Navigability is a Robust Property

0.2 MB·English

by  Dimitris Achlioptas

Navigability is a Robust Property
DimitrisAchlioptas∗ Paris Siminelakis†
DepartmentofComputerScience DepartmentofElectrical Engineering
UniversityofCalifornia, SantaCruz Stanford University
[email protected] [email protected]
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Abstract
n
a TheSmallWorld phenomenonhasinspiredresearchersacrossa numberoffields. A breakthrough
J
initsunderstandingwasmadebyKleinbergwhointroducedRankBasedAugmentation(RBA):addto
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each vertexindependentlyan arc to a randomdestinationselected froma carefullycraftedprobability
2
distribution. KleinbergprovedthatRBAmakesmanynetworksnavigable,i.e.,itallowsgreedyrouting

] to successfully deliver messages between any two vertices in a polylogarithmicnumber of steps. We
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S provethatnavigabilityisaninherentpropertyofmanyrandomnetworks,arisingwithoutcoordination,
. orevenindependenceassumptions.
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∗ResearchsupportedbyaEuropeanResearchCouncil(ERC)StartingGrant(StG-210743)andanAlfredP.SloanFellowship.
†SupportedinpartbyanOnassisFoundationScholarship.
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1 Introduction
TheSmallWorldphenomenon referstothefactthatthereexistshortchainsofacquaintances betweenmost
pairs of people in the world, popularly known as Six Degrees of Separation [17]. Milgram’s famous 1967
experiment[16]showedthatnotonlysuchchainsexist,buttheycanalsobefoundinadecentralizedmanner.
Specifically, each participant in the experiment was handed a letter addressed to a certain person and was
told ofsome general characteristics ofthe person, including their occupation and location. Theywerethen
asked to forward the letter to the individual they knew on a first-name basis who was most likely to know
the recipient. Based on the premise that similar individuals have higher chance of knowing each other
(homophily), the participants typically forwarded the message to their contact most similar to the target, a
strategythatyielded remarkablyshortpathsformostlettersthatreachedtheirtarget(manydidnot).
Kleinberg’s groundbreaking work, formulated mathematically the property of finding short-paths in a
decentralized mannerasnavigability [8,10]. Sincethen,muchprogress hasbeenmade[9]andtheconcept
of navigability has found applications in the design of P2P networks [5, 19], data-structures [4, 15] and
searchalgorithms[14,18]. Keytodecentralization issharedknowledgeintheformofgeometry,i.e.,shared
knowledgeofa(distance) function onpairsofvertices(notnecessarily satisfying thetriangle inequality).
Geometry. A geometry (V,d) consists of a set of vertices V and a distance function d : V V IR ,
+
× →
whered(x,y) 0,d(x,y) = 0iffx = y,andd(x,y) = d(y,x),i.e.,dmustbeasemi-metric.
≥
Given a graph G(V,E) on a geometry (V,d), a decentralized search algorithm is any algorithm that
given a target vertex t and current vertex v selects the next edge v,u E to cross by only considering
{ } ∈
the distance of each neighbor uofv to thetarget t, i.e., d(u,t). Theallowance of paths of polylogarithmic
lengthinthedefinitionofnavigability, below,ismotivatedbythefactthatinanygraphwithconstantdegree
thediameterisΩ(log(n)),reflectinganallowanceforpolynomiallossduetothelackofglobalinformation.
Navigability. A graph G(V,E) on geometry (V,d) is d-navigable if there exists a decentralized search
algorithm whichgivenanytwos,t V willfindapathfromstotoflengthO(poly(log V )).
∈ | |
In his original work on navigability [8, 10], Kleinberg showed that if G is the 2-dimensional grid then
adding a single random edge independently to each v V results in a navigable graph (with d being the
∈
L1 distance on the grid). The distribution for selecting the other endpoint u of each added edge is crucial.
Indeed,ifitcanonlydependond(v,u)anddistinctverticesareaugmentedindependently,Kleinbergshowed
that there is a unique suitable distribution, the one in which the probability is proportional to d(v,u)−2
(and, more generally, d(v,u)−r for r-dimensional lattices). The underlying principle behind Kleinberg’s
augmentation schemehasbynowbecomeknownasRankBasedAugmentation (RBA)[11,13].
RankBasedAugmentation. Givenageometry(V,d),avertexv V,andℓ 0,letN (ℓ)bethenumber
v
∈ ≥
ofvertices withindistance ℓfromu. InRBA,theprobability ofaugmenting v withanedgetoanyu V is
∈
1
P(v,u) . (1)
∝ N (d(v,u))
u
The intuition behind RBAis that navigability is attained because the added edges provide connectivity
across all distance scales. Concretely, observe that for any partition of the range of the distance function
d into intervals, the probability that the (distance of the) other endpoint of an added edge lies in a given
interval isindependent of the interval. Therefore, by partitioning the range of dinto O(logn)intervals we
see that whatever the current vertex v is, there is always Ω((logn)−1) probability that its long-range edge
2
istoavertex atadistance atthesame“scale” asthetarget. Ofcourse, thatalone isnotenough. Inorderto
shrink the distance to the target by a constant factor, we also need the long-range edge to have reasonable
probability to go “in the right direction”, something which is effortlessly true in regular lattices for any
finite dimension. In subsequent work [11], aiming to provide rigorous results for graphs beyond lattices,
Kleinbergshowedthatthegeometricconditions neededforRBAtoachievenavigability aresatisfiedbythe
geometries induced by set-systems satisfying certain conditions when the distance between two vertices is
thesizeofthesmallestset(homophily) containing both(seeDefinition1inSection5).
Another canonical setting for achieving navigability by RBA is when the distance function d is the
shortest-path metricofaconnected graph G (V,E )withlargediameter Θ(poly(n)). Inthatsetting, ifE
0 0 d
istherandomsetofedgesaddedthroughRBA,thequestioniswhetherthe(random)graphG(V,E E )is
0 d
∪
d-navigable. WorksofSlivkins[15]andFraigniaudetal.[7]haveshowntheexistenceofathreshold,below
whichnavigability isattainable andabovewhich(intheworstcase)itisnotattainable, intermsofthedou-
bling dimension ofthe shortest path metric of G . Roughly speaking, the doubling dimension corresponds
0
to the logarithm of the possible directions that one might need to search, and the threshold occurs when it
crosses Θ(loglogn). Thus, wesee that even when dis a(shortest path) metric, very significant additional
constraints ondneedtobeimposed.
TheremarkablesuccessofRBAinconferringnavigabilityrestscruciallyonitsperfectadaptationtothe
underlyinggeometry. Thisadaptation,though,notonlyrequiresperfectindependenceandidenticalbehavior
ofallvertices,butalsoaveryspecific,indeedunique,functionalformfortheprobabilitydistributionofedge
formation. ThisexactfinetuningrendersRBAunnatural greatlyweakeningitsplausibility. Ourgoalinthis
paper is to demonstrate that navigability is in fact a robust property of networks that emerges from very
basicconsiderations withoutadaptation, coordination, orevenindependence assumptions.
2 Our Contribution
Asmentioned,atthefoundationofnavigabilityliessharedknowledgeintheformofgeometry. Ourstarting
premise is that geometry imposes global constraints on the set of feasible networks. Most obviously, in a
physical network where edges (wire, roads) correspond to a resource (copper, concrete) there is typically
an upper bound on how much can be invested to create the network. More generally, cost may represent
a number of different notions that distinguish between edges. We formalize this intuition by (i) allowing
different edges to have arbitrary costs, i.e., without imposing any constraints on the cost structure, and (ii)
taking asinputanupper bound onthetotalcost offeasible graphs, i.e.,abudget. Weremainfully agnostic
in all other respects, i.e., we study the uniform measure on all graphs satisfying the budget constraint. So,
forexample,ifalledgeshaveunitcostwerecovertheclassic Erdo˝s-Re´nyiG(n,m)random graphs(except
nowmisarandomvariable, sharply concentrated justbelowthebudget.)
As one can imagine, the set of all graphs feasible within a given budget may contain wildly different
elements. Ourcapacitytostudytheuniformmeasureonsuchgraphscomesfromaveryrecentgeneraltheo-
remwedevelopedin[1],ofwhichthisworkisthefirstapplication. Atahighlevel,themaintheoremof[1]
assertsthatifasubsetS ofthesetofallundirectedsimplegraphs onnverticesissufficientlysymmetric,
n
G
thentheuniformmeasureonS canbewell-approximatedbyaproductmeasureontheedges,i.e.,ameasure
where each edge is included independently with different edges potentially having different probabilities.
Formally, a product measure on is specified succinctly by a symmetric matrix Q [0,1]n×n of prob-
n
G ∈
abilities where Q = 0 for i [n]. We denote by G(n,Q) the measure in which possible edge i,j is
ii
∈ { }
included independently withprobability Q = Q . Themainresult of[1]then allowsonetoapproximate
ij ji
theuniformmeasurebyaproductmeasureinthefollowingverystrongsense.
3
Sandwichability. TheuniformmeasureU(S)overanarbitrarysetofgraphsS is(ǫ,δ)-sandwichable
n
⊆ G
if there exists a n n symmetric matrix Q such that the two distributions G± G(n,(1 ǫ)Q), and the
× ∼ ±
distribution G U(S)canbecoupledsothatG− G G+ withprobability atleast1 δ.
∼ ⊆ ⊆ −
As discussed above, navigability requires some degree of structure in the underlying geometry. It is
fromthisstructurethatwewillextractthesymmetryneededtoapplythetheoremof[1]andderiveaproduct
form approximation forgraphs withabounded totalcost. Armedwithsuchanapproximation, establishing
navigability becomes dramatically easier, allowing us to demonstrate its robustness and ubiquity. Roughly
speaking, weisolatethreeingredients thatsufficefornavigability onageometry(V,d):
Asubstrateofconnections betweennearby pointsonV,allowingthewalktonevergetstuck.
•
Somedegreeofcoherence ofthedistance function d.
•
Sufficient,andsufficiently uniform,edgedensity acrossalldistance scales.
•
The first two ingredients are generalizations of existing work and, as we will see, fully compatible with
RBA.ThethirdingredientisalsomotivatedbytheRBAviewpoint,butwewillprovethatitcanbeachieved
infarmore-light handed, and thus natural, manner thanRBA.Moreover, inthecourse ofdoing so, wewill
giveitaverynaturaleconomical interpretation asthecostofindexing.
2.1 The two BasicRequirements anda Unifying FrameworkforRBA
Substrate. AsetofedgesE formsasubstrateforageometry(V,d),ifforevery(s,t) V V withs= t,
0
∈ × 6
thereisatleastonevertexv suchthat s,v E andd(v,t) d(s,t) 1.
0
{ } ∈ ≤ −
The existence of the substrate implies that (very slow) travel between any two vertices is possible, so
thatadecentralized algorithm nevergetstriviallystuck.
Coherence is anotion that comes withan associated scale factor γ > 1. Specifically, given ageometry
(V,d)wewillrefertotheverticeswhosedistancefromagivenvertexv V lieintheinterval(γk−1,γk]as
∈
theverticesinthek-th(distance)γ-scalefromv. Also,forafixedλ < 1andanytargetvertext = v,wewill
6
saythatavertexuist-helpfultovifd(v,u) γk (uiswithinthesameγ-scaleast),andd(u,t) < λd(v,t)
≤
(reduces thedistance byaconstant).
Coherence. LetK = log V . Ageometry(V,d)isγ-coherent ifthereisλ < 1suchthatforallv V:
⌈ γ| |⌉ ∈
–Forallk [K],thenumberofvertices inthek-thdistance scalefromv isP (v) = Θ(γk).
k
∈
–Forallt = v,aconstantfractionoftheverticeswhosedistancescalefromvisnogreaterthanthedistance
6
scaleoftaret-helpfultov.
The two conditions above endow the, otherwise arbitrary, semi-metric d with sufficient regularity and
consistencytoguidethesearch. Althoughourdefinitionofcoherenceisfarmoregeneral,inordertoconvey
intuition about thetwo conditions, think for amoment ofV as asetof points inEuclidean space. The first
condition guarantees that there are no “holes”, as the variance in the density of points is bounded in every
distance scale. The second condition guarantees that around any vertex v the density of points does not
change muchdepending onthedirection (target vertext)anddistance scale. Besides those twoconditions,
wemakenofurther assumptions ondand,inparticular, wedonot assumethetriangleinequality.
Coherentgeometriesallowustoprovideaunifiedtreatmentofnavigabilitysincetheyencompassfinite-
dimensionallattices,hierarchicalmodels,anyvertextransitivegraphwithboundeddoublingdimensionand,
asweproveinSection5,Kleinberg’s setsystems[11](seeDefinition1).
4
Theorem1. Everysetsystemsatisfying theconditions of[11]isaγ-coherent geometryforsomeγ > 1.
Theorem2. Let(V,d)beanyγ-coherentgeometryandletE beanysubstrateforit. IfE isthe(random)
0 d
setofedgesobtained byapplying RBAto(V,d),thenthegraphG(V,E E )isd-navigable w.h.p.1
0 d
∪
Theorem 2 subsumes and unifies a number of previous positive results on RBA-induced navigability.
Ourmaincontribution, though,liesinshowingthatgivenasubstrateandcoherence,navigabilitycanemerge
withoutcoordination fromtheinterplay ofcostandgeometry.
2.2 Navigabilityfrom OrganicGrowth
Asmentionedearlier, thesuccessofRBAstemsfromthefactthattheedge-creation mechanismisperfectly
adapted to the underlying geometry so as to induce navigability. In contrast, we will not even specify an
edge-creation mechanism, but rather focus only on the set of graphs feasible with a given budget. Our
requirement ismerelythatthecostfunctionisinformed bythegeometry.
γ-consistency. Given a γ-coherent geometry (V,d), a cost function c : V V R is γ-consistent if c
× →
takesthesamevaluec foreveryedge u,v suchthatd(u,v) (γk−1,γk].
k
{ } ∈
In particular, note that we make no requirements of the values c , not even a rudimentary one, such
k
{ }
as being increasing in k. All that γ-consistency entails is that the partition of edges according to cost
is a coarsening of the partition of the edges by γ-scale. In fact, even this requirement can be weakened
significantly, as long as some correlation between the two partitions remains, but it is technically much
simpler to assume γ-consistency as it simplifies the exposition greatly. One can think of consistency as
limited sensitivity with respect to distance. As an example, it means that making friends with the people
next door might be more likely than making friends with other people on the same floor, and that making
friendswithpeopleonthesamefloorismorelikelythanmakingfriendswithpeopleinadifferentfloor,but
itdoesnotreallymatterwhichfloor.
Cost-geometries. We say that Γ = Γ(V,d,c) is a coherent cost-geometry if there exists γ > 1 such that
(V,d)isaγ-coherent geometryandcisγ-consistent costfunction.
RandomGraphsofBoundedCost. Givenacoherentcost-geometry Γ(V,d,c)andarealnumberB 0,
≥
letG (B) = E V V : 1 c(e) B ,i.e.,G (B)isthesetofallgraphs (edgesets)onV with
Γ { ⊆ × n e∈E ≤ } Γ
totalcostatmostBn. AuniformlyrandomelementofG (B)willbedenotedasE = E (B).
Γ Γ Γ
P
Applyingthemaintheoremof[1],inSection3wewillprovethatrandomgraphsofboundedcost(ona
consistent cost-geometry) haveaproductmeasureapproximation, inthefollowingsense.
Theorem 3. Given a coherent cost-geometry Γ, there exist a a unique function λ(B) 0 and constant
≥
B (Γ) > 0 such that for every B B (Γ) the uniform measure on G (B) is (δ,ǫ)-sandwichable by the
0 0 Γ
≥
productmeasureinwhichtheprobability ofeveryedgewithcostc is
k
1
, (2)
1+exp(λ(B)c )
k
and(δ,ǫ) = 24 , 2V −5K . Thenumberλ(B)> 0canbeexplicitly definedintermsof c .
log|V| | | { k}
(cid:16)q (cid:17)
1Throughoutthepaper,allasymptoticswillbewithrespecttothenumberofvertices|V|=n.Thus,withhighprobabilitywill
alwaysmean“withprobabilitythattendsto1asn→∞.”
5
The regularizer λ = λ(B) in Theorem 3 corresponds to the derivative of entropy with respect to the
budgetB (energy), i.e.,isaninversetemperature, anddependsonB inasmoothone-to-one manner.
Theorem 3 will give us a great amount of access to the uniform measure on G (B). In particular, the
Γ
upperapproximation G(V ,(1+ǫ)Q)willallowustoboundthetotalnumberofedgespresentinatypical
| |
graph, establishing sparsity for all sufficiently small budgets. On the other hand, the lower approximation
G(V ,(1 ǫ)Q) will allow us to establish a lower bound on the number of edges incident to each vertex
| | −
of each distance scale. Combined with the spatial uniformity afforded by independence, this will allow us
to prove that navigability emerges as soon as the total number of edges within each scale is large enough,
establishing navigability forallsufficiently largebudgets.
Theorem4. Foreverycoherentcost-geometryΓ(V,d,c),where V = n,thereexistnumbersB±suchthat
| |
ifE isauniformlyrandomelementofG (B)then:
Γ Γ
– ForallB B+,w.h.p. E = O(n poly(logn)). (Sparsity)
Γ
≤ | | ·
– ForallB B−,foranysubstrateE ,w.h.p.thegraphG(V,E E )isd-navigable. (Navigability)
0 0 Γ
≥ ∪
NotethatTheorem4showsthatnavigability ariseseventually, i.e.,forallB B−,withoutanyfurther
≥
assumptions onthecostfunction orgeometry. Thecaveat, ifwethink ofB asincreasing from0, isthatby
the time there are enough edges across all distance scales, i.e., B B−, the total number of edges may
≥
be much greater than linear. This is because for an arbitrary cost structure c , by the time the “slowest
k
{ }
growing” distance scale has the required number of edges, the other scales may be replete with edges,
possibly manymorethantherequired Ω(n/polylog(n)). Thisisreflected intheordering betweenB− and
B+ that determines whether the sparsity and navigability regimes are overlapping. Inparticular, wewould
likeB− B+ and,ideally, theratioR = B+/B− > 0tobelarge. Whetherthisisthecaseornotdepends
≤
precisely onthedegreeofadaptation ofthecost-structure c tothegeometryaswediscussnext.
k
{ }
2.3 Navigabilityasa Reflection ofthe CostofIndexing
Recall that for every vertex v in a γ-coherent geometry and for every distance scale k [K], the number
∈
of vertices whose distance from v is in the k-th γ-distance scale is P (v) = Θ(γk). At the same time, (2)
k
asserts that the probability of each edge is exponentially small in its cost. Thus, reconciling sparsity with
navigability boilsdowntobalancingthesetwofactors. Wewillexhibitaclassofcostfunctionsthat(i)have
an intuitive interpretation as the cost of indexing, (ii) achieve a ratio R = B+/B− > 0 that grows with n,
i.e.,averywiderangeofbudgetsforwhichwehavebothnavigability andsparsity, and(iii)recoverRBAas
aspecialcasecorresponding toaparticular budgetchoice.
Consideravertexv thatneedstoforwardamessagetoaneighboruatthek-thdistancescale. Todoso,
v needs todistinguish uamong allother P (v)vertices atthek-thdistance scale, i.e.,v needs tobeableto
k
index intothatscale. Todoso,itisnatural toassertthatv mustincuracostofΘ(log P (v)) = Θ(k)(due
2 k
tocoherence) bitstostoretheuniqueIDofuamongtheothermembersofitsequivalence class(intheeyes
ofv). Motivatedbythisweconsider costfunctions whereforsomeα > 0,
1
c∗ = k .
k α
Theorem5. Foranycoherentcost-geometry Γ(V,d,c∗),where V =n,thereexistB− B+ suchthat:
| | ≤
1. B+/B− = ω(polylogn).
6
2. ForallB [B−, B+],w.h.p.:
∈
E (B) = O(n polylogn)).
Γ
• | |
ThegraphG(V,E E (B))isd-navigable.
0 Γ
• ∪
3. Thereexists B [B−, B+]suchthat intheapproximation ofE byG(V ,Q),forevery u,v
a Γ
∈ | | { } ∈
E,Q∗ = Θ(N (d(u,v))−1),i.e,RankBasedAugmentation isapproximately recovered.
ij u
Part 1 of Theorem 5 is equivalent to a scaling window of Θ(loglogn)for the exponent λ, within which
logn
navigability holds with poly-logarithmic average degree. This corroborates Kleinberg’s work that gave a
unique exponent of β = 1 in the context of RBA for the scaling (1) of probability. Nevertheless, under
−
our framework this vanishing window for the highly sensitive paramater λ produces a diverging range for
values of B, explaining the purported fragility of RBA to looking at perturbations in the wrong scale. In
fact, we can use this feature of our model to provide the first theoretical explanation for the discrepancy
betweentheoreticalresultsandempiricalevidence[2,13,6]showingthatrealnetworksexhibitanexponent
βˆ 0.8 < 1. Inoursetting, exponents smallerthan1correspond tohigher averagedegreeandthuswecan
≈
attribute thisdiscrepancy tofinitesizeeffects(finiten)andthedensification [12]ofnetworks.
3 Deriving a Product Measure Approximation: Proof of Theorem 3
Westartwithsomedefinitionsthatwillallowustostatethemaintheoremof[1]. AsetofgraphsS is
n
⊆ G
symmetric with respect toapartition of theset of allpossible n edges, if thecharacteristic function of
P 2
S dependsonlyonthenumberofedgesfromeachpartof butnotonwhichedges.
P (cid:0) (cid:1)
Edge Profile. Given a partition = ( ,..., ) of the set of all possible n edges, for a set of edges
P P1 PK 2
E and foreach k [K],letm (E)denote thenumber ofedges inE from . Theedge profileofE
n k k
∈ G ∈ (cid:0) (cid:1)P
ism(E) := (m (E),...,m (E)).
1 K
We denote the image of a symmetric set S under the edge-profile as m(S). As before let P = =
k k
|P |
1 P (u)bethetotalnumberofedgesinpartk ofpartition .
2 u∈V k P
K
P
EdPgeProfileEntropy. Givenanedgeprofilev = (v1,...,vk)theentropyofvisENT(v) = log k .
v
k=1 (cid:18) k(cid:19)
X
The edge-profile entropy is used to express the number of graphs with a particular edge profile v as
exp(ENT(v)). Given any symmetric set S n, the probability of observing an edge profile v when
⊆ G
sampling an element uniformly at random from S is then given by P (v) = 1 eENT(v). Thus, in order to
S |S|
analyze the distribution of a random edge-profile, and consequently of a random element of G (n,B), we
c
aregoingtoexploitanalyticproperties oftheentropy onthesetoffeasible edgeprofilesm(S).
Convexity. LetConv(A) denote the convex hull of a set A. Say that a -symmetric set S is convex
n
P ⊆ G
ifftheconvexhullofm(S)containsnonewinteger points,i.e.,ifConv(m(S)) Nk = m(S).
∩
EntropicOptimizer. GivenasymmetricsetS,letm∗ = m∗(S) IRk betheuniquesolution to
∈
K
v P v
max v log k +(P v )log k − k . (3)
k k k
v∈Conv(m(S))−k=1(cid:20) (cid:18)Pk(cid:19) − (cid:18) Pk (cid:19)(cid:21)
X
7
Giventhemaximizerm∗(S),thematrixQ∗ = Q∗(S)isgivenbylettingforallk [K]theprobability
∈
ofanedgee beQ∗ := m∗/P . Tostate thetheorem, weneedthe following parameters thatquantify
∈ Pk e k k
theconcentration oftheuniformmeasurearounditsmode.
ThicknessandConditionNumber. Givenapartition anda -symmetricsetS,wedefine
P P
Thickness: µ = µ(S) = min min m∗,P m∗ (4)
k∈[K] { k k − k}
5Klogn
Conditionnumber: τ = τ(S) = (5)
µ(S)
Wenowstatethemaintheorem employedintheproof.
Theorem 6 ([1]). Let be any edge-partition and let S be any -symmetric convex set. For every ǫ >
P P
12τ(S),theuniformmeasureoverS is(ǫ,δ)-sandwichable forδ = 2exp µ(S) ǫ2 τ(S) .
− 12 −
p h (cid:16) (cid:17)i
In our setting, S is the set G (B) := E V V : 1 c B of graphs with bounded
Γ { ⊂ × n e∈E e ≤ }
average cost and is the partition induced by the coherent cost function c. The set m(S) isthen given by
m(S) = v NPk : 1 K c v B . Hence, it is easy to sPee that G (B) is convex and symmetric,
{ ∈ n k=1 k k ≤ } Γ
according totheprevious definition, forallvaluesofB. ToproveTheorem3,weneedtofind:
P
(i) ananalyticexpression forthevectorm∗ asafunction ofB
(ii) therangeofvaluesofB forwhichapplying Theorem6giveshighprobability bounds.
3.1 Finding the Entropic OptimalEdgeProfile
Westartbyintroducing aslightreparametrization intermsoftheaverage-degree profile. ForanedgesetE,
definethevectora(E) := m(E)/n,whereasbeforemistheedge-profile. Inthesamespirit,letp = P /n
k k
denote the average number of edges in part (scale) k. Using this parametrization and by explicitly writing
Conv(a(S)),wecanequivalently expresstheoptimization problem (3)as:
K
max H(a) = [(p a )log(p a )+a log(a )]
k k k k k k
a − − −
k=1
X
K
subjectto a c B
k k
≤
k=1
X
0 a p , k [K] .
k k
≤ ≤ ∀ ∈
Wewillrefertotheaboveoptimizationproblemas(Λ)andtoitssolutionasa∗ = a∗(B). Towardsobtaining
ananalyticexpressionfora∗,wefirstshowthatnocoordinatek [K]liesonthenaturalboundary 0,p .
k
∈ { }
Lemma1. Theoptimalprofilea∗ (B):= a (0,p ) ... (0,p ) : Kc a B .
∈ D { ∈ 1 × × K k k k ≤ }
Proof. We prove the lemma by contradiction. We show that if a∗ is a solutioPn of (Λ) such that a∗ / ,
∈ D
then there is an aˆ∗ for which objective function f takes a higher value. Specifically, for ǫ > 0
∈ D
assume thatthere areindices 1 i,j K such that a∗ = 0and a∗ > δ(ǫ)2,whereδ(ǫ) = ǫc /c . Define
≤ ≤ i j i j
2ForanynontrivialvaluesofBsuchanindexcanalwaysbefound.
8
aˆ∗(ǫ) = (a∗,...,a∗+ǫ,...,a∗ δ(ǫ),...,a∗ ). Ifh(ǫ) = H(aˆ∗) H(a∗)isthedifferenceintheobjective
1 i j− K −
function betweentheassumedoptimala∗ andtheperturbation aˆ∗,then
c
h′(ǫ) = log(ǫ)+log(p a ǫ)+ i (log(a δ(ǫ)) log(p a +δ(ǫ))) .
i i j j j
− − − c − − −
j
Observe, that lim h′(ǫ) = + , since we have assumed that a∗ > 0. This shows that every maximizer
ǫ→0 ∞ j
satisfiesa∗ > 0. Thesameargumentestablishesthata∗ < p forallk [K]. Combiningthetwostatements
k k ∈
wegetthatanymaximizerbelongsin .
D
As a consequence, since they are inactive at the optimum, we can omit separable inequalities from the
formulation. Further,defineB¯ := 1 K p c theaveragecostofthesolutiontotheunconstrainedversion
2 k=1 k k
of(Λ), i.e.,wherea¯ := p /2. IfB > B¯ then theabsolute maximum entropic point a¯ isstill in (B)and
k k
P D
thusthesolution willbealwaysa∗ = a¯ foreverysuchB.
k k
Lemma 2. There is a unique function λ(B) that is one-to-one for all 0 B B¯ and λ(B) = 0 for all
≤ ≤
B B¯,suchthattheuniquesolution of(Λ)isgivenby:
≥
p
a∗(B)= k , k [K] . (6)
k 1+exp[λ(B) c ] ∀ ∈
k
·
Proof. Uniqueness ofthesolution follows easilyfrom convexity ofthedomain andconcavity oftheobjec-
tivefunction. Further, byLemma1,wecanreducetheoptimization problem(Λ)tothefollowing:
K
max [(p a )log(p a )+a log(a )]
k k k k k k
a − − −
k=1
X
K
subjectto a c B .
k k
≤
k=1
X
Toobtainananalytical solution, weformtheLagrangianofthereducedproblem
K K
L(a,λ) = [(p a )log(p a )+a log(a )]+λ B a c .
k k k k k k k k
− − − −
!
k=1 k=1
X X
withtheadditional constraint thatλ 0. TheKarush-Kuhn-Tacker conditions read
≥
∂L a
= 0 log k = λc (7)
k
∂a ⇐⇒ p a −
k (cid:18) k − k(cid:19)
K
∂L
= 0 a c = B . (8)
k k
∂λ ⇐⇒
k=1
X
Solvingthefirstequationfora (λ)weget
k
p
a∗ = k ,
k 1+exp(λc )
k
Substituting thisexpression in(8),wegetthefollowingfunction ofλ:
K
p
g(λ) = c k (9)
k
· 1+exp(λc )
k
k=1
X
9
and the second constraint can now be written as g(λ) = B. The domain of g is the set of non-negative
numbers on which g is continuous and infinitely differentiable. Under positive costs c , it is easy to see
k
{ }
that g′(λ) < 0 for all B < B¯ , hence, g is strictly decreasing in the interval [0, ) and g(0) = B¯. Thus,
∞
g : [0, ) [0,B¯]is1-to-1 andthusinvertible. Thismeansthateverybudget in[0,B¯]isfeasible andthat
∞ →
for each such budget there is a unique λ(B) := g−1(B). ForB B¯, λ(B) = 0. Therefore, we conclude
≥
thatthemaximizerisalwaysuniqueforanyfeasibleB andimplicitly givenbyg(λ) = B.
3.2 Thickness µ(B) ofG (B) andSandwiching
Γ
Ournextstepistousetheanalyticalsolutiontotheoptimizationproblemtoinstantiatethethicknessparam-
eterµdefinedin(4). Using(6),wecanwrite:
p
µ(B)= min m∗ = n min k (10)
k∈[K] k ·k∈[K]1+exp[λ(B)ck]
where we have used the facts that that a∗ = m∗/n and a∗(B) 1/2 m∗ P m∗. To get a more
k k k ≤ ⇒ k ≤ k − k
convenient expression, since 0 < c < we can write the cost as c = 1 log(p ) where 0 < β <
k ∞ k βk k k ∞
when p 1. Thus, approximately3 for large p (eq. k) we have µ(B) n min p1−λ(B)/βk .
k ≥ k ≈ · k∈[K] k
Theorem 6, gives strong (non-constant) probability bounds as long as τ(B) 1. For ourhpurposes wie
≪
are going to consider that the maximum τ(B) (respectively minimum B) that we allow is τ = log−1(n)
0
(respectively B ). Substituting the above expression for µ(B)in (5), weget that the condition τ τ can
0 0
≤
berewrittenasλ(B) λ ,where
0
≤
nlogp β
k k
λ = λ ( p , β ) := min log . (11)
0 0 { k} { k} k∈[K](cid:20) (cid:18)5Klog2(n)(cid:19) logpk(cid:21)
Usingthefunctiong(λ)definedin(9),wecanexpressthisconstraint asB B := g(λ ).
0 0
≥
ToconcludetheproofofTheorem3weseethatµ(B) 5Klog2(n)andτ(B) 1 ,forallB B .
≥ ≤ log(n) ≥ 0
ApplyingTheorem6,forǫ = 24 thatisgreaterthan√12τ ,wegetthatδ 2exp µ(B) ǫ20 τ(B) .
0 logn 0 ≤ 12 −
Theproofisconcluded bysubsqtituting theboundsinthelastexpression. h (cid:16) (cid:17)i
4 Navigability via Reducibility
Inthissection weproveourresults aboutnavigability oncoherent geometries. Westartbygivingaslightly
more formal definition of coherence. Recall that given a geometry (V,d) and a fixed (scale factor) γ > 1,
P (v) denotes the number of vertices in V at “distance” (γk−1,γk] from v. Further, for fixed λ < 1 and
k
all t = v V , let k be the non-negative integer such that d(v,t) (γkvt−1,γkvt] and D (v,t) be the
vt λ
6 ∈ ∈
vertices inV whose distance from v isat mostγkvt andwhose distance from t isatmost λ d(v,t). Thus,
·
D (v,t) isthenumberofnodesthatcouldfacilitate greedy routing (t-helpful), i.e.,reducethedistance to
λ
| |
tbyaconstant factorλ < 1.
Coherence. Fixγ > 1andletK = log (V ) . Ageometry(V,d)isγ-coherent if:
γ
⌈ | | ⌉
(H1)BoundedGrowth: A> 1,α > 0suchthatP (v) γk[α,A],forallv V andk [K].
k
∃ ∈ ∈ ∈
(H2)Isotropy: φ> 0,1 > λ > 0suchthat D (v,t) φγkvt,foralls = t V.
λ
∃ | | ≥ 6 ∈
3Whentheapproximationdoesnotholditmeansthatµ(B) = Ω(n)whichtriviallysatisfiesalltherequirementsweneedfor
“sandwiching”andnavigability.
10