Networks with arbitrary edge multiplicities

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by  Vinko Zlatic

epl draft
Networks with arbitrary edge multiplicities
Vinko Zlatic´1,2, Diego Garlaschelli3 and Guido Caldarelli1,4,5
1 ISC-CNR, Dipartimento di Fisica, Universita` “La Sapienza”, P.le Moro 5, 00185 Roma, Italy
2 2 Theoretical Physics Division, Rudjer Boˇskovi´c Institute, P.O.Box 180, HR-10002 Zagreb, Croatia
1 3 Lorentz Institute for Theoretical Physics, University of Leiden, Niels Bohrweg 2, NL-2333 CA Leiden, The Nether-
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lands
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4 LINKALAB, Via San Benedetto 88, 09129 Cagliari, Italy
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5 London Institute for Mathematical Sciences, 22 South Audley Street, Mayfair, London W1K 2NY, United Kingdom
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J

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PACS 89.75.-k–

] PACS 89.20.-a–
h PACS 05.20.-y–
p
- Abstract–Oneofthemaincharacteristics ofreal-world networksistheirlargeclustering. Clus-
c
tering is one aspect of a more general but much less studied structural organization of networks,
o
i.e. edge multiplicity, defined as the number of triangles in which edges, rather than vertices,
s
. participate. Here we show that the multiplicity distribution of real networks is in many cases
s
c scale-free, and in general very broad. Thus, besides the fact that in real networks the numberof
i edgesattachedtoverticesoftenhasascale-free distribution,wefindthatthenumberoftriangles
s
y attached to edges can have a scale-free distribution as well. We show that current models, even
h whentheygenerate clusterednetworks, systematically fail toreproducetheobservedmultiplicity
p distributions. We therefore propose a generalized model that can reproduce networks with ar-
[ bitrary distributions of vertex degrees and edge multiplicities, and study many of its properties

analytically.
2
v
5
3
4
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. Introduction. – Realnetworks,wherenodes(orver- Moreover, it has been shown that clustering is only one
1
tices) are intricately connected by links (or edges), are aspect of a more general topological organization which
0
characterized by complex topological properties such as is best captured by edge multiplicity [4,5,10], i.e. the
1
1 a scale-free distribution of the degree (number of edges number of triangles in which edges, rather than vertices,
: reachinga vertex),degree-degreecorrelations,and nonva- participate. Besides being more informative than vertex-
v
nishing degree-dependent clustering (density of triangles basedclustering,edgemultiplicitystronglydeterminesthe
i
X reaching a vertex) [1]. Understanding the structural and percolationproperties of networks [11] andtheir commu-
r dynamical properties of complex networks strongly relies nity structure [12].
a
on the possibility to investigate theoretical models which
are both realistic and analytically solvable. Several ana- Amodelwith arbitrary edge multiplicities.– In
lytically solvable models reproducing the most important order to overcome these limitations, here we propose an
local property of real networks, i.e. the degree distribu- analyticallysolvablemodelofnetworkswithnorestriction
tion, have been proposed [1]. However, models repro- on their clustering properties, and able to generate edges
ducing higher-order properties including clustering (also ofanymultiplicity. Letusdenotebym(i,j)themultiplic-
called transitivity) are only a few and are either entirely ity of the edge (i,j), i.e. m(i,j) = k6=i,jaikakj where
computational [2,3] (i.e. not analytically solvable) or aij = 1 if a link between vertices iPand j is there, and
solvable only for particular cases, e.g. when triangles are aij = 0 otherwise. In our model we allow each vertex i
non-overlapping [4–7] or when the network is made by to have k(0) edges of zero multiplicity, k(1) edges of mul-
i i
cliques [8] or other subgraphs [9] embedded in a tree- tiplicity 1 and so on, up to k(M) edges of multiplicity M,
i
like skeleton. Unfortunately, real networks generally vio- whereM =N−2isthemaximumpossiblemultiplicityin
latethe aboveparticularconditions,asempiricalanalyses a networkwith N vertices. Thus eachvertex i is assigned
haverevealedandaswewillfurther showinwhatfollows. a (N − 1)-dimensional vector k ≡ (k(0),...,k(M)), that
i i i
p-1
V. Zlati´c et al.
we denote as the generalized degree, specifying the multi-
plicity structure in the neighborhood of i. The ordinary
degree of vertex i is k = M k(m). Accordingly, we
i m=0 i
considertheensembleofranPdomnetworkswithaspecified
distribution P(k) ≡ P(k(0),k(1),...,k(M)) of generalized
degrees.
Our approach reduces to various previously proposed
models in special cases, but is more generaland allows to
a) b)
analytically investigate more realistic regimes which have
not been explored so far.
Fig.1: a)Maximallyclusteredconfiguration(c=1/3)allowed
• If ki =(ki(0),0,...,0), our model reduces to the con- forforthetopvertex(k=4)innetworkswithnon-overlapping
figuration model [13,14] where each vertex i has a triangles (weak transitivity) such as Newman’s model [6]. b)
specified degree k = k(0), and the network is locally Maximally clustered configuration (c = 1) for the top vertex
i i
tree-like (edges have zero multiplicity). This model (k = 4) in networks with overlapping triangles (strong transi-
has vanishing clustering in the thermodynamic limit tivity),whichisachievedinourmodelbyassigningthatvertex
a generalized degree k=(0,0,0,4,0,...).
N → ∞, and is thus inadequate to reproduce clus-
tered networks.
• If k = (k(0),k(1),0,...,0), Newman’s clustered thenumberandsizeofthesubgraphsconsidered. The
i i i
empiricalresultsthatwewillshowinamomentmake
model [6] is recovered, where each vertex i has at-
tached k(0) = s “single” edges and k(1) = 2t edges this approachinadequatetoreproducerealnetworks.
i i i i
belonging to t triangles. Although this model has a
i
finite clustering for N →∞, it canonly produce net- Edge multiplicity in real networks. – In all the
worksintheweak transitivity regime [4,5],i.e. where abovemodels,thefractionΦ(m)ofedgeswithmultiplicity
the clustering coefficient of a vertex with degree k is misfullyconcentratedonthesmallestpossiblevalues,i.e.
c(k)≤(k−1)−1 (see figure 1a). m = 0,1,2 depending on the particular model (except in
Gleeson’smodel,whereabroaderdistributionofmultiplic-
• If ki =(0,ki(1),0,...,0), we recoverthe model by Shi itiescanbe obtainedwithasuitablechoiceofclique sizes,
etal. [7]wherealltrianglesareclosed. Thismodelis however losing an important degree of freedom required
the maximally clustered version of Newman’s model, in order to fit other properties of real networks [8]). It is
i.e. c(k)=(k−1)−1, but still cannot produce strong importanttocomparethispredictionwiththemultiplicity
transitivity. structure of real networks. In fig.2 we show the cumula-
tive edge multiplicity distribution Φ (m) ≡ Φ(n)
• If ki = (ki(0),0,...,0,ki(c−2),0,...,0) we recover for various real networks. We find t>hat sparsPe nn≥etmworks,
Gleeson’s model [8] where eachvertex i belongs to a such as the Internet and metabolic networks, display a
clique of c vertices (and thus has k(c−2) =c−1 links power-law distribution of edge multiplicities (with simi-
i
of multiplicity c−2) and has k(0) = k −c+1 addi- lar exponents). Denser networks suchas the WorldTrade
i i
tionalexternallinksofzeromultiplicity,thusforming Web show a distribution which is peaked at some very
a network where cliques are embedded in a tree-like large value (see inset).
structure. Althoughthismodelcanproducenetworks Intheseandallothercasesshown,the distributionsare
withstrongtransitivity,itforcesanyvertextobelong broadandextendovermanyordersofmagnitude,insharp
to only one clique. Thus it fails to reproduce net- contrast with the predictions of the above models. In
workswithoverlappingcommunities ofdenselyinter- particular,scale-freemultiplicitydistributionsimplythat,
connected vertices [12]. in models with modules embedded in tree-like structures,
subgraphsofanysizeshouldbeattachedtoverticesinor-
• Finally,ifk =(k(0),k(1),k(2),0...,0)werecoverthe der to reproduce the observed multiplicity structure. In
i i i i
model recently proposed by Karrer and Newman [9] thissituation,suchmodelsbecomeanalyticallyintractable
where, in addition to single edges and edges belong- andtheirveryphilosophybecomesinappropriate. Indeed,
ing to triangles, edges belonging to diamonds (thus the empirical results shown above suggest that network
with multiplicity 2) are also introduced. More gen- formation is much more decentralized than assumed by
erally, that model allows to embed any type of small locally generating non-overlapping modules of fixed size
subgraphsinto ahigher-ordertree-likestructure,and and sparsely connecting them to one another. The con-
can thus produce strong transitivity as in Gleeson’s cept of module itself appears vague, due to the lack (or
model. However, the model can only be applied as to the unreasonable largeness) of a typical scale for the
long as the set of specified subgraphs is fixed a pri- subgraphs required to describe the network. Remarkably,
ori, and its analytical complexity grows rapidly with besidesthe factthatinrealnetworksthe number ofedges
p-2
Networks with arbitrary edge multiplicities
0 so that the transitivity, which is defined as T =3N /N ,
10 C. elegans △ 3
slope =1.51 does not disappear when N → ∞. Therefore, as ex-
H. pylori pected, our model successfully produces networks with
slope =1.60
AS data non-vanishing overall clustering. It can also generate any
slope =1.57
10-2 WTW desired clustering spectrum, i.e. the average clustering
m) CPGolPla borations c¯(k) of vertices with degree k. The latter is
( Actors
Φ> 300 1 N 2N△(i)
c¯(k)= δ (6)
10-4 200 Np(k)Xi=1 k(k−1) ki,k
100
where N (i) is the number of mutually connected neigh-
△
0 bors of vertex i. This leads to
0 100 200
-6 k(k−1)
10 10-1 100 101 102 m103 104 105 106 107 2 c¯(k)p(k)=1X·k=kI·kP(k). (7)
The above relations hold for every network. It is thus
Fig. 2: Cumulative edge multiplicity distributions Φ>(m) for
possible to choose P(k) in order to reproduce both p(k)
various real networks. Inset: histogram of edge multiplici-
and c¯(k) as in other models [2,3,8].
ties (non-cumulative distribution) for the World Trade Web
(WTW), as an example of network with unusually high den-
Percolation properties. – Importantly, we can
sity.
studythepercolationpropertiesofourmodelanalytically,
thus extending previous results [3,6,8,11] to more gen-
attachedto vertices often has a scale-freedistribution, we eral cases. Let D(s|k) be the probability that a vertex of
found that the number of triangles attached to edges can generalized degree k is a member of a set of s mutually
have a scale-free distribution as well. reachablevertices. Similarly, let d(s|k) be the probability
thatavertexconnectedtoavertexv ofgeneralizeddegree
Generating functions and clustering. – Our
kcanreachsothervertices,excludingthevertexvandits
model, by allowing k to have a more general structure,
neighborhood. The relationbetween D(s|k) and d(s|k) is
can span the entire multiplicity spectrum without explic-
itly introducing subgraphs, overcoming the limitations of D(s|k)= d(s |k)···d(s |k)δ . (8)
1 k s,1+s1+...+sk
the aforementioned models (see figure 1b). The ordinary s1X,...,sk
degree distribution is
We can also write a recursion relation for d(s|k) as
p(k)= P(k)δk, M k(m) (1) min(h,k)−1
Xk m=0 d(s|k) = p(h,m|k)
P
Xh mX=0
The generating function of the probability P(k) is
× d(s |h)···d(s |h)δ (9)
1 hr s,1+s1+...+shr
g(z)= (z∧k)P(k) (2) s1,X...,shr
Xk where p(h,m|k) represents the probability to select,
arounda vertexof generalizeddegreek, anedge ofmulti-
wherez∧k= Mm=0zmk(m) andg(z)=g(z0,...,zM). The plicity m leading to a vertex ofgeneralizeddegree h. The
generating funcQtion of the degree distribution is reduced degree h is the number of vertices attached to
r
the destination vertex except itself and the neighborhood
∞
G(z)= zk P(k)δk, M k(m) =g(z,z,...,z) (3) of the first vertex i.e. hr = h−m−1. If degree-degree
Xk=0 Xk m=0 correlations can be neglected, p(h,m|k) reads
P
Wecannowcomputethetransitivityofthenetwork. First k(m)h(m)P(h)
p(h,m|k)= . (10)
we need to count the triangles: k hk(m)i
M The first fraction in eq.(10) represents the probability to
∂g(z)
3N△ =N em·I =NI·∇g(z)|z=1 (4) leavea vertexofgeneralizeddegreek followinganedgeof
(cid:20) ∂z (cid:21)
mX=0 m z=1 multiplicity m. The second fraction is the probability to
reachavertexofgeneralizeddegreehfollowingthatedge.
where e is a unit vector of multiplicity m (i.e. e ≡
m 0 We can also use eq.(9) to write the generating functions
(1,0,...,0), e1 ≡(0,1,0,...,0), etc.) and I = Mm=0mem. dˆ(z|k)= zsd(s|k) of the probabilities d:
s
The total number of connected triples is P
P
N ∂2G(1) dˆ(z|k)=z min(h,k)−1p(h,m|k) dˆ(z|h) hr. (11)
N = (5)
3 2 ∂z2 Xh mX=0 h i
p-3
V. Zlati´c et al.
Ifeq. (11)hasastablesolutiondˆ(z =1|k)≤1thenetwork randomized ensembles computationally, and generate so-
percolates. In order to study the stability of eq. (11) called microcanonical ensembles [19–21] of networks with
around z =1 we can study a perturbative solution dˆ(z = sharp constraints. Finally, other approaches aim at de-
1|k)≈1+χ(k)ǫ in the limit ǫ→0, which yields scribing random networks with given properties analyt-
ically, and generate (grand)canonical ensembles of net-
min(h,k)−1 works with soft constraints [22–27].
χ(k) = p(h,m|k)(h−m−1)χ(h) (12) Ifourmodelisusedasanullmodelforaparticularreal
Xh mX=0 network, it gives predictions about the ensemble of ran-
min(h,k)−1k(m)h(m) dom graphs having the same generalized degree sequence
= (h−m−1)P(h)χ(h) {k }N as the real network. This generalizes the config-
hk(m)ik i i=1
Xh mX=0 uration model [13,14] where only the ordinary degree se-
= [α·β(h−1)−α·(I ∗β)]P(h)χ(h) quence{ki}Ni=1 isspecified. Inthelattercase,if{ki}Ni=1 is
Xh taken from a real network, one is sure that it is a graphic
sequence. Otherwise, if one generates it artificially, one
where α = k/k, β = h(m) e and I ∗ β ≡ mustenforcespecificconditions,givenbytheErdo˝s-Gallai
m hk(m)i m
mImβmem is a vector.PThe percolation transition [28]andHavel-Hakimi [29]theorems,ensuringthatthese-
oPccurs when the maximum eigenvalue of the matrix in quence is graphic. In our case,the realizabilityof {ki}Ni=1
eq.(12) is larger than 1 i.e Λ > 1. Thus we have ob- is much more complicated than in the case of ordinary
max
tained an analytical expression for the percolation tran- graphicdegree sequences, but we now show how it canbe
sition, more general than the one known for networks in related to the standard problem. For convenience, we de-
theweaktransitivityregime [5],andvalidforanylevelof fine the N ×(N −1) matrix Q with entries Qij ≡ki(j−1).
clustering and multiplicity. The row and column sums (i.e. the margins) of Q are
the degree sequence and the (unnormalized) multiplicity
Rich-club effect. – As another example of the ef-
distribution respectively:
fects of broad edge multiplicities, we now consider the
rich-club coefficient R(k), defined as the observed num- N−1 N−2
ber of edges E>(k) among the N>(k) = Np>(k) vertices Qi+ ≡ Qij = ki(m) =ki (14)
with degree larger than k (where p>(k) is the cumula- Xj=1 mX=0
tivedegreedistribution),dividedbythemaximumallowed N N
number N (k)(N (k)−1)/2 ≈ N2p2(k)/2 [15–17]. In Q ≡ Q = k(j−1) =2E(j−1) (15)
> > > +j ij i
random networks with given degree distribution, the rich Xi=1 Xi=1
club behaves approximately as R(k) ∼ k2 [16], so
Rand hkiN where E(m) denotes the total number of edges with mul-
that the measured R(k) must be compared to this non-
tiplicity m. Therefore, as a first condition we find that
constant value. We now consider the case when, as in
the marginal (ordinary) degree sequence {k }N must be
our model, one also specifies a multiplicity distribution i i=1
graphic. There are however strong additional constraints.
Φ(m). Sinceeveryedge(i,j)withmultiplicitym(i,j)≥k
First note that, since we can always partition the edges
surely connects two vertices i,j with degrees k ,k > k,
i j in disjoint sets (each with given multiplicity), each of the
the expected value of E (k) now receives a contribution
> M sequences {k(m)}N must be separately graphic. This
EΦ (k) from edges with multiplicity m ≥ k (where E is i i=1
> introduces constraints along each column of Q. More-
the total number of edges), and the standard approxima-
over,sinceedgemultiplicitiesmustbeconsistentwitheach
tion can only be applied to the remaining E(1−Φ (k))
> other, there are also constraints along each row of Q.
edges. Following [16],weobtainthemodifiedexpectation
A useful mapping allowsus to solve the problem. For a
hki k2 givenvertexi,weconsiderthesubgraphΓi whosevertices
R(k)Rand ∼Φ>(k)Np2(k) +[1−Φ>(k)]hkiN (13) are the neighbors of i and edges are their mutual connec-
>
tions. An example is shown in Fig.2 (note that Γ does
i
If, as in some of the networks considered above, the cu- not contain vertex i itself). If we denote by [x] the value
i
mulative distributions Φ (k) and p (k) are power laws of a topological property x (e.g. the number E of edges,
> >
with exponents −α and −γ respectively, the asymptotic or the link density D = 2E/[N(N −1)]) when measured
behavior of the first summand is ∼ k2γ−α thus reducing on the subgraph Γ , we find important relations, e.g.
i
or increasing the predicted scaling ∼k2.
[N] =k ; [D] =c ; [k ] =m(i,j). (16)
i i i i j i
Graphic generalized degree sequences. – There
have been many attempts in the literature to generate
null models of real networks by generating ensembles of In other words, the number of vertices and link density
random graphs with given properties. Some of these ap- of Γ coincide with the degree and clustering coefficient
i
proaches make use of generating functions [4,5,14], as in of vertex i measured on the whole network respectively.
the present paper. Other approaches aim at constructing Similarly, the degree of vertex j in Γ coincide with the
i
p-4
Networks with arbitrary edge multiplicities
a a physics, Leiden, Netherlands) with funds from benefi-
ciaries of Duyfken Trading Knowledge BV, Amsterdam,
b e b e Netherlands.
(cid:0)(cid:1)(cid:0)(cid:1)i
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∗∗∗
ThisworkwassupportedbyFETOpenprojectFOCnr.
255987. V.Z. also acknowledges support from the Croat-
ian Ministry of Science, Education and Sports project nr.
098-0352828-2863. D.G. also acknowledges support from
the Dutch Econophysics Foundation (Stichting Econo-
p-5