# Exact calculations of first-passage quantities on recursive networks.

**ABSTRACT** 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 in characterizing transport in complex media. We explicitly perform calculations for different classes of recursive networks [finitely ramified fractals, scale-free (trans)fractals, nonfractals, mixtures between fractals and nonfractals, nondecimable hierarchical graphs] of arbitrary size. Our approach unifies and significantly extends the available results in the field.

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**ABSTRACT:**We consider degree-biased random walkers whose probability to move from a node to one of its neighbours of degree k is proportional to k^{\alpha}, where \alpha is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely: i) the time the walker needs to come back to the starting node, ii) the time it takes to pass from a given node, and iii) the time it takes to visit all the nodes of the network. We consider a large database of real-world networks and we show that the value of \alpha which minimizes the three characteristic times is different from the value \alpha_{min}=-1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, assortative networks have preferentially a value of \alpha_{min} in the range [-1,-0.5], while disassortative networks have \alpha_{min} in the range [-0.5, 0]. When only local information is available, degree-biased random walks can guarantee smaller characteristic times by means of an appropriate tuning of the motion bias.Physical Review E 01/2014; 89:012803. · 2.31 Impact Factor - SourceAvailable from: Junhao Peng[Show abstract] [Hide abstract]

**ABSTRACT:**Efficiently controlling the diffusion process is crucial in the study of diffusion problem in complex systems. In the sense of random walks with a single trap, mean trapping time (MTT) and mean diffusing time (MDT) are good measures of trapping efficiency and diffusion efficiency, respectively. They both vary with the location of the node. In this paper, we analyze the effects of node's location on trapping efficiency and diffusion efficiency of T-fractals measured by MTT and MDT. First, we provide methods to calculate the MTT for any target node and the MDT for any source node of T-fractals. The methods can also be used to calculate the mean first-passage time between any pair of nodes. Then, using the MTT and the MDT as the measure of trapping efficiency and diffusion efficiency, respectively, we compare the trapping efficiency and diffusion efficiency among all nodes of T-fractal and find the best (or worst) trapping sites and the best (or worst) diffusing sites. Our results show that the hub node of T-fractal is the best trapping site, but it is also the worst diffusing site; and that the three boundary nodes are the worst trapping sites, but they are also the best diffusing sites. Comparing the maximum of MTT and MDT with their minimums, we find that the maximum of MTT is almost 6 times of the minimum of MTT and the maximum of MDT is almost equal to the minimum for MDT. Thus, the location of target node has large effect on the trapping efficiency, but the location of source node almost has no effect on diffusion efficiency. We also simulate random walks on T-fractals, whose results are consistent with the derived results.The Journal of Chemical Physics 04/2014; 140(13):134102. · 3.12 Impact Factor - SourceAvailable from: Elena Agliari[Show abstract] [Hide abstract]

**ABSTRACT:**On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous structures this property does not hold and, although a single random walker will certainly return to its starting point, two moving particles may never meet. This striking property has been shown to hold, for instance, on infinite combs. Due to the huge variety of natural phenomena which can be modeled in terms of encounters between two (or more) particles diffusing in comb-like structures, it is fundamental to investigate if and, if so, to what extent similar effects may take place in finite structures. By means of numerical simulations we evidence that, indeed, even on finite structures, the topological inhomogeneity can qualitatively affect the two-particle problem. In particular, the mean encounter time can be polynomially larger than the time expected from the related one particle problem.05/2014;

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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

Page 2

2

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

Page 3

3

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.

Page 4

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

Page 5

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}

,

Page 6

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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[6] D.AldousandJ.Fill, Reversible

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