A class of models for random hypergraphs Marc Barthelemy Universit e Paris-Saclay CEA CNRS Institut de Physique Th eorique 91191 Gif-sur-Yvette France and

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A class of models for random hypergraphs

Marc Barthelemy∗

Universit´e Paris-Saclay, CEA, CNRS, Institut de Physique Th´eorique, 91191, Gif-sur-Yvette, France and

Centre d’Analyse et de Math´ematique Sociales (CNRS/EHESS) 54 Avenue de Raspail, 75006 Paris, France

Despite the recently exhibited importance of higher-order interactions for various processes, few

ﬂexible (null) models are available. In particular, most studies on hypergraphs focus on a small

set of theoretical models. Here, we introduce a class of models for random hypergraphs which

displays a similar level of ﬂexibility of complex network models and where the main ingredient

is the probability that a node belongs to a hyperedge. When this probability is a constant, we

obtain a random hypergraph in the same spirit as the Erdos-Renyi graph. This framework also

allows us to introduce diﬀerent ingredients such as the preferential attachment for hypergraphs,

or spatial random hypergraphs. In particular, we show that for the Erdos-Renyi case there is

a transition threshold scaling as 1/√EN where Nis the number of nodes and Ethe number of

hyperedges. We also discuss a random geometric hypergraph which displays a percolation transition

for a threshold distance scaling as r∗

c∼1/√E. For these various models, we provide results for the

most interesting measures, and also introduce new ones in the spatial case for characterizing the

geometrical properties of hyperedges. These diﬀerent models might serve as benchmarks useful for

analyzing empirical data.

INTRODUCTION

Complex networks became increasingly important for

describing a large number of processes and were the sub-

ject of many studies for more than 20 years now [1–3].

Recent analysis of complex systems [4–7] however showed

that networks provide a limited view. Indeed networks

(or graphs) describe a set of pairwise interactions and

exclude any higher-order interactions involving groups of

more than two units. With the increasing amount of data

many higher-order interactions were observed in a large

variety of context, from systems biology [8], face-to-face

systems [9], collaboration teams and networks [10, 11],

ecosystems [12], the human brain [13, 14], document

clusters in information networks, or multicast groups in

communication networks, etc. [15, 16]. Modeling these

higher-order interactions with graphs might lead to er-

roneous interpretations, calling for the need of a more

ﬂexible framework. In addition, these higher-order inter-

actions are highly relevant for all possible processes that

take place in these systems [6]. These processes include

contagion where a disease can spread in a non-dyadic way

[17–19], diﬀusion [20, 21], cooperative processes [22], or

synchronization [23–25]. Models for hypergraphs - null,

or spatial for example, are then much needed for analyz-

ing processes that involve the interaction between more

than two nodes.

In order to go beyond usual graphs, a natural extension

consists in allowing edges that can connect an arbitrary

number of nodes. These ‘hyperedges’ constructed over a

set of vertices deﬁne what is called a hypergraph [26, 27].

More formally, a hypergraph H= (V, E) where Vis a

set of elements (the vertices or nodes) and Eis the set

of hyperedges where each hyperedge is deﬁned as a non-

empty subset of V(a simple example is shown in Fig. 1).

For a directed hypergraph, the hyperedges are not sets,

but an ordered pair of subsets of V, constituting the tail

and head of the hyperedge [28–30].

Hyperedge 0

Hyperedge 1

Hyperedge 2

FIG. 1. Classical representation of a hypergraph. We have

here |V|=N= 10 vertices and E= 3 hyperedges. The

hyperedges are e0={0,1},e1={2,3,4,5,8}, and e2=

{0,6,7,9,10}, with sizes 2, 5, and 5, respectively. All the

nodes have degree k= 1, except for nodes 0,1,2,8 which

have a degree k= 2.

The number N=|V|of vertices is called the order of

the hypergraph, and the number of hyperedges M=|E|

is usually called the size of the hypergraph. The size

of an hyperedge |ei|is the number of its vertices. The

degree of a vertex is then simply given by the number

of hyperedges to which it is connected. A simpler hy-

pergraph considered in many studies is obtained when

all hyperedges have the same cardinality dand is then

called a d-uniform hypergraph (the rank of a hypergraph

is r= maxE|e|and the anti-rank r= minE|e|, and when

both quantities are equal the hypergraph is uniform). A

2-uniform hypergraph is then a standard graph.

Many measures are available for hypergraphs. Walks,

paths and centrality measures can be deﬁned and other

arXiv:2210.12698v3 [cond-mat.stat-mech] 17 Dec 2022

measures such as the clustering coeﬃcient can be ex-

tended to hypergraphs [5, 16, 31–33]. A recent study

investigated the occurence of higher-order motifs [34] and

community detection was also considered [35, 36]. New

measures can however be deﬁned and we will discuss in

particular the statistics of the intersection between two

hyperedges as being the number of nodes they have in

common [36]. For spatial hypergraphs, the geometrical

structure of hyperedges is naturally of interest and we

will discuss some quantities that characterize it.

It was already been noted in [16] that few ﬂexible null

models were proposed in the context of interactions oc-

curing within groups of vertices of arbitrary size. In

[16, 37], a null model for hypergraphs at ﬁxed node de-

gree and edge dimensions is proposed. This generalizes

to hypergraphs the usual conﬁguration model [38]. Other

models for higher-order interactions were described in the

review [5] such as bipartite models, exponential graph

models, or motifs models, but in general hypergraph

models are less developed. Most of these models are in-

troduced in the mathematical literature and are usually

thought as immediate generalizations of classical graph

models. In many of these models, it is usually assumed

that all hyperedges have the same size, which is a strong

constraint. We note that an interesting hypernetwork

growth model was proposed in [39] where both the idea

of hyperedge growth and hyperedge preferential attach-

ment were introduced.

Here, the goal is to present a mechanism that can gen-

erate (in a simple way and with low complexity) a fam-

ily of diﬀerent hypergraphs and which displays the same

level of ﬂexibility of complex network models. We pro-

pose here such a framework and introduce a class of mod-

els that relies on a single ingredient: the probability that

a node belongs to a hyperedge. We will consider vari-

ous illustrations of this class of models. We will start

with the simplest one that could be seen as some sort of

‘Erdos-Renyi hypergraph’, and also discuss preferential

attachment. We then introduce space in diﬀerent ways

and in particular discuss in more detail a random geo-

metric hypergraph where the probability that a vertex

belongs to a hyperedge is 1 if its distance is less than

a threshold, 0 otherwise. For all these illustrations, we

analyse simple measures (such as the degree of vertices or

sizes of the hyperedges), the structural transition for the

giant component, and also introduce new measures. In

particular, for spatial hypergraphs, we characterize the

spatial properties of hyperedges.

A CLASS OF HYPERGRAPH MODELS

Hypergraphs can be represented as bipartite graphs

between the nodes and hyperedges (see Fig. 2 for an ex-

ample). In this representation the links between nodes

and hyperedges indicate a membership relation: there is

a link between node iand hyperedge eif i∈e. The main

Vertices

Hyperedges

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FIG. 2. Example of a small hypergraph with N= 4 vertices

and E= 5 hyperedges represented as a bipartite graph. The

degree of the vertices are k1= 2, k2= 1, k3= 3, k4= 2 and

the sizes of the hyperedges are |e1|= 2, |e2|= 1, |e3|= 3,

|e4|=|e5|= 1. There are two connected components in this

hypergraph {1,2,3}and {4}. The intersection between some

of the hyperedges is: e1∩e2={3},e1∩e3={1,3}, etc. The

sum Pikiis equal to the sum of hyperedge sizes Pj|ej|and

equal to the number of links.

idea of the class of models discussed here is to introduce

the connection probability P(v∈e) that a node vbe-

longs to a hyperedge e. This is directly related to the

incidence matrix Iof the hypergraph which is a N×E

matrix with elements Iie equal to 1 if i∈eand 0 other-

wise. This connection probability can be written under

the form

P(v∈e) = F(e, d(v, e), ...) (1)

where Fis in general a function of the hyperedge e, its

vertices, or some distance between vand e(we will see

various examples below). We note here that in contrast

with a remark in [35] that it is ‘unclear how to impose

properties on a hypergraph when a bipartite represen-

tation is used’, we actually believe that this representa-

tion provides a simple framework for introducing various

mechanisms. We could even think of a generalization of

Eq. 1 in the spirit of the hidden-variable model for net-

works. In these models, some characteristics are assigned

to nodes such as ﬁtnesses [40, 41], or coordinates in a la-

tent space [42–44]. The connection function Fin Eq. 1

could in principle depend on these hidden variables. For

example, if each vertex vhas a ﬁtness ηv, a hyperedge

composed of vertices vi1, vi2, . . . , vimhas then a ﬁtness

which will be a function of the ﬁtnesses of all its vertices

η(e) = G(ηi1, ηi2, . . . , ηim), and the connection function

could then be chosen as

P(v∈e) = F(η(e)) (2)

In this article, we will essentially consider the case where

the function depends on some properties of the hypere-

dege e, including its position in space.

This deﬁnition (Eq. 1) is obviously very general and

we will focus here on diﬀerent functions. First, we will

consider the constant case F=p∈[0,1] which reminds

us of the Erdos-Renyi model [45]. We will then consider

the preferential attachment case where the function F

depends on the size of the hyperedge. We then end this

paper by considering cases where the nodes are in a 2d

space (we will mainly consider the d= 2 case but the

generalization to larger dimensions is trivial) and where

the function Fdepends on a distance (to be deﬁned)

between the vertex vand the edge e. For all these models,

we will present the result for usual measures but also

develop new measures tailored to spatial hypergraphs.

Throughout this work, we denote by Nthe order of

the hypergraph (which is the number of nodes) and E

the number of hyperedges. For all the models considered

here, we will assume that the number Eof hyperedges is

given. Once an initial set of hyperedges is given (in cases

studied here, the initial hyperedges comprise single nodes

chosen at random), we can iterate over all nodes and ap-

ply Eq. 1. Once all nodes are tested, we get the ﬁnal hy-

pergraph that we measure. The simplest measure is the

degree kiof a node i(which is the number of hyperedges

to which it belongs). The degree can also be computed

as the sum of row elements of the incidence matrix Iie:

ki=PeIie (see for example the review [5]). The size

ml=|el|of the hyperedge lis the number of nodes it

contains. Naturally, the average degree and size (and the

distribution of kand m) are important quantities, and it

should be noted that they are not independent. Indeed,

if we use the links in the bipartite representation of the

hypergraph (i.e. there is a link between node iand hy-

peredge eif i∈e, see Fig. 2), their number Lcan be

expressed in two diﬀerent ways as L=Piki=Pj|ej|.

This relation can be rewritten as

Nhki=Ehmi(3)

where hkidenotes the average degree and hmithe aver-

age size of hyperedges. The distributions P(k) and P(m)

are then natural objects to study. We will also focus

on other quantities: the intersection between two hyper-

edges (a quantity that is trivial for graphs and equal to

one), and for spatial hypergraphs the spatial extension

s(e) of a hyperedge e. We will also discuss connectivity

properties of these hypergraphs and in particular we will

focus on abrupt structural changes characterized by the

emergence of a giant component (that will need to be

deﬁned) scaling with N.

THE SIMPLEST RANDOM HYPERGRAPH

Deﬁnition

There is no unique deﬁnition of random hypergraphs

and various generalization of the classical Erdos-Renyi

graph were proposed (see for example [11]). In particular,

mathematicians like to think of a set of all possible hyper-

graphs - given some parameters (number of nodes, etc) -

and to consider a uniform distribution over this set [46].

More speciﬁcally, many papers focus on k-uniform hy-

pergraph for which the size of hyperedges is constant and

equal to k. Given a set of Vvertices and a set of subsets

of these vertices we can construct the natural analogue

of Erdos-Renyi graphs [45]: each k-tuple of vertices is a

hyperedge with probability p[47] (more formally, the set

Hk(n, p) denotes the random k-uniform hypergraph with

vertex set [n] = {1,2, ..., n}in which each of the n

kpos-

sible edges is present independently with probability p).

In other studies, every hypergraph of Ehyperedges on N

nodes has the same probability [48] (the classical Erdos-

Renyi random graph is then recovered in the case k= 2).

This type of model was discusssed by mathematicians in

particular about the phase transition for the giant com-

ponent [4, 46, 47, 49–51]. For k= 2, we recover usual

graphs and from Erdos and Renyi [45], we know that

there is a transition for E=cN with c= 1/2 (which

corresponds to an average degree hki= 2E/M = 1).

This transition is characterized by an abrupt structural

change with the emergence of a giant component scal-

ing. The general result for hypergraphs obtained in [49]

is similar and states that a giant component appears for

c= 1/k(k−1). We refer the reader interested in some

mathematical aspects of this problem to the book [52].

Here, we will construct a ‘Erdos-Renyi hypergraph’ in

the context of Eq. 1. For the usual Erdos-Renyi graph,

the connection probability between two nodes is constant

and similarly for the hypergraph we will then assume that

for each vertex v∈Vand for each hyperedge e, there

is a constant probability pthat vbelongs to e. If this

probability is constant and equal to p, we can then write

P(v∈e) = p(4)

where p∈[0,1]. Starting from a set of Nnodes, we

choose a random set of Ehyperedges (which are Enodes

chosen at random), and add recursively nodes to these

hyperedges following the rule Eq. 4.

Degree and size

This random hypergraph is a very simple model and

most properties are trivial. In particular, the degree k

and size mdistributions are easy to compute and are

binomials

P(ki=k) = E

kpk(1 −p)N−k(5)

P(|ei|=m) = N

mpm(1 −p)E−m(6)

and the average degree is then hki=pE and the average

hyperedge size hmi=pN.

If nodes are in space - in the disk of radius r0for exam-

ple (we will see below in more details examples of spatial

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AclassofmodelsforrandomhypergraphsMarcBarthelemyUniversiteParis-Saclay,CEA,CNRS,InstitutdePhysiqueTheorique,91191,Gif-sur-Yvette,FranceandCentred'AnalyseetdeMathematiqueSociales(CNRS/EHESS)54AvenuedeRaspail,75006Paris,FranceDespitetherecentlyexhibitedimportanceofhigher-orderinteractionsforvariou...

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A class of models for random hypergraphs Marc Barthelemy Universit e Paris-Saclay CEA CNRS Institut de Physique Th eorique 91191 Gif-sur-Yvette France and

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