e.g. the graph with vertex set \([n]\), the entries of \(X\) as weights, and \(\ell \) equal to the identity. However, typically more efficient realisations exist, in the following sense. A weighted, \([n]\)-labelled graph \((\varGamma =(V,E),w,\ell )\) is called an *optimal realisation* of \(X\) if the sum \(\sum _e w(e)\) is minimal among all realisations [13]. We will, moreover, require that no edges get weight \(0\) (since such edges can be removed and their endpoints identified), and that no vertices in \(V \setminus \ell ([n])\) have valency \(2\) (since such vertices can be removed and their incident edges glued together). Optimal realisations of any \(X \in D_n\) exist [13], and there is an interesting question concerning the uniqueness of optimal realisations for generic \(X\) [8, Conjecture 3.20].

Our first step in describing the cones of \(G_3\) and \(G_4\) is to find weighted labelled graphs that realise the elements of \(D_3,D_4\), as follows (for much more about this see [8, 9]). We write \(J_0\) for the matrix of the appropriate size with all entries \(0\).

### *Example 3.1*

- (1)
An element of \(D_2 \setminus \{J_0\}\) is optimally realised by the graph on two vertices having one edge with the right weight. The choice of labelling is inconsequential as long as it is injective. The matrix \(J_0\) is optimally realised by the graph on one vertex.

- (2)
Any matrix in \(D_3\) is realised by the top labelled graph of the poset depicted in Fig. 2 with suitable edge weights (note that we allow these to be zero), but only the matrices in the relative interior of the cone \(D_3\) are optimally realised by it. Matrices on the boundary are optimally realised by some graph further down the poset, depending on the smallest face of \(D_3\) in which the matrix lies.

- (3)
The case of \(D_4\) is similar to that of \(D_3\) in the sense that there exists a single graph \(\varGamma \) which, appropriately labelled and weighted, realises any \(X \in D_4\). However, unlike for \(D_3\), three distinct labellings are required. The labelled graphs are depicted in Fig. 3. For graphs in the relative interior of \(D_4\), the given realisation is optimal (and in fact the unique optimal realisation). \(\square \)

*detour*from \(i\) to \(j\) in an \([n]\)-labelled weighted graph is simply a walk \(p\) starting at \(\ell (i)\) and ending at \(\ell (j)\) that has larger total weight than the path of minimal weight between \(\ell (i)\) and \(\ell (j)\). Such a walk is allowed to traverse the same edge more than once. The data specifying the detour is the triple \((i, j, p)\). A

*labelled weighted graph with detours*is a tuple consisting of a labelled weighted graph and a finite set of detours between distinct ordered pairs \((i,j)\).

Let \((\varGamma , w, \ell , \mathcal {D})\) be an \([n]\)-labelled weighted graph with set of detours \(\mathcal {D}\). It gives rise to a matrix \(A(\varGamma , w, \ell , \mathcal {D})\) whose entry at position \((i, j)\) equals the weight of the detour from \(i\) to \(j\), if there is any, or the weight of a path of minimal weight between \(i\) and \(j\), if there is no detour between \(i\) and \(j\) in \(\mathcal {D}\). In particular, \(A(\varGamma , w, \ell , \mathcal {D})\) need not be symmetric, but its diagonal entries are \(0\). Again, if \(X \in {\mathbb {R}}_{\ge 0}^{n \times n}\) and \(X = A(\varGamma , w, \ell , \mathcal {D})\), then \((\varGamma , w, \ell , \mathcal {D})\) is said to realise \(X\). Any non-negative matrix with zeroes on the diagonal is realised by some labelled weighted graph with detours. Observe also that replacing all detours \((i,j,p)\) by the detours \((j,i,p')\), where \(p'\) is the opposite of \(p\), corresponds to transposing the realised matrix.

### *Example 3.2*

By Lemma 2.7, the Kleene star of a matrix \(A\) in \(G_n\) lies in \(\overline{D_n}\). Thus it makes sense to look for a realisation of \(A\) by a labelled weighted graph with detours that, when forgetting the detours, realises \(A^*\). This is what we will do in the next two sections for \(n=3\) and 4.

## 4 Three Gossipers

## 5 Four Gossipers

*surplus length*of a detour from \(i\) to \(j\) is defined as the difference between the length of the detour and the minimal distance between \(i\) and \(j\) in the graph. Two detours from \(i\) to \(j\) and from \(k\) to \(l\) have the same colour if their surplus lengths are equal.

Next, the group \({\mathbb {Z}}/2{\mathbb {Z}}\) acts on \(G_4\) by transposition. Taking orbit representatives under the larger group \({\text {Sym}}(4) \times ({\mathbb {Z}}/2{\mathbb {Z}})\) from among the \(16\) yields 11 cones. Among these, 9 are simplicial (have six facets), the cone \(D_4\) has 12 facets, and the remaining cone has 9 facets. The cone \(D_4\) is the union of three simplicial cones (see Fig. 3), which are permuted by \({\text {Sym}}(4)\), so we need only one. This is \(C_5\) in Fig. 6. The cone with \(9\) facets turns out to be the union of two simplicial cones. Splitting this up yields \(C_{11}\) and \(C_{12}\) in the figure. It turns out that each \(C_i\) is the image of \({\mathbb {R}}_{\ge 0}^6\) under a linear map into \({\mathbb {R}}_{\ge 0}^{4 \times 4}\) with non-negative integral entries with respect to the standard bases, and that these maps can be realised using weighted, labelled graphs with detours. These are the graphs in the picture. The graphs without the detours realise the Kleene star \(A^*\) with \(A \in C_i\).

### **Theorem 5.1**

The cones realised by the graphs of Fig. 6 give a polyhedral fan structure on \(G_4\). This polyhedral fan is pure of dimension \(6\) and connected in codimension \(1\). Its intersection with a sphere around the origin is a simplicial spherical complex. Moreover, every element of \(G_4\) is the product of (at most) \(6\) lossy phone call matrices.

### *Remark 5.2*

The spherical complex of Fig. 5 clearly has trivial homology. This phenomenon persists for \(n=4\): a computation using +polymake+ shows that all homology groups of the intersection of \(G_4\) with the unit sphere in \(16\)-dimensional space are zero. We do not know whether this is true for general \(n\).

## 6 Five Gossipers

Numbers of subspaces spanned by full-dimensional cones and their numbers of orbits under \(\mathrm{Sym}(n)\)

\(n\) | # spans | # orbits | Orbit size distribution |
---|---|---|---|

2 | 1 | 1 | 1\(\times \)1 |

3 | 7 | 2 | 1\(\times \)1, 1\(\times \)6 |

4 | 289 | 16 | 1\(\times \)1, 6\(\times \)12, 9\(\times \)24 |

5 | 91151 | 787 | 1\(\times \)1, 2\(\times \)20, 1\(\times \)30, 48\(\times \)60, 735\(\times \)120 |

The situation for \(n = 5\) is more complicated than that for smaller \(n\) in that it is no longer true that the subspace spanned by a polyhedral cone of maximal dimension intersects \(G_5\) in a convex cone (recall that for \(n=4\) this did hold, and that we used this in the proof that \(G_4\) has a unique coarsest fan structure).

### *Example 6.1*

## 7 Tropicalising Matrix Groups

In the previous sections we have established Theorem 1.2 through explicit computations. We do not know of any systematic, combinatorial description of a polyhedral structure on \(G_n\) for larger \(n\). However, we will now establish that \(G_n\), which is the support set of *some* finite polyhedral fan by Lemma 2.4, has dimension \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \). Clearly, since \(G_n\) contains \(D_n\), we have \(\dim G_n \ge \dim D_n = \left( {\begin{array}{c}n\\ 2\end{array}}\right) \). So the difficulty of Theorem 1.1 is in proving that its dimension does not exceed \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \).

For this, we make an excursion into tropical geometry. Recall that if \(K\) is a field with a non-Archimedean valuation \(v:K \rightarrow {\mathbb {R}}_\infty :={\mathbb {R}}\cup \{\infty \}\) and if \(I \subseteq K[x_1,\ldots ,x_m]\) is an ideal, then the tropical variety associated to \(I\) is the set of all \(w \in {\mathbb {R}}_\infty ^n\) such that for each polynomial \(f = \sum _\alpha c_\alpha x^\alpha \in I\) the minimum \(\min _\alpha (v(c_\alpha )+w \cdot \alpha )\) is attained for at least two distinct \(\alpha \in {\mathbb {N}}^n\). We denote this tropicalisation by \(\mathrm {Trop}(X)\), where \(X\) is the scheme over \(K\) defined by \(I\). For standard tropical notions we refer to [17]. If \((L,v)\) is any valued extension of \(K\), then the coordinate-wise valuation map \(v:L^n \rightarrow {\mathbb {R}}_\infty ^n\) maps \(X(L)\) into \(\mathrm {Trop}(X)\). If, moreover, \(v:L \rightarrow {\mathbb {R}}_\infty \) is non-trivial and \(L\) is algebraically closed, then the image of the map \(X(L) \rightarrow \mathrm {Trop}(X)\) is dense in \(\mathrm {Trop}(X)\) in the Euclidean topology. Together with the Bieri–Groves theorem [4], this implies that the set \(\mathrm {Trop}(X)\) is (the closure in \({\mathbb {R}}_\infty ^n\) of) a polyhedral complex of dimension equal to \(\dim X\).

*tropical determinant*

### **Proposition 7.1**

The tropicalisation \(\mathrm {Trop}(\mathrm {SL}_n)\) is a monoid under tropical matrix multiplication.

### *Proof*

In general, it is not true that the tropicalisation of a matrix group (relative to the standard coordinates) is a monoid under tropical multiplication.

### *Example 7.2*

### **Proposition 7.3**

The lossy gossip monoid \(G_n\) is contained in \(\mathrm {Trop}(\mathrm {O}_n)\).

### *Proof*

The dimension claim in Theorem 1.1 follows from Proposition 7.3, the Bieri–Groves theorem, and the fact that \(\dim \mathrm {O}_n=\left( {\begin{array}{c}n\\ 2\end{array}}\right) \).

For \(n=1,2,3\), we can say a little bit more about \(\mathrm {Trop}(\mathrm {O}_n)\).

### *Example 7.4*

In general, if a variety is stable under a coordinate permutation, then its tropicalisation is stable under the same coordinate permutation. Consequently, \(\mathrm {Trop}(O_n)\) is stable under permuting rows, under permuting columns, and under matrix transposition.

*not*form a tropical basis. For example, the four-dimensional cone of matrices

*do*suffice to prove that \(\mathrm {Trop}(\mathrm {O}_3) \cap [0,\infty ]^{3 \times 3}\) is equal to \({\text {Sym}}(3) \cdot G_3\), i.e. obtained from \(G_3\) by permuting rows. Indeed, let a \(3 \times 3\)-matrix \(A\) in \([0,\infty ]^{3 \times 3}\) satisfy the tropicalisations of these equations. Then \(\mathrm {tdet}(A)=0\), hence after permuting rows \(A\) has zeroes on the diagonal. Now we distinguish two cases. First, assume that \(A\) is symmetric:

### *Remark 7.5*

We do not know whether the equality \({\text {Sym}}(n) \cdot G_n=\mathrm {Trop}(\mathrm {O}_n) \cap [0,\infty ]^{n \times n}\) (where the action of \({\text {Sym}}(n)\) is by left multiplication) holds for all \(n\). If true, then this would be interesting from the perspective of algebraic groups over non-Archimedean fields: it would say that the image under \(v\) of the compact subgroup \(\mathrm {O}_n(L^0) \subseteq \mathrm {O}_n(L)\), where \(L^0\) is the valuation ring of \(L\), is (dense in) the lossy gossip monoid. But we see no reason to believe that this is true in general. A computational hurdle to checking this even for \(n=4\) is the computation of a polyhedral fan supporting \(\mathrm {Trop}(\mathrm {O}_n)\). For \(n=3\) this can still be done using +gfan+, and it results in a fan with \(f\)-vector \((580,1698,1143)\). Among the \(1143\) three-dimensional cones, \(1008\) are contained in the positive orthant, as opposed to the \(6 \cdot 7=42\) found by applying row permutations to the cones in \(G_3\). This suggests that +gfan+ does not automatically find the most efficient fan structure on \(\mathrm {O}_n\), and at present we do not know how to overcome this.

## 8 Ordinary Gossip

*ordinary gossip monoid*\(G_n(\{0,\infty \})\), which is the submonoid of \(G_n\) of matrices with entries in \(\{0,\infty \}\). Note that there is a surjective homomorphism \(G_n \rightarrow G_n(\{0,\infty \})\) mapping non-\(\infty \) entries to \(0\) and \(\infty \) to \(\infty \), which shows that the length of an element of \(G_n(\{0,\infty \})\) inside \(G_n\) is the same as the minimal number of non-lossy phone calls \(C_{ij}(0)\) needed to express it. A classical result says that length of the all-zero matrix is exactly \(1\) for \(n=2\), \(3\) for \(n=3\), and \(2n-4\) for \(n \ge 4\) [1, 6, 12, 19], and this result spurred a lot of further activity on gossip networks. But the all-zero matrix does not necessarily have the largest possible length—see Table 2, which records sizes and maximal element lengths for \(G_n(\{0,\infty \})\) with \(n \le 9\). The first \(8\) rows were computed by former Eindhoven Master’s student Jochem Berndsen [3].

Sizes and maximal lengths of \(G_n(\{0,\infty \})\) for \(n = 1, \ldots , 9\)

\(n\) | \(|G_n(\{0,\infty \})|\) | Max. length |
---|---|---|

1 | 1 | 0 |

2 | 2 | 1 |

3 | 11 | 3 |

4 | 189 | 4 |

5 | 9152 | 6 |

6 | 1,092,473 | 10 |

7 | 293,656,554 | 13 |

8 | 166,244,338,221 | 16 |

9 | 188,620,758,836,916 | 19 |

While we do not know the maximal length of an element in \(G_n(\{0,\infty \})\) for general \(n\), we do have an upper bound, namely, the maximal number of factors in an irredundant product. This number, in turn, is bounded from above by \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \), as we now prove.

### *Proof of Theorem 1.4 and Corollary 1.5*

Consider \(n\) gossipers, initially each with a different gossip item unknown to all other gossipers. They communicate by telephone, and whenever two gossipers talk, each tells the other all he knows. We will determine the maximal length of a sequence of calls, when in each call at least one participant learns something new. The answer turns out to be \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \).

That \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) is a lower bound is shown by the following scenario: Number the gossipers \(1,\ldots ,n\). All calls involve gossiper \(1\). For \(i=2,\ldots ,n\) he calls \(i,i-1,\ldots ,2\), for a total of \(1+2+\cdots +(n-1) = \left( {\begin{array}{c}n\\ 2\end{array}}\right) \) calls. There are many other scenarios attaining \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \), and it does not seem easy to classify them.

We now argue that \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) is an upper bound. Although we will not use this, we remark that it is easy to see that \(2 \cdot \left( {\begin{array}{c}n\\ 2\end{array}}\right) =n(n-1)\) is an upper bound. After all, each of the \(n\) participants must learn \(n-1\) items, and in each call at least one participant learns something.

Let \(I_1,I_2,\ldots ,I_\ell \) be a sequence of unordered pairs from \([n]\) representing phone calls where in each call at least one participant learns something new. To each \(I_a\) we associate the homomorphism \(\phi _a:=\mathrm {SO}_2({\mathbb {C}}) \rightarrow \mathrm {SO}_n({\mathbb {C}})\) that maps a \(2 \times 2\)-matrix \(g\) to the matrix that has \(g\) in the \(I_a \times I_a\)-block and otherwise has zeroes outside the diagonal and ones on the diagonal. For each \(k \le \ell \) we obtain a morphism of varieties (not a group homomorphism) \(\psi _k:\mathrm {SO}_2({\mathbb {C}})^k \rightarrow \mathrm {SO}_n({\mathbb {C}})\) sending \((g_1,\ldots ,g_k)\) to \(\phi _1(g_1) \cdots \phi _k(g_k)\). Let \(X_k\) be the closure of the image of \(\psi _k\); this is an irreducible subvariety of \(\mathrm {SO}_n({\mathbb {C}})\). The \((i,j)\)-matrix entry is identically zero on \(X_k\) if and only if gossiper \(j\) does not know gossip \(i\) after the first \(k\) phone calls. Since some gossiper learns something new in the \(k\)-th phone call, some matrix entry is identically zero on \(X_{k-1}\) which is not identically zero on \(X_k\). Consequently, we have \(0=\dim X_0<\dim X_1<\cdots <\dim X_\ell \). But all \(X_k\) are contained in the variety \(\mathrm {SO}_n({\mathbb {C}})\) of dimension \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \), so we conclude that \(\ell \le \left( {\begin{array}{c}n\\ 2\end{array}}\right) \).

This concludes the proof of Theorem 1.4. Corollary 1.5 follows because, in any irredundant product of phone calls, every initial segment must be a sequence of phone calls in each of which at least one party learns something new.\(\square \)

Maximum length \(l_n\) of an irredundant product of phone calls

\(n\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

\(l_n\) | 0 | 1 | 3 | 5 | 8 | 12 | 16 | \(\ge 21\) |

## 9 Open Questions

In view of the extensive computations in Sects. 4–6 and the rather indirect dimension argument in Sect. 7, the most urgent challenge concerning the lossy gossip monoid is the following.

### **Question 9.1**

Find a purely combinatorial description of a polyhedral fan structure with support \(G_n\). Use this description to prove or disprove the pureness of dimension \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) and the connectedness in codimension one.

The following question is motivated on the one hand by the fact that \(G_n\) has dimension \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) and on the other hand by Theorem 1.4, which implies that elements of the ordinary gossip monoid \(G_n(\{0,\infty \})\) have length at most \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \).

### **Question 9.2**

Is the length of any element of \(G_n\) at most \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \)?

Once a satisfactory polyhedral fan for \(G_n\) is found, the somewhat ad hoc graphs in Sects. 4 and 5 lead to the following challenge.

### **Question 9.3**

Find a useful notion of optimal realisations of elements of \(G_n\) by graphs with detours, and a notion of *tight spans* of such elements.

For the relation between tight spans and optimal realisations of metrics by weighted graphs see [8, Thm. 5].

We conclude with two questions concerning tropicalisations of orthogonal groups (Sect. 7).

### **Question 9.4**

Is \(\mathrm {Trop}(\mathrm {O}_n)\) a monoid under tropical matrix multiplication\(?\) This is evident for \(n \le 2\), we have checked it computationally for \(n=3\), and it is open for \(n \ge 4\).

### **Question 9.5**

Is it true that \(\mathrm {Trop}(\mathrm {O}_n) \cap [0,\infty ]^{n \times n}\) equals \({\text {Sym}}(n) \cdot G_n?\) Here the action of \({\text {Sym}}(n)\) is by permuting rows. This is true for \(n \le 3\), and open for \(n \ge 4\).

## Notes

### Acknowledgments

We thank Tyrrell McAllister for discussions on the tropical orthogonal group many years ago, and Peter Fenner and Mark Kambites for pointing out problems with an earlier, purely combinatorial proof of Theorem 1.4. JD is supported by a Vidi Grant from the Netherlands Organisation for Scientific Research (NWO) and BJF by an NWO free competition Grant.

## References

- 1.Baker, B., Shostak, R.: Gossips and telephones. Discrete Math.
**2**, 191–193 (1972)CrossRefzbMATHMathSciNetGoogle Scholar - 2.Berman, K.A., Hawrylycz, M.: Telephone problems with failures. SIAM J. Algebraic Discrete Methods
**7**, 13–17 (1986)CrossRefzbMATHMathSciNetGoogle Scholar - 3.Berndsen, J.: Three problems in algebraic combinatorics. Master’s thesis, Eindhoven University of Technology (2012). http://alexandria.tue.nl/extra1/afstversl/wsk-i/berndsen2012
- 4.Bieri, R., Groves, J.R.J.: The geometry of the set of characters induced by valuations. J. Reine Angew. Math.
**347**, 168–195 (1984)zbMATHMathSciNetGoogle Scholar - 5.Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Randomized gossip algorithms. IEEE Trans. Inform. Theory
**52**, 2508–2530 (2006)CrossRefzbMATHMathSciNetGoogle Scholar - 6.Bumby, R.T.: A problem with telephones. SIAM J. Algebraic Discrete Methods
**2**, 13–18 (1981)CrossRefzbMATHMathSciNetGoogle Scholar - 7.Butkovič, P.: Max-Linear Systems, Theory and Algorithms. Springer Monographs in Mathematics. Springer, London (2010)CrossRefGoogle Scholar
- 8.Dress, A.W.M.: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. Math.
**53**, 321–402 (1984)CrossRefzbMATHMathSciNetGoogle Scholar - 9.Dress, A., Huber, K.T., Lesser, A., Moulton, V.: Hereditarily optimal realizations of consistent metrics. Ann. Combin.
**10**(1), 63–76 (2006)CrossRefzbMATHMathSciNetGoogle Scholar - 10.Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes—Combinatorics and Computation. DMV Seminars, vol. 29, pp. 43–74. Birkhäuser, Basel (2000)Google Scholar
- 11.Haddad, R.W., Roy, S., Schäffer, A.A.: On gossiping with faulty telephone lines. SIAM J. Algebr. Discrete Methods
**8**, 439–445 (1987)CrossRefzbMATHGoogle Scholar - 12.Hajnal, A., Milner, E.C., Szemerédi, E.: A cure for the telephone disease. Can. Math. Bull.
**15**, 447–450 (1972)CrossRefzbMATHGoogle Scholar - 13.Imrich, W., Simões Pereira, J.M.S., Zamfirescu, C.M.: On optimal embeddings of metrics in graphs. J. Combin. Theory Ser. B
**36**(1), 1–15 (1984)CrossRefzbMATHMathSciNetGoogle Scholar - 14.Izhakian, Z., Johnson, M., Kambites, M.: Tropical matrix groups (2012). http://arxiv.org/abs/1203.2449
- 15.Jensen, A.N.: Gfan, a software system for Gröbner fans and tropical varieties. http://home.imf.au.dk/jensen/software/gfan/gfan.html (2005–2011)
- 16.Koolen, J., Lesser, A., Moulton, V.: Optimal realizations of generic five-point metrics. Eur. J. Combin.
**30**(5), 1164–1171 (2009)CrossRefzbMATHMathSciNetGoogle Scholar - 17.Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence, RI (2015)Google Scholar
- 18.Sturmfels, B., Yu, J.: Classification of six-point metrics. Electron. J. Combin.
**11**(1), R44 (2004)MathSciNetGoogle Scholar - 19.Tijdeman, R.: On a telephone problem. Nieuw Arch. Wiskd. III Ser.
**19**, 188–192 (1971)zbMATHMathSciNetGoogle Scholar - 20.Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, Berlin (1995)CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.