Table Of ContentAssortative and disassortative mixing investigated using the spectra of graphs
Sarika Jalan 1,2 and Alok Yadav1
1. Complex Systems Lab, Indian Institute of Technology Indore,
M-Block, IET-DAVV Campus, Khandwa Road, Indore 452017, India and
2. Centre for Biosciences and Biomedical Engineering, Indian Institute of Technology Indore,
M-Block, IET-DAVV Campus, Khandwa Road, Indore 452017, India
Weinvestigatetheimpact ofdegree-degree correlations on thespectraof networks. Eventhough
densitydistributions exhibit drastic changes dependingon the (dis)assortative mixing and thenet-
work architecture, the short range correlations in eigenvalues exhibit universal RMT predictions.
5
The long range correlations turn out to be a measure of randomness in (dis)assortative networks.
1
The analysis further provides insight in to the origin of high degeneracy at the zero eigenvalue
0
displayed by majority of thebiological networks.
2
n PACSnumbers: 89.75.Hc,02.10.Yn,89.75.-k
a
J
7 I. INTRODUCTION of complex systems. This paper presents a systematic
2 analysis of impact of degree-degree correlations on the
spectralpropertiesofvariousnetworksundertherandom
] Last two decades have witnessed a rapid advancement
h matrix theory (RMT) framework. Since its introduction
in the field of complex networks [1–3]. The prime idea
p in1960s,inthecontextofnuclearspectra,thetheoryhas
- governing this framework is to consider a system made beensuccessfullyappliedtoawiderangeofcomplexsys-
c of interacting units. To categorize and understand real
o tems rangingfromthe quantumchaosto galaxy[31,32].
world systems based on interacting units, many mod-
s Recently,withaspurtinthe activitiesofnetworkframe-
s. els have been proposed, among which Erdo¨s-R´enyi(ER) work, the RMT got its extension in analysis of spectral
c random [4], scale-free (SF) [5] and small world [6] are properties of various model networks [33, 34] as well as
i the most popular ones. Further, degree-degree correla-
s those arising from real world systems [35, 36].
y tions have also been used as one of the key properties
h of networks characterization [2, 7–18] and is known to
p confer robustness to biological networks [19]. The ten-
[ dencyof(un)likedegreenodestosticktogetheristermed II. METHODS AND TECHNIQUES
1 as (dis)assortativity. Various social networks are known
v to be assortative while few of the biological and tech- To quantify the degree-degree correlations of a net-
4 nological networks have been reported to be disassorta- work, we consider the Pearson (degree-degree) correla-
7 tive [13–18]. Despite its importance for real networks, tion coefficient, given as [7, 9]
3
(dis)assortativity does not appear in any of the model
1 networks discussed above, and is driven by some other [M−1 M j k ]−[M−1 M 1(j +k )2]
0 r = i=1 i i i=1 2 i i ,
. mechanism,forexamplereshufflingalgorithm[20]. While [M−1 MP1(j2+k2)]−[MP−1 M 1(j +k )2]
2 spectral behaviour of uncorrelated networks have been i=1 2 i i i=1 2 i i
0 P P (1)
quite well understood [21], despite real world systems
5 where ji,ki are the degrees of nodes at both the ends of
1 being highly correlated [9], such understanding for the theithconnectionandM representsthetotalconnections
correlated networks still needs to be developed.
: in the network.
v
i Spectralgraphtheoryisanestablishedbranchofmath- The randomnetworkof size N and averagedegree hki
X
ematics,andeigenvaluesofcorrespondingadjacencyma- is constructed using the ER model by connecting each
r tricesareknownasfingerprintsoftheunderlyinggraphs pair of nodes with the probability p = hki/N [4]. These
a
[22–25]. With recent advancement in the network the- networks have assortativity coefficient (r) being close to
ory, the spectral graph theory, traditionally used in in- zero or exactly zero. To generate the networks with var-
vestigationsof randomandregulargraphs,gotextended ious assortativity, we use the reshuffling algorithm [20].
to studies of graphs motivated by real world systems. In this algorithm, after selecting two pairs of nodes ran-
These spectralstudies,apartfrompresenting bounds for domly,wesortthemaccordingtodegree. Thehighestde-
extremaleigenvalueshighlighttheirimportancebyrelat- greenode is thenconnectedto the secondhighestdegree
ing them with the various structural as well as dynam- node with the reshuffling probability p , which governs
r
ical properties of the networks [26, 27]. The studies of the (dis)assortative mixing, i.e. we reconnect a high de-
networks further reveal a key impact of assortativity on greenodetoa(low)highdegreeoneandlowdegreenode
theextremaleigenvalues[28],whichhasbeenexploredin to a (high) low degree one. With the probability 1−p ,
r
context of disease spreading [29] and diffusion processes we rewire them randomly. If new connection resulting
[30], thereby exhibiting the importance of spectral stud- from this rewiring already exists, it is discarded and the
ies of networks for a more comprehensive understanding previous steps are performed. The process is carriedout
2
until a steady value of r is attained. For assortativenet- Brodyparameterliesintherange(0≤β ≤1). Thevalue
works,thek degreenodestoformacompletegraphwith ofβ being 0,indicatesthe Poissondistribution,whereas
thevalueofrbeingone,thenetworkshouldhaveatleast β =1correspondstotheGOEdistribution. Othervalues
(k+1+2n) nodes, where n can be any integer starting of β indicates that the distribution lies intermediate to
from 0. As this condition is not satisfied for all the de- these two.
grees present in the network, the network takes a value The NNSD provides a correlation measure of subse-
lesserthanone. Similarly,thedisassortativenetworkcan quent eigenvalues, whereas the ∆3(L) statistic measures
have the value of r less than −1.0. how the eigenvalues which are L distance apart are cor-
We make a further note that at high assortativity val- related, and can be estimated using the least-square de-
ues, all the similar degree nodes being connected among viationofthespectralstaircasefunctionrepresentingav-
themselves form groups [20]. As we decrease the assor- erage integrated eigenvalue density N¯(λ) from the best
tativity, the connections within the groups of similar de- fitted straight line for a finite interval of length L of the
gree nodes decrease and the connections between differ- spectrum given by [32]:
entgroupsofsimilardegreenodesincrease. Fordisassor-
tative networks, connections between different groups of 1 x+L
similar degree nodes exist giving rise to a bipartite-like ∆3(L;x)= Lma,ibnZx [N(λ¯)−aλ¯−b]2dλ¯ (4)
structure [20].
The SF networks of size N and averagedegree hki are where a and b are regression coefficients obtained after
generated using the Baraba´si-Albert algorithm by start- least square fit. Average over several choices of x gives
ingwithacompletelyconnectednetworkseedandadding the spectral rigidity, the ∆3(L). For the GOE statistics,
new nodesone byone whichconnectwithexisting nodes the ∆3(L) depends on L in the following manner:
using the preferential attachment method [1].
1
Thenetworksarerepresentedinthe formofadjacency
matrix by defining A = 1, if i and j nodes are con- ∆3(L)∼ π2 lnL (5)
ij
nected otherwise A = 0. For an undirected and un-
ij For the network spectra considered in this paper, there
weighted network with N nodes, the adjacency matrix is is no analytical form of N¯, and we perform unfolding
N ×N symmetric square matrix entailing all real eigen-
bynumericalpolynomialfittingusingthesmoothpartof
values. We denote the eigenvalues as λ ,i = 1,2..N and
i the spectra by discarding eigenvalues towards both the
λi ≤λi+1 and analyse them under the RMT framework. ends as well as degenerate eigenvalues, if any [31, 32].
The random matrix studies consider two properties of a
This renders the dimension of the unfolded eigenvalues
spectra: (1) global properties such as spectral distribu-
less than the dimension of the network.
tionoftheeigenvaluesρ(λ),and(2)localpropertiessuch
as eigenvalue fluctuations around λ¯. In RMT, calcula-
tionsofspectralfluctuationsaredoneusingtheunfolded
III. RESULTS
eigenvalues λ¯ = N¯(λ ), where N¯(λ) = λ ρ(λ´)dλ´is
i i λmin
the average integrated eigenvalue densityR[37]. By using
The bulk part of the spectra of ER random net-
these unfolded eigenvalues, nearest neighbour spacings
works with r value being close to zero, follow the well
are calculated as si =λ¯i+1−λ¯i. For symmetric random known semi-circular law [39, 40] (Fig. 1(f)). The ex-
matrices with the mean zero and the variance one, the
tremal eigenvalues deviate from the random matrix pre-
nearest neighbour spacing distribution (NNSD) follows
dictions and indeed provide various information about
GOE statistics given as:
structural and dynamical properties of corresponding
π πs2 systems [11, 26, 27, 29, 41]. In the following, we present
P(s)= sexp(− ), (2)
results pertaining to the impact of assortativity on the
2 4
spectral properties of networks. It turns out that with
which shows a level repulsion at small spacing values
an increase in the assortativity, the semi-circular distri-
with an exponential fall for larger spacings indicating
bution,asobservedfortheuncorrelatedERrandomnet-
that nearest neighbour eigenvalues are correlated [37].
works, remains unchanged (Fig. 1(a)-(e)). The largest
Whereas the spacing distribution of a matrix whose di-
eigenvalue exhibits an increasing trend as already dis-
agonal elements are Gaussian distributed random num-
cussed in [28, 29]. As network is rewired entailing dis-
bers and rest of the elements are zero exhibit Poisson
assortativity,spectraldistribution (ρ(λ)) acquiresa very
statistics (P(s) = exp(−s)) indicating that eigenvalues
differentstructurethanthoseoftheassortativenetworks.
are uncorrelated [37].
The networksstartexhibiting ahigh degeneracyatzero,
The intermediate of these two distributions can be
with overall spectra resembling a double humped struc-
characterized using the Brody equation [38]:
ture (Fig. 1(h)), which becomes more pronouncedas the
Pβ(s)=Asβexp −αsβ+1 , (3) disassortativity becomes higher or value of r becomes
(cid:0) (cid:1) more negative (Fig. 1(i)). This increase in disassorta-
where A and α are determined by the parameter β as
β+1 tivity isalsoaccompaniedwith morenumber ofdegener-
A = (1 + β)α and α = Γ ββ++12 . The value of ate eigenvalues at zero. There could be various reasons
h (cid:16) (cid:17)i
3
for this high degeneracy,few of them, appropriatein the (a) (b) (c)
present context are: First, as discussed that disassorta- 0.6 χ2=0.020.6 χ2=0.07 0.6 χ2=0.09
) β=0.36 β=0.68 β=0.82
tivity supports bipartite-like structure [20] and a com- (s0.4 r=0.9930.4 r=0.992 0.4 r=0.991
plete bipartite network has all zero eigenvalues except P
0.2 0.2 0.2
two. Hence bipartite-like behaviour of the disassorta-
tive networks presents one of the reasons for the occur- 00 1 2 3 4 00 1 2 3 00 1 2 3
renceofhighdegeneracyatzero. Second, tree-likestruc- 1
(d) (e) (f)
ture has been demonstrated to yield degeneracy at zero 0.8 χ2=0.09 0.8 χ2=0.10 0.8 χ2=0.10
eigenvalue [42] and disassortativity encourages tree-like s)0.6 rβ==10..0699 0.6 rβ==10..165 0.6 rβ==10..070
structure [20], which in turn indicates high degeneracy P(0.4 0.4 0.4
at zero. We remark that for large N, the limiting shape 0.2 0.2 0.2
of ρ(λ) is known for various cases, which for sufficiently
0 0 0
dense matrices, tend to follow the Wigner semi-circular 0 1 2 3 0 1 2 3 0 1 2 3
law typical for the Gaussian matrix ensembles [39, 40], 0.8 χ2(=g0).090.8 χ2=(0h.)07 0.8 χ2=0(.i1)0
whereas an ensemble of sparse random matrices of finite )0.6 β=1.12 0.6 β=0.98 0.6 β=1.03
s
size are known to yield states beyond the semi-circular P(0.4 r=-0.5 0.4 r =-0.7 0.4 r=-0.98
law in the tails of the distribution [43–45]. For sparse
0.2 0.2 0.2
random graphs, i.e matrices with 0 and 1 entries having
0 0 0
smaller p values, while the density distribution ρ(λ) of 0 1 s 2 3 0 1 s 2 3 0 1 s 2 3
an ensemble exhibit singularities, with the height of the
peaks being the corresponding multiplicities, the bulk is
FIG. 2: (Color Online) The NNSD for Erd¨os-R´enyi random
still shown to comply with random matrix predictions networks with different values of assortativity coefficient r.
of Wigner’s semicircular law [46, 47]. Moreover, investi- All graphs are plotted for the networks with size N = 1000
gations of various model networks mimicking real world andconnectionprobabilityp=0.01. Histogramsarefromthe
properties have revealed that the spectra of these net- datapointsandsolidlineisforfittingwithBrodydistribution
works exhibit degeneracy at zero [37], as observed for (Eq.3).
the sparse random matrices. On that account, despite
degeneracy at zero, the bulk of the assortative networks
followingthesemi-circulardistribution,isnotsurprising.
haviour of eigenvalues, in order to get insight into lo-
cal fluctuations, we further analyse the short range and
As the spectral density only provides a global be- long range correlations in eigenvalues. The NNSD fol-
lows GOE statistics of RMT (Eq. 2) for all the values
of r except for very high values corresponding to the
(a) (b) (c) highly assortative networks (Fig. 2). What is interest-
0.1 0.1 0.1 ing that the values of r for which ρ(λ) exhibits a very
r=0.993 r=0.992 r=0.991
λ) similar behaviour, except a change in the value of the
(
ρ 0.05 0.05 0.05 largesteigenvalue, the NNSD captures crucial structural
changesreflectedthroughthe value ofthe Brody param-
0-5 0 5 10 15 0-5 0 5 10 15 0-5 0 5 10 15 eter. Forthehighestachievablevalueoftheassortativity
coefficientfortheparticularnetworkparameterforwhich
(d) (e) (f)
0.1 r=0.9 0.1 r= 0.5 0.1 r=0.0 resultsarepresented,thevalueofβ comesouttobeclose
) to 0.3 (Fig. 2(a)), and as the assortativity decreases we
λ
ρ( 0.05 0.05 0.05 witness a smooth transition to the GOE statistics with
valueoftheβ turning one. Depending uponthe network
0-5 0 5 10 15 0 -5 0 5 10 0 -5 0 5 10 size, average degree and degree sequences, the highest
achievable value of r for that network may be different
0.15 (g) 0.2 (h) (i) (asdiscussedintheSectionII),whichmightleadtoadif-
r=-0.5 r=-0.7 0.4 r=-0.98
) 0.1 ferent value of β. Fig. 1(a)-(d) depict that a very small
λ
ρ( 0.05 0.1 0.2 change in the value of r is capable of entailing a pro-
found changein the statistics, in-fact it approachesfrom
0 0 0 thePoissontotheGOE.Sinceaverysmallrandomnessis
-5 0 5 10 -5 0 5 10 -10 -5 0 5 10
λ λ λ knowntobeenoughinintroducingtheshortrangecorre-
lationineigenvalues[48],fora verysmalldeviationfrom
FIG.1: (ColorOnline)SpectraldensityforErd¨os-R´enyiran- the highest assortativity entails GOE statistics. Since
domnetworkswithdifferentvaluesofassortativitycoefficient the assortativity in network supports a the groups hav-
r. AllgraphsareplottedforthenetworkswithsizeN =1000 ing similar degree nodes and as assortativity decreases,
and connection probability p = 0.01, averaged over twenty thesedistinctgroupsofnodesobservedforveryhighval-
different realizations of thenetworks. ues of r gets destroyed leading to a transition from the
4
Poisson to the GOE statistics. As soon as the value of rameter increases with an increase in the rewiring prob-
r is decreased, and sufficient random connections among ability and becomes one at the onset of the small-world
the groupsofsimilardegreenodesareinduced, the value transition, demonstrating that nearest neighbour eigen-
ofBrodyparameterβ becomesoneandnofurthersigna- values are correlated [48]. For such a small change in
ture of structural changes on value of β is found with a the network structure there is no visible change in the
further decrease in the assortativity. density distribution, but the Brody distribution detects
For disassortative networks which are characterized even such a small change in the number of random con-
with negative values of r, what is remarkable is that de- nections, and hence has been proposed to be used as a
spite these networks displaying distinguishable spectral measure of randomness at a fine scale [48]. After the
distributions than those of the assortative networks, the Brody parameter attains a value one, the ∆3(L) statis-
NNSD yields the value of the Brody parameter (β = 1) tic hasbeenshownto measurethe randomness(interms
bringingthemintothe universalityclassofGOE.This is of L0 in this paper), in the underlying network [49].As
not surprising as NNSD is analysed by taking the non- rewiringprobabilityincreasesfurther,thevalueofL0 for
degenerate part of the spectra, and high degeneracy at which∆3(L)statisticfollowsRMTpredictionsincreases,
a particular value, for instance at zero,does not account demonstratingthattheeigenvalueswhichareL0distance
for any effect in the NNSD. As long as the underlying apart are also correlated. Since L0 provides a measure
network has some random connections, the NNSD dis- of randomness in a network [49], for the networks under
plays the GOE statistics [48]. We remark that all the investigation in the present work, it turns out that the
networksconsideredhere formasingle connectedcluster highest assortative network is least random, as value of
as for disconnected networks, even though each individ- L0 is least for that particular r value (Fig. 3(a)). As as-
ualsub-networkfollowsGOEstatistics,thespectrataken sortativity of the network is decreased, the randomness
together may lead to a different spacing statistics [33]. of the network increases reflected in the higher value of
In order to get a further insight to the structural L0. This increase in the size of L0 continues up to r
changes arising due to the changes in r values, we probe being zero, supporting the fact that network reaches to
for the long range correlations in eigenvalues for those themaximumrandomness. ThevalueofL0thenremains
sets which yield the β value one. We find that for all steadyfor a further decreasein the value ofassortativity
these values of r, the long range correlations, measured totheminimumpossiblevalueofr,i.e. tothemaximum
using the ∆3(L) statistic (Eq. 4) follow the universal disassortativity (Fig. 3(c)). As most of the real world
GOE statistics as given by Eq. 5 for a certain value of networks have been reported to posses certain level of
L (denoted as L0) and deviates from this universality disassortativity [9], based on the ∆3(L) results we can
afterwards (Fig. 3). arguethatrealworldsystemsattemptto havemoreran-
Note that a regular network, for instance 1-d lattice
with a periodic boundary condition, follows Poisson dis-
tribution. Asconnectionsarerewired,therebyincreasing 0.8 0.8 0.8
(a) (b) (c)
the randomness in the network, value of the Brody pa- χ2=0.04 χ2=0.06 χ2=0.07
) ) )
(s0.4 β=0.4 (s0.4 β= 0.5 (s0.4 β=0.9
P r=0.999P r=0.998P r=0.997
0.4
0.4 0.4 0 0 0
0 2 4 0 2 4 0 2 4
s s s
3
∆ 0.2 0.2 0.2 (d) (e) (f)
r=0.98 r=0.5 r=0.0 0.4 0.4 0.4
3 3 3
00 20 40 60 00 50 100 00 50 100 150 ∆ 0.2 r=0.98∆ 0.2 r=0.5 ∆ 0.2 r=0.0
0 0 0
0.4 0 20 40 0 20 40 0 20 40
0.4 0.4 L L L
0.2 0.2
3 (g) (h) (i)
∆ 0.2 r=-0.5 0.2 r=-0.7 0.2 r=-0.98 λ) 0.1 r=-0.5 λ) 0.1 r=-0.7 λ) 0.4 r=-0.98
ρ( ρ( ρ( 0.2
0 0 0
0 50 100 150 0 50 100 150 0 50 100 150
L L L 0 0 0
-5 0 5 10 -5 0 5 10 -10 0 10
λ λ λ
FIG. 3: (Color Online) The ∆3(L) statistic for Erd¨os-R´enyi
random networks with different values of assortativity coef- FIG. 4: (Color Online) (a)-(c) represent the NNSD, (d)-(f)
ficient r. All graphs are plotted for the networks with size present the ∆3(L) statistic and (g)-(i) depict the spectral
N = 1000 and connection probability p = 0.01. Solid line is density distribution of ER random networks. All graphs are
the prediction from GOE statistics (Eqs. 4 and 5) and open plotted for the networks with size N =2000, hki=10 and for
circle are calculated from thenetwork. average over twentydifferent realizations of thenetwork.
5
(a) (b) (c) PPI networks N r0 N0(PPI) N0(r=0) N0(r=r0)
0.1 0.1 H.pylori 709 -0.243 317 115 152
λ) 0.1 r=0.35 r=0.3 r=0.2 C.elegans 2386 -0.183 1354 465 1124
ρ( 0.05 0.05 0.05 S.cerevisiae 5019 -0.088 976 717 1149
H.sapiens 2138 -0.084 864 423 643
0-15 0 15 30 0-10 0 10 20 30 0-10 0 10 20 30 D.melanogaster 7321 -0.083 2311 1389 1975
0.15 E.coli 2209 -0.012 487 487 497
(d) (e) (f)
0.1 0.1
) 0.1
λ r=0.1 r=0.0 r=-0.1
ρ( 0.05 0.05 0.05 TABLE I: Comparison of number of zero eigenvalues of PPI
networksofdifferentspeciesandtheircorrespondingconfigu-
0 0 0 rationmodels. r0denotesthevalueoftheassortativitycoeffi-
-10 0 10 20 -10 0 10 20 -10 0 10 20 cient for thePPI networks. N0(PPI)denotes the numberof
0.2 (g) (h) (i) zeroeigenvaluesinthespectraofthePPInetworks. N0(r=0)
0.3
stands for degeneracy at zero for configuration model with
λ) r=-0.2 0.2 r=-0.3 0.4 r=-0.36 r =0, whereas N0(r =r0) denotes the same for the configu-
ρ( 0.1 0.2 rationmodelstakingr valuesequaltothecorrespondingPPI
0.1
network.
0 0 0
-10 0 10 -10 0 10 -20-10 0 10 20
λ λ λ
distributethemselvessymmetricallyandadoptadouble-
hump shape for highly disassortative networks, clearly
FIG. 5: (Color Online) Spectral density for scale-free net-
visible in Fig. 5(i) which is accompanied by a high peak
works with different values of assortativity coefficient r. All
at the zero eigenvalue similar to that of the ER random
graphs are plotted for the networks with size N = 1000,
hki=10, for twentydifferent realizations. networks. It is noteworthy that for highly disassortative
networks, the spectral density of ER and SF model net-
works behave similarly, deviating from their respective
signature distributions. Further, the β value exhibits a
domness,thereby leadingto be disassortative. What fol- transition from the Poisson to the GOE statistics with
lowsthatasvalue ofr increases,bykeepingnetworksize a decrease in the r value. Despite the overall spectral
andaveragedegreesame,thevalueofL0forwhich∆3(L) density being different from that of the ER networks,
statisticfollowsRMTpredictionsincreases,indicatingan the NNSD and ∆3(L) statistic display similarity in be-
increased amount of randomness in the underlying net- haviour, which is in line of the argument that the eigen-
work. Fig. 4 demonstrates that the behaviour of various values fluctuations are calculated from the smooth ho-
spectralpropertiesremainunchangedasnetworksizein- mogeneous part of the spectra by not taking degeneracy
creases. Figs. 4(a)-(c) indicate that the value of Brody into account and density is not known to be a real test
parameter β becomes one with a very small decrease in of GOE statistics [50].
the value of r. With a further decrease in the value of r,
We would like to remark here on the impact and re-
the valueofL0 for whichthe ∆3(L)statisticfollowGOE liability of network size considered in the present inves-
statistic increases indicating an increase in the random-
tigation. In RMT, different quantities are calculated by
ness as discussed earlier. With a further decrease in the
averaginganensemble ofmatrices. Howeverfor realsys-
value of r in the disassortativity regime, there occurs a
tems,calculationsaremadeasrunningaveragesoverpart
peakatzeroeigenvaluewhichbecomesmorepronounced
of the whole spectrum. The random matrix predictions
as network becomes more dissociative which is also ac-
can be applied to real world systems if above two are
companiedwiththedeviationfromthe semi-circulardis-
equivalent, a property knownas ergodicity. More explic-
tribution at very low value of r.
itly, it means that all members of the ensemble, except
Further, inorder to demonstratethe robustness ofthe for a set of measure zero, satisfies the above equivalence
universal RMT predictions against changes in the net- [51, 52]. Due to the ergodicity, one can construct ma-
workarchitecture,wepresentresultsfortheSFnetworks trix ensembles in different ways: (a) large dimensional
forvariousvaluesofr. Forr beingclosetozero,theden- randommatrices with less number of realizations,or (b)
sity distribution of SF networks exhibit the triangular smaller dimensional matrices with large number of real-
shape [40], which, with an increase in the assortativity, izations. We consider an ensemble of twenty network re-
tends to display flattening of the peak. The range of alizationswithalargedimension,whichisalreadyshown
thedistributionalsoshrinksastheassortativityincreases tobe goodenoughto study variousstructuralproperties
(Fig. 5(a)-(e)). On the other hand, as we decrease as- of networks, such as degree distributions, clustering co-
sortativity, i.e. make the network more disassortative, efficients etc. [6]. Moreover, individual entities of each
theshapeofdensitydistributionstartschangingfromits ensemblefollowRMT predictionsforNNSD with agood
signature triangular distribution, with the peak at zero accuracy,characterizedby χ2 values. As we increase the
eigenvalue being more pronounced (Fig. 5(f)-(g)). As realizations, accuracy increases (Fig. 2 and Appendix).
we further increase the disassortativity, the eigenvalues Consideration of an ensemble consisting of many more
6
number of network realizations would not lead to sig- the increase in L0 of ∆3(L) statistic after attainment
nificant betterment or difference in the following prop- of β value one have several implications, one of them
erties of the network spectra: (1) the Brody parameter that concerns the present work is that the value of β
smoothly turning one with a decrease in the value of r distinguishestwonetworksbasedonrandomconnections
at a very fine scale; (2) a further decrease in the values present, while the other is that more assortativity in the
of r leading to an increase in the value of L for which network corresponds to less randomness. Decreasing the
spectra follows GOE statistics; and (3) increasing height assortativity leads to an increase in randomness which
of the peak at zero eigenvalues with an increase in the continuesuptothe valueofr =0,forwhichthe network
disassortativity, owing to the bipartite-like structure of is most random (L0 value being maximum). Then the
the network. value ofL for which∆3(L) statistic followsGOE predic-
Next, in order to investigate if the degree-degree cor- tion starts decreasing and remains steady for a further
relations in real world system have different spectral decreaseinthe value ofassortativityup to the minimum
behaviour than those of the model networks discussed possible value of r (i.e. up to the maximum disassorta-
above, we consider the protein-protein interaction (PPI) tivity case). The SF networks also exhibit the similar
networks of six different species. These networks have statistics of eigenvalue fluctuations as for ER random
alreadybeen shownto follow universalRMT predictions networks, where density distribution for r = 0 and for
of GOE statistics [53]. We concentrate here on the oc- lower |r| values show triangular distribution instead of
currenceof highdegeneracyatthe zeroeigenvalue. The semi-circular. Both the networks, however, exhibit high
assortativitycoefficientandfractionofdegenerateeigen- degeneracy at zero for the disassortative networks. By
values are tabulated in Table I. As all the PPI networks considering different PPI networks, we further demon-
possess negative value of r as well as have a high de- strate the role of disassortativity governing the appear-
generacy at zero, we expect disassortativity to be one of ance of degeneracy at zero eigenvalue.
the factors governing the degeneracy in the real world In spectral graph theory, most of the works concen-
networks. In order to probe more into the correlation trate on extremal eigenvalues [22]; whereas the RMT re-
betweendisassortativityanddegeneracyatzero,wecom- searchfocuses ondistribution ofvariousspectralproper-
pare the corresponding configuration model for all PPI tiesofrandommatriceswithanextensiontotherandom
networks presented above (Table I). It is clearly indi- graphs, largely ignoring many graphs properties exist-
cated that as soon as the value of r takes a negative ing in real world systems. The analysis carried here is a
value (close to the corresponding PPI network), while step towards bridging this gap by considering two most
keepingallotherparametersofthe systemsame,thereis popular tools of random matrix theory, i.e. density and
an increase in the degeneracy at zero eigenvalue. spacing distributions, to understand the impact of one
of the important properties of graphs i.e. assortativity.
This property has been increasingly realized as a char-
IV. DISCUSSION AND CONCLUSIONS acteristic of a system [54–56]. Our analysis is another
demonstration of the importance of spacing analysis in
The density distribution of the random networks for r understandingimpactofdegree-degreecorrelationonthe
valuebeingzerofollowtheWignersemi-circulardistribu- networkdetected throughthe spectra as for very minute
tion. Even with change in the assortativity (0 ≤r <1), changes in r, there are no visible changes in the spectral
thebulkpartofthespectrakeepsdisplayingsemi-circular density, but this leads to a very drastic changes in the
distribution(Fig.1(a)-(f)),whereasincreaseinthedisas- eigenvalues fluctuations demonstrating the impact of r
sortativity(−1≤r <0)leadstothedoublehump,which values on randomness in a network.
issymmetricallydistributedaroundapeakatzeroeigen- Furthermore, ∆3(L) statistic provides an insight into
value (Fig. 1(i)). The height of the peak increases with the reason why social networks tend to be assortative
the increase in the disassortativity of the network. while biological and technological networks tend to be
The NNSD of the networks with the various disassortative. As randomness, measured in terms of L0
(dis)assortativity values (1 < r < −1), reveal that there forwhichthe∆3(L)statisticfollowsRMTprediction,in-
isasmoothtransitionintheβ-valuearoundtheveryhigh creases with a decrease in r. A direct implication of this
assortativity regime. For very high assortativity values, result can be witnessed in case of social networks where
β values lie close to zero, and as network becomes less entities are known to be associated in ordered fashion
assortative β progresses to one. It might be due to the (people with similar age or educational profile are often
reason that the networks with the highest assortativity more connected) [57], thus providing a probable reason
hasgroupsofsimilardegreenodeswhichgetperturbedas as to why social networks tend to assume an assortative
r decreases by making random connections among these topology. On the other hand, it has been reported that
different groups. For rest of the assortativity values, the mostofthebiologicalandtechnologicalnetworkspossess
β remains fixed at 1, which corresponds to the universal a certain level of disassortativity [7, 9, 18]. Also the bi-
GOE distribution as r value goes to negative end. ological networks, for instance the PPI networks exhibit
Further, the property of Brody parameter being able varying amounts of randomness in their underlying net-
todetectchangesinnetworkstructureatafinescaleand works detected through different values of L0 for which
7
the∆3(L)statisticfollowsGOEstatistics[53]. Thisran- 1 (a) 1 (b) 1 (c)
domness has been attributed to mutations occurring in χ2 =0.03 χ2 =0.03 χ2 =0.05
20 20 20
β = 1 β=1 β=1
course of evolution [58]. Relating the disassortative na- λ) r=0.98 r=0.00 r=-0.98
ture of the PPI networks and the randomness they pos- ρ( 0.5 0.5 0.5
sess, with the results obtained from our analysis of the
model networks, suggests that biological networks tend
0 0 0
to become more disassortative in order to comply with 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
their underlying randomness.
1 (a’) 1 (b’) 1 (c’)
To conclude, we present a systematic analysis of χ2 =0.23 χ2 =0.25 χ2 =0.38
1 1 1
the spectral properties of the networks with varying χ2 =0.08 χ2 =0.08 χ2 =0.08
(dis)assortativity. We find that assortativity has a pro- ρ(λ) 0.5 βr3==10.98 0.5 βr3==10.00 0.5 βr5==1-0.98
foundimpactonthespectralpropertiesoftheunderlying
networks. Ata very highassortativityregime,evenwith
a slight decrease in the value of r, the Brody parame- 0 0 0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
ter smoothly turns one. A further decrease in the values λ λ λ
of r leads to an increase in the value of L0 of ∆3(L)
statisticforwhichthespectrafollowsGOEstatistics. As FIG. 6: (Color Online) (a), (b) and (c) plot average NNSD
for an ensemble of twenty realizations for different values of
Brodyparameterβ capturesthe changesinassortativity
r,whereas (a´),(b´)and (c´)plot theensemblehavingdifferent
coefficient at a fine scale [48] and L0 at large scale [49],
number of the network realizations. The histogram is drawn
which further suggest that when r decreases, random-
usingthedatafrothenetworks,andthesolidlineisthefitted
ness increases. With a further decrease in r, at around
Brody distribution. For all the graphs N = 1000 and hki =
r = 0, the density distribution start exhibiting peak at 10.
zeroeigenvaluewhichbecomesmorepronouncedasr de-
creasesfurther. Interestingly,mostofthe studiesonnet-
work spectra report that the bulk part of the spectra ear Algebra PACKage) subroutines into the FORTRAN
of the networks having Gaussian and scale-free degree code. Thecalculationofspacingsandpolynomialfittings
distribution follow semi-circular and triangular distribu- are done using MATLAB.
tions [39, 40] respectively, but for highly disassortative Wepresenttheχ2valuesasameasureofgoodnessoffit
networks, the spectral density of both the degree distri- ofthemodeltodata,alowerofχ2 indicatingabetterfit-
butions can have entirely different behaviour. ting. As depicted from Fig. 6, the χ2 values consistently
Recently,therealmofassortativityhasbeenrealizedin decreasewith anincreasein the number ofnetworkreal-
understandingadaptivesynchronization[55],whichcom- izationsintheensembleimplicatingincreaseintheaccu-
bined with our results of varying amount of randomness racyreachingtothevalueofχ2 beinglessthanonelying
for various values of r can be explored further to under- intheacceptablerange[63]. Forassortativenetworks,as
stand dynamical processes on networks. Further, Table less as three realizations in the individual ensemble are
I indicates that disassortativityis one of the factorscon- good enough to bring χ2 within the acceptable range,
tributing to degeneracy at zero. Prevalence of zero de- whereas for r taking negative values, the number of re-
generacyhasbeenimplicatedintermsofgeneduplication alizations in the ensemble increases little bit more (five
[59]. This, along with the impact of change in topology as depicted in Fig. 6(c´)) in order to bring χ2 within the
of a network, brought upon by assortativity, leading to acceptablerange. Thishappensasfordisassortativenet-
a profound change in the spectral density, provides a di- works,thereishighdegeneracyatzeroeigenvalueleading
rection to explore the evolutionary origin of real world to less effective dimension of the unfolded spectra (refer
systems [60, 61]. Lastly, since randomness or random discussions in the Methods and Techniques section) and
connections in a network has already been emphasized hence more number of realizations of the networks are
forproper functioning ofcorrespondingsystems[62], the required.
profoundroleofassortativityparameterrevealedthrough
the sophisticated random matrix technique is not only
important for network community attempting to model VI. ACKNOWLEDGEMENTS
complex systems, but is interesting for random matrix
communities at the fundamental level as well.
SJ thanks Department of Science and Technology
(DST), Govt. of India grant SR/FTP/PS-067/2011and
Council of Scientific and Industrial Research (CSIR),
V. APPENDIX
Govt. of India grant 25(0205)/12/EMR-II for funding.
It is pleasure to acknowledge Dr. Kunal Bhattacharya
Numerical calculations pertaining to assortative mix- (BITS PILANI) and Igor M. Sokolov (Humboldt-
ing,eigenvaluescalculationsand∆3(L)statisticaredone Universit¨at zu Berlin) for help with the Sokolov’s algo-
using FORTRAN code written by the Authors. The rithm. AY acknowledges Council of Scientific and In-
eigenvalues are calculated by calling LAPACK (Lin- dustrial Research (CSIR), Govt. of India for financial
8
support.
[1] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. 74, 47 [28] P. Van Meighem, H. Wang, X. Ge, S. Tang, and F.A.
(2002). Kuipers, Eur. Phys. J. B 76, 643 (2010).
[2] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D. [29] A. V. Goltsev, S. N. Dorogovtsev, J. G. Oliveira and J.
U.Hwang, Phys. Rep.424, 175 (2006). F. F. Mendes, Phys.Rev.Lett. 109, 128702 (2012).
[3] A.-L.Barab´asiandZ.N.Oltvai,Nat.Rev.Genet.5,101 [30] G.DA´gostino,A.Scala,V.Zlati´candG.Caldarelli,EPL
(2004). 97, 68006 (2012).
[4] P. Erd¨os and A. R´enyi, Publ. Math. Debrecen 6, 290 [31] T. Guhr, A.M.-Groeling and H.A. Weidenmu¨ller, Phys.
(1959). Rep. 299, 189 (1998);
[5] A.-L.Barabasi and R.Albert, Science 286, 509 (1999). [32] T.PapenbrockandH.A.Weidenmu¨ller,Rev.Mod.Phys.
[6] D.J.WattsandS.H.Strogatz,Nature(Landon)393,440 79, 997 (2007).
(1998). [33] G. Palla and G. Vattay,New J. Phys. 8, 307 (2006).
[7] M. E. J. Newman, Phys.Rev.Lett. 89, 208701 (2002). [34] S. Jalan and J. N. Bandyopadhyay, Phys. Rev. E 76,
[8] M.T.Rivera,S.B.SoderstromandB.Uzzi,Annu.Rev. 046107 (2007).
Sociol. 36, 91 (2010). [35] R. Potestio, F. Caccioli and P. Vivo, Phys. Rev. Lett.
[9] M.E.J. Newman, Phys.Rev.E 67, 026126 (2003). 103 268101 (2009).
[10] M. Barth´elemy, EPL 63, 915 (2003). [36] S. Jalan, C. Sarkar, A. Madhusudanan and S. K.
[11] M. E. J. Newman, SIAMRev.45, 167 (2003). Dwivedi, PloS one9, e88249 (2014).
[12] V. Colizza, A. Flammini, M. A. Serrano and A. Vespig- [37] M.L. Mehta, Random Matrices, second ed.(Academic
nani, Nat.Phys.2, 110 (2006); S. Boccaletti, V.Latora, Press, New York 1991).
Y. Moreno, M. Chavez and D. U. Hwang, Phys. Rep. [38] T.A.Brody,LettereAlNuovoCimento7,482(1973);T.
424, 175 (2006); R. Cohen and S. Havlin, Phys. Rev. A.Brody,J.Flores,J.B.French,P.A.Mello,A.Pandey
Lett. 90, 058701 (2003); S. N. Soffer and A. V´azquez, and S.S. Wong, Rev.Mod. Phys.53, 385 (1981).
Phys.Rev.E 71, 057101 (2005). [39] I. J. Farkas, I. Der´enyi, A.-L. Barab´asi and T. Vicsek,
[13] K. I. Goh, E. Oh, B. Kahng, and D. Kim, Phys. Rev. E Phys. Rev.E 64, 026704 (2001).
67, 017101 (2003); M. Catanzaro, G. Caldarelli and L. [40] M. A.M. De Aguiar and Y.Bar-Yam, Phys. Rev.E71,
Pietronero, Phys.Rev. E70, 037101 (2004). 016106 (2005).
[14] M. E.J. Newman andJ. Park, Phys.Rev.E68, 036122 [41] H. Sompolinsky, A. Crisanti, and H. J. Sommers, Phys.
(2003); D.Lusseau and M.E. J. Newman, Proc. R.Soc. Rev. Lett. 61, 259 (1988); K. Rajan and L. F. Abbott,
London B 271, S477 (2004). Phys. Rev. Lett. 97, 188104 (2006); R. M. May, Nature
[15] D.P.Croft,R.James,A.J.W.Ward,M.S.Botham,D. 238, 413 (1972); S. Chauhan, M. Girvan and E. Ott,
Mawdsley and J. Krause, Oecologia 143, 211 (2005). Phys. Rev.E 80, e056114 (2009).
[16] J. Bollen, B. Gonc¸alves, G. Ruan and H. Mao, Artificial [42] S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes and
life 17, 237 (2011). A. N.Samukhin Phys.Rev. E68, 046109 (2003).
[17] G. Bagler and S. Sinha, Bioinformatics 23, 1760 (2007). [43] G. J. Rodgers and A. J. Bray, Phys. Rev. B 37, 3557
[18] D.S.Bassett, E.Bullmore, B.A.Verchinski,V.S.Mat- (1998).
tay,D.R.WeinbergerandA.M.-Lindenberg,J.Neurosci. [44] Y.V.FyodorovandA.D.Mirlin,J.Phys.A:Math.Gen.
28, 9239 (2008). 24, 2219 (1991).
[19] D. A. Pechenick, J. L. Payne and J. H. Moore, PLoS [45] S. N. Evangelou, J. Stat. Phys. 69, 361 (1992).
Comput. Biol. 10, e1003780 (2014). [46] T. Rogers, I.P.Castillo, R.Ku¨hn and K. Takeda,Phys.
[20] R. Xulvi-Brunet and I.M. Sokolov, Phys. Rev.E, 70, Rev. E78, 031116 (2008).
066102 (2004); ibis, Acta Physica Polonica B 36, 1431 [47] I. Dumitriu and S.Pal, Ann.Probab. 40, 2197 (2012).
(2005). [48] J.N. Bandyopadhyay and S. Jalan, Phys. Rev. E, 76,
[21] J. Berg and M. L¨assig, Phys. Rev. Letts. 89, 228701 026109 (2007).
(2002). [49] S.JalanandJ.N.Bandyopadhyay,EPL87,48010(2009).
[22] P. Van Meighem, Graph Spectra for Complex Networks [50] A.PandeyandM.L.Mehta,Commun.Math.Phys.87,
(Cambridge University Press, 2011). 449 (1983).
[23] G.Akemann,J.Baik,P.D.Francesco,TheOxfordHand- [51] O.BohigasandM.J.Giannoni,AnnalsofPhys.89,393
bookofRandomMatrixTheory(OxfordUniversityPress, (1975).
2011). [52] J. B. French,P. A. Mello, and A.Pandey, Phys. Lett. B
[24] F.R.K.Chung,SpectralGraphTheory(AmericanMath- 80, 17 (1978).
ematical Society,USA 1997). [53] A.Agrawal,C.Sarkar,S.K.Dwivedi,N.Dhasmana,and
[25] D. Cvetkovi´c, P. Rowlinson and S. Simi´c, An introduc- S. Jalan, Physica A 404, 359 (2014).
tiontothetheoryofgraphspectra(CambridgeUniversity [54] J. N. Tenenbaum, S. Havlin and H. E. Stanley, Phys.
Press, 2010). Rev. E86, 046107 (2012).
[26] P. Van Mieghem, J. Omic, and R. Kooij, IEEE/ACM [55] V.A.-Gayta´n,J.A.Almendral,D.Papo,S.E.Schaeffer
Trans. Netw. 17, 1 (2009). and S.Boccaletti, Phys. Rev.E 86, 015101(R) (2012).
[27] J.G.Restrepo,E.OttandB.R.Hunt,Phys.Rev.E71, [56] F. Papadopoulos, M. Kitsak, M.A. Serrano, M. Boguna´,
036151 (2005). and D.Krioukov, Nature489, 537 (2012).
9
[57] J. A. Smith, M. McPherson, and L. S.-Lovin, American e87042 (2014).
Sociological Review, 79, 432 (2014). [61] J. Alstott, S. Madnick, and C. Velu, PloS one 9, e95140
[58] D.A.Petrov, Genetica 115, 81 (2002). (2014).
[59] S. A. Teichmann and M. M. Babu, Nat. Gen. 36, 492 [62] G. Longo and M. Mont´evil, Springer-Verlag, Berlin Hei-
(2004). delberg 7160, 289 (2012).
[60] C. B. Victoria, T. Smieszek, H. Jianping, G. Cao, J. J. [63] M.C.SeilerandF.A.Seiler,RiskAnalysis9,415(1989).
Rainey, H. Gao, A. Uzicanin, M. Salath´e, Plos one, 9,
