Impact of community structure on information transfer

0.23 MB·English

by  Leon Danon

Impact of community structure on information transfer
Leon Danon,1,2 Alex Arenas,2 and Albert D´ıaz-Guilera1
1Departament de F´ısica Fonamental,Universitat de Barcelona, Marti i Franques 1, 08028 Barcelona, Spain
2Departament d’Enginyeria Inform`atica i Matem`atiques,
Campus Sescelades, Universitat Rovira i Virgili, 43007 Tarragona, Spain
The observation that real complex networks have internal structure has important implication
for dynamic processes occurring on such topologies. Here we investigate the impact of community
structure on a model of information transfer able to deal with both search and congestion simul-
taneously. We show that networks with fuzzy community structure are more efficient in terms of
packet delivery that those with pronounced community structure. We also propose an alternative
8
packet routing algorithm which takes advantage of the knowledge of communities to improve in-
0
formation transfer and show that in the context of the model an intermediate level of community
0
structureisoptimal. Finally,weshowthatinahierarchicalnetworksetting,providingknowledgeof
2
communitiesatthelevelofhighestmodularitywillimprovenetworkcapacitybythelargestamount.
n
a PACSnumbers: 89.75.Hc,87.23.Ge
J

9
I. INTRODUCTION abletohandlethesetwoimportantissuessimultaneously

] [8,10,15,16]. Inthesemodels,agentsarenodesofanet-
h
p The continuing intensity that accompanies the study workandcaninterchangeinformationpacketsalonglinks
- of complex networks has led to many important contri- in the network. Eachagent has a certain capability that
c
o butions in a variety of scientific disciplines (for a recent decreases as the number of packets to deliver increases.
s review see [1]). Specifically, the study of transportprop- The transition from a free phase to a congested phase
s. ertiesofnetworksisbecomingincreasinglyimportantdue has been studied for different network architectures in
c to the constantly growing amount of information and [8, 10], whereasin [15]the costofmaintaining communi-
si commodities being transferred through them. A partic- cation channels was considered.
y ular focus of these studies is how to make the capacity Thetopologyofthenetworkalsoplaysacentralpartin
h of the network maximal while minimising the delivery communication processes. In [16] the problem of finding
p
time. Both network packet routing strategies and net- optimal network topologies for both search and conges-
[
work topology play essential parts in traffic flow in net- tion for a fixed number of nodes and links was tackled.

2 works. It was found that in the free regime, highly centralised
v Traditionallyroutingstrategieshavebeenbasedonthe topologies facilitating search are optimal, whereas in
3 ideaofmaintainingroutingtablesofthebestapproxima- the congested regime decentralised topologies which dis-
2
tionoftheshortestpathsbetweennodes. Inrealisticset- tribute the packet load between nodes are favoured. It
1
2 tings, however, the knowledge that any one of the nodes has been shown that shortest path routing algorithms
. has about the topology of the network will be incom- are not optimal for scale free networks due to the pres-
9
plete. So, much of the focus in recent studies has been ence of communication bottlenecks [17] and several al-
0
on searchability. In particular, distributed search using ternative routing strategies have been proposed to take
7
0 only local information has been shown to be efficient in advantage of the scale-free nature of complex networks
: spatiallyembeddednetworks[2,3]. Networkswithscale- [11, 18, 19, 20, 21].
v
free degree distributions are particularly navigable using On the other hand, many networks found in nature
i
X local searchstrategies due to the presence of highly con- have been observed to have a modular or community
r nected hubs [4]. structure. Communities are those subsets of nodes that
a However,whenthenumberofsearchproblemsthenet- are more densely linked internally than to the rest of
work is trying to solve increases,it raises the problem of the network. Identifying communities in networks has
congestion at central nodes. It has been observed, both become a problem which has been tackled by many re-
in real world networks [5] and in model communication searchersinrecentyears(seeforexample[22,23,24],and
networks [6, 7, 8, 9, 10, 11, 12], that the networks col- for reviews see [25, 26]). Furthermore, communities are
lapse when the load is above a certain threshold and the often organisedin a hierarchical way [16, 27, 28, 29, 30].
observed transition can be related to the appearance of Thatis,largecommunitiesareoftencomprisedofseveral
the1/f spectrumofthefluctuationsinInternetflowdata smallercommunities. Despitealltheseefforts,theimpact
[13, 14]. that community structure has on information transfer
These two problems, searchand congestion, that have has not been considered.
sofarbeenanalysedseparatelyintheliteraturecanbein- The aim of this paper is two-fold: firstly, we will in-
corporated in the same communication model. Previous vestigate the effect that community structure has on the
workhas contributedacollectionofmodels that capture modelofsearchandcongestion,andsecondlywewillpro-
theessentialfeaturesofcommunicationprocessesandare pose an alternativerouting strategy anddemonstrate its
2
impact in the presence of community structure. In the ets move from their current position, i, to the next node
next section we will describe the model and recall the in their path, j, with a probability p . This probability
ij
most important analytical results. In Section II we will p iscalledthequalityofthechannelbetweeniandj. In
ij
consider the effect that a modular structure of varying thispaper,wetakethespecialcasethateachnodeisable
strengthhasonthebehaviourofthemodel. Wewillthen to send one packet at each time step. It is important to
showhow knowledgeofthis communitystructure canbe note,however,thatthemodelisnotdeterministic. Here,
takenadvantageoftoimprovetransportprocessesinnet- a packet which is waiting at a particular node, will be
works. And inthe finalsection,wegivesome concluding sent with equal probability as any other packet waiting
remarks. at the same node.
Thepacketsinthepresentmodelhavealimitedradius
ofknowledge,thatis,theyareabletodeterminewhether
II. COMMUNICATION MODEL anodewithinacertaindistanceristhedestinationnode.
In this case, the packet takes the shortest possible route
to the destination, otherwise, it travels down a link cho-
Thecommunicationmodelconsidersthattheinforma-
sen at random. In this paper we set r = 1, so that only
tion flowing through the networks is formed by discrete
nearest neighrbours are recognised. It has been shown
packets sent from an origin node to a destination node.
in previousworkthat in the free phase, there is no accu-
Each node is an independent agent that can store as
mulation at any node in the network and the number of
many packets as necessary. However, to have a realis-
packetsthatarriveatnodej is,onaverage,ρB /(S−1),
tic picture of communication we must assume that the j
where B is the effective betweenness of node j which
nodes have a finite capacity to process and deliver pack- j
is defined as the fractional number of paths that pack-
ets. That is, a node will take longer to deliver two pack-
ets take though node j and S is the number of nodes
ets than just one. A particularly simple example of this
in the network. A particular node will collapse when
would be to assume that nodes are able to deliver one
ρB /(S−1)>1 and the critical congestion point of the
(or any constant number) information packet per time j
network will be
step independent of their load, as in the model of decen-
tralised information processing in firms of Radner [31] S−1
and in simple models of computer queues [6, 7, 9, 15], ρc = B∗ (1)
but note that many alternative situations are possible.
∗
In the present model, each node has a certain ability where B is the maximum effective betweenness in the
todeliverpacketswhichislimited. Thislimitationinthe network, that corresponds to the most central node
ability of agentsto deliver informationcanresultin con- If the routing algorithm is Markovian, which is the
gestionofthenetwork. Whentheamountofinformation case here, it is possible to estimate Bj analytically. The
is too large,agentsare notable to handle allthe packets search and congestion process can be formulated as a
andsome of them remainundeliveredfor extremely long Markov chain, which is dependant on the packet transi-
periods of time. The maximum amount of information tion probability matrix. This matrix is derived from the
that a network can manage before collapse gives a mea- adjacency matrix of the network, the radius of knowl-
sureofthe quality ofits organisationalstructure. Inthis edger,andthesearchalgorithm. Usingthisformulation,
study, the interest is focused on when congestion occurs Bj of each node can be calculated analytically for any r
depending on the topology of the network [15]. [10]. In these cases, the paths the packets take will not
The dynamics of the model is as follows. At each be shortest paths. As the radius of knowledge increases,
time step t, an information packet is created at every Bj converges to shortest path betweenness and will be
node with probability ρ. Therefore ρ is the control pa- equal to it when r is greater or equal to the diameter of
rameter: small values of ρ correspond to low density of the network.
packets and high values of ρ correspond to high density .
of packets. When a new packet is created, a destina-
tion node, different from the origin, is chosen randomly
in the network. Thus, during the following time steps III. PACKET DYNAMICS OF
t+1, t+2,..., t+T, the packet travels toward its des- COMMUNICATION MODEL ON NETWORKS
tination. Once the packet reaches the destination node, WITH COMMUNITY STRUCTURE
it is delivered and disappears from the network.
The time that a packet remains in the network is re- The model from [10] can be further exploited to look
lated not only to the distance between the source and attheeffectsthatcommunitystructurehasondynamics.
the target nodes, but also to the amount of packets in To this end we need to be able to construct networks
its path. Nodes with high loads —i.e. high quantities of with controllable community structure. We choose to
accumulated packets— will take longer to deliver pack- use a family of pseudo-random networks since all other
ets or, in other words, it will take more time steps for properties (such as node degree and clustering) will be
packets to cross regions of the network that are highly equivalenttofullyrandomnetworks. Theonlythingthat
congested. In particular,at each time step, all the pack- we will vary is the strength of community structure.
3
First we employ the networks proposed in [32]. These each end of the bridge links will also be increased. As a
networksarecomprisedof128nodes which aresplit into result, the capacity of the network to deliver packets is
fourcommunitiesof32nodeseach. Pairsofnodesbelong- reduced in function of how fuzzy the community struc-
ing to the same community are linked with probability ture is.
p ,whereaspairsbelongingtodifferentcommunitiesare
in
joinedwithprobabilitypout. Thevalueofpinischosenso 105 0.2
that the averagenumber of links a node has to members
of any other community, Z , can be controlled. While
in
pin (and therefore Zin) is varied freely, the value of pout ets104 0.15
k
is chosen to keep the total average node degree, k, con- c
a
stant, and set to 16. As Zin is increased from zero, the g p103
communitiesbecomebetterdefinedandeasiertoidentify. n
To address the question of hierarchical structure we oati ρc 0.1
use a generalisation of the model of generation of net- of fl102
works with community structure that includes two hier- er
b
archicallevels of communities as introduced in [27]. The um 0.05
graphs are generated as follows: in a set of 256 nodes, N101 Z = 4
in
Z = 8
16 compartments are prescribed that will represent our in
Z = 12
first community organisational level. Each of these sub- Zin = 15
communities contains 16 nodes each. Furthermore, four 1000 0.05 0in.1 00 5 10 15 20
ρ
Z
secondlevelcommunitiesareprescribed,eachcontaining in
four sub-communities, that is 64 nodes each. The inter- (a) (b)
naldegreeofnodesatthefirstlevelZ andtheinternal
in1 FIG.1: (colour online) Each point represents thenumberof
degree of nodes at the second level Z are constrained
in2 packets averaged over 100 realisations in the steady state of
tokeepanaveragedegreeZ +Z +Z =18. From
in1 in2 out the dynamics. (a) Number of floating packets as a function
now on, networks with two hierarchical levels are indi-
of ρusingtheoriginal search algorithm in networkswith one
cated as Z - Z , e.g. a network with 13-4 means 13
in1 in2 level of community structure. The different colours denote
links withthe nodesofits firsthierarchicallevelcommu- varyinglevelsofcommunitystrengthascontrolledbythepa-
nity(moreinternal),4linkswiththerestofcommunities rameterZinandtheverticallinescorrespondtotheanalytical
that form the second hierarchical level (more external) predictionof theonsetof congestion (Section II,equation 1).
and 1 link with any community of the rest of the net- (b) Onset of congestion ρc for varyingZin.
work.
Asasimplemeasureofstructuralefficiencyofthenet-
From Fig. 1 we can see that the analyticalcalculation
work in terms of packet transport, we can consider the
fromSectionII, of the onsetof congestionρ agreesvery
c
number of packets present in the network. We allow the
wellwiththepointatwhichthenumberoffloatingpack-
dynamics to reach a steady state, which we detect by
ets diverges. As the strength of community structure is
considering the rate at which the number of packets in-
increased by raising Z , ρ is reduced. This seems logi-
in c
creases in the system. Once this rate becomes small,
cal,sincetheoriginanddestinationofpacketsarechosen
fluctuating around 0, we have reached the end of the
at random. It follows that the probability of creating a
transient. It is important to note that when ρ > ρ the
c packet with both origin and destination in one commu-
system never reaches a steady state, the mean number
nityis1/4. Allotherpacketswillnecessarilyhavetopass
ofpacketskeepsgrowinglinearlywith time, andthe rate
throughatleastonecentral,”bridge”nodethatconnects
never becomes very small. We also average over several
twocommunities. Thisleadstoanincreaseinthenumber
realisations,sincethe number ofpacketsinthe systemis
of packets that pass through bridge nodes, increasing its
subject to statistical fluctuations.
effective betweenness. As a result of receiving a dispro-
portionate amount of packets, these nodes will collapse
at lower values of ρ, leading to a cascade of collapses
A. Original communication model throughout the network. This effect becomes more and
more pronounced as Z increases, so, the stronger the
in
First of all we simulate the dynamics of the model de- community structure, the lower ρc.
scribed above, in which the packets have no knowledge Inthecaseofhierarchicalnetworks,weconcentrateon
ofthe topologyof the networkat the levelof community three different network topologies which are particularly
structure. Introducing community structure in the net- instructive, 13-4, 14-3 and 15-2. Once again the ana-
work topology over which the dynamics occur increases lytical calculation corresponds very well to the point at
thetrafficloadonthenodeswhichconnectcommunities. whichthenumberoffloatingpacketsdiverges,seeFig. 2.
This is in agreement with the finding that cutting links It is worth noting that these three networks have almost
withthehighestbetweennessseparatescommunities[22]. the same ρ . This is due to the fact that the average
c
It follows that the effective betweenness of the nodes at numberoflinkspernodebetweencommunitiesofsize64
4
isconstantandsetto1. Whatisvariedisthestrengthof ets105 105
k
theintermediateandinnermostlevelofcommunitystruc- ac104 104
p
ttuhrise.shIonwtshleittclaeseeffoefctt.he original communication model, oating 103 103
kets104 Number of fl111000201 ZZiinn == 44 om 111000201 ZZiinn == 88 om
c 0 0.02 0.04 0.ρ06 0.08 0.1 0.12 0 0.05 0ρ.1 0.15 0.2
ng pa103 packets110045 110045
oati102 ating 103 103
er of fl 101 mber of flo110021 ZZiinn == 1122 om 110021 ZZiinn == 1155 om
Numb100 111345---432 Nu1000 0.05 0.1 0.ρ15 0.2 0.25 0.3 1000 0.05 0.1 0.ρ15 0.2 0.25 0.3
0 0.005 0.01 0.015
ρ
FIG. 3: (colour online) Comparison of original search al-
gorithm with the modified for the same networks with com-
FIG.2: (colouronline)Numberoffloatingpacketsasafunc- munity structure. The number of floating packets is plotted
tion of ρ using the original search in networks with hierar- against ρ. Thefourpanelsdepict fournetworkswith varying
chical community structure on two levels. The vertical lines communitystrengthcontrolledbytheparameterZin,andthe
correspondtoanalyticalpredictionoftheonsetofcongestion. red points show the original search algorithm and the black
points denote the modified algorithm incorporating commu-
nity information.
B. Modifying the communication model destination node. Indeed, for lower values of Z , this
in
information is detrimental to efficiency, since the pre-
Clearly,networkswithstrongcommunitystructureare defined partitions of the network actually contain fewer
less efficient at delivering packets which are oblivious to internal links, compared to external ones. In this sce-
the underlying topology. But, what happens when we nario, packets are often sent to regions of the network
givethe searchprocesssomeinformationaboutthe com- whicharelesslikelytocontainthedestinationnode. This
munity structure? To address this question we propose is highlighted in Figure 4b. where we see that for very
a simple modification of the way packets are transferred low values of Z the original search algorithm collapses
in
between nodes. thenetworkatmuchhighervaluesofρthanthemodified
Let us consider a packet generated at node i in com- algorithm.
munity c with destination node j in community c . At When the strength of the community structure is in-
i j
each step in its path, the packet is given information of creased,the modified searchalgorithmimproves the effi-
thecommunityofneighbouringnodes. Shouldthepacket ciency ofthe networkconsiderably. For Z >8 [34], the
in
destination communitybe the same as that ofany of the onset of congestion in terms of ρ is considerably higher
neighboursofthe node thatis processingthe packet,the forthe modified searchalgorithm,andthe same network
packet is sent to one of those neighbours, otherwise it is is much more efficient at delivering packets for all values
sent down a link chosen at random. In this way, packets of Z > 8. In other words, the modified algorithm is
in
are able to arrive at the destination community without able to find more efficient routes to deliver packets and
necessarily arriving at the destination node. The idea is the network is able to handle a much higher load.
that once within the destination community, finding the In the modified searchalgorithm,the calculation from
destination node is easier. SectionII(equation1)isstillvalid,however,theanalytic
∗
In Figure 3 we plot the number of floating packets in calculation of B is more involved than in [10]. Never-
the network at the steady state againstthe packetinjec- theless we can estimate ρ of the network by looking at
c
tion rate ρ. The dynamics are performed on networks the point where the number of floating packets diverges.
with ad-hoc community structure of varying strength, In Figure 4a, ρ is estimated in this fashion. When the
c
controlled by the parameter Z , the average number of communities are extremely well defined, say Z = 15,
in in
links internal to the community. When Z =4 the net- flow through the network is restricted. So even though
in
workisequivalenttoanErdo¨s-Renyirandomgraphwith the search method of the packets is greatly improved,
128 nodes and 16 links per node. In this scenario, the and they are able to find the correct community in a
originalsearch algorithm performs much better in terms short number of steps, flow is restricted by the forma-
of ability to deliver packets. This seems logical: giving tion of bottlenecks at the interface between two com-
packets information about communities which are not munities. It emerges that an intermediate community
present will not improve the packet’s ability to find the structure strength, Z = 12 shows optimal efficiency in
in
5
104 0.2 105 105 105
s
et
ck104 104 104
a
p
ckets103 0.15 of floating 110023 110023 110023
oating pa102 ρc 0.1 Number 1100010 0.02 0.011143330---444.0 iii6===01406.081100010 0.02 0.011144440---333.0 iii6===01406.081100010 0.02 0.01114555---0222.0 iii===601406.08
of fl ρ ρ ρ
er (a) (b) (c)
umb101 Zin = 4 0.05 FIG. 5: (colour online) Number of floating packets in net-
N ZZin == 68 workswithhierarchicalcommunitystructure. Thenumberof
in
Z = 10 floating packets is show as a function of ρ for three slightly
100 Ziinn = 12 r = 1 differenthierarchicalnetworks,(a)13-4(b)14-3(c)15-2. For
Zin = 15 0 r = 1 com each, the level of community hierarchy information packets
0 0.1 0.2 0 5 10 15
ρ Z are given is varied between no information given, i = 0, in-
in
formation given at thefirst level i=4and information given
(a) (b)
at the second level i = 16. The vertical lines represent the
analytical calculation as in Section II for the original search
FIG. 4: (colour online) Onset of collapse for the modi-
algorithm.
fiedsearch algorithm asappliedtonetworkswithcommunity
structure at a single level. (a) Number of floating packets as
a function of ρ for the modified search. Different colours de-
community information at this level favours information
note varying community strengths, controlled by parameter
diffusion more, with ρ being 0.071, higher than in the
Zinandtheverticallinesnowdenotetheestimateoftheonset c
of congestion ρc. (b) ρc as a function of internal connectiv- i = 16 case. However in the case of the 14-3 network,
ity Zin for both the original model, r = 1, and the modified the opposite is true: giving information at the alterna-
model, r=1 com. tive, i=16 levelis (marginally)more beneficial. For the
15-2 network, giving more precise information causes a
considerable improvement.
terms of ρ . This suggests that for the flow to be opti- It is interesting to compare these results with other
c
mal there must be a balance between internal strength topological characterisations of complex networks. In
of communities and connections to other communities. particular,themostcommonmeasurerelatedtocommu-
For the case of networks with hierarchical community nitystructureis the modularitymeasure,Q,proposedin
structureasdescribedabove,communityinformationcan [32] which measures the quality of a particular partition
now be given at two levels. The packets can be given of a particular network. It is defined as follows:
information about the community structure on the first
level,thatis,theyaregivenknowledgeaboutwhichcom- Q=X(eii−a2i)
munityofthefourcommunitiesofsize64thedestination i
node belongs to. From here on, this is denoted as i=4. where the element i,j of the matrix e represents the
Alternatively, we can give nodes information on the sec- fraction of links between communities i and j and a =
i
ond level of community structure, so that packets know e . This value can also be measured at two levels.
Pj ij
which one of the 16 communities of size 16 the destina- One is at the firstlevelof the hierarchy,where nodes are
tion node belongs to, which we denote i=16. groupedin 4communities of64nodes each,whichcorre-
Once information about community structure is given spondstothei=4case. Theother,correspondingtothe
to the packets, the efficiency of the network to deliver i=16 case, is considering that the nodes are grouped in
these is increased considerably as in the case of single 16 communities of 16 nodes each. In the three networks
level community structure. The level of community in- we are considering, we only vary the strength of the sec-
formationwhichincreasestheefficacyofinformationflow ond level of community structure, so for the i = 4 case,
bythelargestamountisdependentonthetopologyofthe thevalueofmodularityremainsconstant. Forthei=16
network. Comparedwith no community informationbe- case however, the value of Q varies with the strength of
ing given to the packets, i = 16 increases the values of the second level. For the 13-4 network, the first level of
ρc almost fivefold, in all three networks. In the case of community structure is a better partition in terms of Q,
13-4, ρc is increased from 0.0132 to 0.064. A stronger whereas for 14-3 and 15-2 the second level is a better
community structure at the second level, 14-3 and 15-2 partition. See Table I for values.
does not make much of an impact when i = 16, with ρc For 13-4, where the best partition is found at the first
being 0.063 for both. level of community structure i = 4, giving packets in-
However, when information is given at the interme- formation about the same level improves the efficiency
diate level of community structure, i = 4, the differ- of the network more than giving information at the sec-
encesbecomemoreapparent. Forthe13-4configuration, ond level. For 14-3 and 15-2 the opposite is true: in
6
ρc Q of bottlenecks. In fact, the better defined the commu-
Net
i=0 i=16 i=4 i=16 i=4 nities are, the more affected packet transport becomes.
We have also shown that transport can be dramatically
13-4 0.0132 0.064 0.071 0.660 0.695
improved by providing packets with information about
14-3 0.0118 0.063 0.061 0.726 0.695
the community structure. And finally we have shown
15-2 0.0113 0.063 0.053 0.771 0.695
that the largest improvements are found when the par-
tition with the largest modularity is used to provide the
TABLEI:Tableofvaluesoftheonsetofcollapseρc inhierar-
information.
chical networks and the values of modularity of the same for
two levels of grouping. This suggests that it is possible to infer a priori what
kind of information should be given to packets to opti-
mise packet transport, just by identifying the commu-
bothcasesthebestpartitionisfoundatthesecondlevel, nity structure. By finding the communities at the level
i = 16, and the best flow in terms of ρc is found when of highest modularity, and providing information at this
giving information about the same level. In other words level,packettransportappears to be optimal. The ques-
thetwocoincide. Thismeansthatifcommunitiesareor- tion remains: is this is always the case? It certainly
ganisedina hierarchicalfashion,it isalwaysbest togive seems possible to improveinformationtransfer on anar-
informationat the levelwhere the maximum modularity bitrary network just by providing the search algorithm
is found. information about the community structure at the level
of highest modularity. Since maximising the modular-
ity measure is NP hard [33], all community detection
IV. CONCLUSIONS algorithms that depend on maximising modularity are
heuristicapproximationsandassuchdifferentidentifica-
Inthispaperwehavetakenadvantageofamodelincor- tionalgorithmsfinddifferentpartitionswithvaryingval-
porating search and congestion simultaneously to inves- ues of optimal modularity for real networks. The results
tigatetheimpactthatcommunitystructurehasoninfor- here suggest that giving information about the commu-
mationtransport. Wehaveshownthattransportiscom- nity structure as found by the most accurate algorithms
promisedwhencommunitystructureisintroducedinthe would be best. But this remains to be shown in the case
network since community structure implies the presence of real networks.
[1] S. Boccaletti, V. Latora, Y. Moreno, V. Chavez, D.- [16] R. Guimer`a, A. D´ıaz-Guilera, F. Vega-Redondo,
U.Hwang, Physics Reports 424 175-308 (2006). A.Cabrales,andA.Arenas,PhysicalReviewLetters89,
[2] J. Kleinberg, Nature 406, 845 (2000). 248701 (2002).
[3] H.P.Thadakamalla,RAlbertandS.R.TKumara,New [17] S. Sreenivasan, R. Cohen, E. Lopez, Z. Toroczkai, and
Journal of Physics. 9, 190 (2007). H. E. Stanley,Physical ReviewE 75, 036105 (2006).
[4] L.Adamic,R.M.Lukose,A.R.Puniyani,andB.A.Hu- [18] B. Tadi´c, G. J. Rogers, and S. Thurner, Review, Int. J.
berman, Physical Review E 64, 046135 (2001). Bifurcation and Chaos 17 (2007).
[5] V.Jacobson, inProceedings of SIGCOMM (ACM,Stan- [19] B. Danila, Y. Yong, A. Marsh, and K. E. Bassler, Phys-
ford, CA, 1988). ical Review E 74, 046106 (2006).
[6] T. Ohira and R. Sawatari, Physical Review E 58, 193 [20] B.Danila,Y.Yu,S.Earl,J.A.Marsh,Z.Toroczkai,and
(1998). K. E. Bassler, Physical ReviewE 74, 046114 (2006).
[7] A.Tretyakov,H.Takayasu,andM.Takayasu,PhysicaA [21] Z. Toroczkai and K.E. Bassler, Nature 28, 716 (2004).
253, 315 (1998). [22] M. Girvan and M. E. J. Newman, Proceedings of the
[8] A. Arenas, A. Diaz-Guilera, and R. Guimera, Physical National Academy of Sciences USA99, 7821 (2002).
ReviewLetters 86, 3196 (2001). [23] J. Reichardt and S. Bornholdt, Physical Review Letters
[9] R.Sole and S.Valverde,Physica A 289, 595 (2001). 93, 218701 (2004).
[10] R. Guimer`a, A. Arenas, A. D´ıaz-Guilera, and F. Giralt, [24] R.Guimer`a,S.Mossa,A.Turtschi,andL.A.N.Amaral,
Physical Review E 66, 026704 (2002). Proceedings of the National Academy of Sciences, USA
[11] B.Tadi´c,S.Thurner,andG.J.Rogers,PhysicalReview 102, 77947799 (2005).
E 69 036102(2004). [25] M.E.J.Newman,EuropeanPhysicalJournalB38,321
[12] L.Zhao, Y-C.Lai, K.Park,and N.Ye,Physical Review (2004).
E 71 026125(2005). [26] L. Danon, A. D`ıaz-Guilera, J. Duch, and A. Arenas, J.
[13] M.Takayasu,H.Takayasu,andT.Sato,PhysicaA233, Stat. Mech p. P09008 (2005).
824 (1996). [27] A. Arenas, A. Diaz-Guilera, and C. J. Perez-Vicente,
[14] J. Duch and A. Arenas, Physical Review Letters 96, Physical Review Letters 96, 114102 (2006).
218702 (2006). [28] A. Arenas, A. Diaz-Guilera, and C. J. Perez-Vicente,
[15] R.Guimer`a, A.Arenas, andA.D´ıaz-Guilera, Physica A Physica D:Nonlinear Phenomena 224, 27 (2006).
2001, 247 (299). [29] S.FortunatoandM.Barth´elemy,ProceedingsoftheNa-
7
tional Academy of Sciences, USA 104, 36 (2007). [33] U. Brandes, D. Delling, M. Gaertler, R. Goerke,
[30] M.Sales-Pardo,R.Guimera,A.Moreira,andL.Amaral, M. Hoefer, Z. Nikoloski, and D. Wagner, preprint p.
Proceedings of the National Academy of Sciences, USA arxiv.org:physics/0608255v2(2006).
104, 15224-15229 (2007). [34] Zin = 8 corresponds to Zout = 8, at which point most
[31] R.Radner,Econometrica 61, 1109 (1993). community identification algorithms are able to detect
[32] M.E.J.NewmanandM.Girvan,PhysicalReviewE69, the correct structure, [26].
026113 (2004).