Designer Nets from Local Strategies

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by  Hernan Rozenfeld

Designer Nets from Local Strategies
Hern´an D. Rozenfeld∗ and Daniel ben-Avraham†
Department of Physics, Clarkson University, Potsdam NY 13699-5820
(Dated: February 2, 2008)
Weproposealocal strategy forconstructingscale-free networksofarbitrary degreedistributions,
4 based on the redirection method of Krapivsky and Redner [Phys. Rev. E 63, 066123 (2001)]. Our
0 method includes a set of external parameters that can be tuned at will to match detailed behavior
0 at small degree k, in addition to the scale-free power-law tail signature at large k. The choice of
2 parameters determines other network characteristics, such as thedegree of clustering. The method
is local in that addition of a new node requires knowledge of only the immediate environs of the
n
(randomly selected) node to which it is attached. (Global strategies require information on finite
a
J fractions of the growing net.)

2
PACSnumbers: 02.50.Cw,05.40.−a,05.50.+q,89.75.Hc
1

]
n Recently much effort has been devoted to the study
n of large networks that surroundus in everyday life, such
- as the Internet and the World Wide Web, the electricity
s
i powergrid,flightconnections,socialnetworksofcontacts y x
d
or collaborations, networks of predator-prey, of neurons
.
t in the brain, etc., [1, 2, 3]. An important realization is
a
m that a majority of these networks share some character-
istic properties: A small diameter — a small number of n
-
d links connects between any two nodes on the net; clus-
n tering, or the smallworldproperty — the neighbors of a
o node tend to be connected to one another; and a scale- FIG. 1: The KR redirection model. A new node n (solid)
c free degree distribution — the distribution of the num- is connected to a randomly selected node x with probability
[ 1−r (dashed arrow), or thelink is redirected to node y,the
ber of links emanating from a node (the degree k) has a
ancestor of x, with probability r (solid thick arrow). After
1 power-law tail of the form
Krapivskyand Redner[7].
v
6 P(k)∼k−λ. (1)
9
1 The scale-free property gives rise to exotic behavior of Hereweproposealocalstrategyforconstructingscale-
1 free networks, based on the redirection technique of
the networks, such as resilience to random dilution (the
0 Krapivsky and Redner (KR) [7]. Our method includes
percolation transition does not take place for λ < 3),
4
a set of external parameters that enable fine tuning of
on the one hand, and high vulnerability to removal of
0
additionalproperties (degree distribution at small k, de-
/ the most connected nodes, on the other hand, and has
t gree of clustering, etc.), while guaranteeing a scale-free
a become a principal focus of attention [4, 5].
m Several growth models that produce scale-free net- tail with a given degree distribution exponent, as in the
original KR method.
- works have been suggested. However,most growth tech-
d niques rely on global properties of the network. Such is In the KR redirection model, a new node n is con-
n nectedtoapre-existingnodex,selectedatrandom(with-
the case, for example, for the seminal model of Baraba´si
o out regard to its degree), with probability 1−r. If the
and Albert (BA) [6], where new nodes are connected to
c connection is established, x is said to be the ancestor of
: an existing node with a probability proportional to its
v n (the link may be regarded as directed, from n to x).
degree. The BA algorithm requires global knowledge
i With probability r, the connection is not realized, but
X of the degree of all present nodes. While global algo-
instead it is redirected to y, the ancestor of x (Fig. 1).
rithmshavecontributedimmenselytoourunderstanding
r
a of how scale-free degree distributions might emerge, in Theredirectionstepisessentiallywhatturnstheglobal
most common situations it is more likely that networks BA strategy of selecting a node with a probability pro-
evolve by a set of local rules. (One does not typically portional to its degree, to a local one. (See [8], for an
conduct a survey of the Internet for deciding where to early introduction of this basic principle.) Each node
connectanewrouter,neitherdoesonestudyawholenet- has but one ancestor, so a node of degree k has k −1
work of social contacts for selecting new acquaintances.) incoming links. It follows that an ancestor of degree k
is reached with a probability proportional to k−1, but
purelythroughalocalprocess. TheKRtechniquebuilds
scale-freegraphswithdegreeexponentλ=1+1/r (that
∗Electronicaddress: [email protected] is, 2 ≤ λ ≤ ∞). The BA net corresponds to the special
†Electronicaddress: [email protected] case of r = 1/2 (λ = 3). However, the obtained graphs
2
arealwaystrees—they possessnocycles (aunique path In the long time asymptotic limit N(l)/N → p , hence
l
connects between any two nodes) — and not much else M ∼N(hki−a)∼at [9].
can be controlledduring the growth process,beyond the Due to the time dependence of the rates in Eq. (5),
degree distribution exponent. analytic integration is not feasible. Instead, we argue
We generalize the redirection algorithm as follows. At that the transient time does not affect the ultimate dis-
eachtimestepthenewnodenisconnectedtomdifferent tribution appreciably, and replace M and M by their
0
pre-existing nodes, with probability p , m = 1,2,..., long time asymptotic limits. From here we proceed as
m
( p =1). Eachofthe m nodes is selectedrandomly in [7]. The asymptotic equations admit the generalsolu-
Pm m
and independently by the same redirection trick as for a tion N(l)(t)=n(l)t, where
k k
single node: direct connection to the selected node with
probability 1−r, or redirection to a random ancestor, n(l) =a(1−r)[n(l) −n(l)]
with probability r. The KR model corresponds to the k k−1 k
choice p = δ . The presence of an initial network +r[(k−l−1)n(l) −(k−l)n(l)]}+p δ , k ≥l;
m m,1 k−1 k l k,l
seed, of N nodes, is implied. The seed allows for the
0 n(l) =0, k<l. (7)
process to get started. k
A new node is added at each time step, so the total
(l)
Note that n is in fact normalized (since N ∼ t) and
number of nodes increases as k
represents the probability that a randomly selected node
N(t)=N +t. (2) be of degree k and have l ancestors.
0
Thesolutionoftherecursions(7)isn(kl) =Πk′[1−(1+
In the long time asymptotic limit of t ≫ N0, N(t) ∼ t. r)/(rk′+1+a(1−r)−rl)]. Writing the product as the
Becauseeachnewnodehasmancestorswithprobability exponentialofasum,expandingtofirstorderin1/k′(for
pm, the average number of ancestors of a node at time t largek′),andapproximatingthe sumby anintegral,one
is finally obtains
hmi(t)= N0hmi0+tPmpm, (3) n(l) ∼k−(1+1/r). (8)
N +t k
0
Thesameistruefortheoveralldegreedistribution(with-
where hmi is the initial average number of ancestors
0 out reference to the number of ancestors),
present in the seed. In the long time asymptotic limit,
hmi ∼ mp ≡ a. The average degree of a node (in-
P m k
cluding both incoming and outgoing links) is P(k)=nk =Xn(kl) ∼k−(1+1/r). (9)
l=1
hki=2hmi∼2a. (4)
This follows from Eqs. (7), summing over l, noting that
ConsidernowNk(l)(t),the numberofnodesofdegreek Plln(kl) =a, and proceeding as for n(kl).
that possess exactly l ancestors. It evolves according to We now have at our disposal a simple local algorithm
that produces scale-free graphsof any desired degree ex-
dNdk(lt)(t) =Xmpm{(1M−r)[Nk(l−)1−Nk(l)] p{pone}n)ttλha=t 1ca+n1b/er,tawseawkeeldl aastawsieltl otfopaacrhaimeveetearsw(tihdee
0 m
m
r gamut of additional attributes, without affecting λ. Ob-
+M[(k−l−1)Nk(l−)1−(k−l)Nk(l)]}+plδk,l, k ≥l; vious examples include the average number of outgoing
links from a node, hmi = mp = a, and the average
N(l) =0, k <l. (5) P m
k node degree, hki = 2a. (Note that in the original KR
model a=1 is the only possible outcome.)
The first term inside the curly brackets describes gains
The degree distribution at small k is determined by
andlossesduetodirectconnections(occurringwithprob-
the particular choice of the {p }. This is dramatically
ability1−r), whilethe additionaltermsrefertochanges m
illustrated in Fig. 2, where we show the degree distribu-
due to redirection: since the nodes in question have l
tion of a network grown with r = 1/2 and p = 1/100
ancestors, the probability to reach a node of degree k m
for m = 1,2,...,100 (p = 0 for m > 100). In addition
by redirection is now proportional to k − l, instead of m
to the expected k−3 tail, we see a very distinct distribu-
k−1, as in the original KR model. Note that the num-
tion, for k < 100. The resulting distribution is perfectly
ber of ancestors of a node is fixed at birth, and does not
reproduced, analytically, using Eqs. (7).
change subsequently. The rate for directed connections
More importantly, the inverse problem is easily solved
is normalized by M = N(l) = N ∼ t, while for
0 PlPk k as well. Given a specific distribution P(k), with a scale-
redirected events, free tail ∼ k−λ, it can be produced by the redirection
method in the following way. Set r = 1/(λ−1) for the
N(l)
M =XX(k−l)Nk(l) =N(cid:0)hki−X N l(cid:1). (6) rfreedeirteacitl.ioCnopmropbuatebialit=y,(t1h/u2s)enskuPri(nkg) tthhaetdiessiimrepdlisecdabley-
l k l P
3
0.01 1000 A A A A A A A A
A
A
0.001 100
CRE
k)0.0001 C/ 10
(
P
1
1e-05
0.1
0 0.2 0.4 0.6 0.8 1
1e-06 p
1 100 10000
k
FIG.4: Clustering index,C,fornetsconstructedintheredi-
FIG. 2: Degree distribution of network constructed with rectionmethod,withr=0.1,0.2,...,0.9(bottomtotop)and
pm = 1/100, m = 1,2,...,100, and r = 0.5 (◦). The ana- pm = (1−p)δm,1+pδm,2. Plotted is C/CER, where CER is
lytical prediction from Eqs. (7) and (9) (solid line) is shown theclustering indexof theequivalent ER graphs [1, 10].
for comparison.
1 tribution of loops, or cycles, in the growing nets. One
exact 1 measureofcyclesisprovidedbytheclustering index,de-
4mers
8mers fined for site i as [11]
0.1 12mers
m0.01
p Ei
C = . (11)
(k) 0.01 0.0001 i 12ki(ki−1)
P
1 10 100 It denotes the ratio between E , the actual number of
m i
0.001 links between the ki neighbors of the node, to the max-
imum number of links that would result had all the k
i
nodes been connected to one another. For trees, the
0.00011 100 10000 clustering index is zero at all nodes. In contrast, the
k proposedgeneralizedKRprocesscanproduceloops,and
hence nonzero clustering, whenever a new node is con-
FIG. 3: Convergence to a target distribution (◦) for nets
nected to more than one site: p < 1 (or p > 0 for
grown with the analytically computed pm’s (inset). Shown some m>1). 1 m
are distributions obtained for networks grown with just the
To test this issue we focused on the KR model, but
first 4, 8, and 12 pm’s (solid curves). Convergence to the
where a newly added node is further connected to a sec-
target distribution improves with increasing range of m.
ond node with probability p: p = 1−p, p = p (and
1 2
p =0 for m>2). Several networks were grown in this
m
the desired distribution. According to Eq. (7), way,withdifferentvaluesofrandp. Thenumericaldata
strongly support the predicted relation λ = 1+1/r and
pm =[1+a(1−r)]n(mm). (10) the independence from the pm.
The results for clustering are summarized in Figs. 4
Therequiredn(m)arecomputediteratively,fromEqs.(7) and 5. In Fig. 4, we plot the ratio of the global cluster
m
coefficient (averaged over all the sites of the network),
(1)
and (9). From (9) we have n =P(1). Using this, and
1 C,tothe clustercoefficientofanequivalentErdo˝s-R´eniy
Eqs. (7), for k = 2, l = 1, one can compute n(21), then (ER) graph [1, 10], CER, as a function of p, for various
obtain n(2) =P(2)−n(1) from (9), etc... values of r. As might be expected, the clustering in-
2 2
We have followed this procedure to compute the p dex increases with p, though the effect saturates rather
m
needed to grow a network with the target distribution quickly. Amorepronouncedeffectisachievedbyincreas-
P(k) = c,2c,c,2c, for k = 1,2,3,4, respectively, and ingr, whichchangesC by ordersofmagnitude (note the
P(k) = 2c(k/4)−2.5, for k ≥ 4, (c = 1/10.4359, is de- logarithmicscale). This hastodo withthe influence ofr
termined from normalization: P(k) = 1), shown in onthedegreedistribution. Toseethat,inFig.5weplot,
P
Fig.3. As mightbe expected, the p decayrapidly with for the same data, the ratio of C to C — the clus-
m rand
m(inthisparticularcase,p ∼m−2.5)andareasonably tering index in equivalent networks,with identical scale-
m
well-fitting distribution is obtained with just m≤4. In- free degree distributions, but where all the connections
creasing the range of m yields a better fit to the target. have been redistributed randomly. The dependence on
Alternatively, a better fit can be achieved, for a fixed r seems a lot weaker by this comparison. Fig. 5 demon-
range of m, by treating a as a variable whose value is strates, however, significant differences between random
then optimized. scale-free networks and nets constructed by the general-
Another interesting property is the number and dis- ized redirection method.
4
2
the p to vary with time. Consider, for example, the
m
r=0.5 “self-consistent” networks grown with p (t) = P(m,t),
r=0.6 m
1.5 rrr===000...789 r=0.1 that is, where the pm reflect the existing degree distri-
bution at time t. The degree distribution that results
d
Cran 1 fhraovme nthoitsastttreamtepgtye,dusainnaglyrz=ing1/t2h,isisdsihstorwibnuitnioFn,igb.e6c.aWusee
C/
the equations involved are quite more complicated, and
r=0.9
0.5 we cannot think of any serious applications to this cu-
rious model. Still, it remains an amusing problem that
0 mighttestthelimitsofthegeneralapproachofevolution
0 0.2 0.4 0.6 0.8 1
p equations.
We have presented a local strategy — the generalized
FIG. 5: Clustering index for the same data as in Fig. 4, but redirection method — for the growth of scale-free net-
compared to equivalent scale-free random graphs (see text).
works. Local algorithms require information about only
the immediate environs of a visited node, and are more
0.004
likely to reflect real growth processes than global algo-
0.001 rithms, where addition of new nodes requires informa-
0.003 k) tion about the whole network. The generalized redirec-
(
P tion technique includes a class of external parameters,
k)0.002 1e-06 the pm, that may be tuned at will, without affecting the
P( 1000 10000 degreeexponent (which is controlledby the independent
k
parameterr), toachieveavarietyofeffects. We haveex-
0.001 plicitly demonstrated how to design nets with arbitrary
degree distributions, at small degree k, and power-law
(scale-free) tails, at large k. We have also shown that
0
0 500 1000 1500 2000 2500 additional attributes, such as the degree of clustering,
k
can be manipulated by a proper choice of the model’s
FIG. 6: Degree distribution of “self-consistent” net grown parameters. However, how to do this effectively remains
with pm = P(m,t). The distribution peaks at a value of an open question. We anticipate that analogous algo-
k that increases with time, while its tail might be possibly rithms to the one presented for the degree distribution
interpretedasscale-free(inset): λ=1+1/r=3isconsistent can be designed for other attributes as well, by studying
with our noisy data. the relevant rate equations.
As a final remark, we note that we have tacitly as-
Acknowledgments
sumed that the set {p } is finite. Indeed, were the set
m
infinite, there would be no way to even start the growth
process, unless one had an infinitely large seed as well, We thank Erik Bollt for useful discussions, and Sid
sinceconnectinganodetomrandomnodesrequiresthat Redner for helpful discussions and for providing us with
therebeatleastmnodespresenttobeginwith. Oneway Fig. 1. We are grateful to NSF Grant No. PHY-0140094
to get around the limitation to finite sets, is by allowing for partial financial support of this work.
[1] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. 74, 47 Krapivsky, B. Kahng, and S. Redner, Phys. Rev. E 66,
(2002). 055101(R) (2002).
[2] S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, [8] Am. J. Sociology 96, 1464 (1991). See, also: M. E. J.
1079 (2002). Newman, Social Networks 25, 83 (2003).
[3] S. Bornholdt and H. G. Schuster, Handbook of Graphs [9] In the KR model the normalization factor M → 1, in
and Networks, (Wiley-VCH,Berlin, 2003). the long time asymptotic limit, and can be justifiably
[4] R. Albert, H. Jeong, and A.-L. Barab´asi, Nature (Lon- neglected.
don) 406, 378 (2000); 406, 6794 (2000). R. Cohen, K. [10] AnequivalentERgraphisanErd˝os-R´eniyrandomgraph
Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. withthesamenumberofnodes,N,andthesamenumber
85, 4626 (2000). of links, M, as in the original scale-free graph. It is easy
[5] R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, to show that CER =2M/N(N −1). That is, in our case
Phys. Rev. Lett. 86, 3682 (2001). CER =2(1+p)/(N−1).
[6] A.L. Barab´asi, R.Albert, Science, 286, 509 (1999). [11] D. J. Watts and S. H. Strogatz, Nature (London), 393,
[7] P.L. Krapivsky and S. Redner, Phys. Rev. E 63, 066123 440 (1998).
(2001); J. Phys. A 35, 9517 (2002); J. Kim, P.L.