Structure and function in flow networks

0.81 MB·English

by  Nicolás Rubido

Structure and function in flow networks
Nicola´s Rubido,1,2 Celso Grebogi,1 and Murilo S. Baptista1
1Institute for Complex Systems and Mathematical Biology,
University of Aberdeen, King’s College, AB24 3UE Aberdeen, UK
2Instituto de F´ısica, Facultad de Ciencias, Universidad de la Repu´blica, Igua´ 4225, Montevideo, 11200, Uruguay
(Dated: January 15, 2013)
This Letter presents a unified approach for the fundamental relationship between structure and
function in flow networks by solving analytically the voltages in a resistor network, transforming
the network structure to an effective all-to-all topology, and then measuring the resultant flows.
3 Moreover, it defines a way to study the structural resilience of the graph and to detect possible
1 communities.
0
2 PACSnumbers: 41.20-q,95.75.Pq,89.40.-a,89.75.Hc,89.75.Fb
Keywords: ResistorNetworks,MaximumFlows,GeneralizedInverseLaplacianMatrix,Communities.
n
a
J
The relation between structure and function is one of vectors of this matrix.
0
1 the most studied topics in the theory of complex net- Second, we show that the equivalent resistance (also
works [1–7]. The structure is a topological representa- known as two-point distance resistance) formula [13] is

] tion of the interacting elements forming a network and retrieved[Eq. (7)] without further assumptions from the
h
p it is rigorously described by the theory of graphs. Be- analyticalexpressionforthe voltagesinducedby the sin-
- haviour is a functional observable of the network and gle s-t pair. The equivalent resistance between any two
c canbemeasuredbyavarietyofdifferentapproachesand nodesisatopologicalmeasurethatgivesfurtherinforma-
o
s methodologies, e.g., it can be quantified as a statistical tion about the structure of any graph. It is shown here
. descriptionoftheweightsofconnectionsofthe structure howitconstitutesahighlyusefultooltoreducethecom-
s
c of the network [1–4] or depend on the dynamics of inter- plexity of any network structure: an arbitrary topology
si actions among the elements. is transformed to an effective all-to-all complete graph.
y
This work deals with flow networks that satisfy con- Third,usingboththevoltagedifferencesandtheequiv-
h
servationlaws(Kirchhoff’slaws). Themodelfortheflow alent resistance, an effective functional flow adjacency
p
[ networkisstatedin termsofresistornetworks(weighted (EFFA) between any two points in the circuit is con-
symmetric graphs), which have a source node s and a structed[Eq.(8)]. Theseeffectiveflowsarethe realmea-
1
sink node t feeding the system, and a linear relationship surable observables that are obtained when a probe is
v
4 between voltages and currents [8]. The problem models placedbetweenanytwonodesinthenetwork. TheEFFA
0 a wide variety of physical phenomena, such as river flow matrixexpressesthelevelofconnectivityintermsofflow
6 channels,transportnetworks,ionchannels,andelectrical transmission; the functional relationship among the net-
2 circuits. Its solutions establish a clear relationship be- work components in the effective all-to-all topology. In
.
1 tween the topologicalstructure of the networks(namely, particular, if two nodes are not directly connected, an
0 adjacencymatrixandedgeweights,assumedknown)and effective current between them may still exist.
3
the functional flows passing through nodes and edges
Fourth, as an application of our approach, we show
1
(that are a consequence of solving the flow model). The
: howtoidentify the nodesthataremainlyresponsiblefor
v foundationofnetworkflowtheoryrootsbacktoKirchhoff
thetransmissionofflow,howtodetectqualitativelycom-
i
X [9], answering what are the current flows in an electrical munities of the network, and how to infer maximal node
circuit when a set of voltages is applied. The solution is
r outflow for different graph topologies and resistor distri-
a then achieved by solving Kirchhoff’s equations [9–11].
butions. This reveals topological issues that are related
In this Letter, we present a unified approach for the to node maximum capacities in a natural manner.
fundamentalrelationshipbetweenstructureandfunction The starting point for our unified approach is the cal-
inflow networksby calculatingvoltages(loads)whenin- culationofvoltagesinanetworkbysolvingtheVFprob-
put currents (flows) are given. The main results of this lem. The VF problem is set by interpreting the graph
approach are the following. structure as a linear resistor network circuit, which then
First, a novel formula for the voltages in the flow receives an input flow I at node s and leaves at node t
network model, i.e., the Voltage-Flow (VF) problem (a current I enters the circuit at a source and leaves at
[Eq.(2)],isanalyticallyderived. Thesolution[Eq.(6)]is a sink). The network structure is given by a connected
expressedintermsofthe knowngraph’sweightedLapla- strict graph G = {V, E} (where V and E are the sets
cianmatrix eigenspacebase. Inparticular,we showthat of nodes and edges, respectively) with symmetric edge
the behaviour of the flows is not only dependent on the weights W ≡A /R for i, j = 1,..., N (N being the
ij ij ij
matrix spectra, but intrinsically connected to the eigen- number of nodes in V, A the ij-th element of the adja-
ij
2
cency matrix, and R the edge’s resistance). However,removingthenulleigenspaceofthespanning
ij
Imposing conservation of charge in every node of the setofeigenvectors,therankofthenewGinEq.(4)isN
network, i.e., a graph where at each node the current and inversion can be directly performed. Consequently,
arriving at it equals the amount leaving it (first Kirch- eachelement of the generalizedinverseLaplacianmatrix
hoff’s law), the flow vector is (F~(st))i = I (δis−δit) for Zij ≡(G−1)ij is found from
i=1, ..., N, with δ being the Kronecker delta. Thus,
ij N−1
using the laws of Kirchhoff and Ohm, the net current in 1
Z = (~v ) (~v ) , (5)
any node is given by ij nX=1 n i λn n j
N with the new index n = k−1 and i, j = 1,...,N. It is
A
Fi(st) =I (δis−δit)= Rij Vi(st)−Vj(st) , (1) easy to show that Nk=1GikZkj =δij −1/N, where the
Xj=1 ij (cid:16) (cid:17) extra constant factPor comes from the incompleteness of
the restricted eigenspace and it cancels out when calcu-
where V(st) is the voltage potential at node i given the lating voltage differences.
i
particular s-t pair. Equation (1) is rearrangedsuch that Returning to Eq. (2), the solution for the VF problem
the VF problem is expressed in matrix form atagivennodei∈V is V~(st) = N Z F~(st) =
j=1 ij
(cid:16) (cid:17)i (cid:16) (cid:17)j
GV~(st) =F~(st), (2) I (Zis−Zit), P
N−1
where the upper-indexes indicate that the problem de- V~(st) =I (~vn)i [(~v ) −(~v ) ] . (6)
wpeenigdhsteodnLthaepllaocciaatniomnaotfrtixh.eTs-hteneondtersieosnoVf GanadreG is the (cid:16) (cid:17)i nX=1 λn n s n t
This formula is the backbone of our unified approach.
N W if i=j, It is applicable to any connected weighted graph and is
G = k=1 ik (3)
ij (cid:26)P−Wij if i6=j. extendible to many sources and sinks with different in-
puts and outputs as long as flow conservation is fulfilled
Then, the VF solution is achieved once the voltages in ( F(st) = 0 and F = 0 ∀i 6= s or t). Furthermore, it
each node are found from inverting G. However, the aPlloiwsitoderivetheeiquationfortheequivalentresistance
rank of G is N −1 (det(G)= Nn=−01λn =0) and direct between any two nodes, to construct the effective flows,
inversion is not possible [11, 12Q]. and later apply them to, e.g., community detection.
TheLaplacianinversionisaddressedherebyadifferent Thederivationoftheequivalent resistance ρ between
ij
interpretationofthederivationofthegeneralizedinverse any two nodes [13] is done as follows. Set the source at
matrix, also known as Moore-Penrose matrix [11, 13]. a starting node (i = s) and the sink at an ending point
The eigenvalue problem for the Laplacian is given by (j = t), then Eq. (6) provides an exact closed formula
GP= PΛ, where Λij = δijλn (n = i−1 = 0,...,N − for ρij in terms of eigenvalues and eigenvectors of the
1) is a diagonal matrix containing all eigenvalues and weighted Laplacian matrix. The reason for doing this is
P={~v0,~v1, ...,~vN−1}isthematrixwhosecolumnsare thatthevoltagedifferencebetweenthesetwonodesgives
the eigenvectors of G. We prove here that to express the incoming currenttimes the resistancebetween them,
any element of G, the only elements needed from the hence, (V~(st)) −(V~(st)) = Iρ . Therefore, using the
s t st
spectral decomposition are the non-null eigenvalues and same procedure for every pair of nodes in the graph all
corresponding eigenvectors. This results in the following ρ ’s are determined,
ij
decomposition of G
N−1
1 2
G=PrΛrPTr , (4) ρij = λ (~vn)i−(~vn)j , (7)
nX=1 n h i
where T indicates transpose, P ∈ RN×(N−r), Λ ∈
r r providinga solutionthat is independent of where the s-t
R(N−r)×(N−r),andrbeingthenumberofconnectedcom-
pair is placed.
ponents of G, which in this Letter is fixed at r =1.
The matrix element ρ is a topological measure that
ij
Equation (4) is derived by analyzing each Laplacian
weighs all paths between any pair of nodes i and j. All
matrix entry using the full spectral decomposition. In
paths between these nodes that are allowed by the adja-
particular, G = N P λ P−1, where for k = 1,
ij k=1 ik k−1 kj cency structure of the graph are collapsed to one equiv-
λ0 = 0, so no contPribution is added to the sum. Thus, alent link with weight ρij. The result is that a single
observing that the inverse of an orthogonalmatrix is its
effective link is created between i and j. The important
transpose (P−1 = P = (~v ) ), the ij’s entry of G
kj jk k−1 j contribution is that this quantity presents the possibility
N
is given by: G = (~v ) λ (~v ) , which is to diminish the complexity of any graph to a simpler all-
ij k=2 k−1 i k−1 k−1 j
easily extendible toPvarious connected components mak- to-all equivalent topology with a betweenness score given
ing the lower bound for the summation index k=r+1. by ρ .
ij
3
As aworkingexample,results ofapplying the solution Fig. 1 (found from Eq. (7)).
givenby Eqs.(6)and(7)to aring graph,namely CN,of Havingthevoltagedifferenceandequivalentresistance
N =16 nodes with resistors equal to one (Rij =1∀i,j) given by Eqs. (6) and (7), the third result of our unified
and input current I = 1 are shown in Fig. 1. For this approach is addressed: the construction of the effective
network, the set of unordered eigenvalues is given by edge flows, namely, φ(st). We introduce these flows by
ij
λn =2−2cos(2πn/N), with n=0,...,N −1 [12]. finding the voltage differences among any two nodes in
The top left panelinFig.1 showsthat for the N(N− the effective topology givenby ρ . Thus, the EFFA ma-
ij
1) = 240 possibilities of s-t pairs in CN the maximum trix elements for each s-t configuration are given by
voltages achieved in the ring do not surpass 2 (arbitrary
units). Thisisthemaximumdifferencethattheeigenvec- φ(st) = 1 V~(st) − V~(st) , (8)
tor coordinates can achieve in the non-normalizedframe ij ρij (cid:20)(cid:16) (cid:17)i (cid:16) (cid:17)j(cid:21)
(toprightpanelinFig.1),hencegivingamaximumvolt-
age difference of 4. This result shows that not only the where(V~(st)) >(V~(st)) suchthataneffectivecurrentis
i j
eigenvalues of the Laplacian are important to infer the flowing fromnode i to j, otherwise φ(st) =0. Hence, the
ij
network’sfunctionalbehaviour,buteigenvectorsarealso
s-t dependent EFFA matrix defines an effective directed
needed. Also, the maximum (minimum) is periodic, in
network, which breaks the symmetry of the effective all-
the sense that the largest difference happens every time
to-all topology structure given by ρ . In particular, the
ij
the source and sink are separated by 8 edges.
physicalcurrentoneachedgeij ∈E ofthegraphisgiven
by I(st) = Aij [(V~(st)) −(V~(st)) ].
st()) 2 15 1 ThijevoltaRgijedifferenicesdefinejafunctionalrelationship
Vx(i 1 0.5 among nodes and the equivalent resistance an effective
st()V,n()mai−−021 ~v()ni150 −00.5 tbtohoputhos,lsotigrtuyecsnttuarrbuelcetasunrdteo.fufiTnnhcdteioifnuntnarclotcdihouancratailochnteuorbifsstthitcehsaEotFfmFthAiegrhgetrlaanptoehst,
mi be there in the topological structure, and detect highly
0 50 100 150 200 −1
(s-t) 5 10 15 connected areas (communities) in a functional way. In
n
2 4 this sense, the s-t analysis acts similarly as how a dye
200 doeswhenintroducedtoacirculatorysystemofaperson:
1 5 3
150 it highlights a certain part of the network structure to
)
st(-100 0 i10 2 detect what is being sought, e.g., clog blocking, arteries
50 −1 1 stiffness, etc.
15
−2 0
5 10 15 5 10 15 1
i j
0.15
5 5
FIG. 1: The top left panel has the maximum (black) and
minimum (red) voltages that are achieved in an unweighed i10 0.1 i10 0.5
(Rij =1 i,j) ring graph (CN) with N =16 nodes for every 0.05
sto-tppraiigrhitn∀puanniteslosfhIow=s1thfoeretihgeeninvpecuttocrsur(r√enNt~vant)nocdoeorsd.inTahtee 15 5 10 15 0 15 5 j10 15 0
values (color scale) of the corresponding weighed Laplacian j
matrix. Thebottom left panelexhibitsincolor coded,forall
thesepossibleinput-outputconfigurations,thevoltages V(st) FIG.2: Theleft panelshowstheeffectiveflowfunctionalad-
of every node. The bottom right panel shows the equivalient jacency(EFFA)matrixfortheunweighedCN graphofFig.1,
resistor matrix (ρij) for every pair of nodes in CN. where all source-sink pairs havebeen averaged, i.e., φij. The
right panel exhibits the same EFFA matrix for a particular
source-sink pair, namely, s = 2 and t = 10 (φ(2,10)). Each
ij
The bottom panels in Fig. 1 show the voltage solu- matrix element’s magnitude is associated with a color scale.
tions for all the s-t pairs (left) and all equivalent resis-
tances (right). The resulting values show the functional
The mean EFFA matrix is defined to detect statisti-
and structural symmetry that this simple network has.
cally relevant functional observables. It is calculated by
As all edge weights are unity (R = 1), the equivalent
ij all the effective currents (Eq. (8)) among the different
resistanceρ maximumcanonlybe4. Thisiseasilyseen
ij pairsaveragedoverallpossiblerealizationsofsource-sink
when calculating it directly by means of series and par-
pairs. Its matrix elements are
allel resistor calculations. For instance, for nodes i = 1
and j = 9 there are two parallel paths with 8 edges in N N φ(st)
ssaermiees,vahleunecea:s ρel−1e,19m=ent81ρ+1,918in⇒theρb1o,9tt=om4,riwghhticphainselthoef φij ≡Dφ(ijst)E(st) =Xs=1t=1X(t=6 s)N(Nij−1), (9)
4
whichdefine amatrix that constitutes aneffective struc- 100
tural flow quantity and is related to the equivalent resis- 100 1
F
tance matrix. The EFFA matrix and its averaged coun- i 0.5 PD10−5
terpart for the example of the ring graph are shown in 400
0
Fig.2,wheretheleftpanelshowsthemeanEFFAmatrix 6 100
100
and the right panel exhibits the symmetry breaking for 4 F
D
a particular s-t pair. i 2 P10−5
400
With the three quantities analytically expressed(volt- 0
ages in Eq. (6), equivalent resistances in Eq. (7), and 0.2 100
100
F
tdhoemeffgercatpihves c(RurNre)n[t1s5i]n, sEmqa.l(l-8w))o,rwlde(iSnWve)stnigeattweo[r1k4s][r1a6n]-, i400 0.1 PD10−5
scale-free(SF)topologies[17],andnumericallygenerated 100 400 100 400 0 10−2 10−1 100
clustered networks (CN) of N = 29 nodes. The follow- j j φ(st)
ij
ing conclusions constitute the application of our unified
approachto these concrete topologies. In particular, the FIG. 3: The figure shows the following unweighted graphs:
differences between the respective adjacency and effec- a random (top row), a scale-free (centre row), and a small-
world(bottomrow)networkofN =29 nodesandconnecting
tive topologies (equivalent resistance) are shown in the
probabilityp=0.05[14]. Theadjacencymatrix(leftcolumn),
left and centre columns of Figs. 3 and 4.
equivalent resistance matrix (centre column in color coded),
To study other features of the EFFA matrix that are and effective functional flow adjacency PDFs (right column)
notrepresentedinitsaveragedversionandtoanswerhow foralls-tconfigurations(grayscale)areshownforeachgraph.
the flow is effectively distributed in any given topology,
the probability density function (PDF) of the effective
edgeflowsφ(st) foreveryparticulars-tiscalculated. This
ij
doesnotonlytellshowdiffusiveorlocalizedthetransport
of currents is as a function of the effective topology, but
also gives the maximum flow values a particular graph
can have, i.e., the maximum edge capacity.
Results show that the EFFA PDFs for RN, SW, and FIG.4: Thefigureshowstheadjacencymatrix(leftcolumn),
SF networks are roughly invariant under changes of the equivalentresistance (middlecolumn),and theaverageeffec-
s-t pair. However, SW and SF networks produce more tive functional adjacency matrix (right column) for an un-
(st) weighed clustered network formed by 4 random graphs of
high flow magnitudes (φ ≃ 1) than RN for a broad
range of parameters [14],ijhence, RNs distribute flows in Nc = 27 [14]. The network has a total of N = 29 nodes
andeachmatrixelementmagnitudeisassociatedwithacolor
a more uniform way than SF or SW networks. These
scale.
observations are seen in the example shown on the right
column of Fig. 3.
The detection of communities is done by analysis of the possibility of using voltages as a betweenness mea-
the meanEFFAmatrixandthe equivalentresistance. In sure score is pointed out in Refs. [1–3] and is related to
general, the application of the EFFA analysis on numer- shortest-paths and random walks. With our unified ap-
ically generated CNs qualitatively exhibits the communi- proach, and the four results that are addressed in this
tiesthatthenetworks have. Thisisseeninthecentreand Letter, we show the practicality of it. Moreover,the lin-
rightpanelsoftheexampleexhibitedinFig.4,wherethe ear relationship between loads and flows, and conserva-
communities are spotted as dark areas, both in ρij and tion of charge, are restrictions needed to define the flow
φij, though the contrast in the mean EFFA matrix is networkontopofthe structurebut notfor the structure
higher than in the equivalent resistance, implying that itself. Thus, the flow analysis is applicable to any net-
communities are better identified by φij. The φij matrix work structure outside the domain of flow networks as
notonlydetects thecommunities butalsoshows that the long as the graph to be analysed is connected.
effective flows between communities are larger than the An estimate of the computational time that our ap-
ones within each community. Hence,theintra-edgescon- proach requires is as follows. To calculate the voltages
necting communities must have a high flow capacity as one needs the eigenvectors and eigenvalues of the Lapla-
they represent the vulnerable links in the network. Re- cian matrix. This is calculated in time O(Nα), with α
moval of any of these edges generate drastic changes in being an exponent that nowadays rounds 2 and N the
the flows, leading to the isolation of communities. number of nodes in the network. Then, if all possible
Theobservationsonflowtransmissionandcommunity source-sink pairs are analysed, the flows require O(N3)
detectionhavebeenpreviouslyshownbydoingstructural to be found. This means that the algorithm is not as
analysis of networks, e.g., in Refs. [1–7]. In particular, effective as other methods [18] but its mathematical for-
5
mulasstillprovidegreatinformationforobtainingupper implicationstothestructureandfunctionofthenetwork.
andlowerboundsforcurrentswithouttheneedforsimu- [9] G. Kirchhoff, Vorlesungen u¨ber Mechanik (Wien, Wil-
lations. Furthermore,communitydetectioncanbequan- helm, Ed.1864-1928).
[10] RavindraK.Ahuja,ThomasL.Magnanti, andJamesB.
titatively described if other techniques are used, such as
Orlin,Network Flows: Theory, Algorithms, and Applica-
PDF analysis of the EFFA matrix elements.
tions (Prentice-Hall, 1993).
Authors acknowledge the Scottish University Physics
[11] B. Bollob´as, Modern Graph Theory (Springer-Verlag,
Alliance (SUPA). New York,1998, chap. 9).
[12] Fan R. K. Chung, Spectral Graph Theory (Am. Math.
Soc. and CBMS 92, 1997).
[13] F.Y.Wu,J.Phys.A:Math.Gen.37,6653-6673 (2004);
Arpita Ghosh, Stephen Boyd, and Amin Saberi, SIAM
[1] M. Girvan and M. E. J. Newman, PNAS 99(12), 7821- Rev. 50(1), 37-66 (2008); and references therein.
7826 (2002). [14] Starting with a ring graph of N = 29 nodes, a tuning
[2] M. E. J. Newman and M. Girvan, Phys. Rev. E 69, parameter p controls the addition of links. For the RN,
026113 (2004). SW,andSFgraphstheprocessisstraightforward (taken
[3] M. E. J. Newman, Eur. Phys.J. B 38, 321-330 (2004). from Refs. [15–17]) and p is related to the probability
[4] A.E. Motter, Phys. Rev.Lett. 93(9), 098701 (2004). of adding an edge. The RN process follows the strategy
[5] Rui Yang, Wen-Xu Wang, Ying-Cheng Lai, and Guan- of [15]. The SF graph is constructed as in the rewiring
rong Chen, Phys.Rev.E 79, 026112 (2009). procedure of Ref. [17]. The SW graph is constructed as
[6] Wen-Xu Wang and Ying-Cheng Lai, Phys. Rev. E 80, in Ref.[16]. FortheCN,fourRNofNc =27 are created
036109 (2009). withequalstatistics(probabilityp=0.01),thenapprox-
[7] RoniParshani, Sergey V.Buldyrev,and ShlomoHavlin, imately 10 links are added between them randomly. In
Phys.Rev.Lett. 105, 048701 (2010). addition, weights are included from theuniform random
[8] In particular, the approach that we take is derived from distribution such that Rij (0,1] i,j.
themaximumflowproblem,whichcorrespondstofinding [15] P. Erdos and A.Renyi,Pu∈bl. Mat∀h. 6, 290 (1959).
afeasibleflowthroughasingle-sourcesandasingle-sink [16] D. J. Watts and S. H. Strogatz, Nature 393, 440-442
tthatismaximumwhencostsandcapacitiesattheedges (1998).
are given. The solution for this problem is equal to the [17] A.-L. Barab´asi and R.Albert, Science 286, 509 (1999).
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hand, if the s-t flow is fixed, then the problem turns
intofindinghowisthetransmissiongivenandwhatisits