Raising Graphs From Randomness to Reveal Information Networks

2.2 MB·

by  Róbert Pálovics

Raising Graphs From Randomness to Reveal
Information Networks
Róbert Pálovics1,2 András A. Benczúr1
1InstituteforComputerScienceandControl,HungarianAcademyofSciences
2BudapestUniversityofTechnologyandEconomics
[email protected], [email protected]
ABSTRACT We study the growth of information networks by consid-
ering processes where each node and edge is added to the
We analyze the fine-grained connections between the aver-
7 network only once, and no node or edge is deleted from the
age degree and the power-law degree distribution exponent
1 network. Our key finding is related to how the average de-
in growing information networks. Our starting observation
0 gree and the power-law degree distribution of the network
is a power-law degree distribution with a decreasing expo-
2 evolves over time. More specifically,
nentandincreasingaveragedegreeasafunctionofthenet-

n work size. Our experiments are based on three Twitter at- • the exponent of the power-law degree distribution in the
a mentionnetworksandthreemorefromtheKoblenzNetwork network decreases down to two over time, and
J Collection. Weobservethatpopularnetworkmodelscannot • theaveragedegreegrowsasa+cnb,wherenisthenumber

2 explain decreasing power-law degree distribution exponent of nodes in the network.
and increasing average degree at the same time. For example, as seen in Fig. 1, in graphs generated by the
] Weproposeamodelthatisthecombinationofexponential Baraba´si-Albertmodel[3],thedegreedistributionexponent
I
S growth, and a power-law developing network, in which new stays very close to constant. The starting point of our ob-
. “homophily”edgesarecontinuouslyaddedtonodespropor- servations is that in a real network, the degree distribution
s
tionaltotheircurrenthomophilydegree. Parametersofthe log-log plot lines get flattened as the networks grow.
c
[ average degree growth and the power-law degree distribu- Weemphasizetheimportanceoftheconstantaintheav-
tion exponent functions depend on the ratio of the network erage degree formula. The constant was considered negligi-
1 growth exponent parameters. Specifically, we connect the bleintheexperimentsofLeskovecetal.[15]. Inourresults,
v growth of the average degree to the decreasing exponent of however, the constant helps capture the mixture of edges
6 the power-law degree distribution. Prior to our work, only that appear at random vs. as a result of common interest,
0
oneofthetwocaseswerehandled. Existingmodelsandeven and fit to the actual measurements as seen in Fig. 2.
4
theircombinationscanonlyreproducesomeofourkeynew To our best knowledge, there is no graph model yet that
0
observations in growing information networks. captures both effects simultaneously. Leskovec et al. [15]
0
observe densification, a power-law growth for the average
.
1 degree. Their models apply to graphs where the exponent
0 1. INTRODUCTION of the degree distribution is less than 2 and remains con-
7 stant over time. They predict densification for networks
1 While information spread is a main effect in the world- with power law exponent larger than 2 that is the case for
v: wesipdeecisaolclyiahlanredtwtoorakn,atlhyezea,catsuanlomteedchamanoincsgoofththerisspbyroLceibssenis- all of our real networks, however they give no models.
i Ourexponentialgrowthandpreferentialattachmentbased
X NowellandKleinberg[16]. Onlinesocialnetworksarewidely model results in networks with initially slowly growing av-
used for information network analysis [2, 5, 6, 14]. While
r erage degree in addition to decreasing power law exponent.
a manyonlinesocialservicesexist,itishardtofinddatasets Themaindifferenceofourmodelcomparedtoearliermodels
thatrevealdetailedtemporalrecordsofthenetworkgrowth.
can be summarized in three points.
Theconceptof growth isablendoftwomechanismsthat
• The power law exponent, as in all our real networks, is
can be hardly separated. First, information networks grow
greater than 2 that could not be modeled in [15].
naturallybynewlyestablishedconnectionsbetweenindivid-
• Ourmodelexplainstheinitialbehaviorofthedegreesasa
uals. Second, onecanviewthegrowthasanetworkdiscov-
natural mixture of influence and preferential attachment
ery process. Interactions of users observed via information
edges, also correctly predicting the ratio of these edges.
flow partly reveal the hidden social network of them.
• Ourmodelgeneratestheeffectsofbothincreasingaverage
degree and decreasing power law exponent.
As a general overview of the possible models based on
ACMacknowledgesthatthiscontributionwasauthoredorco-authoredbyanem- ourobservations,networksstarttogrowatrandom,likean
ployee,contractororaffiliateofanationalgovernment. Assuch,theGovernment
Erdo˝s-R´enyi graph. Then certain rule such as preferential
retainsanonexclusive,royalty-freerighttopublishorreproducethisarticle,ortoal-
lowotherstodoso,forGovernmentpurposesonly. attachment [3] intensifies during the growth process, and
WSDM2017,February06-10,2017,Cambridge,UnitedKingdom causesscale-freedegreedistributionwithadecreasingexpo-
nent. The stronger the rule is, the closer the exponent of
(cid:13)c 2017ACM.ISBN978-1-4503-4675-7/17/02...$15.00
DOI:http://dx.doi.org/10.1145/3018661.3018664 the degree distribution gets down to two in a more coher-
1.1 Reproducibility
Alldatasetsusedinourmeasurementsareavailableinan
anonymized format at our website1. Measurements covered
inthenextsectionsarefurtherdetailedandillustratedwith
morefiguresinJupyterNotebooks2 thatareavailableinour
GitHub repository3.
2. RELATEDRESULTS
InformationspreadinsocialnetworkslikeTwitterorFace-
book have been analyzed in [2, 5, 6, 14]. Several properties
Figure 1: Degree distribution snapshots of grow-
ofthesenetworksarewidelyconsideredtobeapproximately
ing networks at different sizes. Left: The Barab´asi-
distributed as power law, with a few exceptions found in
Albertmodelyieldsfixedexponent. Right: TheOc-
[19]. Newman reviews the theoretical background of power
cupy data set (see Section 4) with flattening slope
lawfunctionsanddistributionsfoundinempiricaldatasets
as the network grows.
in [8, 17]. We rely on these calculations in Section 4, where
weestimatetheexponentofseveralpower-lawdegreedistri-
butions.
Alargenumberofmodelsdescribinggrowingnetworksop-
erate by assuming a constant average degree. For instance,
themodelofBaraba´sietal.[3]iscapableofgeneratingnet-
works with power-law distributions, but the average degree
of the network remains fixed during the growth of the net-
work. Anothermodelthatassumesconstantaveragedegree
is the popular copying model [12].
However,severalmeasurementsfromthepastreportthat
theaveragedegreeofagrowingnetworkincreasesovertime
[4, 11, 21, 22, 24]. Some state that the average degree is a
power-law function of the number of nodes. This effect has
Figure 2: Growth of the average degree in the Oc-
beennamedacceleratedgrowthbyDorogovtsevetal.[9],and
cupy data set (see Section 4).
densification law by Leskovec et al. [15].
Densification Laws. Leskovec et al. [15] mention that
the larger the exponent of the densification law, the denser
the network over the growing process. They demonstrate
densification law on six different networks, and introduce
thecommunityguidedattachmentmodelandtheforestfire
model. They claim that both can lead to densification law,
and give a theoretical explanation for community guided
attachment.
Related to our work, they derive a formula connecting
the exponent of densification law and the exponent of the
power-law degree distribution of the network. Their model
worksifboththedegreeandthedensificationexponentsare
constantovertime. Theyactuallypredictthatifthedegree
distribution exponent is fixed and larger than 2, then there
isnodensificationandtheaveragedegreeremainsconstant.
They observe that for degree distribution exponents larger
Figure 3: The Occupy data set when the number than 2 and changing over time, densification is possible in
of nodes is 1,000, with several disconnected com- the network, however they give no models in this case.
ponents, random node pairs, and some high degree In contrast to [15], we find that the increasing average
nodes. Isolated nodes are not drawn. degreeisnotasimplepower-lawfunctionofthenetworksize.
We connect this growth to the decreasing power-law degree
entnetwork. Asthedegreedistributionlog-logplotflattens,
distribution exponent. We propose models that reproduce
the chance for very high degree nodes in a strongly skewed
both increasing average degree and decreasing power law
distributionincreasesthatusuallyactasthemainorganizer
exponent in evolving networks.
of the network structure. The intermediate stage in the life
Work of Dorogovtsev et al. In [9], the notion of“ac-
of a growing network is illustrated in Figure 3, with the
celeratedgrowth”forpower-lawdegreedistributionwithin-
numberofnodesaround1,000. Onecanobserveseveraldis-
creasing average degree is introduced. Similar to Leskovec
connected random edges and some larger components with
etal.,theirformulasconnecttheexponentofthedegreedis-
high degree nodes that have already started evolving their
tribution and the accelerated growth exponent. The main
internalstructure. Asourmainresult,weaimtomodelthe
transitionfromarandomandmostlydisconnectedgraphto 1https://dms.sztaki.hu/en/letoltes/networkgrowth
a highly organized and very skew degree distribution net- 2http://jupyter.org/
work. 3https://github.com/rpalovics/networkgrowth
differenceisthattheypredictdensificationfornetworkswith of Dorogovtsev et al. and Va´zquez, vertex copying models
fixed degree distribution exponents above 2 as well. How- generate networks with fixed degree distributions.
ever,incontrasttoourkeyobservation,theexponentofthe Summary. We can conclude that numerous network
power-law degree distribution remains fixed in all models models result in power-law degree distributions where the
proposedin[9]. Furthermore,wefindadifferentgrowthfor exponentofthedistributionisthefunctionofthemodelpa-
theaveragedegreeandconnectthistothedecreasingpower rameters. However, it is common in all of the listed models
law exponent. thatforagivenparametersetting,thedegreedistributionre-
Another work of Dorogovtsev et al. is a growing network mainsthesameduringthegrowththenetwork. Besidesthe
model in [10]. The concept of the model is that while ran- aboveclassofpopularmodels,severalpapersreportpower-
dom users continuously connect to the network, edges ap- law growth for the average degree. The proposed models
pear at constant c rate between already existing nodes via forthiseffectmisstocapturethefactthatwhilethedegree
preferential attachment. The model results in a network increases,theexponentofthepower-lawdegreedistribution
with power-law degree distribution where the exponent can decreases in real networks when the exponent is above 2.
be derived from the only model parameter c (see [10]). A Furthermore,inourmeasurementswereportthattheaver-
similar model has been analyzed by Baraba´si et al. [4], and agedegreegrowsasa+c·nb,andthatisdifferentfromthe
byChungin[7]. Notethatwhilethemodelleadstodifferent densification law.
degree distribution exponents for different values of c, for a
givengrowthprocesscisalwaysfixedinthecitedworksand 3. POWER-LAWDEGREEDISTRIBUTIONS
the degree distribution remains the same.
In this section, we summarize the main properties and
Models of V´azquez. In contrast to densification law
the measurement of the exponent and the average degree
and accelerated growth, in [22] the average degree is mea-
in graphs with power-law degree distributions. We follow
sured to increase as a logarithmic function in the network
Newman [17] for properties of power-law distributions and
size. Va´zquez investigates network models that generate
[8] for fitting the exponent. Our main goal is to review and
power-law degree distributions with various exponents and
emphasize seemingly simple facts that often become prob-
introduces the triangle closing model. This dynamic model
lematic or misleading in experimentation. We also extend
has only one parameter u, and at each time step (i) with
the discussion to prepare the background for our new mod-
probability 1−u we add a new node to the network and
els.
connect it to one uniform randomly chosen previous node,
and(ii)withprobabilityuwecloseapossible,previouslynot 3.1 NotationsforDegreeDistributions
existing triangle in the graph. An in-depth analysis of the
A graph has power-law degree distribution, if the proba-
modelcanbefoundin[23],wheretheauthorclaimsthatthe
bilitythatagivennon-zerodegreenodehasdegreek inthe
processresultsinanetworkwithpower-lawdegreedistribu-
network, p(d(i)=k) is
tion and the exponent of the distribution depends on the
model parameter u. While the model is capable of generat- p(d(i)=k)=Ck−α, α>2.
ing networks with various degree distribution exponents, in
a fixed parameter setting, the exponent of the distribution Iftheexponentαisgreaterthan1,asinallofourexample
remains constant over time. networks, we can compute the value of C by
Random Edge Sampling by Pedarsani et al. They (cid:90) ∞ (cid:90) ∞
questiondensificationlawin[18]andstatethatdensification p(d(i)=k)dk=1, C =1/ k−αdk=α−1.
can be a result of the fact that measurements on networks 1 1
areusuallycarriedoutonedgessampledfromanunderlying The number of degree zero nodes in real networks is less
fixed network. In other words, they conclude that densifi- explored in the literature. For simplicity, we introduce the
cation may arise as a feature of the common edge sampling notation NZ for the fraction of nodes with non-zero degree.
procedure to measure dynamic networks. They show that
networkgrowthcanbeadirectconsequenceofthesampling 3.2 RelationofExponentandAverageDegree
process, therefore the sampling process itself is a plausible Byasimplecomputationoftheaveragedegreeinnetworks
explanation of network densification law. In Section 4.6 we with power-law distribution [8],
introduce a simple experiment indicating that the growth
of the average degree can not be explained by random edge d(n)=NZ·(cid:82)∞(α−1)·k−α·kdk=NZα−1 =NZ(cid:104)1+ 1 (cid:105). (1)
1 α−2 α−2
sampling. We simply shuffle our time series that contains
timestamped information about the growth of the network. Let∆:=α−2. NotethatbothNZand∆candependon
We notice that the uniform shuffling procedure yields sig- the size of the network n:
nificantly different average degrees at different sizes of the
d(n)=NZ(n)[1+1/∆(n)], (2)
network.
Vertex Copying Models. Another popular model or equivalently
thatiscapableofgeneratingpower-lawdegreedistributions
(cid:2) (cid:3)
∆(n)=1/ d(n)/NZ(n)−1 . (3)
is the vertex copying model. Here at each time step we
add a new node that chooses an ambassador node uniform
(2) indicates that a size dependent average degree in a net-
randomly. Next, we copy each of the connections of the
work with power-law degree distribution is caused by
ambassador node to the new node with probability q, or
• NZ(n), if the fraction of non-zero degree nodes signifi-
attachanewedgetothenewnodeuniformatrandomwith
cantly changes with n,
probability 1−q. The model leads to a power-law degree
• ∆(n), if the exponent of the degree distribution changes
distributionwithexponentα= 1−q. Similarlytothemodels
q during the growth of the network.
3.3 Maximum Likelihood Estimate of Expo- network numberofnodes numberofedges
nentandGoodnessofFit TwitterOccupy [oc] 364,649 1,004,961
Twitter15O [15] 55,976 120,632
Estimating the exponent of a power-law distribution can TwitterMH17 [mh] 1,366,691 3,489,068
be problematic in case of shifts in the distribution [20], or Facebookwallpost [fb] 46,947 183,335
when exponential cutoffs [17] modify the shape of the dis- DBLPcoauthorship [db] 1,314,050 5,362,414
tribution. To overcome these problems, specific estimates Wikipedia [wi] 1,870,709 36,532,531
have been proposed [8, 20] and reviewed in [19]. We follow
Table 1: Summary of the data sets used.
and extend the methodology of [8] by applying the maxi-
mum likelihood approximation for the exponent based on
the subset of nodes with a degree at least a certain x,
 −1
αˆ=1+|{i:xi ≥x}| (cid:88) lnxxi . (4)
i:xi≥x
Foreachx,wemaydefinetheKolmogorov-Smirnovstatistic
KS(x)betweentheobserveddistributionx andagenerated
i
power-law distribution of the corresponding αˆ.
A common method to fit the exponent is to select x
opt
that yields the lowest KS(x). However, in our experiments
inSection4.4,themaximumvalueturnsouttobeunstable:
forseveralxvalueswithverydifferentαˆ,KSisverycloseto
thatofx . Forthisreason,wewillcomparethreedifferent
opt
methods to fit the power law exponent in our experiments.
1. We compute the estimate αˆ with all i, i.e. x=1.
all
2. We select x that yields minimum KS and use the cor-
opt
responding αˆ .
opt
3. We use the entire set of estimates
αˆ :={αˆ for x with |KS(x)−KS(x )|<0.05)}. (5)
0.05 opt
4. MEASUREMENTSONREALNETWORKS
In this section, we measure the degree distribution in six
realnetworks. Allthesenetworksaregrownintimebypro-
cesses where each node and edge is added to the network
only once, and no node or edge is ever deleted. Our first
task is to measure the connection between the power law
exponent, the average degree, and the fraction of nonzero
degree nodes as in equations (2) and (3) as the networks
Figure4: Numberofedges(top)andaveragedegree
grow. We will also investigate the importance of time in
(bottom) as the function of the number of nodes.
how different types of new nodes and edges may appear in
these graphs. user. Otherwise,weaddedtheauthoruserofthetweetasa
new node without any connection to the network.
4.1 DataSets
KONECT is the Koblenz Network Collection of large
We use three Twitter mention networks and three com- network data sets [13]. We selected networks from the col-
pletelydifferentnetworksfromtheKONECTdatacollection lection with timestamped edges evenly distributed in time,
in our experiments. In Table 1 we summarize the sizes of anddegreedistributionclosetopowerlaw. Forexample,we
thedatasets,byshowingforeachnetworkthefinalnumber discarded networks that were crawled for several weeks but
of nodes n and the number of edges e. a significant part of their edges appeared within one day.
Twitter, a rich information source, lets us define several
4.2 AverageDegree
graphs: the follower network, the retweet graph formed by
retweetcascades,andthementionnetwork. Inourwork,we Inourmeasurementswefirstmeasurethenumberofedges
will analyze mention networks defined by authors mention- eandtheaveragedegreed:=2e/nasthenetworkgrows,as
ing other users by their Twitter user names in their tweets. thefunctionofthenumberofnodesn. Fig.4(top)showsthe
We collected data sets from Twitter that contain tweets growthofnumberofedgeseforthesixdifferentnetworksin
aboutacertaintopic,i.e. containagivenhashtag. Weused log-logplot. Forallnetworks,thenumberofedgesincreases
the OccupyWallstreet and the 15O tweet collection of [1]. super-linearly and apparently fit well to a power function,
Furthermore, we used our own crawl, where we collected as in [9, 15].
tweetsabouttheMalaysiaAirlinesFlight17. Westartedto Analyzingeasthefunctionofnmayhoweverbemislead-
download tweets immediately after the first news reports. ing, since e already grows at least linearly with n and the
Wedefinedourgraphsbysortingthetweetsbytime. Ifa functionwillautomaticallybeveryclosetolinearaslongas
tweetcontainedamentioninourdata,weaddedadirected the average degree is small. Indeed, as seen in Fig. 4 (bot-
edge to the network between the author andthe mentioned tom), initially the average degree is around 1, and it slowly
a b ln(c) c RMSE
oc 0.8 0.29 -2.02 0.132 0.0528
15 0.72 0.21 -1.26 0.284 0.073
mh 0.3 0.24 -1.1 0.33 0.078
fb 1.1 0.93 -7.2 0.00075 0.023
db 1.1 0.41 -3.9 0.02 0.091
wi 1.4 0.6 -5.4 0.00452 0.099
Table 2: Parameters of the fitted a+c·nb curves for
the average degree in case of the six data sets.
∆n ∆e
Z 1 0
R 2 1
I 1 1
H 0 1
Table 3: Growth events. For each event, the orange
part is added to the network.
data. For our six networks at different sizes, we estimate
Figure5: Fittedcurvesforthegrowthoftheaverage
∆(n)=α(n)−2 and d(n).
degree for the six different networks.
WefirstinvestigateNZ(n),thefractionofnodeswithnon-
starts to increase in all of our networks, which results in a zerodegreeinourdata,sinceequation(2)alsoinvolvesthis
lineargrowthofefor3–4ordersofmagnitude,withoutany function. The networks from the Koblenz collection do not
structural reason. containzerodegreenodes,NZ(n)=1. TheTwittermention
We find that the increase of the average degree as the networks include users that have adopted the certain hash-
function of n can be best modeled by tag, but have not been mentioned. For this reason, NZ(n)
is typically less than one for the Twitter data sets, as seen
d(n)=a+cnb =a+eb·ln(n)+ln(c). (6) in Figure 7. Interestingly, NZ(n) is roughly constant in all
three cases.
In Fig. 5 we plot the average degrees as the function of
Figure 8 shows d(n) and ∆(n) for all the six data sets at
log n. We show the fit to equation (6) for each network
10 exponentially growing size of the networks. On the left, we
separately. Our results indicate good fit. We summarized
show the observed d(n), while on the right, the fitted ∆(n)
the parameters of our fitted curves in Table 2, where we
by the methods of Section 3.3, i.e. the values αˆ , αˆ and
indicated the root mean squared error (RMSE) for each fit. all opt
the set αˆ as in equation (5). Note that for all networks
0.05
the exponent is always above 2.
4.3 Evolving Exponent of Power-law Degree
Next, for both sets of plots of in Figure 8, we show the
Distributions
transformationbetweend(n)and∆(n). Ontheleft,weplot
Next we analyze the degree distribution of the observed the result of equation 2 when applied to all the fitted α
networks that we compute at exponentially scaled sizes. In values from the corresponding plot on the right. And on
Fig.6,weplotthedegreedistributionwhennreaches2i,i= the right, we show the result of equation (3) applied to the
5,6,7,...forthesixdifferentnetworks. Thedistributionsare observed average degree. The exponent is always above 2,
exponentially binned and shown on log-log scale. andasitgetscloserto2,∆takesverysmallvaluesshownon
For all data sets, we may visually observe that the expo- logarithmic scale, and d(n) increases in the network. Since
nent of the distribution decreases as the network grows that the calculation for the average degree tends to infinity as
we will confirm by the methods of Section 3.3 in the next thepowerlawexponenttendsto2in(2),thevalueishighly
subsection. We may conclude that the degree distribution sensitive to the variation of the fitted exponent when it is
changes in time such that the probability of larger degree close to 2. Keeping the sensitivity of the estimates near
nodes increases while the power law exponent decreases. exponent2inmind,weobservethattheconnectionbetween
theaveragedegreeandtheexponentestimatesareconsistent
4.4 DegreeDistributionandAverageDegree
with the equations (2) and (3).
In this section we connect two previous observations, the AsthemainconclusionofthemeasurementsinSections4.2–
growthoftheaveragedegreeandthedecreaseofthedegree 4.4, in all of the six data sets we observe that
distribution exponent. Recall from Section 3.2 that d(n) is • theexponentofthepower-lawdegreedistributiondecreases
the average degree and α(n) is the degree exponent of the – i.e. nodes with higher degree appear more likely – but
networkasfunctionsofthenumberofnodesn. Thesestatis- always stay above 2;
ticsareconnectedviathenotion∆:=α−2inequations(2) • the average degree grows as a+c·nb;
and (3). In particular, (2) predicts a connection match- • the connection between the growing average degree and
ingourobservationthattheaveragedegreegrowswhilethe theshrinkingexponent,uptotheuncertaintyintheesti-
exponent shrinks. Next we investigate how well the pre- mation of the exponent, is as predicted by the power-law
dicted connection between degree and exponent fits to our distribution.
Figure 6: Degree distributions for different sizes of
the six networks. log n is indicated in the figures.
2
From left to right and top to bottom: oc, 15, mh,
fb, db, wi.
Figure 7: Fraction of nonzero nodes NZ(n) for the
three Twitter data sets.
4.5 Edge Formation: From Random to Joint
Figure 8: Left: Comparison of the average degree
Interest
(red curve) and its estimates computed from the
Thegrowthoftheaveragedegreecouldbeanaturalresult exponentestimatesbyusing(2). Right: Comparison
ofneweventsthatgenerateedgesalreadyhavingoneorboth of the estimates of ∆. The red curve is based on (3).
endpoints in the network. Initially, while the network is
very small, edges that connect two nodes will likely appear can appear without any connection to the already existing
between nodes that have not yet been part of the network, part to the network (Z). Second, a new random edge may
hence the number of edges will increase by one but at the appear with nodes that are both new in the network (R).
sametime,thenumberofnodesmayincreasebyoneoreven Third, one node can appear and immediately join with a
by two. Later however, when many of the eventual nodes new edge to the network (I). This event may be a result
have already joined the network, the events will bring an of influence between the nodes. Finally, two existing nodes
extra edge but no new nodes, thus increasing the average can connect to each other as a result of common interest,
degree. possibly also involving preferential attachment that we call
Inordertocharacterizeeventswithoneortwoendpoints homophily edges (H).
already in the network, we distinguish four different events Tomeasuretheincreaseinaveragedegreeasthepossible
intheprocessasdetailedinTable3. First,anewrootnode result of increased homophily events H, we measure the ra-
actual growth of the network always yields denser graphs
than random sampling from the final snapshot of the net-
work.
5. MODELING
WeshowedinSection4.2thattheaveragedegreeincreases
in evolving networks. Next, in Section 4.3 we saw that the
exponent of the degree distribution of a growing network
decreases. In Section 4.4 we connected the two observa-
tions, and concluded that while the evolving exponent of
thepower-lawdegreedistributiondecreases,theaveragede-
gree grows. Finally we demonstrated in Section 4.5 that as
more and more edges appear between existing nodes, natu-
rallytheaveragedegreeincreases. Weintendtogivenetwork
models that capture these effects simultaneously, i.e. give a
dynamic network model with the following requirements:
• increasing average degree according to a+cnb,
• decreasing power law exponent that tends to 2,
• vanishing fraction of random edges,
• increasing number of homophily edges.
Asfarasweknow,thereexistsnonetworkmodelyetthat
Figure 9: Ratio of different types of edges as the
isabletodescribeallourobservations. Modelsthatexplain
function of n. R, I, and H are defined in Table 3.
increasingaveragedegreeareoftencalleddensificationlaws
[15]. InSection5.1wewillreviewhowexistingdensification
law models predict our other three observations. Then in
Section 5.2 we detail the model of Dorogovtsev et al. [10]
that is closest to our new set of models. In Section 5.3,
we introduce our basic model of growing networks with de-
creasingpower-lawdegreedistribution. Afterextendingthe
model in 5.4, we verify our models with simulations in Sec-
tion 5.5.
5.1 RelationtoDensificationLaws
Modelsgivenby[9,15]foracceleratedgrowthanddensifi-
Figure 10: Result of shuffling the events in the time cationlawareforgraphswithfixeddegreedistributionexpo-
series of the Occupy (left) and MH17 (right) data nents. Moreover,bothofthempredictfortheaveragedegree
sets. The original graph is always denser than the power-law growth, d(n)=cnb. As stated above, in our ex-
randomized one. periments we measured that the average degree first slowly
growsfromaconstantvaluethatisaround1. Furthermore,
tio of different edge types (R,I,H) as the function of the
noneoftheproposedmodelsforacceleratedgrowthandfor
network size. The plots for the six data sets are in Fig-
densificationarecapableofgeneratingpower-lawdegreedis-
ure 9. The most important observation is that the ratio of
tributionwithadecreasingexponent. Finally,inSection4.6
randomedgesRdeclines,andinfluenceeventsI areroughly
weshowedthatthesimpleuniformedgesamplingproposed
constant,andtheratioofH increasesasthenetworkgrows.
by Pedarsani et al. [18] leads to a different network growth
process.
4.6 NoFittoUniformEdgeSampling
5.2 TheConceptofPreferentialEdges
Pedarsani et al. [18] state that the growth of the average
degreecanbearesultofuniformedgesamplingfromahid- ClosesttoourresultisagrowingnetworkmodelofDoro-
den network. We made a simple experiment to investigate govtsev et al. [10]. The model is capable of generating net-
theirassumptions. Werandomizedtheedgesequenceofthe works with power-law degree distribution. In this genera-
network, and measured the growth of the average degree tivemodelonepartoftheedgesareaddedbetweenalready
in the randomized time series. In other words, instead of existing preferentially chosen nodes in the network. More
observing the edges in a temporal order according to their specifically, at each time we perform two steps.
creationtimestamps,weprocessedtheminarandomorder. • A new node is added to the network with c new edges.
InFigure10wecomparethegrowthofd(n)fortherandom- The endpoints of the edges are selected at random with
izedandtheoriginaltimeseriesintheOccupy(left)andthe preferentialattachment. Ifthenumberofedgesiseinthe
MH17(right)datasets. Theresults,accessibleasdescribed network, then the probability that node i with degree d
i
in Section 1.1, are similar for the remaining data sets. gets connected to the new node is d /2e.
i
Accordingtoourmeasurements,thegrowthoftheaverage • cnewedgesareaddedbetweenalreadyexistingnodes. If
degreecannotbecompletelyexplainedbyrandomsampling nodes i and j have degrees d and d , the probability of
i j
from a hidden social web. While the average degree gener- an edge between them is proportional to d d .
i j
ally grows in both cases, it is always larger in the original Themodelresultsinanetworkwithpowerlawexponent,
timeseriesthanintherandomizedone. Inotherwords,the whosevaluecanbederivedfromasinglemodelparameterc,
see[10]. Sinceforafixedc,weobtainafixedexponent,the
model cannot explain the shrinking phenomenon. Similar rn
models have been analyzed by Baraba´si et al. [4], and by
Chung in [7], both of which yield a constant exponent.
2se
5.3 ModelI:RandomandHomophilyEdges h
Next we introduce our new model that relies on prefer-
ential attachment, and can generate growing networks with
decreasing power law exponents. In our first growing net-
n
workmodelweaddateachtimestep(i)randomnewedges
that connect two new nodes in the network, (ii) and edges
betweenalreadyexistingnodesinthenetwork. Morespecif-
ically, as indicated in Figure 11, at time t:
• For some constant r, r·n(t) new random edges appear
that indicate the random growth of the network.
Figure 11: Visualization of Model I. At each time t,
• Eachnodeiselectsothernodestoconnectwithhomophily
we add rn(t) random, and 2se (t) preferential edges.
edges randomly. Theexpectednumberofnewhomophily h
connectionscreatedbynodeiiss·dh(i),wheredh(i)isthe that tends to 1 as t → ∞. (12) and (13) are consistent
homophilydegreeofnodei,i.e. thenumberofhomophily with our measurements in Section 4.5 where we observed
edges connected to node i. For a given new connection thatthenumberofrandomlyappearingRedgesdecreaseas
of node i, the target node is selected by preferential at- the network grows while the number of homophily H edges
tachment. In other words, the probability of selection for increases over time.
node j as a new neighbor of i is d(j). In our model, at the beginning of the growth random
First we analyze the growth of the average degree in the edges dominate and the network is similar to an Erdo˝s-
model. The number of nodes n(t) in the network at time R´enyigraph. Then,theaddedpreferentialedgesslowlystart
t can be derived from the number of created random new to dominate the process as computed in (13). This effect
edges by time t. Each new random edge brings in two new pushes the network towards a scale-free graph. The model
nodes, therefore of Dorogovtsev et al. [10] adds preferential edges between
dn already existing nodes at constant rate c at each time step.
dt =2rn, n(0)=N0, n(t)=N0e2rt, (7) Theirmodelresultsinpower-lawdegreedistributionwitha
fixedexponent. Inourmodelasthenumberofpreferentially
where N is the initial number of nodes in the network.
0 added edges increase during the growth, the exponent will
The number of generated random edges e (t) is half of the
r start to decrease. We verify this assumption that is based
number of nodes,
on the derivations of [10] by simulations in Section 5.5.
e (t)=1/2·n(t)=N /2·e2rt. (8)
r 0 5.4 Model II: Extension with Influenced and
The number of preferentially added edges eh(t) can be RootNodes
derived from a separated equation. At each time step the
In this section we extend our model of random and ho-
expected number of preferentially crated edges by node i is
mophily edges by introducing two additional events.
s·d (i), therefore
h • We add p·n(t) random influenced nodes that newly con-
deh =(cid:88)sd (i)=2se ,e (0)=H ,e (t)=H e2st, (9) nect to the network with one edge. For a new influenced
dt h h h 0 h 0 node, the influencer is selected uniform random from the
i
already existing nodes in the network.
where H0 is the initial number of the homophily edges. • We add q ·n(t) random root nodes that appear in the
As the number of edges e(t) = eh(t)+er(t), the average network with 0 degree.
d(t) is then With similar notations, the number of nodes is
d(t)= 2(er(t)+eh(t)) = N0e2rt+2H0e2st (10) dn =(p+q+2r)n, n(0)=N , n(t)=N e(p+q+2r)t. (14)
n(t) N0e2rt dt 0 0
=1+ 2H0e2(s−r)t =1+ 2H0 n(t)rs−1 (11) The number of influenced nodes ni(t), the number of zero
N0 N0s/r degreerootnodesnz(t),andthenumberofnodesaddedby
random edges n (t) are then
Note that the model generates a growing average degree r
thatincreasesfromconstant1. Theexponentofthepower- n (t)= p N e(p+q+2r)t, (15)
law second term is s/r−1 that is larger than 0 if s>r. i p+q+2r 0
The fraction of added random edges is n (t)= q N e(p+q+2r)t, (16)
z p+q+2r 0
e (t) 1 1
r = = . (12) 2r
er(t)+eh(t) 1+eh(t)/er(t) 1+2H0/N0erst nr(t)= p+q+2rN0e(p+q+2r)t. (17)
If s > r, that tends to 0 as t → ∞. Similarly, the fraction
Hencethenumberofinfluenceedges,rootedgesandrandom
of homophily edges is specified by the sigmoid function,
edges are
e (t) 1
er(t)h+eh(t) = 1+N0/(2H0)erst (13) ei(t)=ni(t), ez(t)=0, er(t)=1/2nr(t). (18)
Figure 12: Results of simulations for Model I. Dif- Figure 13: Comparison of estimated and calculated
ferent values of s correspond to different colors as b values for Model I. Calculated values are based on
noted above. Left: d(n) Right: ∆(n). the setting of s and r, b=s/r-1. The estimated val-
uesarecomputedfromthesimulatedaveragedegree
Thedifferentialequationforthenumberofhomophilyedges
curves.
remains the same,
deh =(cid:88)sd (i)=2se ,e (0)=H ,e (t)=H e2st (19)
dt h h h 0 h 0
i
The average degree based on the previous formulas is
2(e (t)+e (t)+e (t))
d(t)= r i h (20)
n(t)
2 r+p N e(p+q+2r)t+2H e2st
= p+q+2r 0 0 (21)
N e(p+q+2r)t
0
2(r+p) 2H
= + 0e(2s−p−q−2r)t (22)
p+q+2r N
0
= 2(r+p) + 2H0 n(t)p+2qs+2r−1. (23)
p+q+2r N2s/(p+q+2r)
0
Notethattheprocessgeneratesagrowingnetworkwhere
the average degree grows by d(n)=a+c·nb, where
• a= 2r+2p . In other words, the fraction of nodes added
p+q+2r
by influence p, the fraction of nodes added as root q, and
the fraction of nodes added by random edges r sets to-
gether the constant term.
• b= 2s −1 is the exponent of the power-law growth.
p+q+2r
• c= 2H0 is the multiplier of the power-law term.
N2s/(p+q+2r)
0
Themodeliscapableofgeneratingnetworkswithisolated
nodes as in our Twitter data sets. In case of networks from
the Koblenz Network Collection, the average degree is al-
ways greater than 1, and there are no root nodes in the
growing network. By setting the proper fraction of random
edgesrandinfluencednodesp,andsettingq=0,themodel
Figure 14: Results of fitting the model to real data
is also capable of modeling these networks.
sets. Left: Occupy, right: Facebook. Top: Growth
5.5 VerifyingtheModelsbySimulations of the average degree. Middle: Decrease of the ex-
ponent of the degree distribution. Bottom: Visual-
First we investigate the properties of networks generated ization of the degree distributions at different net-
by Model I of Section 5.3. In Fig. 12 we fixed H0 = 2,r = work sizes.
0.05,andransimulationsatdifferenthomophilyratess. On
the left, we show the increase of the average degree, while Next we turn to analyzing how Model II fits to real net-
ontheright,theestimateof∆whilegrowingthesimulated works. We present the experiments for Occupy and Face-
network. Note that different colors correspond to different book; for the other networks, see Section 1.1. For both
values of s, and we ran 40 different simulations in case of data, we compute parameters p,q,r,s,N and H from the
0 0
each parameter setting. The results indicate that s sets the values of a,b and c in Table 2. Then we analyze d(n) and
velocityofthegrowthofd(n). Furthermore,∆(n)decreases ∆(n) in networks simulated by the corresponding parame-
in our simulations at each value of s. ters. Specifically we set p = 0.002, q = 0.022, r = 0.038,
InFig.13,weshowthatsimulationsfitwelltothemodel s = 0.0645, N = 14, H = 2 for Occupy, and p = 0.0089,
0 0
equation (11). We estimated b from the observed function q = 0.0, r = 0.04, s = 0.0857, N = 85, H = 2 for Face-
0 0
d(n) and compared that to s/r−1 that we calculated from book.
the model parameters. Our results indicate the growth of In Fig. 14, we show our measurements for the two data
the average degree in our simulations follows (11). sets in separate columns. By measuring in the real and
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