Anomalous behavior of trapping in extended

Anomalous behavior of trapping in extended dendrimers with a perfect trap Zhongzhi Zhang, Huan Li, and Yuhao Yi Citation: The Journal of Chemical Physics 143, 064901 (2015); doi: 10.1063/1.4927473 View online: http://dx.doi.org/10.1063/1.4927473 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/143/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Anomalous properties of the local dynamics in polymer glasses J. Chem. Phys. 131, 114501 (2009); 10.1063/1.3223279 Viscoelastic properties of dendrimers in the melt from nonequlibrium molecular dynamics J. Chem. Phys. 121, 12050 (2004); 10.1063/1.1818678 Elastic behavior of adsorbed polymer chains J. Chem. Phys. 121, 11481 (2004); 10.1063/1.1818673 Dynamics of dendrimers and of randomly built branched polymers J. Chem. Phys. 116, 8616 (2002); 10.1063/1.1470198 Formation of segmental clusters during relaxation of a fully extended polyethylene chain at 300 K: A molecular dynamics simulation J. Chem. Phys. 110, 8835 (1999); 10.1063/1.478789

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.120.224.18 On: Tue, 11 Aug 2015 01:04:38

THE JOURNAL OF CHEMICAL PHYSICS 143, 064901 (2015)

Anomalous behavior of trapping in extended dendrimers with a perfect trap Zhongzhi Zhang,1,2,a) Huan Li,1,2 and Yuhao Yi1,2 1 2

School of Computer Science, Fudan University, Shanghai 200433, China Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China

(Received 31 May 2015; accepted 15 July 2015; published online 10 August 2015) Compact and extended dendrimers are two important classes of dendritic polymers. The impact of the underlying structure of compact dendrimers on dynamical processes has been much studied, yet the relation between the dynamical and structural properties of extended dendrimers remains not well understood. In this paper, we study the trapping problem in extended dendrimers with generationdependent segment lengths, which is different from that of compact dendrimers where the length of the linear segments is fixed. We first consider a particular case that the deep trap is located at the central node, and derive an exact formula for the average trapping time (ATT) defined as the average of the source-to-trap mean first passage time over all starting points. Then, using the obtained result we deduce a closed-form expression for the ATT to an arbitrary trap node, based on which we further obtain an explicit solution to the ATT corresponding to the trapping issue with the trap uniformly distributed in the polymer systems. We show that the trap location has a substantial influence on the trapping efficiency measured by the ATT, which increases with the shortest distance from the trap to the central node, a phenomenon similar to that for compact dendrimers. In contrast to this resemblance, the leading terms of ATTs for the three trapping problems differ drastically between extended and compact dendrimers, with the trapping processes in the extended dendrimers being less efficient than in compact dendrimers. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4927473] I. INTRODUCTION

As an integral class of macromolecules,1 dendrimers have attracted extensive attention of scientists in the interdisciplinary fields of physics, chemistry, and materials, since they display unusual geometrical, physical, and chemical properties.2–5 These chemical compounds are synthesized by repeating units arranged in a hierarchical self-similar fashion, and are characterized by basic building element, branching of the end groups, as well as the number of generations. Among various dendritic polymers, compact and extended dendrimers constitute two important types of dendrimer families. In the compact family, the length of linear unit (segment) in all generations is identical, while in the extended family segment length is varying, which decreases toward the periphery. The unique structure of two dendrimers makes them potentially promising candidates for a broad range of applications, e.g., artificial antenna systems for light harvesting.5,6 Light harvesting by dendrimers can be described by trapping process,7–11 which is a kind of random walk with a deep trap positioned at a given site, absorbing all particles visiting it. The highly desirable quantity for this paradigmatic dynamical process is trapping time (TT), also known as mean firstpassage time (MFPT),12–18 which is the expected time for a particle starting off from a source point to first reach the trap. The average trapping time (ATT), defined as the average of trapping time over all starting nodes, gives insight to the trapping process, since it provides a quantitative measure of trapping efficiency. In addition to light harvesting, trapping is a)Electronic mail: [email protected]. URL: http://www.researcherid.

com/rid/G-5522-2011. 0021-9606/2015/143(6)/064901/9/$30.00

also closely related to many other significant dynamical processes in diverse complex systems, such as target search,19–25 energy or exciton transport in polymer systems.26–31 In consideration of its direct relevance, trapping in various complex systems has been an active subject of research in the past years, with particular attention focused on determining ATT in diverse systems that display different structural and other properties. Thus far, trapping problem has been extensively studied for many systems, including regular planar and cubic lattices,32–34 T–fractals and their extensions,35–41 Sierpinski gasket,42,43 and Sierpinski tower,44 small-world uniform recursive trees,45–47 scale-free networks,48–56 complex weighted networks,57–59 as well as directed weighted treelike fractals.60 These studies uncovered how the behavior of ATT for trapping is affected by different properties of complex systems, such as topological structure, weight, and direction of edges. Except for the aforementioned systems, concerted theoretical efforts have also been devoted to trapping problem in polymer systems (especially dendrimers), reporting specific properties and phenomena of the trapping process in these macromolecules. The trapping problem in compact dendrimers was first addressed in Refs. 7–11, where the MFPT from the peripheral node to the central node was computed both in the presence and in the absence of a fixed energy bias. Partly inspired by these works, applying the theory of finite Markov chain, the exact expression for MFPT from an arbitrary node to the central node was derived in Refs. 61–63, where the ATT was also obtained when an immobile trap is located at the center. The influence of trap location on ATT for trapping in compact dendrimers was explored in Ref. 64. Moreover, the factors governing the trapping efficiency for compact dendrimers have also been studied in detail.65–69

143, 064901-1

© 2015 AIP Publishing LLC


064901-2

Zhang, Li, and Yi

J. Chem. Phys. 143, 064901 (2015)

In contrast to compact dendrimers, trapping problem in extended dendrimers has received less attention,61 although they are an important class of nanoscale supermolecules and their specific structure is suggested to yield different trapping behavior from that for their compact counterparts. Since extended dendrimers are an important class of nanoscale supermolecules, it is of theoretical and practical importance to address the influences of topology of extended dendrimers on trapping occurring on them. In this paper, we present a comprehensive study of trapping in extended dendrimers, with an aim to explore the impact of their geometrical structure on the trapping efficiency. We first focus on a special case of trapping problem with a single trap located at the central node. We derive closed-form expressions for the MFPT from an arbitrary node to the trap, as well as for the ATT to the trap over all starting points. Then, using these results, we determine an exact solution for the ATT to an arbitrary target. Finally, we attack the case of trapping with the trap distributed uniformly over all nodes. By using two different techniques, we deduce an explicit formula for the ATT. We show that the trap position has a strong effect on the efficiency of trapping occurring on extended dendrimers, since their leading scalings display distinct dependence on the system size, which increase with the shortest distance from the trap to the central node. We also demonstrate that for all the cases of trapping problems, the dominant behaviors for trapping in extended dendrimers are different from those corresponding to compact dendrimers.

FIG. 1. Structure of an extended dendrimer network D 4.

node being at level 0, and the peripheral nodes being at level Mg . Let G i (g) denote the generation (iteration) at which all nodes at level i were created. In order to determine G i (g), we introduce two quantities L i (g) and Si , where L i (g) represents the possible maximal value of levels that nodes created at generation i belong to, and Si is defined to be 1, i = 0,      i   Si =   j, i ≥ 1,  1 + j=1 

II. CONSTRUCTION AND PROPERTIES

In this section, we introduce the construction of extended dendrimers and study their relevant properties.

=

A. Construction method

We concentrate on a particular network for extended dendrimers, with the number of nearest neighbors of any branching site being 3. Let Dg (g ≥ 1) denote the extended dendrimer having generation number g, with the length of segment at generation i being g − i. Then, Dg can be built by g iterations in the following way.65,70 At iteration 0, the dendrimer comprises only one central node; at iteration 1, 3 segments with length g − 1 are created being attached to the central node. At iteration i (1 < i < g), for each branching node of generation i − 1 (i.e., the node generated at iteration i − 1 farthermost from the center), two segments (chains) containing g − i nodes are generated and are linked to the branching node. At the last iteration g, for each node created at generation g − 1, we add a pair of new nodes and link them to it. All the nodes introduced at this stage are called peripheral nodes of Dg . Figure 1 illustrates the structure for a specific extended dendrimer D4. Let Mg denote the length of the shortest path from an arbitrary peripheral node to the center. Then, Mg =

g −1  i=1

1 i + 1 = (g 2 − g + 2). 2

According to their shortest distances to the central node, all nodes in Dg can be divided into Mg + 1 levels, with the central

(2)

Then, we have   Mg − Sg −i−1, 0 ≤ i < g, L i (g) =   Mg , i = g, 

(3)

and  0, i = 0,      j | j ∈ (M − S , M − S ], 0 < i < Mg , G i (g) =  g g−j g g − j−1    g, i = Mg .  (4) By construction, the number of nodes at level i is  i = 0,  1, Ni (g) =   3 · 2G i (g )−1, 1 ≤ i ≤ Mg .  Thus, the total number of nodes in Dg is Ng =

(1)

i2 + i + 2 . 2

Mg 

Ni (g) = 9 · 2g −1 − 3g − 2.

(5)

(6)

i=0

And the total number of edges in Dg is Eg = Ng − 1 = 9 · 2g −1 − 3g − 3.

(7)


064901-3

Zhang, Li, and Yi

J. Chem. Phys. 143, 064901 (2015)

B. Structural properties

We now employ the relation in Eq. (13) to derive Ptot(g). To this end, we look upon Dg as a rooted tree with the central node being the root, and use Bi (g) to denote the number of nodes in the subtree with a node at level i being its root. Then, it follows that 2 Bi (g) = Ng −G i (g ) − 1 + L G i (g )(g) − i + 1 3 1 = gG i (g) + 3 · 2g −G i (g ) − 2g − i − G i (g)2 2 3 + G i (g) − 1. (14) 2 For any edge (i, j) connecting two nodes i and j, we assume that node j is the father of node i in the rooted tree, and that node i is at level l. Then, for the two adjacent nodes i and j, Ni < j (g) = Bl (g) always holds. Combining this fact and the results given by Eqs. (13) and (14), the quantity Ptot(g) can be evaluated as

We proceed to present some important structural properties of Dg , focusing on degree distribution and average path length. 1. Degree distribution

It is obvious that, for nodes in Dg , there are only three possible degree values (1, 2, and 3). The degree of any node at levels L 0(g), L 1(g), . . . , L g −1(g) is 3; the degree of all peripheral nodes (i.e., nodes at level Mg ) is 1; and the degree of all other nodes is 2. Let ∆i (g) represent the number of nodes with degree i. Then, it is easy to obtain ∆1(g) = N Mg (g) = 3 · 2g −1, ∆2(g) =

g −2 

L i (g )−1

(8)

N j (g)

i=1 j=L i−1(g )+1 g

=3·2 −3·2

g −1

− 3g + 1,

(9)

Ptot(g) =

g −1 

NL i (g ) = 3 · 2g −1 − 3,

Ni (g)Bi (g) Ng − Bi (g)

i=1

1( = 81 · 4g g 2 − 189 · 4g g − 189 · 22g +1 8 − 9 · 2g +1g 3 + 45 · 2g +2g 2 + 441 · 2g +1g

and ∆3(g) =

Mg 

(10)

) − 105 · 2g +2 − 52g 3 − 120g 2 + 124g + 816 .

i=1

respectively.

(15)

Inserting Eq. (15) into Eq. (11) gives d¯g =

2. Average path length

The average path length denotes the average of length for the shortest path between two nodes over all node pairs. Assume that each edge in Dg has a unit length. Then the length of the shortest path from node i to j in Dg , denoted by d i j (g), is the minimum length of the path connecting the two nodes. Let d¯g represent the average path length of Dg , defined by Ptot(g) , d¯g = Ng (Ng − 1)/2

(11)

where Ptot(g) is the sum of d i j (g) over all pairs of nodes, i.e.,  Ptot(g) = d i j (g). (12) i< j

We continue by showing the procedure of determining Ptot(g), which just equals the number of edges in the shortest paths between all pairs of nodes in Dg . Instead of counting the edges in the paths, here we count the paths passing through a given edge, and then sum the results of all edges in Dg . Let (i, j) be an edge in Dg linking two nodes i and j, and let Ei j (g) be the number of the shortest paths of different node pairs which pass through (i, j). Let Ni < j (g) denote the number of nodes in Dg lying closer to node i than to node j, including i itself. Then, total shortest-path distance can be computed through71,72   Ptot(g) = Ei j (g) = Ni < j (g)N j 0 corresponding the case when the trap is placed on a node with distance i to the center, the walker, starting from a large group of nodes, must first visit the central node and then proceeds from the center along the path r 0 − r 1 − · · · − r i−1 − r i until it is trapped. Equation (37) shows that the MFPT Tr i−1r i from a node at level i − 1 to its direct neighbor at level i grows with i. Therefore, the scaling of Tr i (g) grows


064901-8

Zhang, Li, and Yi

with increasing i. In particular, when the trap is moved away from the central node to a peripheral node, the ATT changes from N ln(N) to N ln(N)2. On the other hand, when the trap is uniformly distributed over Dg , the dominant scaling of ATT ⟨T⟩g scales with the network size Ng as Ng (ln Ng )2, which is the same as that of TP (g) for trapping in Dg with a trap fixed at a peripheral node. Thus, the Ng ln Ng behavior of the ATT to the central node is not representative of extended dendrimers, while the scaling of ATT to a peripheral node is a representative property for trapping in extended dendrimers. The equivalence of the dominant scalings of ATTs for TP (g) and ⟨T⟩g can be easily understood. From Eqs. (6) and (10), it is evident that in Dg the fraction of peripheral nodes is about 31 . Furthermore, the ATT is the highest for all trapping problems in Dg with a single immobile trap. Therefore, the TP (g) alone determines the leading behavior of ⟨T⟩g . The aforementioned results also indicate that for the three cases of trapping problems under consideration, the trapping processes are less efficient in extended dendrimers than in compact dendrimers. Actually, trapping in extended dendrimers also displays different phenomena from other complex systems. For example, for trapping in Vicsek fractals64,77 as a model of regular hyperbranched polymers,78–80 the efficiency is identical, despite the location of the trap. Furthermore, trapping process in extended dendrimers is more efficient than in Vicsek fractals, implying that extended dendrimers have a desirable structure favorable to diffusion, as opposed to Vicsek fractals. In addition to compact dendrimers and Vicsek fractals, trapping has also been studied in various other trees, such as T–shape treelike fractals and their extensions,35–41 fractal scale-free trees,54 small-world uniform recursive trees,45–47 non-fractal scale-free trees.54 Previously reported results show that trapping behaviors in the first two network families are similar to those of Vicsek fractals, and that trapping behaviors in the last two network families resemble those of compact dendrimers. Thus far, the dependence relation Ng (ln Ng )2 of TP (g) and ⟨T⟩g on system size Ng has not been observed in other systems, implying that the structure of extended dendrimers is unique. Before closing this section, we should stress that although we only focus on trapping in a particular extended dendrimer, where any branching site has three nearest neighbors, for trapping in other extended dendrimers with the number of nearest neighbors of each branching site greater than 3, the qualitative scalings are similar to those found for the specific extended dendrimer under consideration. IV. CONCLUSIONS

We have performed an extensive study of the trapping problem on extended dendrimers with varying lengths of the molecular branches. We first addressed a particular case with the trap fixed at the central node; we then considered the case that the trap is positioned at an arbitrary node; and we finally studied the case with the trap uniformly distributed over all nodes in the systems. For these three cases of trapping problems, we studied the ATT as an indicator of the trapping efficiency, and obtained closed-form formulas for the ATTs, as

J. Chem. Phys. 143, 064901 (2015)

well as their leading terms. We found that the location of trap has an essential influence on the trapping efficiency, with the ATT increasing with the shortest distance between the central node and the trap. We showed that the behavior of trapping in extended dendrimers is unusual, since the leading scalings of the ATTs are unique, which are not observed in compact dendrimers and even other complex networked systems. ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant No. 11275049. 1A.

A. Gurtovenko and A. Blumen, Adv. Polym. Sci. 182, 171 (2005). A. Tomalia, A. M. Naylor, and W. A. Goddard, Angew. Chem., Int. Ed. Engl. 29, 138 (1990). 3J. Frechet, Science 263, 1710 (1994). 4D.-L. Jiang and T. Aida, Nature 388, 454 (1997). 5R. Kopelman, M. Shortreed, Z. Y. Shi, W. Tan, Z. Xu, J. S. Moore, A. BarHaim, and J. Klafter, Phys. Rev. Lett. 78, 1239 (1997). 6V. Balzani, S. Campagne, G. Denti, J. Alberto, S. Serroni, and M. Venturi, Sol. Energy Mater. Sol. Cells 38, 159 (1995). 7A. Bar-Haim, J. Klafter, and R. Kopelman, J. Am. Chem. Soc. 119, 6197 (1997). 8A. Bar-Haim and J. Klafter, J. Phys. Chem. B 102, 1662 (1998). 9A. Bar-Haim and J. Klafter, J. Chem. Phys. 109, 5187 (1998). 10A. Bar-Haim and J. Klafter, J. Lumin. 76-77, 197 (1998). 11X. Peng and Z. Z. Zhang, J. Chem. Phys. 140, 234104 (2014). 12L. Lováz, in Paul Erdös is Eighty Vol. 2, edited by D. Miklós, V. T. Só, and T. Szönyi (Jáos Bolyai Mathematical Society, Budapest, 1996), pp. 353-398, http://www.cs.elte.hu/∼lovasz/survey.html. 13S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, 2001). 14J. D. Noh and H. Rieger, Phys. Rev. Lett. 92, 118701 (2004). 15S. Condamin, O. Bénichou, and M. Moreau, Phys. Rev. Lett. 95, 260601 (2005). 16S. Condamin, O. Bénichou, and J. Klafter, Phys. Rev. Lett. 98, 250602 (2007). 17S. Condamin, O. Bénichou, and M. Moreau, Phys. Rev. E 75, 021111 (2007). 18S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, Nature 450, 77 (2007). 19F. Jasch and A. Blumen, Phys. Rev. E 63, 041108 (2001). 20O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and R. Voituriez, Phys. Rev. Lett. 94, 198101 (2005). 21M. F. Shlesinger, Nature 443, 281 (2006). 22E. Agliari, A. Blumen, and O. Mulken, Phys. Rev. A 82, 012305 (2010). 23O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Rev. Mod. Phys. 83, 81 (2011). 24S. Hwang, D.-S. Lee, and B. Kahng, Phys. Rev. Lett. 109, 088701 (2012). 25S. Hwang, D.-S. Lee, and B. Kahng, Phys. Rev. E 85, 046110 (2012). 26I. M. Sokolov, J. Mai, and A. Blumen, Phys. Rev. Lett. 79, 857 (1997). 27A. Blumen and G. Zumofen, J. Chem. Phys. 75, 892 (1981). 28O. Mulken, A. Blumen, T. Amthor, C. Giese, M. Reetz-Lamour, and M. Weidemuller, Phys. Rev. Lett. 99, 090601 (2007). 29E. Agliari, O. Mülken, and A. Blumen, Int. J. Bifurcation Chaos 20, 271 (2010). 30E. Agliari, Physica A 390, 1853 (2011). 31O. Mülken and A. Blumen, Phys. Rep. 502, 37 (2011). 32E. W. Montroll, J. Math. Phys. 10, 753 (1969). 33R. A. Garza-López and J. J. Kozak, Chem. Phys. Lett. 406, 38 (2005). 34R. A. Garza-López, A. Linares, A. Yoo, G. Evans, and J. J. Kozak, Chem. Phys. Lett. 421, 287 (2006). 35B. Kahng and S. Redner, J. Phys. A: Math. Gen. 22, 887 (1989). 36E. Agliari, Phys. Rev. E 77, 011128 (2008). 37C. P. Haynes and A. P. Roberts, Phys. Rev. E 78, 041111 (2008). 38Y. Lin, B. Wu, and Z. Z. Zhang, Phys. Rev. E 82, 031140 (2010). 39Z. Z. Zhang, B. Wu, and G. R. Chen, EPL 96, 40009 (2011). 40A. Julaiti, B. Wu, and Z. Z. Zhang, J. Chem. Phys. 138, 204116 (2013). 41J. Peng and G. Xu, J. Chem. Phys. 141, 134102 (2014). 42J. J. Kozak and V. Balakrishnan, Phys. Rev. E 65, 021105 (2002). 43J. L. Bentz, J. W. Turner, and J. J. Kozak, Phys. Rev. E 82, 011137 (2010). 2D.


064901-9 44J.

Zhang, Li, and Yi

J. Kozak and V. Balakrishnan, Int. J. Bifurcation Chaos 12, 2379 (2002). Z. Zhang, Y. Qi, S. G. Zhou, S. Y. Gao, and J. H. Guan, Phys. Rev. E 81, 016114 (2010). 46Z. Z. Zhang, X. T. Li, Y. Lin, and G. R. Chen, J. Stat. Mech.: Theory Exp. 2011, P08013. 47H. X. Liu and Z. Z. Zhang, J. Chem. Phys. 138, 114904 (2013). 48A. Kittas, S. Carmi, S. Havlin, and P. Argyrakis, EPL 84, 40008 (2008). 49Z. Z. Zhang, Y. Qi, S. G. Zhou, W. L. Xie, and J. H. Guan, Phys. Rev. E 79, 021127 (2009). 50Z. Z. Zhang, W. L. Xie, S. G. Zhou, M. Li, and J. H. Guan, Phys. Rev. E 80, 061111 (2009). 51V. Tejedor, O. Bénichou, and R. Voituriez, Phys. Rev. E 80, 065104(R) (2009). 52E. Agliari and R. Burioni, Phys. Rev. E 80, 031125 (2009). 53E. Agliari, R. Burioni, and A. Manzotti, Phys. Rev. E 82, 011118 (2010). 54Z. Z. Zhang, Y. Lin, and Y. J. Ma, J. Phys. A 44, 075102 (2011). 55B. Meyer, E. Agliari, O. Bénichou, and R. Voituriez, Phys. Rev. E 85, 026113 (2012). 56Y. H. Yang and Z. Z. Zhang, J. Chem. Phys. 138, 034101 (2013). 57Z. Z. Zhang, T. Shan, and G. R. Guan, Phys. Rev. E 87, 012112 (2013). 58Y. Lin and Z. Z. Zhang, Phys. Rev. E 87, 062140 (2013). 59Y. Lin and Z. Z. Zhang, Sci. Rep. 4, 5365 (2014). 60B. Wu and Z. Z. Zhang, J. Chem. Phys. 139, 024106 (2013). 61J. L. Bentz, F. N. Hosseini, and J. J. Kozak, Chem. Phys. Lett. 370, 319 (2003). 62J. L. Bentz and J. J. Kozak, J. Lumin. 121, 62 (2006). 63B. Wu, Y. Lin, Z. Z. Zhang, and G. R. Chen, J. Chem. Phys. 137, 044903 (2012). 45Z.

J. Chem. Phys. 143, 064901 (2015) 64Y.

Lin and Z. Z. Zhang, J. Chem. Phys. 138, 094905 (2013). Raychaudhuri, Y. Shapir, V. Chernyak, and S. Mukamel, Phys. Rev. Lett. 85, 282 (2000). 66S. Raychaudhuri, Y. Shapir, and S. Mukamel, Phys. Rev. E 65, 021803 (2002). 67D. Rana and G. Gangopadhyay, J. Chem. Phys. 118, 423 (2003). 68D. J. Heijs, V. A. Malyshev, and J. Knoester, J. Chem. Phys. 121, 4884 (2004). 69O. Flomenbom, R. J. Amir, D. Shabat, and J. Klafter, J. Lumin. 111, 315 (2006). 70M. A. Martín-Delgado, J. Rodriguez-Laguna, and G. Sierra, Phys. Rev. B 65, 155116 (2002). 71H. Wiener, J. Am. Chem. Soc. 69, 17 (1947). 72A. A. Dobrynin, R. Entringer, and I. Gutman, Acta Appl. Math. 66, 211 (2001). 73D. J. Watts and H. Strogatz, Nature 393, 440 (1998). 74P. Tetali, J. Theor. Probab. 4, 101 (1991). 75P. G. Doyle and J. L. Snell, Random Walks and Electric Networks (The Mathematical Association of America, Oberlin, OH, 1984); e-print arXiv: math.PR/0001057. 76D. J. Klein and M. Randi´ c, J. Math. Chem. 12, 81 (1993). 77J. H. Peng, J. Stat. Mech. 2014, P12018. 78A. Blumen, A. Jurjiu, Th. Koslowski, and Ch. von Ferber, Phys. Rev. E 67, 061103 (2003). 79A. Blumen, Ch. von Ferber, A. Jurjiu, and Th. Koslowski, Macromolecules 37, 638 (2004). 80F. Fürstenberg, M. Dolgushev, and A. Blumen, J. Chem. Phys. 138, 034904 (2013). 65S.
