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Exact calculations of first-passage quantities on recursive networks
B. Meyer,1E. Agliari,1,2O. B´ enichou,1and R. Voituriez1
1Laboratoire de Physique Th´ eorique de la Mati` ere Condens´ ee, CNRS UMR 7600,
case courrier 121, Universit´ e Paris 6, 4 Place Jussieu, 75255 Paris Cedex
2Dipartimento di Fisica, Universit´ a di Parma, Viale Usberti 7/A, 43100 Parma, Italy
(Dated: February 23, 2012)
We present general methods to exactly calculate mean-first passage quantities on self-similar
networks defined recursively. In particular, we calculate the mean first-passage time and the splitting
probabilities associated to a source and one or several targets; averaged quantities over a given set
of sources (e.g., same-connectivity nodes) are also derived. The exact estimate of such quantities
highlights the dependency of first-passage processes with respect to the source-target distance, which
has recently revealed to be a key parameter to characterize transport in complex media. We explicitly
perform calculations for different classes of recursive networks (finitely ramified fractals, scale-free
(trans)fractals, non-fractals, mixtures between fractals and non-fractals, non-decimable hierarchical
graphs) of arbitrary size. Our approach unifies and significantly extends the available results in the
field.
As they account for transport efficiency in complex
media, first-passage processes [1] on fractal or complex
networks have given rise to a lot of interest in the past
few years. Such networks, showing a very broad spec-
trum of topological and thus transport characteristics,
have been opportunely used to model a wide range of
systems, from biology to sociology or computer science
[2–5]. Within this outline, the exact determination of
first-passage quantities has been the topic of many works
[6–13], in a dialog with parallel investigations of general
scalings and asymptotic results [14–17]. The global mean
first-passage time (GMFPT), defined as the mean first-
passage time of a random walk starting from an arbitrary
site (source) in presence of a fixed trap (target), and in
particular its scaling with the number of nodes N, was
meant to encompass the general properties of transport
on fractal and transfractal networks. Yet, recent results
[16] have highlighted the insufficiency of this mere de-
scription and put forward the role of non-averaged quan-
tities, associated to a single starting point. For example,
in the context of diffusion limited reactions in complex
media, the initial position of the reactants has indeed
been shown to be a key parameter that can control the
entire kinetics of the process [18].
In that context, we present a general algebraic method
to calculate exactly first-passage quantities on self-similar
networks, for a given source point. More precisely, we
consider here two first-passage observables:
• the mean first-passage time (MFPT) from a site S
to a site T, or to several sites {Ti}, i.e. the average
time it takes a random walker that starts at S to
reach T or any point in {Ti};
• the splitting probability, defined in presence of sev-
eral targets Tias the probability, starting from S,
to reach a given target Ti0before all the other tar-
gets.
Actually, these quantities both satisfy simple but formal
N ×N linear systems, where N is the number of sites of
the network. The regime of interest is typically N large
and makes the explicit resolution of such linear systems
out of reach. The aim of the present work is to provide a
general method, applicable to a broad class of networks,
which yields explicit and exact formulas for the MFPT
and splitting probabilities even for very large N. To do
so, we make use of the self-similar properties of the net-
works considered and develop a general renormalization
scheme following ideas presented in [11].
More precisely, we consider in this paper self-similar
networks, which can be defined recursively [2] by consid-
ering the result of the action of a similarity transforma-
tion r(Γ) that maps all points i of a network Γ onto new
points i?= r(i). Considering that the transformation r is
homothetic of ratio ρ < 1, a network is called self-similar
if at generation n its iteration Γnis equal to the union of
p replicas of r(Γn−1). As a result, self-similar networks
can be split into a finite number of equal sub-units, those
sub-units being the network at the previous generation.
Explicit examples will be given throughout this paper.
An important point is the relationship between self-
similarity and fractality. A network is called fractal if
one can define a constant df such as N ∼ Rdf, where
N denotes the volume (or number of nodes) and R the
chemical diameter. Note that whereas most examples
of deterministic fractal networks are self-similar, the re-
ciprocal is false: there exists self-similar networks that
do not show the fractal property, such as the so-called
(u,v)-flower networks with u = 1 that will be defined
more precisely below. In the latter example, the diameter
scales as a logarithm of the number of nodes: R ∼ logN,
which is referred to as the small-world property. Such a
network can be formally seen as a fractal network with
infinite fractal dimension, and is sometimes called trans-
finite fractal or transfractal [19].
Beyond fractality, self-similar networks may also ex-
hibit other prominent properties that seem to be com-
mon to real networks, especially biological and social
networks, such as scale-free features or modular struc-
ture [20]. The former implies that the distribution of
arXiv:1202.4903v1 [cond-mat.stat-mech] 22 Feb 2012
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FIG. 1: Examples of self similar networks : 4th generation
Sierpinski gasket and 5th generation T-graph.
the degree of the nodes of the network follows a power
law, while the latter means that the network can be
divided into groups (modules), within which nodes are
more tightly connected with each other than with outside
nodes [21, 22]. In the last part of this article we focus on
a class of hierarchical, non-decimable, recursive networks
able to capture simultaneously scale-free behavior and
modular structure, yet preserving (weak) self-similarity
[20, 21]. As we will see, the calculation of first-passage
quantities on such networks requires an alternative ap-
proach that will be presented in the last section of this
article.
This paper is organized as follows. In the first Section,
we introduce general definitions and present in detail
the method of calculation of splitting probabilities and
MFPTs on the example of the Sierpinski gasket. Then,
we extend this method to other deterministic self-similar
networks on the example of the Song-Havling-Makse net-
works (Section II), while further examples (T graph and
(u,v) flowers) are given in Appendix. We stress that this
approach allows one to calculate explicitly the splitting
probabilities and MFPTs for any starting site of the lat-
tice (but for specific targets). In addition, we will show
that it leads to simple expressions of selective averages
over starting sites (that is averages over specific class of
starting sites, to be defined below). In the particular case
where the average is performed over all starting sites, we
recover expressions of global MFPTs recently obtained.
Finally, in Section III, we present an alternative method
of calculation in the case of hierarchical networks.
I. THE GENERAL METHOD ILLUSTRATED
ON THE EXAMPLE OF THE SIERPINSKI
GASKET
A. Definitions
In this section, we first introduce the class of networks
to which our method of calculation can be applied, and
give basic definitions that will be used throughout the
paper. Note that here we do not aim at giving mathe-
matically formal definitions, and will rather largely rely
on explicit examples.
Hierarchical networks.As stated in introduction, we
consider hierarchical networks which can be defined [2] by
considering the action of a rescaling transformation r(Γ)
that maps all points i of a network Γ onto new points
i?= r(i).A self-similar network is then constructed
recursively from an elementary motif (initiator) Γ0 by
writing Γn= ∪i=1..pr(Γn−1). For example, the Sierpin-
ski gasket of generation n is built by joining p = 3 copies,
called subunits, of Sierpinski gaskets of generation n − 1
(see Figs. 1 and 2), where the initiator is the elementary
triangular network with three nodes.
A
B
A
A
CBB
D
CC
F
E
???
?
?
?
r(
??
?
?1
?0
FIG. 2: Sierpinski gasket : renormalization scheme and asso-
ciated crossing times τk.
Levels.
said to belong to the level ξ, if there exists an integer ξ ≤
n such that i0∈ Γξ. We will denote by Lkthe set of nodes
of level k. Note that the networks are constructed such
that a site that belongs to the level ξ also belongs to levels
ξ + 1, ξ + 2, ..., n . Figure 3 illustrates this definition
for the 3rd-generation of a Sierpinski gasket (see legend).
Note that, for the nth-generation of a Sierpinski gasket,
there are3k+1+3
2
sites on the level k (k ∈ 1,...,n).
Labeling of the sites and subunits.
level k in a self-similar network of generation n can be
reached recursively by defining a path, i.e. a sequence
{i0,...,ik} where ij labels the position of each of the
nodes of the initiator Γ0. An example is given in Fig-
ure 4 in the case of the Sierpinski gasket, by assigning
the value ik = 0 to a top subunit, ik = 1 to a left one
and ik = 2 to a right one. With these rules, the path
{i1,i2,i3,i4} = {2,0,1,2} allows to locate the last-level
sub-unit (t(4),l(4),r(4)) within the 4th level.
Transport process. We consider throughout this pa-
per a nearest neighbor Markovian random walker char-
acterized by generic transition probabilities w(r?|r) from
site r to r?. Unless specified we will consider isotropic
A given site i0of a hierarchical network Γnis
Any subunit of
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FIG. 3: Color online. Third-generation Sierpinski gasket :
levels. Blue circles : level 3 ; green squares: levels 2,3 ; ma-
genta stars: levels 1,2,3 ; orange diamonds: levels 0,1,2,3.
l(0)
r(0)
t(0)
l(1)
r(1)
t(1)
l(2)
r(2)
t(2)
l(2)
r(2)
t(2)
l(3)
r(3)
t(3)
l(4)
r(4)
t(4)
i1= 2 (≡ right)
i2= 0 (≡ top)
i3= 1 (≡ lef t)
i4= 2 (≡ right)
FIG. 4: Color online. Sierpinski gasket : example of labels.
random walks such that w(r?|r) = 1/κ(r) where κ(r) is
the connectivity of node r.
B.Splitting probabilities
In this section we wish to calculate the splitting prob-
ability PrT|rAi(r), defined as the probability for a Marko-
vian random walker starting at r to reach the target rT
in the presence of other absorbing sites {rAi}i, in other
words the probability to reach the site rT before all the
other absorbing sites. For the sake of readability, we will
make use of the following notation:
PrT|rAi(r) = P(r).
We will present the method on the example of the Sier-
pinski gasket and show later on how it can be generalized
to any self-similar network. Let us consider a general sub-
unit Λk−1at a given level k −1, which is depicted in the
right hand side of Fig. 4. We assume that the splitting
probability P(r) is known for all starting sites of level
k − 1 in Λk−1(in the Sierpinski gasket there are only 3
such sites which correspond to the summits A,B,C of
the main triangle defining Λk−1), and that the absorbing
sites rT and {rAi}iare located outside Λk−1. Here the
subunit Λk−1is the union of p = 3 copies of subunits Λk.
The splitting probabilities for sites r of level k in Λk−1
(namely the sites A,B,C,D,E,F in Fig 2) satisfy the
following backward equation [1]:
0 =
?
r?∈Lk∩Λk−1
π(r?|r)P(r?) − P(r). (1)
Here π(r?|r) is the splitting probability that the random
walker starting from the node r of level k in Λk−1reaches
first the site r?among all sites of level k in Λk−1. The
sites of level k in Λk−1actually form a graph Γ1of gen-
eration 1 up to a rescaling factor (see Fig 2). In the case
of an isotropic walk, it is clearly seen on the example
of the Sierpinski gasket (for which all nodes of the ini-
tiator Γ0 are equivalent) that π(r?|r) = w(r?|r), where
w(r?|r) is the elementary transition probability on Γ1.
More explicitly, on Fig 2, on has π(B|E) = π(D|E) =
π(F|E) = π(A|E) = 1/4. In the remainder of the arti-
cle, we consider isotropic random walks on networks, for
which this property holds. Nevertheless, the calculation
method that we present can be in principle extended to
directed or non-uniformly weighted networks (such that
π(r?|r) ?= π(r?|r)), as long as the scale-invariance hypoth-
esis is fulfilled.
In this example, let us assume that P(A), P(B) and
P(C) are known; the expressions of P(D), P(E) and
P(F) of the splitting probabilities starting from the
points D,E,F of level k in Λk−1can then be obtained
readily by making use of Eq. (1). One has
P(E) =1
4[P(A) + P(F) + P(D) + P(B)],
P(F) =1
4[P(A) + P(E) + P(D) + P(C)],
P(D) =1
4[P(B) + P(E) + P(F) + P(C)],
(2)
(3)
(4)
which can be rewritten as a linear system
P(D)
P(E)
P(F)
=
1/5 2/5 2/5
2/5 2/5 1/5
2/5 1/5 2/5
P(A)
P(B)
P(C)
. (5)
We now proceed iteratively and consider for example
the upper subunit Λk of Λk−1 that contains the nodes
A,E,F of level k. This choice is taken into account by
assigning to a variable ikthe value 0 ≡ top. It is of course
possible to apply the same operation to any of the two
other subunits : the lower-left one (ik= 1 ≡ left) or the
lower-right one (ik= 2 ≡ right). Let us then rename the
nodes of level k − 1 in Λk−1and the nodes of level k in
Λkas follows:
t(k−1)= A, l(k−1)= B, r(k−1)= C,
t(k)= A, l(k)= E, r(k)= F.
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4
Equation (5) can then be rewritten as:
P(r)
P(k)≡
P(t)
P(l)
(k)
=
100
2/5 2/5 1/5
2/5 1/5 2/5
P(t)
P(l)
P(r)
(k−1)
.
(6)
Similarly, one can define three matrices Miksuch that
equation (6) reads for each value of ikcorresponding to
either a top, left or right subunit:
P(k)= MikP(k−1)
(7)
with
M0=
100
2/5 2/5 1/5
2/5 1/5 2/5
,
M1=
2/5 2/5 1/5
01
1/5 2/5 2/5
0
and M2=
2/5 1/5 2/5
1/5 2/5 2/5
001
. (8)
As shown above, any subunit (of level k) in the net-
work can be reached recursively by defining a path, i.e.
a sequence {i1,...,ik} with il ∈ {0,1,2} for 1 ≤ l ≤ k.
Iterating equation (7) then yields straightforwardly:
P(k)= MikMik−1···Mik0+1P(k0).
This shows that as soon as P(k0)is known for a given level
k0, the splitting probability starting from any point of
level k ≥ k0can be obtained exactly and only requires to
compute a product of k−k03×3 matrices. In particular
if the targets are chosen among the 3 sites of level 0 then
P(k0)is calculated trivially and the splitting probability
for any starting point of the network is readily obtained.
Let us consider an explicit example illustrated in Fig.
4 that represents a Sierpinski gasket of generation 4. We
aim at calculating the splitting probability to reach t(0)
before r(0)starting from l(4):
(9)
Pt(0)|r(0)(l(4)) ≡ P(l(4)).
P(l(4)) is the 2nd coordinate of the vector P(4), associ-
ated to the subunit of level 4 (with vertices in red). Since
the three matrices M0, M1and M2are known, all we
need to determine is the path of that subunit and the
value of P(0). As before, one has:
i1,i2,i3,i4 = 2,0,1,2
and clearly
P(0)=
1
1/2
0
.
Thus,
234/625
229/625
89/250
applying
equation(9), weget
P(4)
=
, so that finally:
P(l(4)) =229
625
(10)
C.MFPT
In this section we consider the MFPT of a random
walker to a target site rT starting from a site r, that we
denote T(r). Following the main steps of derivation of
the splitting probability, we first write down a backward
equation for the MFPT starting from a given site r of
level k in a subunit Λk−1of level k − 1 (see Fig. 2):
?
As discussed before, the splitting probability π(r?|r) is
readily given by the transition probability w(r?|r) on the
corresponding graph Γ1 of generation 1 formed by the
sites of level k in Λk−1(sites A,B,C,D,E,F in Fig 2).
In addition we introduced the quantity τkdefined as the
time it takes a random walker to exit a subunit of level
k. On the example of the right hand side of Fig 2, τkis
the mean time to reach either B,D,F,A starting from
E. Note that by symmetry it can also be defined as
the mean time to reach either E or F starting from A.
In this example, all the exit nodes of a given subunit
play a symmetric role, and by construction, all subunits
of a given level are the same.
depend on r in equation (11). This property results here
from the symmetry of the initiator Γ0, in which all nodes
are equivalent, and from the symmetry of the transition
probabilities. A stated previously, we will consider in this
paper only networks having this property.
More explicitly, we rely on the example of Fig. 2 and
assume that the MFPT starting from the sites A,B,C of
level k − 1 is known. Using Eq. (11), one can write :
1
4
− τk= ∆T(r) =
r?∈Lk∩Λk−1
π(r?|r)T(r?) − T(r). (11)
Therefore τk does not
T(E) =
?
(T(A) + τk) + (T(F) + τk)
+(T(D) + τk) + (T(B) + τk)
?
,
and the the 2 similar relations at nodes F and D. Fol-
lowing the derivation of splitting probabilities above, the
MFPT starting from any site of level k can be expressed
linearly in terms of the MFPT starting from sites of level
k − 1. For example, focusing on the top subunit AEF
(corresponding to the choice ik= 0), one obtains:
22/5 1/5 2/5
T(k)= τk
0
2
+
100
2/5 2/5 1/5
T(k−1)
= τkV0+ M0T(k−1),
T(r)
(12)
where T(k)≡
T(t)
T(l)
(k)
denotes the vector of MFPTs
starting from the 3 vertices of level k of a given subunit of
level k located by its path {i1,...,ik}. Similar equations
can be obtained for the left (ik= 1) or the right (ik= 2)
subunit, and yield the following general recursive relation
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5
:
T(k)= τkVik+ MikT(k−1),
where the Mikmatrices are given by Eq. (8) and
2
(13)
V0=
0
2
V1=
2
0
2
V2=
2
2
0
. (14)
Assuming that T(k0)is known for some level k0, the
MFPT for a given level k > k0can then be written as:
T(k)= τkVik+
k−1
?
l=k0
τl
?MikMik−1···Mil+1
?Vil, (15)
where here by definition τik0Vik0≡ T(k0). We give be-
low the example where the target is located at the apex,
for which the determination of T(0)is straightforward.
As we show in the next paragraph, τkcan be calculated
explicitly. In the case where the target is located at the
apex, Eq. (15) therefore provides an explicit and exact
expression of the MFPT starting from any starting site
of the network. Examples will be given below.
Determination of the exit time τk.
refer to are those of figure 2. It is easily seen that τkand
τk−1are related by the following system:
where T(E), T(F) and T(D) denote the mean time to
reach any of the two B or C sites starting from E, F
and D. The solution of this system is τk−1= 5τkwhich
leads to τk= 5n−kτn. At level n the subunit is a simple
triangular graph so that obviously τn= 1. Finally, one
has:
The notations we
τk−1= τk+1
T(E) =1
T(D) =1
T(E) = T(F),
2(T(E) + T(F))
4[(τk+ τk−1) + (τk+ T(F)) + (τk+ T(D)) + τk]
4[(τk+ T(E)) + (τk+ T(E))] +1
2τk
τk= 5n−k.(16)
Example 1.
wish to calculate T(l(4)) when the target is set at the
apex t(0). First, it is necessary to determine T(0):
T(l(0)) = τ0+1
2T(r(0)) and T(l(0)) = T(r(0)),
Here, we refer again to Fig. 4 and we
therefore
T(0)= 2 × 5n
0
1
1
= τ0V0.(17)
The path and the value of τkare already known. Finally
one obtains:
T(4)= 1176(18)
Example 2 : a class of sources.
ample, the aim is to study the MFPT dependence with
respect to the target-source distance. The target is left
at t(0), and the class of sources is the sites located on
the adjacent edge [t(0),l(0)], at the distances rp = 2p,
0 ≤ p ≤ n. Those sources are the lower-left vertices of
the triangles corresponding to the paths {0} (r = 2n),
{0,0} (r = 2n−1), {0,0,0} (r = 2n−2), etc. Thus, the
problem is solved by using formula (15) for k = n − p
and Mil= M0for all l. We find finally:
T(r = 2p) = 5p(3n−p+1− 1).
In the present ex-
(19)
The latter result can be compared to the general
asymptotic expression of the MFPT that has recently
been derived in [17] in the large system size limit. In
the case of a target located at the apex of the Sierpinsky
gasket it reads for N large :
Ta∼ 2Nrdw−df,(20)
where r is the source-target distance, df the fractal di-
mension of the network and dwthe walk dimension, char-
acterizing the power-law behaviour of the mean-square
displacement with respect to time : ?∆r2? ∼ t2/dw. In
the present case, 2N = 3n+1+ 3, df = ln(3)/ln(2) and
dw= ln(5)/ln(2). Thus:
Ta∼ 5p3n−p+1. (21)
As expected this asymptotic regime is recovered by tak-
ing the large volume limit n → ∞ in the exact expression
(19).
D.Averages
In this section, we aim at calculating the MFPT to
a target site averaged over different classes of starting
points. The average can cover either a class of starting
points (all the sites of a given level, all sites of a given
connectivity...) or all the sites of network. In this latter
case the averaged MFPT is often called the Global MFPT
[8, 11]. We will show in this section that averages of the
MFPT over all sites of a given level take simple explicit
forms.
In the case of the Sierpinski gasket, we start from Eq.
(15) that gives an explicit expression of the MFPT start-
ing from any of the three points of level k in a given
subunit of level k. Since each subunit of level k is in one
to one correspondence with a path {i1,··· ,ik}, one has
to calculate?
use of the following identity
{i1,···,ik}T(k). Here we assume that the
target site is at level 0 and that T(0)is known. Making
?
{i1,···,ik}
k
?
l=0
=
k
?
l=0
?
{i1,···,ik}
,
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6
we are back to calculate the expression
?
The variable ikis chosen out of a set of p values (in the
case of the Sierpinski gasket, p = 3), and one has:
?
×(M0+ M1+ ... + Mp)k−l(V0+ V1+ ... + Vp).
{i1,···,ik}
?MikMik−1···Mil+1
?Vil.
{i1,···,ik}
?MikMik−1···Mil+1
?Vil= pl−1
(22)
We then define Mtot= M0+ ... + Mpand Vtot= V0+
... + Vp, and obtain the following general formula:
?
paths
T(k)= (Mtot)kT(0)+
k
?
l=1
τlpl−1(Mtot)k−lVtot.
(23)
We give below explicit examples.
Case of a single target at the apex.
given level k, and denote by S(k)the MFPT to a target
site set at the apex summed over all the starting sites
of level k. Using expressions (8) and (14), and the fact
that p = 3 for the Sierpinski gasket, the diagonalization
of Mtotleads to:
Let us consider a
pl−1(Mtot)k−lVtot =
3k−1
5k−l
3 1 1
1 3 1
1 1 3
k−l
4
4
4
= 4 × 3k−1
1
1
1
. (24)
Then, using T(0)= τ0V0, Eq. (23) yields:
?
+5n+13k−1?1 − 5−k−1?
The first coordinate of the latter expression corre-
sponds to the sum of the MFPTs starting from the top
vertex of each of the subunits of level k. Its 2nd and
3rd coordinates are respectively the sum for the left ver-
tices and for the right vertices. An examination of the
Sierpinski gasket shows that the sum of the three co-
ordinates of the vector defined by Eq. (25) is equal to
2S(k)− T(l(0)) − T(r(0)). Knowing that the number of
sites of level k is
2
, we finally obtain the MFPT
?T?
2S(k)
3k+1+ 1
5n+13k+ 4 × 5n− 5n−k3k
paths
T(k)= 4 × 6k−15n−k
−2
1
1
1
1
1
. (25)
3k+1+1
(k)averaged over all starting sites of level k:
?T?
(k)
=
=
3k+1+ 1
. (26)
For the particular case k = n this quantity is the GMFPT
and is in agreement with the results of [8].
The 3 vertices of level 0 are targets.
that the 3 vertices of level 0 are targets and wish to calcu-
late the MFPT averaged over all starting points of level
k. The only quantity that we need to modify in the cal-
culation of the previous paragraph is T(0). In the case of
3 targets one has straightforwardly
We now assume
T(0)=
0
0
0
. (27)
The other terms of equation (23), that are given by equa-
tion (24), are unchanged. We thus have:
?
paths
T(k)= 5n3k−1?1 − 5−k?
1
1
1
. (28)
We finally obtain the desired quantity:
?T?
(k)=5n3k−1(1 − 5−k)
3k+1+ 1
. (29)
E. Conclusion
In this section we have derived on the example of the
Sierpinski gasket exact expressions of the splitting prob-
abilities and MFPTs. Our method yields simple expres-
sions in the case where the target(s) belong(s) to the level
0 of the network and applies to any starting site. This
method can be readily generalized to other self-similar
networks for which a similar addressing of the subunits
of each level can be defined. The only task is then to cal-
culate the matrices Mi, the vectors Viand the exit time
τk. The example of the T–graph is given in appendix,
and further examples that require slight adaptations of
the method are detailed below.
II.THE SONG-HAVLIN-MAKSE NETWORKS
We now consider the example of the Song-Havlin-
Makse networks, which have been introduced in [23, 24]
in order to build self-similar networks that mix fractal
and non-fractal growing schemes. Here, we focus on the
deterministic version of these networks, and show that
the method developed in the previous section applies
upon minor modifications.
The building scheme of these networks starts from a
single link between two nodes. The next generation is
obtained by attaching m new nodes (sons) to each of
them, then by deleting the original link and by creat-
ing x new links connecting the newly created sons, as
shown on figure 5. These networks are self-similar and
a labeling of each subunit of level k can be defined. As
before we will derive the corresponding matrices Mik,
Page 7
7
x=1
x=2
FIG. 5: Song-Havlin-Makse network renormalization scheme:
starting from a link between two nodes, the next generation
is obtained by attaching m new nodes (sons) to each of them,
then by deleting the original link and by creating x new links
connecting the newly created sons. Example of m = 3 and
x = 1 or x = 2.
the vectors Vikand the exit time τk. It is then possible
to calculate explicitly splitting probabilities and MFPTs
for any starting point on the network; here we will calcu-
late examples of MFPTs averaged over different classes
of sources.
Labeling of subunits and sites.
ble labeling based on 2-dimensional vectors. The scheme
does not depend on m and x (provided that x ≥ 1). This
is due to the symmetric role of the x connecting links (re-
spectively of the m − x free links). Although the results
do depend on m and x, this dependence does not apply
to the matrices and vectors. This can be understood as
follows. By definition (see Eq. (7)), Mkdetermines how
the splitting probabilities P(a(k)) and P(b(k)) at level k
are related to the same quantities at level k−1. It is easy
to see that all trajectories starting from one of the nodes
at level k in one branch will reach sites a(k−1)or b(k−1)
of level k −1 before any other site of other branches. As
a consequence, Mk is independent of the number x of
linking branches (or the number m−x of free branches).
Matrices and vectors.One finds
Figure 6 shows a possi-
M1=
?1 0
1 0
?
, M2=
?0 1
0 1
?
, M3=
?
10
2/3 1/3
?
,
M4=
?2/3 1/3
?0
1/3 2/3
?
, M5=
?1/3 2/3
01
?
,(30)
V1=
?0
1
?
,V2=
1
?
,V3=
?0
2
?
,V4=
?2
2
?
,V5=
?2
(31)
0
?
.
Crossing time.One finds
?
τk= 3 +6m
x
?n−k
. (32)
a(k−1)
b(k−1)
ik = 1
ik = 3
ik = 4
ik = 5
ik = 2
b(k)
a(k)
b(k)
a(k)
b(k)a(k)
b(k)
a(k)
a(k)
b(k)
FIG. 6: Song-Havlin-Makse network : indexes. The indexa-
tion of the points does not depend on m nor x (when x ?= 1),
because all the x connection links play a symmetric role, like-
wise the m − x free links. Note the switch of positions of a
and b in the case ik= 2.
A. MFPT averaged over starting sites of level k
The target is set at a(0). Let us introduce the param-
eter νik, that numbers the repetition of a branch ik:
?ν1= ν2= m − x
Due to the branch repetitions, it is necessary to adapt
equation (23) by using the following notation :
ν3= ν4= ν5= x.
Mtot=
5
?
q=1
νqMq; Vtot=
5
?
q=1
νqVq; νtot=
5
?
q=1
νq.
The analog of equation (23) is then :
?
paths
T(k)= τ0Mk
?0
µ = 3 +6m
totVi0+
k
?
l=1
τlνl−1
totMk−l
totVtot, (33)
with Vi0=
1
?
. Let us define:
x,
λ1= x,λ2= 2m + x
where λ1and λ2are the two eigenvalues of Mtot. Thus:
?
?
?1
?1
−
2λ2
1
paths
T(k)=1
2µn
?
λk
1
?−1
1
?
?−1
?
?
?
+ λk
2
?1
1
??
+
k
?
l=1
µn−lλl−1
2
×
λk−l
1
?
(m − x)
1
?
+ λk−l
?
?1
m + 3x
(µ − 1)λ2
2
(m + 3x)
?1
1
??
=
2
1 +m − x
λ2
λk
1
?−1
λk
2
1
+
2+
m + 3x
(µ − 1)λ2
?−1
1
?
?m − x
+
?1
1
???x
3
?k?
µn.
(34)
Page 8
8
In order to relate the previous formula to ?T?
need to take into account the repetitions of branch ik= 5,
otherwise the nodes belonging to the levels ≤ k−1 would
be counted several times. Let us call Nkthe number of
sites belonging to level k. An examination of the labeling
scheme shows that ?T?
of?
(Nk− 1)?T?
paths
(k), we
(k)is related to the 2nd coordinate
pathsT(k)by:
(k)=
?
T(k)???
2− (x − 1)
k−1
?
l=0
?
paths
T(l)???
2,
(35)
which yields:
• for x ?= 3:
?T?
(k)=
µn
Nk− 1
?1
?m − x
?
1
2
?
1 +m − x
λ2
?λk+1
m + 3x
(µ − 1)λ2
?
+
2+
m + 3x
(µ − 1)λ2
2
− xλk
λ2− 1
?3x − 3 − 2x(x/3)k
2+ x − 1
−
2λ2
+
x − 3
?
,
(36)
• for x = 3: in the latter expression the fraction
3x−3−2x(x/3)k
x−3
needs to be replaced by 1 − 2k.
In expression (36) the volume of level k writes :
(in the third term of the bracket)
Nk= 2mλk
2− 1
λ2− 1+ 2. (37)
B. MFPT averaged over the starting sites of
connectivity 1.
Let us keep the target at a(0), and calculate the MFPT
averaged over the starting sites of connectivity equal to
1. They belong to level n; let us call Nκ=1their number.
One has:
Nκ=1= 2(m − x)λn−1
2
.(38)
An examination of the labeling scheme leads to the fol-
lowing formula, that relates the sum (Nκ=1− 1)?T?κ=1
to the coordinates of?
?
+
pathsT(k), which has been calcu-
lated previously:
(Nκ=1− 1)?T?κ=1=
paths
m T(n−1)???
?
T(n)???
2− T(n)???
1
?
?
paths
?
1− (2x − m) T(n−1)???
2
?
.
(39)
The MFPT averaged over all sites of connectivity 1 can
then be calculated, using expression (34):
?
+2
33
(Nκ=1− 1)?T?κ=1=4λn−1
2
?x
?
3m + 2x +3m2
x
?
?n−1
(3m − 7x)
?µn(m − x)
λ2(µ − 1).
(40)
We show in appendix that this method also applies to
the case of the (u,v)-flower networks introduced in [25]
as examples of deterministic scale-free networks, that are
either fractal or small-world.
III.A DIFFERENT METHOD FOR
HIERARCHICAL GRAPHS
A. Recursivity and Modularity
We now consider a different class of graphs, that is hier-
archical, non-decimable, self-similar networks, which are
built deterministically and recursively in a manner remi-
niscent of exact fractal lattices. More precisely, the graph
of generation g is obtained by properly linking together
a certain number of copies of generation g − 1. Differ-
ently from networks previously analyzed, where different
replicas meet at a single node, here exact renormalization
procedures are not applicable. Yet, we can exploit modu-
larity to detect analogous subgraphs whose nodes satisfy
intrinsic, mutual relations, and self-similarity, which al-
lows to establish recursion relations.
Now, in order to fix ideas we focus on a particular
example of hierarchical network, introduced in [26] and
further investigated in [27–30], (see Fig. 7); other exam-
ples can be found in [22, 31–34].
By denoting as Ggthe graph of generation g, we have
that G0is given by a single node, also called “root”, while
G1is a chain of length three obtained from G0by adding
two more nodes and connecting each of them to the root;
the two nodes added are called “rims” (of level 1). Sim-
ilarly, at the second iteration, one introduces two copies
of G1, whose rims are directly connected to the root: now
the root is connected to the original two rims of level 1
and to four rims of level 2.
Proceeding analogously, at the g-th iteration one in-
troduces two replica of the existing graph, i.e. of Gg−1,
and connects the root with all the new 2 × 2g−1rims,
referred to as rims of level g. Hence, the root turns out
to be a hub connected with 2nrims of level n, where
n ∈ [1,g], in such a way that its coordination number is
zg= 2(2g− 1), on the other hand, rims of level n have a
coordination number equal to n.
Given a rim of level n, here referred to as rn, one can
see that it is not only connected to the root but also
to other “minor hubs” hk,n, namely nodes that work as
main hub for any subgraph Gk, k = 1,...,n, containing
both rnand hk,n; more precisely, we refer to hk,nas the
Page 9
9
hub of height k, with respect to a rim of level n (see
Fig. 7). The root will be also referred to as the main hub
and denoted as H ≡ hg,g. Also, given a node i which is
a rim of level n, we say that the set of rims of the same
level and sharing with i the same hub of height k are rims
shifted by k with respect to i; this set is denoted as {rk,n}
and its cardinality is |{rk,n}| = 2n(see Fig. 7). The total
number of nodes making up Gg is Ng = 3g, while the
total number of rims is?g
for hubs is given by the power law P(k) ∼ k−γ, with ex-
ponent γ = log3/log2 ≈ 1.59, while the remaining nodes
follow an exponential degree distribution P(k) ∼ (2/3)k.
For further details about the topological properties of Gg
we refer to [26–30].
We also notice that each subgraph making up the
whole graph can be looked at as a module; connections
between a module and the remaining graph are few (with
respect to the size of the subgraph itself) and concern
only the pertaining rims. Indeed, it is possible to de-
termine a hierarchy of nodes, based on their degree of
clustering, consistently with [22]: Although for Gg it is
not possible to establish a one-to-one correspondence be-
tween the clustering coefficient of a vertex and its degree,
one can see that the clustering coefficient [38] decreases
with the degree.
The MFPT’s on Gg have already been analyzed for
special target locations [28–30] and, before proceeding,
it is worth recalling some results which may be useful in
the following. In particular, for a simple RW on Gg, the
mean time to first reach the main hub H starting from
an arbitrary rim of level g is [28]
l=12n= 2(2g− 1) = zg.
Furthermore, we mention that the degree distribution
Tg(H,rg) =8
3
?3
2
?g
− 3, (41)
while the mean time to first reach any of the 2grims of
level g starting from the main hub is
Tg({rg},H) =4
3
?3
2
?g
− 1, (42)
where the mean is taken over all possible paths; no-
tice that the asymptotic behavior of Tg(H,rg) and
Tg({rg},H) is the same, namely ∼ (3/2)g, even if the
number of targets is 1 and 2g, respectively.
B. Labeling code and time to main hub
We now introduce a method which allows to calculate
straightforwardly the mean time to first reach the main
hub starting from an arbitrary node. First of all, we need
a proper labeling for nodes which exploits the topological
features of the structure. Basically, we associate to an
arbitrary node i belonging to the graph Gg a code, e.g.
ξi= (lrt..rrt), made up of g letters properly chosen in the
alphabet {t,r,l}, as we are going to explain. The whole
graph can be looked at as the combination of three graphs
of the previous generation: Gg−1(corresponding to t) and
two copies of Gg−1arranged on the right (r) and on the
left (l), respectively. Now, according to which of these
main subgraphs i belongs to, we have that ξ1
either t, r or l. Once detected the main subgraph, one
proceeds analogously distinguishing the three subgraphs
of second order, i.e. Gg−2, and evaluating which contains
the node i, hence determining ξ2
iteration, one is left with the subgraph G1, in such a way
that its subgraphs are simply three nodes, one of them
corresponds to i. For instance, referring to Fig. 7, we
have:
iis equal to
i. Finally, at the g-th
ξ4 = (ttlt),
ξ28 = (lttt),
ξ41 = (llll),
ξ52 = (lrrt).
As anticipated, we focus on arrangements where source
and target belong to different modules; as we will see,
this typically requires the passage through H, in such
a way that we first need to calculate the time T(H,i).
For this aim, our addressing, while able to determine
univocally a node, is somehow redundant, since, due to
the intrinsic symmetry, the distinction between left and
right subgraphs is unnecessary. For this reason, one can
denote any of the two subgraphs in the bottom as b in
such a way that, for a graph of generation g, one could
write ξi= (ti1bi2...tik−1bik), with?k
ξ1
i= b, with i1 and ik possibly zero, while
il> 0 for l ∈ [2,k − 1]. Reading this string from right
to left, we can write a general expression for the MFPT
from i to H. In fact, assuming that i is a rim (ξk
of a certain inner subgraph Gg1, in order to reach H, we
need to pass through the main hub of Gg1itself, where g1
is simply ik. Now, the main hub of Gg1is either H (when
ik= g or when k = 2) or the main hub of a certain inner
subgraph Gg2, where g2turns out to be ik+ ik−1. One
can proceed analogously, bouncing from hub to rim and
from rim to hub of larger and larger subgraphs, in such a
way that the following general expression for the MFPT
from i to H holds
l=1il= g; also notice
that, without loss of generality, we can always assume
i= t and ξk
i= b)
T(H,i) ≡ T(H,ξi) =(43)
=
k
?
l=1
[Tjl(H,rjl) + Tjl+1({rjl+1},H)],
jl = g −
l?
l?=1
il?,
where we used H to indicate the main hub of the
(sub)graph considered (denoted by the index jl) and
T0= 0. Recalling the examples above and using Eqs. 41
Page 10
10
?? ?? 67??
??
??
??
??
??
28
??
??
41
??
??
32
??
??
Rims of level n= 2
??
??
Rims of
level n= 3
??
??
Rims of
level n= 4
??
??
Rims of level n= 1
??
??
52
FIG. 7:
subgraphs/levels). The labeling is complete for the subgraph in the top, and proceed analogously for the other subgraphs for
which only a few labels have been inserted as example. Here 1 represents the main hub, i.e. H ≡ h4,4, nodes {2,3} are rims
of level 1, nodes {5,6,8,9} are rims of level 2 and so on, as specified. Also, focusing on node 14, we notice that 10 and 13 are
h2,3 and h1,3, respectively, while node 15 is r1,3, {17,18} = {r2,3} and {23,24,26,27} = {r3,3}.
(Color on line) The graph Gg of generation g = 4. Darker nodes are hubs and brighter nodes are rims (of different
and 42, we write
T(H,ξ4) = T(t2bt) = T1({r1},H) + T2(H,r2),
T(H,ξ28) = T(bt3) = T3({r3},H) + T4(H,r4),
T(H,ξ41) = T(b4) = T4(H,r4),
T(H,ξ52) = T(b3t) = T1({r1},H) + T4(H,r4).
Of course, summing up such times over the whole set of
nodes, one recovers the global mean first passage time
τg≡?
Now, in order to complete the calculation for the
MFPT from an arbitrary source i to a target j, we need
T(j,H), which can be calculated exploiting the centrality
of H and the self-similarity of the graph, by implement-
ing a set of recursive equations. In order to preserve the
generality of the method we focus on a particular class of
targets, easily identifiable in generic hierarchical graphs,
namely on rims of an arbitrary level n.
i?=HT(H,i)/(N − 1) [28, 31].
C.MFPT’s from hubs
Beyond those discussed before, in order to calculate the
MFPT from H to a rim rn, we need further quantities.
First, let us introduce the following: Tg(rn,H), which
represents the mean time to go from the main hub to a
rim of level n, Tg(rn,hk,n), which represents the mean
time to go to a rim of level n from a hub of height k
with respect to rn, and Tg(rn,rk,n), which represents the
mean time to reach a rim of level n from a rim of the
same level, but “shifted” by k (see Fig. 7) [39].
Then, we can write the set of equations:
Tg(rn,H) =
1
zg
+1
zg
n−1
?
l=0
2l[1 + Tg(rn,rl+1,n)](44)
+
1
zg
n−1
?
?
l=1
2l[1 + Tl(H,rl) + Tg(rn,H)]
+
1
zg
g
l=n+1
2l[1 + Tl(H,rl) + Tg(rn,H)],
Page 11
11
where the first term in the r.h.s. accounts for a direct
jump from the root to the target, the second one accounts
for shifted rims of level n itself and the remaining terms
account for rims of all levels other than n;
Tg(rn,hk,n) =
1
zk
+1
zk
k−1
?
l=0
2l[1 + Tg(rn,rl+1,n)] (45)
+
1
zk
k−1
?
l=1
2l[1 + Tl(H,rl) + Tg(rn,hk,n)],
similarly to the previous case;
Tg(rn,rk,n) =
1
n
n
?
k−1
?
l=k
[1 + Tg(rn,hl,n)](46)
+
1
n
l=0
[1 + Tg(rn,rk,n) + Tl({rl},H)],
where the first term in the r.h.s. accounts for the fact
that to reach rn you need to pass through a common
minor hub, or, possibly H itself, while the second term
accounts for bounces from the starting point to close (non
common) minor hubs.
This system of recurrent equations can be solved
by first focusing on Eqs. 45-46 and building up the
differences between terms for k + 1 and k so to get
rid of the sums.The solutions found for Tg(rn,hk,n)
and Tg(rn,rk,n) are then plugged into Eq. 44 and, ex-
ploiting Eqs. 41-42, as well as proper initial conditions
(e.g. 2T(rn,h1,n) = 2 + T(rn,r1,n) and Tg(rn,n,rn) =
Tg(H,rn) + Tg(rn,H)), one obtains closed form expres-
sions which read as:
?
?3
?3
cn
22k
Tg(rn,H) =
cn
2
1 −
?n−1
?k
1 +1
1
2n+n
2nψ(n)
?
+ 3
− 4
2
+ 22−n(3g− 2g)
?
n
n − k+n
(47)
Tg(rn,hk,n) = 2
2
?
− 1 +cn
2
1 +n
2φ(k,n)
?
, (48)
Tg(rn,rk,n) =
2φ(k,n)
?
, (49)
where
cn= 8(3g− 2g)[2n − 1 − 2n−1nφ(n − 1,n) + nψ(n)]−1,
and
φ(k,n) =
k−1
?
i=0
2−i
n − 1 − i,
ψ(n) =
n−1
?
l=0
?
1
n − l+ 2l−1φ(l,n)
?
.
All these formula have been successfully checked ver-
sus numerical estimates obtained by means of the pseudo
Laplacian [35].
It is convenient to report the asymptotic (N → ∞, i.e.
g large) expressions of previous quantities, which turn
out to be the same for all of them, namely:
Tg(rn,H) ∼ Tg(rn,hk,n) ∼ Tg(rn,rk,n) ∼3g
n,
(50)
so that it is easy to see that the level n plays algebraically:
although the distance hub-rim and rim-rim is equal to 1
and 2, respectively, whatever n,k, rims added at larger
generations are “easier” to be reached. We also notice
that the height of the minor hub considered or the shift
among rims, just provide minor order corrections. In par-
ticular, once generation and level are fixed, the MFPT
decreases with k, the reason is that, although the distance
between starting point and target remains the same, in-
dependently of k, a large k implies the passage through
more connected hubs which are easier to be reached.
Analogous recursive equations can be implemented for
the case of multiple targets (see for examples [28, 29]),
while here we just focus on the case of single target.
D.Examples
The results explained in the previous section, together
with those summarized in the Sec.VA, allow to get an
exact expression for the MFPT between two nodes i and
j, such that the path has to include a hub h. In this way
one first detects the hub h and then calculate T(i,j) as
a sum of the partial MFPT from i to h and from h to j.
In order to clarify the procedure, we now present some
examples where several kinds of situations are considered.
Source and Targets are both rims
Being i a rim of level n and j a rim of level m with n ?= m
[40], it is easy to see that, in order to go from i to j (or
vice versa), one has to pass through H, so that
T(j,i) = T(rm,rn) = Tg(H,rn) + Tg(rm,H)
= Tn(H,rn) + Tg(rm,H) =8
?
?3
3
?3
2
?n
− 3
+
cm
2
1 −
?m−1
1
2m+m
2mψ(m)
?
+ 3
− 4
2
+ 22−m(3g− 2g) ∼3g
m.
(51)
Hence, the leading term is typically Tg(rm,H). This also
implies that the MFPT for the same nodes, but opposite
direction, i.e. T(rn,rm), differs from T(rm,rn) and their
ratio goes like n/m.
For example, let us refer to Fig. 7 and let us fix i = 14
and j = 41 (or, of course, equivalent nodes). Then, we
find
T4(41,14) = T3(H,r3)+T4(r4,H) = 6+1109/12 ≈ 98.42,
Page 12
12
where we used Eqs. 41 and 47. Analogously,
T4(14,41) = T4(H,r4)+T4(r3,H) = 21/2+809/6 ≈ 145.33.
As expected from Eq. 51, T(14,41) > T(41,14) due to
the fact that from the main hub it is easier to reach a
rim which belongs to higher levels.
Source and Targets are a hub and a rim
Being i a hub of height k with respect to a set of rims
of level n and j a rim of level m with n ?= m, again, in
order to go from i to j (or vice versa), one has to pass
through H, so that
T(j,i) = Tg({rn},hk,n) + Tg(H,rn) + Tg(rm,H)
= Tk({rk},H) + Tg(H,rn) + Tg(rm,H) (52)
4
323
?
+ 22−m(3g− 2g) ∼3g
Hence, the leading term is typically Tg(rm,H). This is
rather intuitive as Tk({rk},H) accounts for trapping on
a set of 2knodes, Tg(H,rn) for trapping on the main
hub, while Tg(rm,H) for trapping on a single node with
relatively small (equal to m) coordination number.
As an example, let us consider the graph G4in Fig. 7
and calculate
T4(14,28) =7
2
For comparison, let us also consider G3 for a “rescaled
case”, that is
=
?3
?k
1
2m+m
− 1 +8
?3
2
?n
?
− 3 +cm
2
×
1 −
2mψ(m) + 3 − 4
?3
2
?m−1
m.
(53)
2+21
+809
6
≈ 148.83.
T3(5,10) = 2 + 6 + 54 = 62.
Source and Targets are rims of the same subgraph
Let us consider the subgraph Gg? of Gg, where g?< g
and let us fix i and j as rims of level m < n < g?of the
subgraph, whose main hub has to be crossed in order to
go from i to j; for simplicity, let us assume that j also
belongs to the set {rg} in Gg. Therefore we can write
T(j,i) = Tg?(H,rm) + Tg(rn,hg?,n)
= Tm(H,rm) + Tg(rn,hg?,n)
?3
∼
For example, still referring to the labeling of Fig. 7 let
us fix i = 32 and j = 41. Then, for g = 4 we find
(54)
(55)
=
8
3
3g
n.
2
?m
+ 2
?3
2
?g?
− 4 +cn
2
?
1 +n
2φ(k,n)
?
T(41,32) = T2(H,r2)+T4(r4,h3,4) = 3+887/12 ≈ 76.92,
where we used Eqs. 41 and 48. Analogously, for g = 3
we find
T(14,11) = T1(H,r1)+T3(r3,h2,3) = 1+211/6 ≈ 38.17.
IV.CONCLUSIONS
In this work we introduced general methods to calcu-
late exactly first-passage quantities on self-similar net-
works, which are defined recursively. In particular, we
focused on the mean first-passage time from a source S
to one or to several target sites {Ti}, and on the split-
ting probability, namely the probability to reach, starting
from S, a given target before all other targets. Indeed,
these quantities allow a sound description of a wide range
of dynamical processes such as diffusion limited reactions
or search processes embedded in complex media [36, 37].
In general, the methods introduced strongly rely on
the recursivity of the underlying structure, namely on
the fact that the whole graph can be built according to
a recursive procedure: at the n-th iteration the graph
Gnis obtained by properly combining a finite number of
graphs Gn−1, each corresponding to the graph itself at
the previous iteration.
Recursive networks where replicas meet at a single
node are amenable to exact analysis by renormalization
techniques and we show that the above-mentioned first-
passage quantities can be recast as solutions of simple
matricial equations. We also consider examples where
replicas are connected by links and exact decimation is
no longer accomplishable; then, one can impose a num-
ber of coupled equations, each corresponding to a set of
equivalent nodes.
Hence, a broad range of topologies can be addressed
via a unifying approach: we considered explicitly recur-
sive networks as diverse as finitely ramified fractals (Sier-
pinski gasket, T-fractal), scale-free (trans)fractals ((u,v)-
flowers), non-fractals, mixtures between fractals and non-
fractals (Song-Havlin-Makse networks), non-decimable
hierarchical graphs (Barab´ asi-Ravasz-Vicsek network).
In any case, calculations performed are exact; results
previously obtained for special cases of target arrange-
ments are recovered and extended to account for more
general configurations.
Appendix A: The T-graph
In this section we apply the method that has been pre-
sented in section I to the T-graph (see Fig. 1). We first
give the matrices Mik, the vectors Vikand the exit time
τk. It is then possible to calculate explicitly splitting
probabilities and MFPTs for any starting point on the
network. For example, we calculate explicitly two differ-
ent types of averages: the MFPT averaged over starting
points of a given level and the MFPT averaged over all
starting sites with a given connectivity. In the particu-
lar case where the average is performed over all starting
positions, we recover the result of Ref [10].
Page 13
13
FIG. 8: Renormalization scheme and indexes for the T-graph
1. General results
Quantities P and T are here 2-dimensional vectors,
and will be denoted
b
(but not unique) labeling scheme (p = 3).
Matrices and vectors.One finds
?
M3 =
?a
?(k)
. Figure 8 shows the chosen
M1=
10
1/2 1/2
?
,
M2=
?1/2 1/2
?2
?1/2 1/2
?
?2
01
?
,
1/2 1/2
. (A1)
V1=
?0
2
?
, V2=
0
?
, V3=
3
?
.(A2)
Crossing time.One finds
τk= 6n−k.(A3)
2. Examples of averaged MFPTs
Let us consider the case where the target is located
at the extreme left vertex of the T-graph : T = a(0).
Note that an average MFPT calculated for that target
for a network of generation n is equal to an average for
the next (n + 1) generation with a target at the center.
Using that setting, the equivalent of equation (15) is:
T(k)=
k
?
l=0
τl
?Mik···Mil+1
?Vilwith Vi0=
?0
1
?
(A4)
Follow-
.
MFPT averaged over the sources of level k.
ing the method developed in section ID, we make use of
Eq. (23) and obtain:
?a
6n
5
?
paths
b
?(k)
= 6n−1?4 − 2−k??−1
+
1
?
?
4 × 3k− 3 × 2−(k+1)??1
1
?
. (A5)
As a result of the indexation scheme that has been
chosen, S(k)=?
coordinate of Eq. (A5). Therefore, using the fact that
there are 3ksources, one gets :
pathsb(k); in other words the MFPT
summed over all sources of level k is equal to the 2nd
?T?
(k)=4
56n
?
1 +
5
6 × 3k−
7
2 × 6k+1
?
.(A6)
For the particular case k = n, the latter expression gives
the exact result derived in [10].
MFPT averaged over starting points of given con-
nectivity. The nodes of the T-graph have connectivity
κ = 1 or 3.It is possible to use (A5) to determine
the MFPT averaged over all the Nκ=3 starting sites of
connectivity κ = 3 (which of course gives access to the
similar quantity for the starting sites of connectivity 1).
Indeed, an examination of the labeling scheme applied
to the entire network shows that the first coordinate of
(A5) is equal to 2Nκ=3?T?
Nκ=3=3k− 1
2
(k)
κ=3. Then, using the fact that
, one obtains :
?T?
(k)
κ=3=4
56n
?
1 +
1
2 × 3k+1×1 − 2−k
1 − 3−k
?
.
Appendix B: The (u,v)-flowers
In this section, we consider the case of the (u,v)-flower
networks introduced in [25] as examples of deterministic
scale-free networks, that are either fractal or small-world
depending on the values of the two parameters u and v.
The algorithm to build the (u,v)-flower networks is as
follows: one starts (generation n = 0) with two sites con-
nected by a link; generation n+1 is obtained recursively
by replacing each link by two parallel paths of respec-
tively u and v links. In order to simplify some notations,
we define w ≡ u+v, and take (without loss of generality)
u ≤ v. Examples of (1,3) and (2,2)-flowers are shown in
Figure 9. Let us remark that the connectivity of a site
is determined by its level: κk= 2n−k+1when k ≥ 1 and
κ0= 2n.
We focus on the examples (u = 1,v = 3), (u = 2,v =
2) and (u = 2,v = 3). As will be shown in Section B1,
the (1,3)-flower is a transfractal (small-world) network,
whereas the (1,3) and (2,3)-flowers are fractal networks.
Note that the GMFPT for a target on level 0 was previ-
ously obtained by Zhang et al. [12] in the cases (1,3) and
(2,2) flowers. We show here that the MFPT can be ob-
tained for any (u,v)-flower network for any starting site,
and that averages over different classes of starting sites
can be obtained.
Page 14
14
FIG. 9: Renormalization scheme for two (u,v)-flower net-
works : (a) u = 1,v = 3 ; (b) u = 2,v = 2. Figure re-
produced from [19].
1.General properties of the (u,v) flowers
The number of links of level k is wkand the number
of sites of level k is given by [25]:
Nk=w − 2
w − 1wk+
w
w − 1. (B1)
The value of the parameter u splits the scaling of the
diameter R with respect to n into two distinct cases:
R ∼
?(v − 1)n for u = 1,
for u ≥ 2.un
(B2)
Therefore, when u = 1, the network is a small-world.
When u ≥ 2, it is a fractal network, and its fractal di-
mension is obtained by combining equations (B1) and
(B2):
df=ln(u + v)
lnu
. (B3)
Crossing time.
Section IC, we find a general formula for τk:
Following the method presented in
τk=
?
u + v
2 −u−1
u
−v−1
v
?n−k
.(B4)
This result holds also for u = 1.
a(k−1)b(k−1)
ik= 0
ik= 1
(i) u = 1 , v = 3.
ik= 2
ik= 3
a(k)
b(k)a(k)
b(k)
a(k)
b(k)
a(k)
b(k)
a(k−1)b(k−1)
ik= 0
ik= 1
a(k)
b(k)
a(k)
b(k)
(ii) u = 2 , v = 2.
a(k−1)b(k−1)
ik= 0
ik= 1
ik= 2
ik= 3
ik= 4
a(k)
b(k)
a(k)
b(k)
a(k)
b(k)
a(k)
b(k)
a(k)
b(k)
(iii) u = 2 , v = 3.
FIG. 10: Color online. (u,v)-flower networks : indexes.
2. (1,3)-flower
We first study the case u = 1, v = 3, as an example
of small world network. Figure 10 (i) shows the labeling
scheme that has been chosen. We find that τk = 3n−k
and:
?1 0
M2=
1/3 2/3
?0
It is then possible to calculate exactly the splitting prob-
abilities and MFPTs for any starting site on the network.
MFPT averaged over starting sites of level k.
sume that the target is located at a(0)in level 0, and we
apply the method developed in Sections ID (derivation
of?
titions). We first use formula (23) with T(0)=
derive
?
M0=
?2/3 1/3
?
0 1
?
?
?
,
M1=
?
?1/3 2/3
?2
10
2/3 1/3
?
?
,
,
M3=
01
,(B5)
V0=
0
, V1=
?0
2
,
V2=
2
?
, V3=
?2
0
?
. (B6)
We as-
pathsT(k)) and IIA (substraction of branches repe-
?0
??1
3n
?
??
(B7)
to
paths
T(k)= 3n
?
2k−1
?−1
1
?
+ 4k
?
1 −
1
2 × 3k
1
.
We then focus on the 2nd coordinate of the latter expres-
sion, but we need to take care of the fact that branches
ik= 0 and ik= 3 generate two contributions of the same
source b(k): a way to avoid this repetition is to substract
the same quantity for all the levels lower than k, just like
Page 15
15
in Eq. (35) (with x = 1). We then get:
(Nk− 1)?T?
with Nk=2
latter expression is in agreement with [12].
(k)= 2×3n−14k+3n−k4k−2×3n−1(B8)
?4k+ 2?. For the particular case k = n, the
3
3. (2,2)-flower
In this example we find τk= 4n−kand:
?
V0=
M0=
10
1/2 1/2
?
?
,
M1=
?1/2 1/2
?1
01
?
,(B9)
?0
1
,
V1=
0
?
. (B10)
It is then possible to calculate exactly the splitting prob-
abilities and MFPTs for any starting site on the network.
MFPT averaged over starting sites of level k.Again,
the target is located on a(0)and T(0)=
repeted branches must be taken into account:
method has been given in Section IIA. Here ν0= ν1= 2.
Using formula (33) we get:
?
?0
4n
?
. The
the
?
paths
T(k)= 4n
2k−1
?−1
1
?
+1
6
?
4k+1− 1
??1
1
??
(B11)
.
Then we focus on the 2nd coordinate of the latter expres-
sion, and apply equation (35):
(Nk− 1)?T?
(k)=4n
18
?2 × 4k+1+ 3k + 10?
(B12)
with Nk=2
latter expression is in agreement with [12].
3
?4k+ 2?. For the particular case k = n, the
4.(2,3)-flower
The flower of parameters u = 2, v = 3, is a fractal net-
work characterized by τk= 6n−k. Figure 10 (iii) shows
the chosen labeling scheme. One finds
M0=
?
?1/3 2/3
?
?
V3=
10
2/3 1/3
?
?
?
?
?
,
M1=
?2/3 1/3
1/3 2/3
?
,
M2=
01
, (B13)
M3=
?0
10
1/2 1/2
,
M4=
?1/2 1/2
?2
?1
01
?
,
V0=
2
,V1=
?2
?0
2
,
V2=
0
?
?
,
1
,
V4=
0
.(B14)
It is then possible to calculate exactly the splitting prob-
abilities and MFPTs for any starting site on the network.
MFPT averaged over starting sites of level k.
target is located on a(0). We again apply formula (23)
?0
?
The
with T(0)=
6n
?
and find :
?
paths
T(k)= 6n
2k−1
?−1
1
?
+ 5k−1
?7
2−1
6k
??1
1
??
(B15)
.
Substracting the same quantity for all the levels lower
than k (see Eq. (35)) finally yields:
(Nk− 1)?T?
(k)= 6n
?21
85k−1− 7 × 5k−16−k+15
8
?
(B16)
with Nk=3×5k+5
4
.
Acknowledgement
EA is grateful to the Italian Foundation “Angelo della
Riccia” for financial support. The research belongs to
the strategy of exploration funded by the FIRB project
RBFR08EKEV which is acknowledged.
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