Location functions on finite metric spaces an axiomatic ...szigetiz/SA60/talkspdf/3... · A...

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Median graphsLocation functions, rationality, and consensus axioms

Our two fundamental theorems: studying the connectionsConclusion

Location functions on finite metric spacesan axiomatic approach

B. Novick,H.M. Mulder, F. McMorris, R. Powers

Meeting in honor of Andras SeboApril 24, 2014, Grenoble, France

Beth Novick Location functions on Finite Metric Spaces

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Trees and hypercubesMedian graphs as ternary algebrasThe structure of median graphsA meta-conjecture

Preliminaries

DefinitionLet G = (V ,E ) be a connected simple graph.

I The interval between u and v is the set

I (u, v) = {x | x lies on a shortest u, v -path}.

I A subgraph H = (W ,F ) of G is convex if. for any twovertices x and y in W , we have

I (x , y) ⊆W .


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Now let’s get started with a simple question: what dohypercubes and trees have in common?

Both trees and hypercubes display an ‘Expansion Property’.


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Now let’s get started with a simple question: what dohypercubes and trees have in common?

Both trees and hypercubes display the following property:

For every u, v ,w , the set I (u, v) ∩ I (v ,w) ∩ I (u,w) containsexactly one vertex.

In fact, this is the definition of a ‘median graph’ ...


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A median graph is one in which, for every triple, u, v ,wexactly one vertex lies in I (u, v) ∩ I (v ,w) ∩ I (u,w).

Equivalently,A median graph is a connected, simple graph G (V ,E ) in which∀a1, a2, a3 ∈ V , ∃! x = m(a1, a2, a3), called the median,satisfying

d(ai, x) + d(x, aj) = d(ai, aj), for all i 6= j ∈ {1, 2, 3}.

a 1

a 2

a 3

m (a 1 , a 2, a 3)

Figure : a median graph

a1

a2 a3

a2

a1

a3

Figure : not median graphs


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The ternary operation defined on the previous slide enjoyscertain properties.

On median graphs, the ternary operation, m : V 3 7→ V , satisfies:(m1) m (a, b, c) = m (b, a, c) = m (c, b, a)

(m2) m (a, a, b) = a

(m3) m (m (a, b, c), u, v) = m (a,m (b, u, v),m (c, u, v))

DefinitionA ternary algebra (X , f ) is a set X with a ternary operationf : X 3 7→ X .A ternary algebra satisfying (m1) - (m3) is called a medianalgebra.


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Median algebras are a meeting point for variouscombinatorial structures.

For example:

I In a distributive lattice one may definem(a, b, c) = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c).

I Given a median algebra we can define a semilattice byx ≤a y ⇔ m(a, x , y) = x .

I Congruence class geometries.

I ‘Unique ternary distance’ graphs.

I Median graphs!


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Median graphs have a very beautiful and useful structure.

Every median graph can be obtained from K1 by a series of‘convex expansions’ :

Convex ConvexG'

u'

v'

G1' G0' G2'

Convex Convex

G1 G2

u1 u2

v1 v2


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The role of trees and hypergraphs in median graphs,together with many other results motivated:

Meta-conjecture

(Mulder 90) Any ‘reasonable’ property shared by trees andhypercubes is enjoyed by median graphs.

So how did I become interested in these beautiful mediangraphs? ...


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Location functionsaxiomatizationsOur breakthrough result

A long-standing open question involved finding anaxiomatic characterization of the median function onmedian graphs.

I Let (X , d) be a finite metric space.

I A profile, π = (x1, x2, . . . , xk), of length k , on (X , d) is afinite sequence of elements of X .

I Let X ∗ be the collection of all profiles on X .

DefinitionA Location function on a metric space (X , d) is any function:

L(π) : X ∗ → 2X − ∅


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A location function usually aims to minimize ‘remoteness’in some sense, usually involving distance.

Well-known location functions are:

Cen(π) = {x | x minimizes maximum distance to the xi}

Med(π) = {x | x minimizes∑k

i=1 d(x , xi )}

Mean(π) = {x | x minimizes∑k

i=1 d(x , xi )2}

`p(π) = {x | x minimizes∑k

i=1 d(x , xi )p}

AntiMed(π) = {x | x maximizes∑k

i=1 d(x , xi )}

Even if we restict ourselves to graphs, axiomaticcharacterizations of location functions seems to be hard ...


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Borrowing from the theory of consensus theory in voting,certain axioms seem reasonable:

(F ) Faithfulness: L(x) = {x}, for all x ∈ X .(The case of only one client.)

(A) Anonymity: L(π) = L(πσ), whereπσ = (xσ(1), xσ(2), . . . , xσ(k)).(The ordering of the clients should not matter.)

(C ) Consistency: L(π) ∩ L(ρ) 6= ∅ ⇒ L(πρ) = L(π) ∩ L(ρ).(If two profiles agree on some output x , then x is inthe output of their concatination.)

(B) Betweenness: L(u, v) = I (u, v), for u, v ∈ X .(All locations between exactly two preferred locationsare equally good.)


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An intriguing theorem motivated a body of work:

TheoremThe Median function Med over any finite metric space satisfies(A), (B) and (C ).

For which metric spaces is ABC sufficient for L to be Med?

Theorem(McMorris, Mulder and Roberts, 1998) Let G = (V ,E ) be acube-free median graph and let

L : V ∗ → 2V − {∅}.

Then L = Med iff L satisfies axioms (A), (B) and (C).

Cube-free median graphs contain trees. – So the property inquestion holds for trees! ..


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To exclude the possible Q3 obstruction, McMorris, Mulderand Roberts added an axiom.

Convexity (K): Let π = (x1, . . . , xk) be a profile of length k ≥ 2 in G . If∩ki=1L(π − xi ) = ∅ then

L(π) = Con(∪ki=1L(π − xi )).

Theorem(MMR, 1998) Let G = (V ,E ) be a median graph and letL : V ∗ → 2V − {∅},. Then L = Med iff L satisfies axioms (A), (B), (C)and (K)

The implicit question here: is there a functionL : V ∗ → 2V − {∅} satisfying (A), (B) and (C), and someprofile π on Q3 with L(π) 6= Med(π)?Of note: this work yielded some new structural results for mediangraphs. Also of note: another paper by the same authors yielded astructural result on median semilatices.Beth Novick Location functions on Finite Metric Spaces

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Our quest for such a profile on Q3 and such a consensusfunction lead to two surprising discoveries.

Theorem(Mulder, Novick, 2010) For any positive integer n, a locationfunction L on the graph Qn is the function Med iff L satisfies (A),(B) and (C).

So this gave us hypercubes!And later, motivated by the Meta-conjecture, we established:

Theorem (ABC)

(Mulder, Novick, 2010) Let L be a location function on anymedian graph G. Then L satisfies (A), (B) and (C) iff L = M.

settling the implied conjecture of 1998.Again, a new structural characterization for median graphs was abyproduct.


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A companion theoremRelationships among the axiomsAnonymity: an intriguing axiom

Another characterization involving 3 axioms dates back to2000, but involved an axiom specific to median graphs.

(Cond) 12 -Condorcet: u ∈ L(π) if and only if v ∈ L(π), for each

profile π on G and any edge uv of G with|πuv | = |πvu|.

Convex ConvexG'

u'

v'

G1' G0' G2'

Convex Convex

G1 G2

u1 u2

v1 v2

Theorem (Companion)

(MMR, 2000) Let L be a location function on any median graphG . Then L satisfies (F), (C) and (Cond) iff L = Med.


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So A,B ,C ⇔ F ,C ,Cond ⇔ Med .What are the relationships among the five axioms?

TheoremFor a location function L on a median graph, the only implicationsthat hold are:

◦ (A) and (B) and (C ) ⇒ (F ) and (Cond).

◦ (F ) and (C ) and (Cond) ⇒ (A) and (B).

◦ (B) and (C ) and (Cond) ⇒ (F ) and (A).

The proof is constructive.

One of our examples: the ‘irrational chairman’. (Used to show (B)+ (C ) need not ⇒ (A).)For each example showing non-implication, we generalize toarbitrary median graphs.


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Anonymity seems to be an especially intriguing axiom.

For a set of splits S on median graph G , the spread of v is

[v ]S = ∩{G1 | v ∈ G1, (G1,G2) 6∈ S}.

LemmaThe (distinct) spreads partition V .

Example

Here S = {(Gxy ,Gyx), (Gyz ,Gzy )}.{[x ]S , [y ]S , [z ]S} is a partition of V .

u

wv

u

v wv

wu

u

x y

vz

1

LemmaLet S(π) = {balanced splits of π}. Then (Cond) implies:v ∈ L(π)⇒ [v ]S(π) ⊆ L(π).


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For hypercubes, we have a surprising implication involvingAnonymity.

TheoremIf G is a hypercube, then (C ) and (Cond)⇒ (A).

Proof: Mate property, (C ), and previous lemma force L(π) tocompulsory sides of certain splits based only on Med(π), whichrespects (A).Crucial difference: G 6= Qn ⇒ v can lack a mate:

G1 a

G 2

G1

G 2 x b

a

x b

y

The path P 3 The 2-cube Q 2


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SummaryFuture workReferences

A theme of this work has been the interplay betweenstructural results and axiomatic characterizations.

Summing up:

• Mulder and Novick settled a long standing question bycharacterizing the median function on median graphs: ourABC Theorem.

• We showed that (A), (B) and (C ) are indeed independent.

• We completely resolved relationships among the five axioms inthe ABC Theorem and the ‘Companion Theorem’.

◦ Anonymity proved especially intriguing.◦ We generalized our ‘counter-examples’ to median graphs.


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My hope is that our structural insights will continue toevolve and deepen.

Conjectures/remarks and future work:

◦ Properties of gated metric spaces may yield those (X , d) forwhich (A) + (B) + (C ) ⇒ Med.

◦ It is intriguing that (C ) + (Cond) ⇒ (A) on hypercubes.How can this be generalized?

◦ Re-prove our ABC Theorem in the arena of mediansemilattices.

◦ For various families of graphs, characterize those consensusfunctions satisfying (A), (B) and (C ).


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References

A. W. M. Dress and R. Scharlau.

Gated sets in metric spaces.Aequationes Mathematicae, 34, 1987.

F.R. McMorris, H.M. Mulder, R.C. Powers.

The median function on median graphs and semilattices.Discrete Appl. Math., 101, 2000.

F.R. McMorris, H.M. Mulder and F.S. Roberts.

The median procedure on median graphs.Discrete Applied Mathematics, 84, 1998.

H. M. Mulder.

The structure of median graphs.Discrete Math., 24, 1978.

F.R. McMorris, H.M. Mulder and F.S. Roberts.

The median procedure on graphs.Report 9413/B, Econometrisch Instituut EUR, 1994.

H.M. Mulder, B.A. Novick.

A tight axiomatization of the median procedure on median graphs.Discrete Appl. Math., 161, 2013.

R. Vohra.

An axiomatic characterization of some locations in trees.European Journal of Operations Research, 90, 1996.


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Merci!

THANKS FOR YOUR ATTENTION!


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