Download - Negative correlation properties for graphs
![Page 1: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/1.jpg)
![Page 2: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/2.jpg)
Negative correlation propertiesin graphs or: How I learned to stoplooking for the edge e amongspanning trees already containing fand instead look among allspanning trees.
Alejandro EricksonMaster’s thesis work while at theDepartment of Combinatorics and OptimizationUniversity of Waterloo
April 11, 2010
![Page 3: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/3.jpg)
OutlineOutlineKirchhoff’s law and Rayliegh monotonicityKirchhoff’s LawRayleigh monotonicityThe combinatorics!ExamplePrevious workRayleigh condition for other stuffForest Rayleigh is equivalent to negative correlationEvidence for forest Rayleigh propertyCurrent workSeries Parallel graphs and 2-sums
![Page 4: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/4.jpg)
Kirchhoff’s lawElectrical network of resistors, each with conductanceyg .a
b
ee has resistance re ,conductance ye = 1
re
Kirchhoff’s law gives a formula for the conductancebetween nodes a and b.
![Page 5: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/5.jpg)
Model the network as a graph G (shocking!) and let Tbe the generating polynomial for spanning trees of G .T (G ) = + + ++ + + += yeyf yh + yeyf yg + yeygyh + yf ygyh+ yeyhyi + yf ygyi + yeyf yi + ygyhyi
b
e
f g
h
i
a
![Page 6: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/6.jpg)
Let G /ab be G with nodes a and b identified.Kirchhoff’s LawYab = T
( )T ( ) = T (G )
T (G /ab)
b
e
f g
h
i
a
![Page 7: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/7.jpg)
Rayleigh monotonicityLord Rayleigh (1842-1919)observed that increasingthe conductance of anyedge should not decreasethe effective conductanceof the whole network.Yab = T (G )
T (G /ab)Yab is non-decreasingin the directionof every variable ye .For any edge e , we have
∂∂yeYab ≥ 0.
![Page 8: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/8.jpg)
Three bits of notation:I T g denotes evaluation at yg = 0. T g = T (G \ g ).I Tg denotes partial derivative w.r.t yg . Tg = T (G /g ).I Add an edge, G Î G + f where f = ab.
Now, Kirchhoff’s law is1Yab = T (G )
T (G /ab) = T f
Tfand the Rayleigh property is thatT f
e Tf − T f Tef(Tf )2 ≥ 0,
for each distinct pair of edges e and f and positive ygs
1old G
![Page 9: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/9.jpg)
Three bits of notation:I T g denotes evaluation at yg = 0. T g = T (G \ g ).I Tg denotes partial derivative w.r.t yg . Tg = T (G /g ).I Add an edge, G Î G + f where f = ab.Now, Kirchhoff’s law is1
Yab = T (G )T (G /ab) = T f
Tfand the Rayleigh property is thatT f
e Tf − T f Tef(Tf )2 ≥ 0,
for each distinct pair of edges e and f and positive ygs1old G
![Page 10: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/10.jpg)
Forget all that stuff about electrical networks.
![Page 11: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/11.jpg)
Selecting treesNotice thatI ygTg are the spanning tress containing gI T g generates those ones not containing g .
We select a spanning tree X with probabilityproportional to ∏g∈X yg , with positive ygs.The chances our tree contains e areyeTe
T.
If we restrict ourselves to trees already containing f ,then the chances our tree contains e areyeyf Tef
yf Tf.
![Page 12: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/12.jpg)
Selecting treesNotice thatI ygTg are the spanning tress containing gI T g generates those ones not containing g .We select a spanning tree X with probabilityproportional to ∏g∈X yg , with positive ygs.
The chances our tree contains e areyeTe
T.
If we restrict ourselves to trees already containing f ,then the chances our tree contains e areyeyf Tef
yf Tf.
![Page 13: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/13.jpg)
Selecting treesNotice thatI ygTg are the spanning tress containing gI T g generates those ones not containing g .We select a spanning tree X with probabilityproportional to ∏g∈X yg , with positive ygs.The chances our tree contains e are
yeTe
T.
If we restrict ourselves to trees already containing f ,then the chances our tree contains e areyeyf Tef
yf Tf.
![Page 14: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/14.jpg)
Selecting treesNotice thatI ygTg are the spanning tress containing gI T g generates those ones not containing g .We select a spanning tree X with probabilityproportional to ∏g∈X yg , with positive ygs.The chances our tree contains e are
yeTe
T.
If we restrict ourselves to trees already containing f ,then the chances our tree contains e areyeyf Tef
yf Tf.
![Page 15: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/15.jpg)
Equivalent conditionsObvious but important:T consists of those terms not containing g and thosecontaining g . That is
T = T g + ygTg
Back to the Rayleigh condition(Te )Tf − ( T ) Tef=(T fe + yf Tef )Tf − ( T f + yf Tf ) Tef(Rayleigh) =(T fe )Tf − ( T f ) Tef ≥ 0
![Page 16: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/16.jpg)
Equivalent conditionsObvious but important:T consists of those terms not containing g and thosecontaining g . That is
T = T g + ygTg
Back to the Rayleigh condition(Te )Tf − ( T ) Tef=(T fe + yf Tef )Tf − ( T f + yf Tf ) Tef(Rayleigh) =(T fe )Tf − ( T f ) Tef ≥ 0
![Page 17: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/17.jpg)
So what!?!We showed that
TeTf − TTef = T fe Tf − T f Tef ≥ 0.The missing piece
TeTf − TTef ≥ 0 if and only ifyeyf (TeTf − TTef ) ≥ 0 if and only if
yeTe
T≥ yeyf Tef
yf Tf.
So the chances of selecting a spanning tree with e arenot increased by choosing among those alreadycontaining f !
![Page 18: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/18.jpg)
Time for an exampleb
e
f g
h
i
a
T = + + + + + + +Te = + + + +Tf = + + + +Tef = + +and yeyf (TeTf − TTef ) = yeygyh × yf ygyh = ×
![Page 19: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/19.jpg)
Where is the proof?I The classical proof using electrical networks isprinted in Grimmett’s book.I The most often cited proof is due to Brooks SmithStone and Tutte (1940).I A stronger property was shown by Choe andWagner (2006).I A combinatorical (bijective) proof is given byCibulka, Hladky, LaCroix and Wagner (2008).So if this stuff has been done over at least four times,what’s all the fuss?
![Page 20: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/20.jpg)
Mathematicians love variations!
Let’s replace T by the spanning forests, F .This was proposed in print in the early 90s.Considerable evidence has been published but, as ofyet, no proof thatFeFf − FFef ≥ 0,
for positive ygs and pair of distint edges e and f .
![Page 21: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/21.jpg)
A “weaker” versionSpecial case of Rayeligh, FeFf − FFef
I Set each yg to 1.I ie, choose spanning forest uniformly at random.
Special case ≡ Rayleigh(independently: Cocks and E., 2008)I All graphs are forest Rayleigh iffI all graphs satisfy the special case.proof idea: Suppose a graph is not Rayleigh, then
FeFf − FFef < 0 for certain ygs. Replace edges bycertain disjoint paths to create a graph that is notnegatively correlated.
![Page 22: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/22.jpg)
Evidence for the conjecture– Small graphsare negatively correlated(Grimmett, Winkler, 2004).– Two-sumsof Rayleigh graphsare Rayleigh (Wagner,Semple, Welsh 2008)Ï Smaller graphs areRayleigh (E., Wagner, 2008)ÏSeries parallel graphs areRayleigh (E., Wagner, 2008)
![Page 23: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/23.jpg)
SOS conjecture (Wagner)The spanning forest Rayleigh difference,
∆F{e, f } = FeFf − FFef
is a sum of monomials times squares of polynomials,∆F{e, f } =∑
S
ySA(S)2.
The Rayleigh property, FeFf − FFef ≥ 0 for positive ygs,follows immediately.One major hangup: the signs of the terms in A(S) areunknown.
![Page 24: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/24.jpg)
SOS conjecture (Wagner)The spanning forest Rayleigh difference,
∆F{e, f } = FeFf − FFef
is a sum of monomials times squares of polynomials,∆F{e, f } =∑
S
ySA(S)2.The Rayleigh property, FeFf − FFef ≥ 0 for positive ygs,follows immediately.One major hangup: the signs of the terms in A(S) areunknown.
![Page 25: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/25.jpg)
S-sets and A-sets∆F{e, f } =∑
S
ySA(S)2.I An S-set is a set of edges S so that S ∪ {e, f } iscontained in a cycle.I The A-sets of S are those spanning forests A so that
A ∪ {e, f } contains a unique cycle which contains S .
![Page 26: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/26.jpg)
S-sets and A-sets∆F{e, f } =∑
S
ySA(S)2.I An S-set is a set of edges S so that S ∪ {e, f } iscontained in a cycle.I The A-sets of S are those spanning forests A so that
A ∪ {e, f } contains a unique cycle which contains S .
e f
![Page 27: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/27.jpg)
S-sets and A-sets∆F{e, f } =∑
S
ySA(S)2.I An S-set is a set of edges S so that S ∪ {e, f } iscontained in a cycle.I The A-sets of S are those spanning forests A so that
A ∪ {e, f } contains a unique cycle which contains S .
e f
![Page 28: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/28.jpg)
S-sets and A-sets∆F{e, f } =∑
S
ySA(S)2.I An S-set is a set of edges S so that S ∪ {e, f } iscontained in a cycle.I The A-sets of S are those spanning forests A so that
A ∪ {e, f } contains a unique cycle which contains S .
e f
![Page 29: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/29.jpg)
S-sets and A-sets∆F{e, f } =∑
S
ySA(S)2.I An S-set is a set of edges S so that S ∪ {e, f } iscontained in a cycle.I The A-sets of S are those spanning forests A so that
A ∪ {e, f } contains a unique cycle which contains S .
e f
![Page 30: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/30.jpg)
Given an S-set, S with S ∪ {e, f } contained in a cycle C ,A(S) =∑
A
c(S , e, f , C )yA−S .
There they are! The signs c(S , e, f , C ). And there areMANY of them.
![Page 31: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/31.jpg)
Testing on small graphsWagner had some guesses for the signs and we tested∑
S
ySA(S)2 = FeFf − FFef
in Maple, for graphs up to 7 vertices.He also found signs that worked for thecube and Möbius ladder on 8 vertices.Necessary conditionsNext, we “show” the SOS-conjectureshould hold for two sums and thatit does hold for series parallel graphs.
![Page 32: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/32.jpg)
Series parallel graphs and 2-sumsMy presentation The details
![Page 33: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/33.jpg)
Suppose G = H ⊕g K and let C be a cycle of G . Theneither C is contained in H − g or K − g orC = CH ∪ CK − g for cycles through g in H and K .
Facts about 2 sums and ∆F{e, f } = FeFf − FFef
I If e ∈ H and f ∈ K , then∆F (G ){e, f } := ∆F (H){e, g}∆F (K ){f , g}I If e, f ∈ H , then ∆F (G ){e, f } = F (K )2∆F (H){e, f }The Rayleigh difference factors over the factors of the2-sum.
![Page 34: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/34.jpg)
Suppose G = H ⊕g K and let C be a cycle of G . Theneither C is contained in H − g or K − g orC = CH ∪ CK − g for cycles through g in H and K .Facts about 2 sums and ∆F{e, f } = FeFf − FFef
I If e ∈ H and f ∈ K , then∆F (G ){e, f } := ∆F (H){e, g}∆F (K ){f , g}I If e, f ∈ H , then ∆F (G ){e, f } = F (K )2∆F (H){e, f }The Rayleigh difference factors over the factors of the2-sum.
![Page 35: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/35.jpg)
How does the SOS-form factor?∑S
ySA(S)2is all about the cycles of G , through e and fIf e ∈ H and f ∈ K , these cycles come fromC = CH ∪ CK − g .
∆F (G ){e, f } := ∆F (H){e, g}∆F (K ){f , g}
In the same way, A-sets of G come from A-sets of Hand K , so the SOS-form factors.Ke
g fH
![Page 36: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/36.jpg)
How does the SOS-form factor?Show ∑
S
ySA(S)2 = F (K )2∆F (H){e, f } = F (K )2∑SH
ySHAH (SH )2If e, f ∈ H , we sum over
I S-sets in H not containing g .I careful that A-sets of H and forests of K do notform extra cycles in G .
I S-sets containing g in H and edges of K .I snag! The forests we use from K need to make aunique cycle with g and satisfy another SOS form.
e
f
g
![Page 37: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/37.jpg)
How does the SOS-form factor?Show ∑
S
ySA(S)2 = F (K )2∆F (H){e, f } = F (K )2∑SH
ySHAH (SH )2If e, f ∈ H , we sum over
I S-sets in H not containing g .I careful that A-sets of H and forests of K do notform extra cycles in G .
I S-sets containing g in H and edges of K .I snag! The forests we use from K need to make aunique cycle with g and satisfy another SOS form.
f
e g
![Page 38: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/38.jpg)
How does the SOS-form factor?Show ∑S
ySA(S)2 = F (K )2∆F (H){e, f } = F (K )2∑SH
ySHAH (SH )2If e, f ∈ H , we sum over
I S-sets in H not containing g .I careful that A-sets of H and forests of K do notform extra cycles in G .
I S-sets containing g in H and edges of K .I snag! The forests we use from K need to make aunique cycle with g and satisfy another SOS form.
e
f
g
(K g − Kg )Kg =∑Q yQB(Q)2?
![Page 39: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/39.jpg)
Series parallel graphs.∆-SOS∆F{e, f } = FeFf − FFef =∑
S
ySA(S)2Φ-SOS
ΦF{g} = (F g − Fg )Fg =∑Q
yQB(Q)2.If K is series parallel then it is Φ-SOS.Hope for 2-sumK is ∆-SOS by inductive hypothesis. Can we show thatif K is ∆-SOS, then it is Φ-SOS?This reduces to yet a third “SOS” form (see paper).
![Page 40: Negative correlation properties for graphs](https://reader033.vdocuments.site/reader033/viewer/2022052301/5594c3711a28ab913d8b4688/html5/thumbnails/40.jpg)
SummaryI Goal: Prove ∆F{e, f } = FeFf − FFef ≥ 0
I Method: ProveFeFf − FFef =∑
S
ySA(S)2I Next step: ProveΦF{g} = (F g − Fg )Fg =∑
Q
yQB(Q)2.Many thanks to David Wagner and my classmatesfrom Waterloo for their ideas and encouragements.