Convergence law for hyper-graphs with prescribed degree sequences

0.24 MB·English

by  Nans Lefebvre

Convergence law for hyper-graphs
with prescribed degree sequences
Nans Lefebvre
LIAFA, Université Paris VII
[email protected]
Abstract—We view hyper-graphs as incidence The logicalanalysisis one wayto understanda random
5 graphs, i.e. bipartite graphs with a set of nodes structure, by studying the probabilities that first-order
1 representing vertices and a set of nodes representing
formulas are satisfied. It has been started on random
0 hyper-edges, with two nodes being adjacent if the
2 corresponding vertex belongs to the corresponding graphs independently in [11] and [9], a result celebrated
hyper-edge. It defines a random hyper-multigraph as the zero-one law of random graphs. The whole of the
y
specified by two distributions, one for the degrees of Erdős-Rényimodel was successfully charted [22], and [21]
a
M the vertices, and one for the sizes of the hyper-edges. extended results to k-uniform hyper-graphs with a differ-
We develop the logical analysis of this framework and
ent logicalformalism, using a k-ary predicate. The logical
first prove a convergence law for first-order logic, then
6 structure traditionally used allows only to treat simple
characterise the limit first-order theories defined by a
wideclass ofdegree distributions. Convergencelawsof graphs; following [6], it is shown that using the incidence
]
O other models follow, and in particular for the classical graph as logical structure allows to handle non-uniform
Erdős-Rényi graphs and k-uniform hyper-graphs. hyper-multigraphswithout modifying the logic,using just
L
a binary predicate. This allows to study directly the logic
. I. Introduction
s of random hyper-multigraphs, and since this framework
c
[ Understanding the graphs that occur in nature is an extendsconservativelytheclassicaloneitallowstorecover
important challenge in computer science, as many phe- previous results on classical Erdős-Rényi graphs and k-
3
nomenacanbe seenasdiffusionprocessesongraphs,such uniform hyper-graphs as special cases.
v
asthediffusionofideasinsociety,frommarketingtosocial The main result is a convergence law for a class of ran-
9
2 uproar, or the spreading of epidemics [5]. Most natural domnon-uniformhyper-multigraphs(Theorem1).Besides
4 graphs have a sparse structure yet are locally highly natural conditions to avoid pathological sequences, the
7 clustered, a property that classical random graphs fail to sequencesinthisclassmusthavelimitdistributionswitha
0
model.Thismotivatedtheintroductionofamyriadofnew bounded second moment and a sublinear fourth moment.
.
1 models with more realistic features, with an overview of Thisincludes the caseofDiracdistributionsthatgenerate
0 rigorous results in [8]. hyper-graphs with uniform hyper-edge sizes. It extends a
5
This paper considers the bipartite configuration model resultofLynch[18]thatgaveaconvergencelawforaclass
1
[2], interpreted as the incidence graph of a hyper- ofgraphsdefinedbytheirdegreesequence,andcontributes
:
v
multigraph by considering one set of vertices to be hyper- to the body of works extending classical results to hyper-
i
X edges, and the binary relation to specify that a vertex graphs,suchas[7]forthephasetransition.Twosequences
r belongs to a hyper-edge. A random hyper-graph is speci- of random hyper-multigraphs are contiguous if they share
a fied by two given degree distributions, one for the degrees the same limit theory, i.e. first-order formulas are either
of the vertices and one for the sizes of the hyper-edges almost surely true on both sequences or neither, allowing
(communities),andgeneratestheuniformincidencegraph to compare random sequences and models from a logical
on these degree distributions. The analogy with the real- pointofview.Thepossiblelimittheoriesarecharacterised
world formation of graphs is compelling, since vertices and axiomatised, and from this the contiguity of random
are often linked by some shared community, actors may hyper-multigraphs is deduced (Theorem 2).
be linked because they play in the same film, scientists In the second section, the logical framework and nota-
because they collaborate on a paper [12], [16]. In society, tions are introduced. In the third section, the considered
communitiescanbeofanysize,involvingjusttwovertices class of random hyper-multigraphs is defined, and the
(friendships), a small number of vertices (families), up to convergence law is proved for this class. In the fourth
very large communities (professions, urban areas). Hence section, the limit theories are investigated and a criterion
thisisaflexiblemodelthatgeneratesawideclassofgraphs for the contiguity of random sequences is deduced; then
by allowing a wide range of limit distributions, generates some distributions are compared to other models, such as
graphs with a sparse structure and dense communities, Erdős-Rényi graphs. In the fifth section, some properties
and models complex relations (see [10] for an application of interest are studied, and some remarks on future work
of hyper-graph as abstractions of social networks). conclude.
II. Preliminaries
Inthissection,themostgeneralobjectsaredefinedfirst
v
and more classical structures such as simple graphs are 3
definedasspecificcases.Thisisconsistentwiththelogical e1
model since graphs can be defined by conditioning on the
v v e
1 2 3
validity of a first-order property. Some definitions may be
e
non-standard to fit the context of this paper. 2
Let (n)k denote the k-falling factorial of n, (n)k = v4
k−1(n i). If is a (countable) set, let ℘( ) denote
Qthie=s0et o−f probabSility distributions on , S be aSn element
of℘( ),ands S be arandomvariabSlesampledfromS.
(a)Asmallhyper-multigraph
S ∼
with a multiple edge and a
A. Hyper-multigraphs and the incidence graph
hyper-edge
A (non-uniform, undirected) hyper-multigraph H = e e e
1 2 3
(V = 1,...,n ,M : (V) N) is a set of vertices V
{ } P 7→
equipped with a set of hyper-edges given with multiplici-
ties. For a set e (V), e E denotes that M(e) > 0, v v v v
1 2 3 4
∈ P ∈
andwesupposethatanyewithM(e)>0isofsizeatleast
(b)∈-edgesareimplicitlyoriented
2.A hyper-multigraphis a hyper-graph if all multiplicities
are 1, in which case the hyper-graphcan be considered as Fig. 1: A hyper-multigraph and its incidence graph
a structure H =( 1,...,n ,E). A hyper-multigraphis k-
{ }
uniform ifeveryhyper-edgeisofsizek,andisamultigraph
if it is 2-uniform. A graph is a 2-uniform hyper-graph. of root r and radius l is the subgraph of G defined by
Let the classes of all hyper-multigraphs, hyper-graphs, VB = v VG d(v,r) 6 l and EB is the restriction
{ ∈ | }
multigraphs, and graphs be denoted respectively by HN, of EG to pairs of vertices in VB. A neighbourhood of a
H, GN, and G. Random hyper-multigraphs (elements of graph denotes a ball of finite radius, and its type denotes
℘(HN)) are denoted by , and random hyper-multigraph its isomorphism class.
H
sequences by ( n)n∈N. A bipartite graph is a graph with two sets of vertices
Two verticesHv and v′ are adjacent, denoted by v ¨ v′, V and W, such that for all (v,v′) V2 (respectively
if there is an e E such that v E and v′ E. The W2), v ˚ v′. Classes of bipartite hyp∈ergraphs, etc., can
∈ ∈ ∈
converse notion, that there is no e E such that v e be defined, but in this paper only bipartite simple graphs
and v′ e, is denoted by v ˚ v′. Th∈e degree of a ver∈tex are used as in the next definition:
∈
v,denoted deg(v) is the number ofedges containingv,i.e.
Definition 1. The incidence graph I of a hyper-
G
e e E v e . These two notions can be specialised
|{ | ∈ ∧ ∈ }| multigraphG=(V,M)isabipartitegraph(V,EI, I)with
to (m,k)-adjacencyby specifying respectivelythe number ∈
E = (e,i) i 6 M(e) such that v e v (e,i).
m of hyper-edges,and the size k of the edges,i.e. v ¨m v′ I { | } ∈ ↔ ∈I
k Each hyper-edge is represented by a node in E , and -
if there are m hyper-edges of size k containing v and v′. I ∈
edges link vertices to the hyper-edges they belong to.
The following definitions can be extended to hyper-
multigraphs, but will only be used on simple graphs,
Definition 2. Let I denote the class of incidence graphs,
usually denoted by G. The excess exc(G) of a graph G
that is the set of bipartite graphs with an ordered pair of
is E V .Ahomomorphism ofagraphg =(V ,E)toG
is|an|−inj|ec|tion σ :V V such that for all (v ,gv ) V2, setsof vertices where every vertexin EI has degreeat least
v ¨ v implies σ(vg)7→¨ σ(v ). A graph H is1a s2ub∈grapgh 2.
1 2 1 2
of G if there is a homomorphism from V to a subset The incidence graph of two hyper-multigraphs are iso-
H
of V , and this subset is called a realisation of H. A morphic if and only if the two hyper-multigraphs are
G
subgraphis (asymptotically)realisable in a randomgraph isomorphic, therefore:
if it is (asymptotically) realised probability bounded
G Proposition 1. The class of hyper-multigraphs HN and
away from 0. A homomorphism is an isomorphism if it is
the class of incidence graphs I are in bijection.
bijective and (v ,v ) V2, v ¨ v σ(v ) ¨ σ(v ).
1 2 ∈ g 1 2 ⇔ 1 2
A rooted graph is a graph with a distinguished vertex This last property allows to use the incidence graph as
(the root), and a rooted graph isomorphism is a graph a faithful representation of hyper-multigraphs.
isomorphism that maps the roots together. A graph G
B. The bipartite configuration model
induces a distance between vertices defined as d (x,y)=
G
min t x ,...,x such that x ¨ ¨ x x = The bipartite configuration model (dv,de) is a way
1 t+1 1 t+1 1
{ | ∃ ··· ∧ B
x x =y (by conventionif this number is not defined to generate randomhyper-multigraphsby specifying their
t+1
∧ }
itisconsideredtobe infinite). Aball B(r,l)= V ,E ,r corresponding incidence graphs. To generate an incidence
B B
h i
(dv,de), i.e. v dv and e de. See [1] for details on
D D
e1,1 e1,2 e2,1 e2,2 e2,3 the weak convergence of distributions, it implies that for
all k
lim pv(k)=dv(k).
n
n→∞
Note that there may be no graphthat realises a given de-
v ,1 v ,1 v ,2 v ,1 v ,2
1 2 2 3 3
gree sequence; however with some conditions on the limit
distributions (that distributions considered here satisfy),
(a)Thelinessymbolisethematchingofthenodes
asymptoticallythereisalwaysagraphrealisingthedegree
e e
1 2 sequence [4]. Another coherence condition is assumed,
that the sums kdv(k) and kde(k) must be equal;
Pk n Pk n
howeverthenumberofedgesisadjusteddependingonthe
number of vertices, so the average of dv may be different
v1 v2 v3 from the average of de.
C. Logic
(b)Nodesarecollapsedbyequivalenceclasses
The language of first-order (FO) logic on graphs is
Fig.2:Aconfigurationandcorrespondingincidencegraph
definedastheclosureunderbooleanconnectivesandquan-
tifiersofequalityandabinarypredicate.Usuallythevari-
ablesareinterpretedasverticesandthebinarypredicateis
graphonnvertices,thismodeltakestwodegreesequences, interpreted as the adjacency relation ¨, however doing so
dv = (dv0,dv1,...) and de = (de0,de1,...), such that dv(k) makesinexpressibleanypropertyaboutedgemultiplicities
(respectively de(k)) is the number of vertices in V (re- or hyper-edges. Instead, having as set of variables the
spectively in E) of degree k, with S = Pdv = Pde. A union of vertices and edges and interpreting the binary
configuration C = fv : CV V,fe :CE E,σ is then predicate as the relation on the incidence graph defines
gpiavretnitiboynstawcocosredtisngohftonoddve,sd7→eC,Van=d aCrEan=do7→{m1,p.e.r.m,iSu}t,attiwono aFlOlo∈w.sFtoorhmaanldlyl,eth∈hyepelorg-micualltsigtrruapcthusrweiitshohVut∪mEodI,if∈yiin.gTthhies
σ of 1,...,S (equivalently,arandommatchingofCV to logical language.
{ }
C ). The incidence graph is then obtained by collapsing
E
nodes in C into a vertex ofdegree k accordingto fv and Proposition2. AnygraphpropertyisFO-definableifand
V
collapsingnodesofC accordingtofe,andsettingv eif only if it is FO∈-definable.
E
∈
andonlyifthereexistcv,cesuchthatf(cv)=v,f(ce)=e, See [6]. Therefore all results on the FO-logic of graphs
and σ(cv)=ce. This process may not define an incidence aretrue onthe FO∈-logic ofgraphs(consequently,FO∈ is
graph since there may be multiple edges between pairs denoted FO).
of vertices in V and E - in this case, it does not define A sentence is a formula without free variables, and the
a proper hyper-multigraph since hyper-edges are sets of quantifier-depth of a formula is defined as the maximal
vertices. However, with some conditions on the sequence nestedness of a quantifier in the formula. A structure H
of degree sequences (that the sequences considered here satisfies a formula ϕ, denoted H = ϕ, if ϕ is true on H.
|
satisfy), it generates an incidence graph with positive For a sentence ϕ and a sequence of random structures
probability [2], and the random structure obtained by ( n)n∈N, let µ(ϕ, n) denote Pr[Hn =ϕ] with Hn n,
H H | ∼H
conditioning on the absence of multiple -edges is the and let µ(ϕ, ) denote the limit of this sequence (called
uniform hyper-multigraph with degree sequ∈ences (dv,de). limiting probHability) if it exists. A sequence of random
Figure2showaconfigurationandthecorrespondinggraph structures ( ) has a convergence law (for FO-logic) if
n
with de(2) = 1, de(3) = 1, dv(1) = 1, dv(2) = 2 and every sentenHce ϕ has a limiting probability µ(ϕ, ) (it is
H
all other degrees and edge sizes set to 0. Let denote a a zero-one law if all limiting probabilities are either 0 or
D
sequence of degree sequences, i.e. = (d1,d2,...) where 1). A sentence is (asymptotically) almost surely true if its
D
each dn is a sequence of natural numbers (therefore dn(k) limiting probability is 1, and an axiom is such a sentence.
is the number of vertices of degree k in the n-th graph). The theory of a set of axioms is the set of all sentences
Let ′v (resp. ′e) be the sequence (pv,pv,...) with implied by the axioms.
D D 1 2
pv(k)= 1 dv(k) (resp. pe(k)= 1 de(k)). Then for each
n |V| n n |E| n Definition 3. The limit theory (or almost sure theory)
pair ( v = (dv,dv,...), e = (de,de,...)) of sequences,
( v,De) den1otes2 the raDndom in1cid2ence graph sequence TH of a random structure H is the set of its almost surely
B D D true sentences. A limit hyper-multigraph is a countable
onthecorrespondingdegreesequences.Bydefinition,inci-
hyper-multigraph that is a model of the theory, i.e. it
dencegraphshavenoedgeofsizesmallerthan2,therefore
satisfies all formulas of the limit theory.
it is assumed that de(0) = de(1) = 0 for all n N.
n n ∈
We consider pairs of sequences of sequences that converge A limit theory is consistent and so always has a model,
weaklytoapairofasymptoticdegreedistributions,denoted and unless the theory is complete (which happens only
if there is a zero-one law) there are infinitely many non- of graphs. The main theorem is then that the random
isomorphic models. These definitions can naturally be sequence defined by the three former conditions corre-
specialised to graphs, and bipartite graphs. spond exactly to the theories satisfying the three latter
ToseetheexpressivepowerofFOonhyper-multigraphs, properties.
some syntactic sugar can be defined. For example, even if The considered sequences of random incidence graphs
there is only one domain, vertices can be distinguished satisfy, :
from edges by a formula, i.e. x is a vertex if and only (i) ′v and ′e convergesweaklytodv andde(in℘(N)).
if v(x) is true, with v(x) y. (y x). The formula (ii) DThereexDistsaN suchthat n>N, k N,dv(k)=
characterising edges is natu≡ral∀ly t¬he ne∈gation of v(x), i.e. 0 implies that dv(k)=0 an∀d de(k)=∀ 0∈implies that
n
e(x) y.(y x). This allows to define edge and vertex de(k)=0.
≡ ∃ ∈ n
quantifiers as Vx x.v(x) and Ex x.e(x). That u (iii) For ( ,d) being ( v,dv) or ( e,de),
and v are adja∃cent≡ca∃n be express∃ed by≡u∃¨ v e.(u D D D
e∧v ∈ e), and the size of an edge can be exp≡re∃ssed b∈y lim Xn dn(k)k2 =Xk2d(k)< ( )
e = 2 v v u.(v e v e v = v ((u = n→∞ n ∞ ∗
| | ≡ ∃ 1 2∀ 1 ∈ ∧ 2 ∈ ∧ 1 6 2 ∧ 6 i=1
v u = v ) = u / e)) where v / e denotes (v e).
S1im∧ilar6ly, n2ote t⇒hat |e∈|=k, as well a∈s any ¨k (v1¬,...∈,vk) Xn dn(k)k4 =o(n). ( )
or (m,k)-adjacency relation, is definable by a first-order n ∗∗
i=1
formula.Furthermore,itis possible to define subclassesof
Condition (i) signifies that the degree sequences converge
hyper-multigraphs in first-order logic:
to probability distributions on N. Condition (ii) signifies
Proposition 3. The subclasses GN, G and H of HN are there is a threshold after which all represented degrees
FO-definable. and edge sizes have to be in the support of the dis-
tributions. Condition (iii) signifies that the distributions
A hyper-multigraph is a multigraph if every edge is of
have a bounded second moment ( ) and that their tails
size2,andagraphiftherearenomultipleedges.Thiscan ∗
be expressed by a first-order formula ΦG = ( Ee.(e = are not too heavy (∗∗), i.e. their fourth moment grows
∀ | | slowerthan n.Note thatthere areno requirementsonthe
2)) ( e e uv.(u e v e u e v e e =
eH2N)),∧Hw¬h=i∃leΦ1ΦG2G∃iNfa=nd(∀oEn∈ley.(1if|∧eH| =∈2G)1)..∧AThhy∈epreefr2o-∧mreu,lf∈toigrr2aal∧plhH1is6∈a pinadirepoefnddiesntrtibouftdiovn.s, so the distribution de is completely
hyper-g|raph(isinH)ifthere∈arenomultiple hyper-edges, Examples of distributions satisfying these properties
are distributions with finite support (the case of Dirac
which can be expressed by the formula ( e e v.(e =
1 2 1
¬ ∃ ∃ ∀ 6 distributions generating hyper-multigraphs with uniform
e (v e v e )). It follows that convergence laws
2 1 2
∧ ∈ ↔ ∈ edge sizes or regular degrees) or power law distributions,
can be deduced on subclasses by interdefinability:
i.e. d(k) c k−α with α greater than 3. The power
Proposition4. Let bearandomhyper-multigraphhav- lawswith≈expo·nentsmallerthan3donothaveasublinear
H
ingaconvergencelaw.Thentherandomgraph definedas fourthmomentsodonotsatisfycondition(iii).Asequence
G
the distribution induced by on graphs has a convergence that does not satisfy (ii) may have 0 vertices of degree k
H
law, with for even n and 1 vertex of degree k for odd n, violating
µ(ϕ, )= µ(ϕ∧ΦG,H). the possibility of a convergence law since having degree
G µ(ΦG, ) k is a first-order property (note that such a sequence can
H
III. The random hyper-multigraph with given still satisfy condition (i) and (iii), and therefore still have
degree sequences a local weak limit).
The considered graph sequences satisfy:
In this section, we define some restrictions on the class
of sequences we consider, which allow us to prove that (I) Every acyclic neighbourhood has either no realisa-
this class satisfies properties that imply a convergence tions or an unbounded number of realisations.
law. After the definitions and statements, the proof is (II) The number ofrealisationsofeveryunicyclic neigh-
given first assuming these properties. Then in the fol- bourhood converges to a finite value.
lowing subsection the theorem is proven for trivial pairs (III) Every subgraph with positive excess has asymptot-
of distributions, to simplify subsequent arguments. The ically no realisation.
following subsections prove that the properties hold with Note that these three properties cover all types of sub-
high probability, using a key lemma showing that the graphs,since a connected graphhas excess 1 if andonly
−
expected number of realisations of small subgraphs can if it is a tree, excess 0 if it is unicyclic, and positive excess
be computed from the configurations. if it has more than one cycle.
To get a convergence law, it is necessary to put some These properties shed more light on conditions (i)-
restrictionson the sequences considered.Three conditions (iii). The conditions (i)-(iii) have two natures, the first
encompassing a large class of degree sequences are given, is to avoid pathological sequences (condition (ii)), and
andentailthreepropertiesofthe correspondingsequences the second one is to ensure that properties (I)-(III) hold
r r r r r r r r r anyq the numberofneighbourhoodsofeachtype iseither
unbounded, or converges to a finite value. It follows that
every first-order sentence has a limiting probability.
Fig. 3: Types of 3-spheres
B. Trivial pairs of distributions
To simplify subsequent arguments, in the other sub-
(condition (iii)). It may be proven that there is a con- sections it is assumed that supp(dv) is different from
vergence law even when relaxing conditions of the second 0 , 1 , and 0,1 , i.e a pair of distribution is trivial
sort,but inthatcasethe limittheory willbe qualitatively i{f s}up{p(}dv) {0,1}. Otherwise it would be necessary to
⊆ { }
different. Suppose that for all n > N, the sequence has addparticularcasesto everystatement,so these pairsare
m vertices of degree k for some N,m N (therefore treated as special cases here. Note that if this is the case,
k / supp(dv)). Then condition (I) is vio∈lated, as there there is a convergence law (even a zero-one law), albeit
are∈exactly m neighbourhoods with a vertex of degree k a fairly uninteresting one; in particular, the number of
(and with high probability these neighbourhoods are all unicyclicneighbourhoodsiszeroforeverysuchneighbour-
trees). For condition (iii), note that when the tails of the hood, while for non-trivial distributions it converges to a
distributionsgetheavier,itmeansthattherearemoreand non-trivialvalue.Ifsupp(dv)= 0 ,thenthesetofalmost
{ }
moreverticesofhighdegrees,andatsomethresholdthese sure formulas are simply formulas stating that there exist
verticesaccountforsuchanimportantproportionofedges vertices, and formulas about non-adjacency. The first set
thattheygetconnectedtoeachothers,realisingsubgraphs gives an axiom scheme, and the second set follows from
of positive excess, violating (III). the axiom ( e v.v e). There is, up to isomorphism,
¬ ∃ ∃ ∈
only one countable graph that models these axioms (the
Lemma 1. Let ( v, e) be an incidence graph sequence countable empty graph). Now if 1 supp(dv), there are
B D D
satisfying (i)-(iii). Then it satisfies (I)-(III). ∈
hyper-edges in the graph, but they are all isolated, i.e.
This allows to state the main theorem: ( v e1e2.v e1 v e2 e1 = e2). There is an axiom
s¬ch∃em∃e for ev∈ery e∧dge∈size k∧in t6he support of de, stating
Theorem 1. Let ( v, e) be an incidence graph se-
thatthereareatleastmsuchhyper-edges.Thereis,upto
B D D
quence satisfying (i)-(iii). It defines a sequence of random
isomorphism,only one countable graphthat models these
hyper-multigraphs that has a convergence law.
axioms, a countable collection of isolated hyper-edges of
In the following subsection, the proof is given assuming size k for each k supp(de), along with a countable
that Lemma 1 holds, as the truth of first-order formu- collection of isolated∈vertices if 0 supp(dv). Note that
las is determined from neighbourhoods alone. Then the these arguments hold for every pos∈sible de. From model-
theorem is proven for the special cases of trivial pairs of theoretical arguments, the uniqueness of the countable
distributions, allowing to simplify subsequent arguments. model of the limit theory implies a zero-one law, and
The next subsections prove that properties (I)-(III) hold, therefore convergence.
thereby proving Lemma 1.
C. Proof of property (I): the local weak limit
A. Proof of the convergence law Theproperty(I)isprovedusingthelocalweaklimit;the
idea is to look at the ‘typical’ neighbourhood of a vertex.
To prove the convergence law, first the notion of q-
With high probability this neighbourhood is a tree, and
sphere is defined inductively. It is a finite subclass of
thetreeobservedbypickingavertexuniformlyatrandom
the class of balls of radius q 1 introduced to capture
− convergestoasimple randomobject,abranchingprocess.
the expressive power of first-order logic. There is only
This branching process gives that any tree that can be
one 1-sphere, where the root has no neighbours. Then
realised is realised infinitely often.
a (q +1)-sphere is a root with at most q neighbours of
A graph G induces a probability distribution µ on
each type of q-sphere. By definition there is only a finite G
rootedgraphsbytakingauniformrootamongallvertices,
number of types of q-spheres. See Figure 3, where black
i.e.
vertices mean that the vertex has no other neighbour. 1
Intuitively, each quantifier allows to go further, and The µG = Xδ(G,v),
V
neighbourhoodof a vertexv can be mapped to a q-sphere | |v∈V
by setting (inductively) to q the number of neighbours where δ is the Dirac distribution over the isomorphism
of v that are of some type of (q 1)-sphere if there are class of the rooted graph (G,v). A sequence of random
−
more than q such neighbours. Hanf locality lemma (see graphs ( ) has weak local limit µ if µ µ, i.e. the
Gn Gn
the standardtextbook[15])statesthattwostructuresG , sequence µ converges weakly to µ (for details on weak
1 Gn
G areequivalentupto q-quantifierssentencesiffor every convergenceinmetricspaces,see[1]).Abranchingprocess
2
type T of (3q)-sphere, both G and G have more than q is a random rooted tree, defined by a root (generation 0),
1 2
neighbourhoods of type T or both have the same number with eachvertex in generationi having a randomnumber
of neighbourhoods of type T. By properties (I)-(III), for of offspring vertices in generationi+1. Let BGW(dv,de)
Lemma 3. Let H be a (connected) subgraph with k
Root V v
• vertices, k edges, m -edges, c its number of vertex-
e v
∈
Generation 1 E automorphisms, ce its numberof edge-automorphisms. Let
• • dv dv, de de be random variables, and recall that
Generation 2 V ∼ ∼
• • • • exc(H) = m (kv + ke). Then the expected number of
... realisations of−H is
E[(dv) ] E[(de) ]
Fig. 4: First generations of a bipartite branching process n−exc(H)Q degH(vi) ·Q degH(ej) .
cvce(Edv)m−ke(Ede)ke
Here a vertex (resp. edge) automorphism is an auto-
be a two-sorted Galton-Watson branching process, where morphism that maps vertices to vertices (resp. edges to
each point has an independent number of offsprings. It is edges). Note that this is the expectations of the number
defined inductively by a root of type V (respectively E), of embeddings, so if H has degrees that are not in the
with x dv offsprings of type E (respectively x de support of the distribution but are smaller than degrees
∼ ∼
offsprings of type V). Then a point in generation i > 0 in the support, this number will be positive. Contrarily,if
has x d∗ offsprings, with d being dv if the point is of H contains some degree l bigger than any degree in the
type V∼and de if it is of type E, with support, then the expectation of the l-falling factorialis 0
(x+1)d(x+1) hence the whole expression goes to 0.
d∗(x)= This formula is proven directly from the configuration.
yd(y)
Py TheexpectednumberofrealisationsofH canbewrittenas
(this accounts for the edge already present between the cv1ce Pτ E[ρ(τ(H),G)], where the sum is over all injective
point and its parent). An example is given in Figure 4. maps from VH to VG and EH to EG, and ρ is the number
ofhomomorphismsforagivenmap,i.e.ρispositiveifτ is
Lemma 2. If ( v, e) satisfies (i)-(iii), then µ
BGW(dv,de). B D D B nauhmombeormoofrp-heidsgmesa,ntdhe0reotahreer(wni)se.R(ecSal)lingsuthchatmSapiss,thaes
∈ kv· Ede ke
The proof is based on the exploration of the connected therearen= ESdv vertexequivalenceclasses,and ESde edge
component containing the root, and can be adapted from equivalence classes. The division by the number of auto-
the standard proof of convergence for the configuration morphisms accounts for the repetitions of maps. The first
model (see [19] for details). It allows to deduce property -edge of the homomorphism between vertex v1 and edge
∈
(I) from the following consequence: e1ismatchedwithprobabilitydeg(τ(v1)) deg(τ(e1))/S,as
·
therearedeg(τ(v )),deg(τ(e ))nodesintheconfiguration
Corollary 1. Let ( v, e) satisfy (i)-(iii) and T be a 1 1
thatcanbechosentocreatethe -edgebetweenv ande ,
B D D 1 1
bipartite tree, the following are equivalent: ∈
and it is chosen among S nodes. For the following edges,
• T is asymptotically realised infinitely often. the number of nodes corresponding to vertices that have
• T is realisable. already been matched is correspondingly diminished, and
• v VT (resp. e ET), deg(v) supp(dv) (resp. similarlythe number ofnodes becomes(S i)ifi -edges
∀deg∈(e) supp(de∀)).∈ ∈ have already been matched. Define the pro−duct ∈
∈
Thiscorollaryfollowsdirectlyfromthepreviouslemma, kv
sincetheprobabilitythatB(Dv,De)isisomorphictoatree PV =Y(degG(τ(vi)))degH(vi),
T isgivenbyBGW(T).Ifthisprobabilityispositive,then i=1
there is a linear number of vertices in with neighbour- and similarly P where the sum is over all edges. Now, a
B E
hood isomorphic to T. Since BGW(T) > 0 if only if all uniformly sampled set of k vertices has the same distri-
of the degrees appearing in T are in the support of the bution as dNk, the k-th product of d (to see this, the k
distributions (dv,de), property (I) follows. vertices are distinct with probability (n) /nk which goes
k
to 1 as n goes to infinity). Therefore
D. Subgraph counting lemma
k k
Thissubsectiongivesaformulafortheexpectednumber
ofsmallsubgraphsandprovesit.Itprovestheconvergence EY(di)ki =YE[(d)ki]
i=1 i=1
of the expected number of realisations of subgraphs with
ninoRnth-eneceangllaextthitvaestue(bxnsc)eekcstsdi,oepnnr.ootveisntghtehke-rfaelmlianignifnacgttowrioalporfopne.rTtihees fasoilmlroiawlnainyrlgykPt∈oENrae,swaPnrEi′dt)e.thNPeVorwefaowsreePh(VS′av)ke=l≈imQnnkik→=v(1E∞(dd(vvS))−kd.ekgPH)/u(nvtit)=in(gEanaddlvl
expected number of realisations of a bipartite graph with this together, we have
non-negative excess can be computed from the following 1 nkv+ke(Edv)ke
formula: cvce(S)m X(PV ·PE)= cvce(S)m(Ede)ke(PV′ ·PE′ ).
τ
sequences through the lens of first-order contiguity, a tool
to compare random sequences from the point of view of
logic.Thenclassicalmodelsofrandomgraphsareanalysed
as special cases of this framework.The completions of the
Fig. 5: Bipartite graphs of excess 1 limit theories are studied next, and in the last subsection
the models of the limit theories are investigated.
By rewriting (S) by nm(Edv)m, we obtain a factor A. First-order axioms
m
nkv+ke−m,i.e.n−exc(H).Simplifyingtheexpressionbythis An axiomatisation of the limit theory can be deduced
rewriting yields the expected formula. from properties (I)-(III) and distributions (dv,de). Let
E. Proof of properties (II) and (III): subgraphs of non- Tdv,de be the limit theory of B(Dv,De). First, by prop-
erty (II) this axiomatisation contains no sentence about
negative excess
unicyclic neighbourhoods, since each may be realised or
Now we prove properties (II) and (III). To prove prop- not. Then property (III) gives that every sentence stating
erty (III), note that any graph with positive excess con- the absence of neighbourhoods of positive excess is in the
tainsasubgraphofexcess1whichcanbeofthreedifferent limit theory. For all (r,s) N2, let Z be the formula
r,s
forms,eitheritisconstitutedoftwocyclessharingavertex stating ‘for every set of r v∈ertices and s edges, there are
and its maximal degree is 4, or it is constituted of two notr+s+1 -edges’,andZ betheaxiomschemeofallsuch
cycles linked by a path, or a cycle with a chord, and in sentences.T∈hen,forallk / supp(dv)(resp.k / supp(de)),
both the maximal degree is 3 (see Figure 5). To deduce letYv (resp.Ye)bethefo∈rmulastating‘there∈isnovertex
k k
property(III),itissufficienttoseethatgraphswithexcess of degree exactly k’, and Y denote the axiom scheme of
1 are not realisable: allYv andYe sentences.Note thatthis axiomschemecan
Corollary 2. Let ( v, e) be a random incidence graph be replaced by a single sentence if both supports of the
B D D distributions are finite or cofinite. Then, for all bipartite
sequencesatisfying(i)-(iii) andH beabipartitegraphwith
trees T, and all m N, let XT be the formula stating
exc(H)=1. Then H is not realisable. ∈ m
‘thereareatleastmexactrealisationsofT’,andT denote
First note that by condition (iii), the average degree the axiom scheme of all such sentences. Here, an exact
converges to a finite value, so it suffices to show that for realisation means that for all internal nodes in T, the
a graphwith positive excess, the value given by Lemma 3 formula states that the vertex has exactly the degree it
goes to zero. Since this value is multiplied by n−exc(H) = has in T. For leaves, the formula states that the vertex
n−1, it is sufficient to show that the numerator is in o(n), hasdegree1ifandonly ifthe node hasnooffspringinthe
which is exactly the second part of condition (iii) since branching process (since edges must be of size at least 2,
the maximal degree of a subgraph of excess 1 is either leavescanonly be vertices).Therefore the formula asserts
3 or 4. Then any graph with positive excess contains an the existence of T as induced components if and only if
embedded subgraph of excess 1 which is not realisable, so the branching process can die out isomorphic to T. The
it is not realisable itself and property (III) follows. three axiom schemes X, Y, Z (with X and Z depending
When applied to unicyclic graphs (graphs with excess on the distributions) axiomatise the almostsure theory of
0), the formula given by Lemma 3 gives property (II): ( v, e), i.e. TX,Y,Z =Tdv,de.
B D D
Corollary 3. Let ( v, e) be a random incidence graph
B D D B. First-order contiguity
sequencesatisfying(i)-(iii) andH beabipartitegraphwith
exc(H)=0, then the expected number of realisations of H To compare two sequences of random structures from a
converges. logical point of view, the notion of first-order contiguity
is used, as an analogue to the probabilistic notion of
By the first part of condition (iii), the numerator is
contiguity of [13].
finite, therefore every term in the expression is finite.
There is in fact convergence in distribution to Poisson Definition 4. Two sequences of random structures are
random variables of the number of cycles for each size of first-order contiguous if and only if every sentence has
cycles (there is even joint convergence). Details can be a limiting probability either on the two sequences or on
foundin[4],[14].This concludestheproofofconvergence. neither of them, and both sequences share the same almost
sure theory.
IV. Limit theories and first-order contiguity
In other words, two sequences of random hyper-
In this section, we investigate the limit theories defined
multigraphs are first-order contiguous if and only if for
by the incidence graphs sequences ( v, e). First, an
B D D every first-order sentence ϕ, ϕ is either not almost sure
axiomatisation of the limit theories is given, and the
(in general it may not converge), or it is almost sure on
models of the limit theories are investigated. Then, as
the two sequences.
some sequences have the same limit theories, we look at
Theorem 2. Two sequences of hyper-multigraphs de- on the absence of double edges. Since the number of mul-
fined respectively by ( v, e) and ( v, e) both sat- tiple edgesconvergesin distributionto a Poissonvariable,
B D1 D1 B D2 D2
isfying (i)-(iii) are first-order contiguous if and only if the probability that the graph is simple is the probability
(supp(dv),supp(de))=(supp(dv),supp(de)). of 0 in Poi(λ), the Poisson distribution of parameter λ.
1 1 2 2
In the Erdős-Rényi graph (n,c/n), each edge is inde-
Which means that from the point of view of first-order G
pendently present with probability c/n. The degree of a
logic,thestatisticalinformationgivenbythedistributions
vertex is therefore given by a binomial of parameter n 1
does not matter, only the supports matter. Since there is and c/n, i.e. Pr[deg(v)=k]= n (c)k(1 c)n−k. So −the
a convergence law, the first property holds obviously, and (cid:0)k(cid:1) n − n
average degree is c, and the degree distribution converges
thesecondpropertyisimmediatesincetheaxiomatisation
to the Poisson distribution of parameter c. Therefore in
only depends on (supp(dv),supp(de)). Note that the lim-
the limit there are infinitely many vertices of each degree.
iting probability of sentences that are not in the almost
Then Theorem 2 implies directly:
sure theory may differ. In fact, this is the case exactly if
the limit distributions differ: Corollary 5. Let ( , ) be a sequence of incidence
{2}
graphs satisfying (i)B-(iDii)Dwith supp(d)=N. The sequence
Corollary4. Thereisaformulaϕwithµ(ϕ, ( v, e))=
µ(ϕ, ( v, e)) if and only if (dv,de)=(dvB,dDe)1. D1 6 of graphs it defines, conditioned on being simple, is con-
B D2 D2 1 1 6 2 2 tiguous to (n,c/n).
G
Since all sentence probabilities are only dependent on
There are no multiple edges in (n,c/n) by definition,
thelimitdistributions,the‘onlyif’partistrivial.Toshow G
so it is necessary to condition on simplicity.
the other direction, suppose the two pairs of distributions
have the same pair of supports (otherwise the proof is
trivial),andsupposedv isdifferentfromdv onvaluesiand D. Completions of thelimit theories andlimiting probabil-
1 2
j (the case de being symmetric). A formula that asserts ities
the existence of a cycle has a non-trivial probability; The set of possible limit theories share a strong resem-
then specifying the branching of vertices in the cycle to blance to the theory of the Erdős-Rényi graph (n,c/n),
G
have exactly i neighbours gives a formula satisfying the andtheycanbecompletedinasimilarfashion.Acomple-
requirement. tion τ is a set of sentences such that T +τ is a complete
Asafinalremark,thesetoflimittheoriesdefinedbythe theory, i.e. every formula (or its negation) belongs to the
classofdegreedistributionsconsideredherecanbeordered theory. A convergence law is equivalent to the fact that
bythesupportsofthecorrespondingdistributions.Indeed, there is a set C of completions of the limit theory, such
the set of realised subgraphs of a sequence B1(D1v,D1e)) is that any two elements of C are mutually inconsistent. A
included in the set of subgraphs realised by a sequence q-completion is a sentence that completes the theory only
(tBsh2ue(pDpp2v(a,diDrv)2eo,)fs)usipufppt(phdoeer)tp)s.aioIrnfoBfth2s,iusip.sepe.onr(stsesu,potpfh(eBdr1v1e),iisssuiapnpcml(udade1xe)id)mia⊆nl fiunpitteo, qhoqwueavnetrifiseorms.eIfinntitheefoprrmesuenlatscaarseeqn-coomcopmleptlieotnios:n is
2 2 Corollary 6. Any sentence fixing the number of neigh-
theory for this class correspondingto the pair of supports
(N,N). bourhoods in each realisable unicyclic type of (3q 1)/2-
−
sphere is a q-completion of the theory of ( v, e).
B D D
C. Thecaseofthe(multi)graphwithgivendegreesequence
This observation was made in [17] for Erdős-Rényi
and the Erdős-Rényi graphs
graphs and is given here without a proof, as it can be
Much work has been done on the configuration model, adapted from the original paper without much modifica-
which generates the uniform simple graph on a given tion. The argument follows from properties (I)-(III), the
degree sequence . This is a specific case that can be only difference is that the realisable types depend on the
D
generated by the bipartite graph ( , ) where supports of the distributions.
{2} {2}
B D D D
contains only edges, converging to δ the Dirac distri- Other results of the paper can similarly be extended
{2}
bution with mass on 2. Note that in that case, the local to the bipartite configuration model. In particular, since
weak convergence is a simple Galton-Watson, and that in formulas with non-trivial limiting values are formulas
the Lemma 3 the terms depending on the edges cancel about unicyclic neighbourhoods, it can be shown that
out. Precisely, Ede = E[(de) ] = 2, so for any realisable the limiting values of formulas can be written using ba-
2
subgraph, sic arithmetic operations with the values in Poi(λ) and
ke E[(de) ] BGW(dv,de),foranyλgivenbytheformulaofLemma3.
Qj=1 degH(ej) =1. Conversely,anyrealnumberthatcanbeexpressedinsuch
(Ede)ke
a formis the limit ofa first-orderformula.The automata-
In particular the expected number of multiple edges is theoretic arguments used to show this also imply that
therefore λ = (E[(dv) ])2/(4(Edv)2), and the uniform the asymptotic probability of a formula (or a formula
2
graph with degree sequence d is obtained by conditioning with a given asymptotic probability) can be computed in
polynomial space and that this bound is optimal. Details
can be found in [17].
Fig. 6: A triple edge or a double 3-hyper-edge
E. Models of limit theories
In this subsection, the countable models of the limit
theory are investigated. The idea is the following. To prove the first statement,
By the given axiomatisation, models of the limit theo- suppose there is a finite component in the limit, when 0
and 1 / supp(dv). Let this finite component be called
ries are constituted of tree components and independent
∈
unicyclic components. However, if there are no vertices H and have radius l, then for any q greater than l the
of degree less than 2, then the BGW branching process type of the q-sphere corresponding to H is not generated
neverdies,andtherearenosmallcomponents.Sincethere byBGW(dv,de),contradictingpreviousobservations.For
is a giant component and that with high probability this the second statement, it is sufficient to show that the
componentisunique[20],thegraphisconnectedwithhigh same types of q-spheres are realised independently of
probability.However,thecountablegraphsthatmodelthe the presence of infinite components. Let L1, L2 be two
limit theoryarenot connected.As observedin [22],this is countablemodelsofthelimittheoryof (dv,de).Suppose
B
that H is an infinite tree component in L and that L
a most roundabout way to show that connectivity is not 2 1
a first-order property. does not contain infinite components. For any q, let T be
a type ofq-sphererealisedin H.By hypothesis,L satisfy
Consider for example the limit theory of (δ ,δ ), the 2
2 2
B property (I) and therefore T is realised infinitely often in
2-regular multigraph. The multigraphs it generates are a
L , in components of radius greater than q.
collection of cycles. However, except for a finite number 1
of vertices, all vertices belong to cycles of size larger than V. Other properties of the model
k, for any k. As the length of the cycles grows, formulas
In this section, two properties of interest are studied,
of first-order logic cannot distinguish long cycles from
first the probability that a hyper-multigraph is a hyper-
an infinite alternating path of vertices and edges. Since
graph, and second that, unlike classical models of ran-
theyrealisethesameq-spheres,twocountablegraphsthat
dom graphs (including the graphs with specified degree
are collections of infinite paths satisfy the same set of sequences), the ¨-graph can be a sparse graph with non-
first-order formulas, regardless of the number of infinite
trivial clustering coefficient.
paths in each. Therefore the limit models of (δ ,δ ) are
2 2
B
constitutedofacollectionofinfinitepathsandacollection A. Probability to be a hyper-graph
of small cycles. Let k > 3. Then the incidence graph of a double k-
Nowconsiderthe limittheoryof (δ3,δ2),the 3-regular hyper-edge is a subgraph with k vertices, 2 hyper-edges,
B
multigraph. The multigraphs it generates are connected and 2k -edges. Therefore the excess of this subgraph is
∈
withhighprobability.Howeverasngrowsthesmallcycles 2k (2+k) > 0 (see Figure 6). Similarly, the incidence
−
grow further apart from each others (since two cycles graphofak-multiple edgeisasubgraphwith2vertices,k
at fixed distance realises a subgraph of positive excess). edges,and 2k -edges, with excess 2k (2+k)>0. Since
∈ −
It follows that in the limit the cycles are in isolated subgraphswith positiveexcessarenotrealisable,a hyper-
components,andeachvertexinthecyclesistherootoftwo multigraph is a hyper-graph if and only if there are no
infinite trees with 3-regular vertices and 2-regular edges. double edges. Therefore the property to have no multiple
Therefore the limit models of (δ3,δ2) are constituted of hyper-edge can be computed, as it is just the probability
B
a collection of infinite trees with 3-regular vertices and 2- to have no 2-cycle.
regular edges and a collection of unicyclic components. Using Lemma 3 to get
supLpa(sdtvc)o.nIstidgerentehreatelismhitypthere-omryultoifgrBa(pdhvs,dwei)thwiitnhdu1ce∈d λ= (E[(dv)2])2·(E[(de)2])2,
4(Edv)2(Ede)2
components of finite size. By the local weak limit, every
realised induced component is realised linearly often, and the probability of this property is then the probability of
therefore appears infinitely often in the limit. Then in the 0 in Poi(λ).
limit there may be any number of infinite components
B. Clustering coefficient
(including none) since, from a first-order point of view,
In this subsection, we consider the ¨-graph induced by
infinite components can be simulated by arbitrarily large
theincidencegraph,andshowhowitsclusteringcoefficient
finite components. Therefore we have:
is affected by the choice of the distributions. Recall that
Proposition5. Let (dv,de)beasequencesatisfying(i)- x ¨ y if they belong to a common hyper-edge of any
(iii). If 0,1 / supp(dBv) then all components of the limit size, so the ¨-graph is obtained by replacing every k-
∈
incidence graph are infinite. Otherwise there may be any hyper-edge by a complete graph (a clique) on k vertices.
number of infinite components in the limit. The clustering coefficient is an important property in the
studyofsocialnetworks,asitmeasureshowmuch‘friends of multigraphs. Some models in the literature may have
of my friends are friends’. There are different definitions natural definitions in terms of incidence graphs, while
depending on the context, the one considered here is the some may be contrived.
local clustering coefficient, C(v) = M′(v)/M(v), where
References
M(v) is the number of pairs of neighbour vertices of v
(verticesatdistance2intheincidencegraph)andM′(v)is [1] PatrickBillingsley.Convergenceofprobabilitymeasures,volume
493. JohnWiley&Sons,2009.
the number ofconnected pairsofneighbourvertices(pairs
[2] Jose Blanchet and Alexandre Stauffer. Characterizing optimal
v1, v2 with v1 ¨ v2). If (dv,de) is a graph (de = δ2), sampling of binary contingency tables via the configuration
then as observed earlier tBhe number of ¨-triangles in the model. RandomStructures&Algorithms,42(2):159–184, 2013.
[3] BélaBollobás,SvanteJanson,andOliverRiordan. Sparseran-
graphis givenbyLemma3 andsoisinO(1). Thereforein
domgraphswithclustering. RandomStructures&Algorithms,
that case the average clustering coefficient goes to 0, and 38(3):269–323, 2011.
is 0 for almost all vertices. When there are k-hyper-edges [4] Charles Bordenave. Notes on random graphs
and combinatorial optimization. Lecture notes.
with k>2, the clustering coefficient becomes
http://www.math.univ-toulouse.fr/~bordenave/coursRG.pdf.
|e|−1 [5] Tom Britton, Maria Deijfen, and Fredrik Liljeros. A weighted
Cv = Pv∈eM(cid:0) 2 (cid:1) with M =X|e|−1. cstoantfiisgtuicraaltipohnysmicosd,e1l4a5n(5d):i1n3h6o8m–o1g3e8n4e,o2u0s11e.pidemics. Journal of
(cid:0)2(cid:1) v∈e [6] Bruno Courcelle and Joost Engelfriet. Graph structure and
The average clustering coefficient is therefore given by monadicsecond-orderlogic:alanguage-theoreticapproach,vol-
C = EdvE de−1 /E M with dv dv and de de. This ume138. CambridgeUniversityPress,2012.
(cid:0) 2 (cid:1) (cid:0)2(cid:1) ∼ ∼ [7] Élie de Panafieu. Phase transition of random non-uniform
formula shows that by choosing the correct distributions, hypergraphs. In Combinatorial Algorithms, pages 127–139.
it is possible to generate sparse random graphs with a Springer,2013.
[8] RichardDurrett. Random graph dynamics, volume200. Cam-
specified clustering coefficient.
bridgeuniversitypressCambridge,2007.
[9] Ronald Fagin. Probabilities on finite models. The Journal of
VI. Conclusion and open questions
SymbolicLogic,41(01):50–58, 1976.
Thelogicalframeworkusedhereallowstoextendseveral [10] Gourab Ghoshal, Vinko Zlatić, Guido Caldarelli, and MEJ
Newman.Randomhypergraphsandtheirapplications.Physical
classical results on graphs to the very general structure
ReviewE,79(6):066118, 2009.
of random hyper-multigraphs, solving the problem stated [11] YVGlebskii,DIKogan, MILiogonki,andVATalanov. Range
back in [17]: “Some more interesting problems would be anddegreeofrealizabilityofformulasintherestrictedpredicate
calculus. Kibernetika,5(2):142–154, 1969.
to prove limit laws for structures with relations of degree
[12] Jean-LoupGuillaumeandMatthieuLatapy.Bipartitegraphsas
greater than 2. Our techniques rely heavily on graph- models of complex networks. Physica A: StatisticalMechanics
theoretic concepts, and it is not obvious how to extend anditsApplications, 371(2):795–813, 2006.
[13] Svante Janson. Random regular graphs: asymptotic distribu-
them to relations of higher degree” (edited). The solution
tions and contiguity. Combinatorics, Probability and Comput-
is simple, and relies on no more than graph-theoretic ing,4(04):369–405, 1995.
concepts. [14] Svante Janson. The probability that a random multigraph is
simple,ii. Preprint,2013. arXiv:1307.6344.
The specific class considered here encompasses a wide
[15] LeonidLibkin. Elementsoffinitemodeltheory. springer,2004.
range of possible hyper-multigraphs, but a wide range of [16] DajieLiu,NorbertBlenn,andPietVanMieghem.Modelingso-
distributions remains to be studied. As remarked in [2], cialnetworks withoverlappingcommunitiesusinghypergraphs
andtheirlinegraphs. arXivpreprintarXiv:1012.2774, 2010.
the bipartite configuration model allows to use a larger
[17] James F Lynch. Probabilities of sentences about very sparse
classofdistributionsthanthe simpleconfigurationmodel, random graphs. Random Structures& Algorithms, 3(1):33–53,
since the maximal degree can be as big as n o(n), 1992.
− [18] James F Lynch. Convergence law for random graphs with
whereas for the simple configuration model the maximal
specifieddegreesequence.ACMTransactionsonComputational
degree must be in O(√n) (otherwise the probability that Logic(TOCL),6(4):727–748, 2005.
a random configuration is a graph goes to zero). However [19] MarcMezardandAndreaMontanari.Information,physics,and
computation. OxfordUniversityPress,2009.
forsuchdistributionsthemethodsusedhere(inparticular
[20] Michael Molloyand BruceReed. The sizeofthe giant compo-
the local weak convergence) cannot be used. When ( ) nent of arandomgraph withagivendegree sequence. Combi-
∗∗
does not hold, as the distributions get heavier tails, sub- natorics,probabilityandcomputing,7(03):295–305, 1998.
[21] NicolauCorçaoSaldanha. Someexamplesofasymptoticcombi-
graphswithpositiveexcessappear,andwhen( )doesnot
∗ natorial behavior, zero-oneandconvergence results onrandom
hold the expected number of cycles becomes unbounded. hypergraphs. arXivpreprintarXiv:1411.5290, 2013.
When considering such distributions, there will be non- [22] Joel Spencer. The strange logic of random graphs, volume 22.
Springer,2001.
contiguous sequences defined by distributions with the
same supports,andcontiguity willalsodepend on the tail
of the distributions.
In another direction, only sparse graphs are considered
here, but other regimes of interest could be studied, such
as sublinear or dense regimes of bipartite graphs. Finally,
other models of hyper-graphs can be considered in this
setting, such as the model considered in [3], or models