Lossy Gossip and Composition of Metrics
Lossy Gossip and Composition of Metrics
Andries E. Brouwer 0
Jan Draisma 0
Bart J. Frenk 0
0 Imagine travelling between three locations such as Eindhoven (E , a mediumsized town in the Netherlands), a parking lot P on the border of the Dutch capital Amsterdam, and the city centre A of Amsterdam. In Fig. 1, the travel times by car between these locations are depicted by the leftmost triangle, while the travel times by bike are
We study the monoid generated by n n distance matrices under tropical (or minplus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension n , and we compute 2 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.
Minplus matrix multiplication; Finite metrics; Gossip; Tropical algebraic groups

Fig. 1 Composing the car metric with the bike metric
depicted by the second triangle. The large distances between E and either P or A
are covered much faster by car than by bike. On the other hand, because of crowded
streets, the short distance between P and A is covered considerably faster by bike than
by car. As a consequence, an attractive alternative for travelling from E to A by car
is to travel by car from E to P and continue by bike to A. In other words, to get from
E to A we first do a step in the car metric and then a step in the bike metric, where
we optimise the sum of the two travel times. Computing this carbike metric for the
remaining ordered pairs leads to the picture on the right in Fig. 1. The corresponding
matrix computation is
where is tropical or minplus 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 3dimensional cone of symmetric matrices
satisfying all triangle inequalities. Hence one might think that such tropical products
sweep out a 3 + 3 = 6dimensional set. However, if we perturb the travel times in
the two metric matrices slightly, then their minplus product moves only in a
threedimensional 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.
While keeping this minplus product in the back of our minds, we next contemplate
the following different setting. Each of the three gossipers, Eve, Patricia, and Adam,
has an individual piece of gossip, which they can share through onetoone phone
calls in which both callers update each other on all the gossip they know. Record the
knowledge of E , P, A in a threebythree uncertainty matrix with entries 0 (for i s
gossip is known by j ) and (for the other entries). Then initially that matrix is the
tropical identity matrix, with zeroes along the diagonal and outside the diagonal. A
phone call between E and P, for example, corresponds to tropically rightmultiplying
that tropical identity matrix with
resulting in this very same matrix. A second phone call between P and A leads to
Note the resemblance of this computation with the carbike metric computation above.
This resemblance can be made more explicit by passing from gossip to lossy gossip,
where each phone call between gossipers k and l comes with a parameter q [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 pi j [0, 1] of i s gossip.
Assume the (admittedly simplistic) procedure where k updates his knowledge of gossip
i to q pil if this is larger than pik and retains his knowledge pik of gossip i otherwise,
and similarly for gossiper l. In this manner, the fractions pi j are updated through a
series of lossy phonecalls. Passing from pi j to the uncertainty ui j := log pi j
[0, ] of gossiper j about gossip i and from q to the loss a := log q [0, ] of the
phone line in the call between k and l, the update rule changes uik into the minimum
of uik and uil + a, and similarly for uil . This is just tropical rightmultiplication with
the matrix Ckl (a) having 0s on the diagonal, 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 Wkl there are reminiscent of our matrices Ckl ).
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 = Dn be the set of all metric n n matrices, i.e. matrices
with entries ai j R0 satisfying aii = 0 and ai j = a ji and ai j + a jk aik . For
standard notions in polyhedral geometry, we refer to [20].
Throughout the paper, we give [0, ] the topology of the onepoint
compactification of [0, ), i.e. the topology of a compact, closed interval.
Theorem 1.1 The set { A1 Ak  k N, A1, . . . , Ak Dn} is the support of a
(finite) polyhedral fan of dimension n2 , whose topological closure in [0, ]nn (with
product topology) is the monoid generated by the matrices Ckl (a) with k, l [n] :=
{1, . . . , n} and a [0, ].
We will denote that monoid by Gn, and call it the lossy gossip monoid with n
gossipers. The most surprising part of this theorem is that the dimension of Gn is not
larger than n2 . We will establish this in Sect. 7 by proving that Gn is contained in the
tropicalisation of the orthogonal group On.
Theorem 1.2 For n 5 the fan in the previous theorem is pure and connected in
codimension 1. Moreover, for n 4, there is a unique coarsest such fan. This coarsest
fan has D2, D3, D4 among the 1, 7, 289 fulldimensional cones; and in total it has
1, 2, 16 orbits of fulldimensional cones under the groups Sym(2), Sym(3), 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 5 every element of Gn is the tropical product of at most n2
lossy phone call matrices Ckl (a), but not every element is the tropical product of fewer
factors.
We conjecture that the restriction n 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 n2 , and this bound is attained.
This implies a bound on the length of irredundant products of matrices Ckl (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 Ckl (0), k, l [n] every
irredundant product of such matrices has at most n2 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 (R {})nn; a
general theory of such submonoids also seems very worthwhile. Note that subgroups
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 polyhedralcombinatorial proof
of Theorem 1.1 is known. In Sect. 8 we study the monoid generated by the ordinary
gossip matrices Ckl (0), k, l [n]: using the ordinary orthogonal group we prove
Theorem 1.4, and for n 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 Dn to be the topological closure of Dn in
[0, ]nn, and we denote by Gn the monoid generated by Dn under minplus matrix
multiplication. We call Gn the lossy gossip monoid with n gossipers. This terminology
is justified by the following lemma.
Lemma 2.1 The lossy gossip monoid Gn is generated by the lossy phone call matrices
Ckl (a) (k, l [n], a [0, ]) having zeroes on the diagonal and everywhere else
except for values a on positions (k, l) and (l, k).
Proof Lossy phone call matrices lie in Dn, so the monoid that they generate is
contained in Gn. For the converse it suffices to show that every element A of Dn is the
product of lossy phone call matrices. We claim that, in fact, A = k<l Ckl (akl ) =: B,
where the akl 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 ai0,i1 +ai1,i2 + +ais1,is
where s n2 , i0 = i , is = j , and where the Ci0,i1 , . . . , Cis1,is with s n2 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 ai, j .
Although elements of Gn need not be symmetric, they have a symmetric core.
Lemma 2.2 Each element A of Gn satisfies ai j = 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 K and an l L such that akl = alk . Write A = Ci1, j1 (b1)
Cis, js (bs ) with b1, . . . , bs R0. If there is no r such that ir and jr lie in different
sides of the partition, then akl = alk = for all k K and l L. Otherwise, among
all r for which ir and jr lie in different parts of the partition, choose one for which br
is minimal. Then air , jr = a jr ,ir = br .
Lemma 2.3 Every connected graph on [n] occurs as symmetric core of some element
of Gn.
Proof Number the edges of that subgraph I1, . . . , Im and consider the tropical product
CIm 1 + 2m
where the product stabilises once all edges have acquired a finite length. For {k, l}
equal to some Ii we have akl = alk = 1 + 2i , where we use that the sum of two of
these m numbers is larger than any third. For any other {k, l} the value akl is a sum
of some number m of distinct negative powers of 2, the integer m itself, and some
number of distinct positive powers of 2. This sum uniquely determines the sequence
of factors contributing to it, of which there are at least two. Hence the sum determines
the ordered pair (k, l). In particular, we have alk = akl .
Observe that Ckl (a) Ckl (b) = Ckl (a b), where denotes tropical addition
defined by a b = min(a, b). Thus Lemma 2.1 exhibits Gn as a monoid generated by
certain oneparameter submonoids, reminiscent of the generation of algebraic groups
by oneparameter subgroups. This resemblance will be exploited in Sects. 7 and 8.
We define the length of an element X of Gn as the minimal number of factors in
any expression of X as a tropical product of lossy phone call matrices Ckl (a). A rather
crude but uniform upper bound on the length of elements of Gn 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 Gn is at most n2(n 1)/2. In particular, the length of every element
of Gn is bounded by this number.
Proof Let A be an element of Gn and write
A = CI1 (a1)
where a j are nonnegative real numbers and I j are unordered pairs of distinct numbers
in [n]. The entry at position (h, i ) of A, if not equal to , is the minimum of expressions
ak1 + + aks , where (Ik1 , . . . , Iks ) is a path from h to i in the complete graph on [n]
and k1 < < ks . Choose such a path with s minimal, and call this the minimal path
from h to i . Since it is never cheaper to visit a vertex twice, we have s n 1. This
shows that for each of the n(n 1) pairs (h, i ) only at most (n 1) of the factors are
necessary, and this gives an upper bound of n(n 1)2 on the quantity in the lemma.
The sharper bound in the lemma comes from the fact that if j = h, i lies on a minimal
path from h to i , then i does not lie on a minimal path from h to j . Hence the total
number of ordered pairs (i, j ) with i, j unequal to h and j on the minimal path from
h to i is at most (n 1) + n 21 = n2 , and this bounds the number of factors essential
for the hth row of A. This gives the bound.
Proof We proceed by induction on n. For n = 1 there are no factors. Let Wn1 be an
irredundant expression over Gn of length n3 not involving the index 1. Let Ph be the
product
Ph = C12(bh1)
(of length h) and put
Wn = Wn1
Pn1
Pn2
Proposition 2.6 The closure of Dn under tropical matrix multiplication is the support
of some finite polyhedral fan in Rnn and equals Gn Rn0n. Its topological closure
0
in [0, ]nn equals Gn.
Note that this is Theorem 1.1 minus the claim that the dimension of that fan is (not
more than) n2 ; this claim will be proved in Sect. 7.
Proof By Lemma 2.1 and the proof of Lemma 2.4 the closure of Dn under tropical
matrix multiplication is a finite union of images of orthants Rk
under piecewise linear maps. Such an image is the support of0sowmitehpkolyhne(dnralf1a)n2.
The remaining two statements are straightforward.
A := I A A 2
= I A A 2 A (n1) = (I A) (n1),
where I is the tropical identity matrix [7, p. 21]. The (i, j )entry of A records the
length of the shortest path from i to j in the directed graph on [n] with edge lengths
ai j . From this interpretation it follows readily that for A1, . . . , As [0, ]nn with
zero diagonal, and Sym(s), we have ( A1 As ) = ( A(1) A(s)).
Lemma 2.7 The Kleene star maps Gn into its subset Dn.
Proof Let A Gn be the tropical product of lossy phone call matrices C1, . . . , Ck .
Note that CiT = Ci . We have
A = (C1
= (C1
Ck ) = (Ck
Ck )T = ((C1
C1) =
C T
where we have used the remark above, the fact that transposition reverses
multiplication order, and the fact that Kleene star commutes with transposition. Thus A is a
symmetric Kleene star and hence a metric matrix.
3 Graphs with Detours
In the next two sections we will visualise elements of the lossy gossip monoids G3
and G4, 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 Dn (see, e.g. [8,13]).
Let = (V , E ) be a finite, undirected graph and w : E R0 be a function
assigning lengths to the edges of . The weight of a path in ( , w) is the sum of the
weights of the individual edges in the path. A map : [n] V is called a labelling,
or [n]labelling, if we need to be precise, and the pair ( , ) is referred to as a labelled
graph, or an [n]labelled graph.
A weighted [n]labelled graph gives rise to a matrix A( , w, ) in Dn whose entry
at position (i, j ) is the minimal weight of a path between (i ) and ( j ). We say that the
weighted labelled graph ( , w, ) realises the matrix A( , w, ). Any matrix X Dn
has a realisation by some weighted, [n]labelled graph, e.g. the graph with vertex set
[n], the entries of X as weights, and equal to the identity. However, typically more
efficient realisations exist, in the following sense. A weighted, [n]labelled graph
Fig. 2 Minimal realisations of threepoint metrics
( = (V , E ), w, ) is called an optimal realisation of X if the 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 \ ([n]) have valency 2 (since such vertices can be removed and their
incident edges glued together). Optimal realisations of any X Dn 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 G3 and G4 is to find weighted labelled
graphs that realise the elements of D3, D4, as follows (for much more about this see
[8,9]). We write J0 for the matrix of the appropriate size with all entries 0.
Example 3.1 We give optimal realisations of the elements of Dn, for n = 2, 3, 4. For
the cases n = 5, 6 see [16,18].
(1) An element of D2 \ { J0} 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 J0 is optimally realised by the graph on one vertex.
(2) Any matrix in D3 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 D3 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 D3 in which the matrix lies.
(3) The case of D4 is similar to that of D3 in the sense that there exists a single graph
which, appropriately labelled and weighted, realises any X D4. However, unlike
Fig. 3 Minimal realisations of fourpoint metrics. The parallel sides of the middle rectangle have equal
weight
Fig. 4 Examples of labelled weighted graphs with detours. (a) Path with a single detour from 1 to 2. (b)
Graph with 4 detours
for D3, three distinct labellings are required. The labelled graphs are depicted in
Fig. 3. For graphs in the relative interior of D4, the given realisation is optimal
(and in fact the unique optimal realisation).
We now extend realisation of metric matrices by graphs to realisations of arbitrary
matrices in Rnn with zeroes on the diagonal. For this we need an extension of the
0
concept of a labelled weighted graph. Let i and j be the distinct elements of [n]. A
detour from i to j in an [n]labelled weighted graph is simply a walk p starting at
(i ) and ending at ( j ) that has larger total weight than the path of minimal weight
between (i ) and ( 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 ( , w, , D) be an [n]labelled weighted graph with set of detours D. It gives
rise to a matrix A( , w, , 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 D. In particular, A( , w, , D) need not be
symmetric, but its diagonal entries are 0. Again, if X Rnn and X = A( , w, , D),
0
then ( , w, , D) is said to realise X . Any nonnegative 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 We give two examples of labelled weighted graphs with detours. First,
the graph in Fig. 4(a) has a single detour from 1 to 2, and realises the matrix
Except when a = 0, this matrix is not in G2. The example in Fig. 4(b) is more
interesting. It has detours between the ordered pairs (1, 4), (2, 4), (4, 1), (4, 2). The
weights a, b, c, d, e, f are nonnegative. By varying this sixtuple in R60 this graph
with detours realises the 6dimensional cone of all matrices of the form
A =
a + d
d
0
f
Observe that both (1, 4) and (4, 1) are detours, and their lengths are restricted by the
inequality a14 a41 (indeed, the difference equals 2b). This 6dimensional cone is
one of the maximal cones in G4, namely, cone C10 in Fig. 6. The graph with detours
in Fig. 4(b) represents these inequalities in a visually attractive manner, but one also
sees in one glance that the cone of all matrices of the form is simplicial: it is the image
of R60 under an injective linear transformation into R404. This motivates our choice
for graphs with detours to represent cones of G3 and, more importantly, G4.
By Lemma 2.7, the Kleene star of a matrix A in Gn lies in Dn. 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
Since G3 is a pointed fan, no combinatorial information is lost by intersecting that fan
with a sphere centred around the allzero matrix. The resulting spherical polyhedral
complex is depicted in Fig. 5. Detour graphs realising the maximal cones can be
constructed by realising the arrows in an arbitrary manner as detours in the undirected
graph. The middle cone is (the topological closure of) D3, with its three
codimensionone faces corresponding to the second layer in Fig. 2 and its three codimensiontwo
faces corresponding to the third layer.
The computations to show that Fig. 5 gives all of G3 are elementary and can be done
by hand. We use pictorial notation and write A( ) for the matrix realised by a labelled
weighted graph with detours . Here, instead of drawing a detour as a walk, we draw
it as an arrow whose length is assumed to exceed the distance in the undirected graph.
First, to prove that the matrices A( ) with as in the figure are indeed in G3 we
observe that
Fig. 5 Representation of the spherical complex of G3. The labelled graphs with detours corresponding
to the maximal cells are indicated. The middle triangle represents the cone of distance matrices, and on
its codimensionone faces one of the three points ends up between the other two points. The remaining
codimensionone faces of the remaining six cones are where one of the edge lengths in the Kleene star
becomes zero
for any c a + b (and a, b 0 as always). Together with the fact that Ci j (a)
Ci j (d) = Ci j (a d) this implies that
Cij (d) and Cjk (d)
are contained in the complex of Fig. 5 for all choices of a, b, c, and d with c a + b.
Next we compute
and, for m := max(a b, b a),
It follows by transposition that the products
Cik (d), and A (
are also contained in one of the cones of Fig. 5. This concludes the proof of Theorem 1.2
for n = 3.
5 Four Gossipers
The computations for G4 are too cumbersome to do by hand. Instead we used
Mathematica to compute a fan structure on G4. Figure 6 gives graphs with detours
realising all the cones of G4, up to transposition and the action of Sym(4). The 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.
These graphs were obtained as follows. First, generate all 66 possible piecewise
linear affine maps [0, ]6 G4 of the form
(a1, . . . , a6) CI1 (a1)
where I1, . . . , I6 are unordered pairs of distinct indices. Among the image cones,
select only the sixdimensional ones and compute their linear spans. There are 289
different linear spans. Compute the Sym(4)orbits on these spans; this yields 16 orbits.
Choose a representative for each of these orbits on spans, and for each representative
select all cones with that span. It turns out that, for each representative span, one
of the cones contains all other cones. To show that the orbits of these 16 maximal
cones give all of G4, leftmultiply each of these 16 cones with all possible lossy
phone call matrices and show that the resulting unions of cones are contained in the
union of the 289 maximal cones; this is facilitated by the fact that each of these
cones is the intersection of G4 with (the topological closure in [0, ]nn of) a
sixdimensional subspace. Then we check that the faces of these 289 sixdimensional
Fig. 6 Orbit representatives of labelled weighted graphs with detours realising a polyhedral fan structure
on G4 with simplicial cones. The white vertices are the labelled vertices
cones do indeed form a polyhedral fan, i.e. that the intersection of any two of these
faces is a common face of both. In the process of this check, which we performed both
with Mathematica and (more rapidly) with polymake [10], we find that the fan
*
4
Fig. 7 Walking from maximal cones to maximal cones by edge contraction, except in case C7C12. The
edge to be contracted is indicated by an asterisk. This shows that the cones in the grey boxes intersect in
a cone of dimension 5. The intersection between C7 and C12 is obtained by setting equal certain surplus
lengths in the graphs representing C7 and C12
has f vector (43, 327, 1042, 1560, 1092, 289). This latter check yields the statement
about the unique coarsest fan structure in Theorem 1.2.
Next, the group Z/2Z acts on G4 by transposition. Taking orbit representatives
under the larger group Sym(4) (Z/2Z) from among the 16 yields 11 cones. Among
these, 9 are simplicial (have six facets), the cone D4 has 12 facets, and the remaining
cone has 9 facets. The cone D4 is the union of three simplicial cones (see Fig. 3), which
are permuted by Sym(4), so we need only one. This is C5 in Fig. 6. The cone with 9
facets turns out to be the union of two simplicial cones. Splitting this up yields C11
and C12 in the figure. It turns out that each Ci is the image of R60 under a linear map
into R44 with nonnegative integral entries with respect to the standard bases, and
0
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 Ci .
Finally, connectivity in codimension 1 is proved by Fig. 7. It shows that any maximal
cone can be connected to D4 by passing through (relatively open) codimensionone
faces; note the specified labelling. Most intersections in Fig. 7 are of a simple type,
where one of the edge weights becomes zero to go from one cone to the
neighbouring cone; these contracted edges are then marked with an asterisk on both sides.
The only exception is the connection from C7 to C12. Although (suitable elements in
the Sym(4)orbits of) these cones intersect in a fivedimensional boundary cone, the
boundary cone is obtained from the parameterisations specified by the graphs with
detours by restricting the parameterisation to a hyperplane where two of the weights
are equal. This leads to the following theorem.
Theorem 5.1 The cones realised by the graphs of Fig. 6 give a polyhedral fan structure
on G4. 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 G4 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 G4 with the unit sphere in 16dimensional space are zero.
We do not know whether this is true for general n.
6 Five Gossipers
More extensive computations establish the claimed facts about G5. Since 66 is a small
number, but 1010 is not, the computation requires many refinements. We omit the
details. It turns out that every element has an expression as a tropical product of at
most 10 lossy phone call matrices. The set G5 is the support of a polyhedral fan which
is pure of dimension 10, and connected in codimension 1. Some statistics are given in
Table 1. The single orbit of size 1 is that of Dn.
Table 1 Numbers of subspaces
spanned by fulldimensional
cones and their numbers of
orbits under Sym(n)
Orbit size distribution
1
7
289
91151
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 G5 in a convex cone (recall that for n = 4 this did hold, and that we used
this in the proof that G4 has a unique coarsest fan structure).
Example 6.1 Consider the open, 10dimensional cone P consisting of all matrices
C45(a)C34(b)C45(c)C24(d)C45(e)C14( f )C12(g)C23(h)C13(i)C15(j)
where we have left out the sign for brevity, and where the parameters a,..., j
satisfy the inequalities
a > c, e > c, f > d + g, b + d > h, h > g + i, c + d + g + i > a + b,
b + i > d + g, c + d + g > j, i + j > b + c, c + j > d + g, g + j > d + e.
Similarly, consider the open, 10dimensional cone Q consisting of all matrices
C45(a)C34(b)C45(c)C24(d)C45(e)C15( f )C12(g)C24(h)C23(i)C13(j)
a > c, e > c, c + d + g > f, c + f > d + g, f + g > c + d,
h > d, b + d > i, i > g + j, f + j > a + b, b + j > d + g.
In matrix form, these matrices are
The linear spans of P and Q are the same, and the closures of P and Q cover all
10dimensional cones in G5 with this span. The latter matrix becomes the former after
the substitution f j, h f g, i h, j i, and this substitution turns its
inequalities into
a > c, e > c, c + d + g > j, c + j > d + g, g + j > c + d,
f > d + g, b + d > h, h > g + i, i + j > a + b, b + i > d + g.
We see that both cones satisfy
a > c, e > c, f > d + g, b + d > h, h > g + i, b + i > d + g, c + j > d + g,
c + d + g > j, i + j > b + c, c + d + g + i > a + b, g + j > c + d.
In addition, P satisfies g + j > d + e and Q satisfies i + j > a + b. It follows that
P and Q have a 10dimensional intersection and their union is not a convex cone.
7 Tropicalising Matrix Groups
is in proving that its dimension does not exceed n2 .
For this, we make an excursion into tropical geometry. Recall that if K is a field with
a nonArchimedean valuation v : K R := R {} and if I K [x1, . . . , xm ]
is an ideal, then the tropical variety associated to I is the set of all w Rn such
that for each polynomial f = c x I the minimum min(v(c) + w ) is
attained for at least two distinct Nn. We denote this tropicalisation by 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 coordinatewise valuation map
v : Ln Rn maps X (L) into Trop(X ). If, moreover, v : L R is nontrivial
and L is algebraically closed, then the image of the map X (L) Trop(X ) is dense
in Trop(X ) in the Euclidean topology. Together with the BieriGroves theorem [4],
this implies that the set Trop(X ) is (the closure in Rn of) a polyhedral complex of
dimension equal to dim X .
We now specialise to matrix groups. As a warmup, consider the special linear group
SLn, defined over Q by the single polynomial det(x ) 1, where x is an n nmatrix of
indeterminates. Since there is only one defining polynomial and all its coefficients are
1, the valuation does not matter and Trop(SLn) equals the set of all A = (ai j )i j
Rnn for which the tropical determinant
tdet( A) := mSyimn(n)(a1(1) + + an(n))
is either zero, or else negative and attained at least twice.
Proposition 7.1 The tropicalisation Trop(SLn) is a monoid under tropical matrix
multiplication.
Proof For A, B Trop(SLn) set C := A B. A straightforward
computation shows that the tropical determinant is (tropically) submultiplicative, so that
tdet(C ) tdet( A)+tdet(B) 0+0 = 0. Hence it suffices to show that if tdet(C ) < 0,
then there are at least two permutations realising the minimum in the definition of
tdet(C ). Let Sym(n) be one minimiser of the expression c1(1) + + cn(n).
For each i [n] let (i ) [n] be such that ci(i) = ai (i) + b (i)(i). Now there
are two cases: either is a permutation or there exist i, j with (i ) = ( j ). In the
latter case, also the permutation (i, j ) = is a minimiser, and we are done. In the
former case, write = . Then we have
0 > tdet(C ) = c1(1) + + cn(n)
= a1 (1) + + an (n) + (b1 (1) + + bn (n)),
so that at least one of tdet( A) and tdet(B) is negative. If tdet( A) < 0, then since
A Trop(SLn), there exists a permutation = such that
a1 (1) + + an (n) a1 (n) + + an (n),
and we find that := = is another minimiser. The argument for tdet(B) < 0
is similar.
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 Let G denote the group of 4 4matrices of the form
where x runs through the field K . This is a onedimensional algebraic group isomorphic
to the additive group, whose tropicalisation consists of all matrices
0
x
x ,
1
aa ,
0
where a R . But we have
0
which for a, b < does not lie in Trop(G).
Now consider the orthogonal group On consisting of all matrices g that satisfy
gTg = I . We do not know whether Trop(On) is a monoid under tropical matrix
multiplication, but we shall see that this tropicalisation does contain the lossy gossip
monoid. For this, we take L to be the field C{{t }} of Puiseux series in a variable t ,
and v to be the order of a Puiseux series at 0. Motivated by the analogy between the
lossy phone call matrices Ci j (a), a R0 and oneparameter subgroups of algebraic
groups (see Sect. 2), we introduce the oneparameter subgroups gi j (x ) of On by
where the 1s stand for identity matrices, the cosines and sines are in the {i, j } {i, j
}submatrix, and the empty entries are 0. For any choice of x in the field L whose order
v(x ) at zero is positive, the matrix gi j (x ) is a welldefined matrix in the orthogonal
group On(L).
Proposition 7.3 The lossy gossip monoid Gn is contained in Trop(On).
Proof First note that v(gi j (x )) = Ci j (v(x )), so the statement would be
immediate if we knew that Trop(On) were closed under tropical matrix multiplication. We
prove something weaker. Let a1, . . . , ak be strictly positive rational numbers and let
(i1, j1), . . . , (ik , jk ) be pairs of distinct indices. Then for a vector (c1, . . . , ck ) Ck
outside some proper hypersurface, no cancellation takes place in the expression
in the sense that
v[gi1, j1 (c1t a1 ) gik, jk (ck t ak )] = v[gi1, j1 (c1t a1 )]
Here the righthand side equals Ci1, j1 (a1) Cik , jk (ak ) and lies in Trop(On)
since the lefthand side does. Since Trop(On) is closed in the Euclidean topology, all
of Gn is contained in it.
The dimension claim in Theorem 1.1 follows from Proposition 7.3, the Bieri
n .
Groves theorem, and the fact that dim On = 2
For n = 1, 2, 3, we can say a little bit more about Trop(On).
Example 7.4 For n = 1, Trop(On) consists of the single 1 1matrix 0. Next, for a
2 2matrix
x
g = u
to lie in O2 we need that x 2 + u2 1 = y2 + v2 1 = 0 = x y + uv = 0, and
these equations generate the ideal of O2. Tropicalising these equations yields that
min{a, c, 0}, min{b, d, 0}, min{a + b, c + d} are all attained at least twice. This is not
sufficient to characterise Trop(O2); indeed, for any negative a, b the matrix
satisfies all tropical equations above, but (unless a = b) not the tropicalisation of
the equation x v yu = 1 which expresses that O2 SL2. Imposing this additional
condition, i.e. that min{a + d, b + c, 0} is attained at least twice, we find that Trop(O2)
consists of three cones:
Trop(O2) =
The first cone is G2, the second cone is G2 with the columns reversed, and the third
cone makes the fan balanced.
In general, if a variety is stable under a coordinate permutation, then its
tropicalisation is stable under the same coordinate permutation. Consequently, Trop(On) is stable
under permuting rows, under permuting columns, and under matrix transposition.
For n = 3, a computation using gfan [15] shows that the quadratic equations
expressing that columns and rows both form orthonormal bases, together with the
equation det 1, do not form a tropical basis. For example, the fourdimensional cone
of matrices
a
a
c
with a b c 0 d is contained in the tropical prevariety defined by the
corresponding tropical equations, and for dimension reasons cannot belong to the
threedimensional fan Trop(O3).
However, these quadratic equations do suffice to prove that Trop(O3) [0, ]33
is equal to Sym(3) G3, i.e. obtained from G3 by permuting rows. Indeed, let a 3
3matrix A in [0, ]33 satisfy the tropicalisations of these equations. Then tdet( A) = 0,
hence after permuting rows A has zeroes on the diagonal. Now we distinguish two
cases. First, assume that A is symmetric:
0
A = a
b
b
c .
0
0
d
e
Then we claim that A lies in D3. Indeed, suppose that a > b + c. Then the
tropicalisation of the condition that the first two columns are perpendicular does not hold for
A. Hence a b + c and similarly for the other triangle inequalities; we conclude that
A D3. Next, assume that A is not symmetric. After conjugation with a permutation
matrix, we may assume that A is of the form
with a > d. Then the tropical perpendicularity of the first two columns yields
d = e + f , that of the last two columns yields c = f , and that of the first two
rows yields d = b + c. So A looks like
b
c ,
0
which is one of the cones in G3.
Remark 7.5 We do not know whether the equality Sym(n) Gn = Trop(On)
[0, ]nn (where the action of Sym(n) is by left multiplication) holds for all n.
If true, then this would be interesting from the perspective of algebraic groups over
nonArchimedean fields: it would say that the image under v of the compact subgroup
On(L0) On(L), where L0 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 Trop(On). 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 threedimensional cones, 1008 are
contained in the positive orthant, as opposed to the 6 7 = 42 found by applying row
permutations to the cones in G3. This suggests that gfan does not automatically find
the most efficient fan structure on On, and at present we do not know how to overcome
this.
8 Ordinary Gossip
In this section we study the ordinary gossip monoid Gn({0, }), which is the
submonoid of Gn of matrices with entries in {0, }. Note that there is a surjective
homomorphism Gn Gn({0, }) mapping non entries to 0 and to , which shows
that the length of an element of Gn({0, }) inside Gn is the same as the minimal
number of nonlossy phone calls Ci j (0) needed to express it. A classical result says that
length of the allzero matrix is exactly 1 for n = 2, 3 for n = 3, and 2n 4 for n 4
[1,6,12,19], and this result spurred a lot of further activity on gossip networks. But
the allzero matrix does not necessarily have the largest possible lengthsee Table 2,
Gn({0, })
While we do not know the maximal length of an element in Gn({0, }) 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 n2 , 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 n2 .
That n2 is a lower bound is shown by the following scenario: Number the gossipers
1, . . . , n. All calls involve gossiper 1. For i = 2, . . . , n he calls i, i 1, . . . , 2, for a
total of 1 + 2 + + (n 1) = n2 calls. There are many other scenarios attaining
n , and it does not seem easy to classify them.
2
We now argue that n2 is an upper bound. Although we will not use this, we remark
n
that it is easy to see that 2 2 = 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 I1, I2, . . . , I be a sequence of unordered pairs from [n] representing phone
calls where in each call at least one participant learns something new. To each Ia we
associate the homomorphism a := SO2(C) SOn(C) that maps a 2 2matrix g to
the matrix that has g in the Ia Ia block and otherwise has zeroes outside the diagonal
and ones on the diagonal. For each k we obtain a morphism of varieties (not a group
homomorphism) k : SO2(C)k SOn(C) sending (g1, . . . , gk ) to 1(g1) k (gk ).
Let Xk be the closure of the image of k ; this is an irreducible subvariety of SOn(C).
The (i, j )matrix entry is identically zero on Xk 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
kth phone call, some matrix entry is identically zero on Xk1 which is not identically
zero on Xk . Consequently, we have 0 = dim X0 < dim X1 < < dim X . But all
Xk are contained in the variety SOn(C) of dimension n2 , so we conclude that n2 .
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.
We computed the longest irredundant products of phone calls for small n, see
Table 3.
Table 3 Maximum length ln of an irredundant product of phone calls
9 Open Questions
In view of the extensive computations in Sects. 46 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 Gn . Use this description to prove or disprove the pureness of dimension
n2 and the connectedness in codimension one.
Question 9.2 Is the length of any element of Gn at most n2 ?
Once a satisfactory polyhedral fan for Gn 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 Gn by graphs
with detours, and a notion of tight spans of such elements.
n 4.
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 Trop(On ) a monoid under tropical matrix multiplication? This is
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.
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.
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