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Resistance Distance in Potting Networks
Journal Pre-proof Resistance Distance in Potting Networks Jiaqi Fan, Jiali Zhu, Li Tian, Qin Wang
PII: DOI: Reference:
S0378-4371(19)31725-X https://doi.org/10.1016/j.physa.2019.123053 PHYSA 123053
To appear in:
Physica A
Received date : 29 August 2019 Please cite this article as: J. Fan, J. Zhu, L. Tian et al., Resistance Distance in Potting Networks, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123053. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Highlights (for review)
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Highlights
1. A class of self-similar and symmetric networks named potting networks are
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constructed. 2. The elimination principle of electric circuits are used to calculate the resistance
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distance.
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3. The recursion formula of two-point resistance on potting networks is established.
Journal Pre-proof *Manuscript Click here to view linked References
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RESISTANCE DISTANCE IN POTTING NETWORKS
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
Abstract. On a resistor network, the resistance distance is the effective resistance ruv between two nodes u, v of the network. In this paper, we study a family of self-similar and symmetric networks named potting networks and obtain a recursion formula to calculate resistance distances.
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1. Introduction
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For decades the study of complex networks has received the attentions of researchers from different scientific fields. For example Watts and Strogatz [1] studied small-world network model, Barab´ asi and Albert [2] investigated scale-free network model, Newman [3] studied the structure and function of complex networks and many researchers discussed various fractal networks ([4]-[12]). As a class of complex networks, the resistor network is also of great importance and attracts increasing attention from researchers. In 1847, Kirchhoff [13] formulated the study of electric networks as an instance of a linear analysis. In electric circuit theory and graph theory, the computation of two-point resistances in resistor networks is a classical problem. Many scholars are devoted to the study of it. In [14], Wu obtained the resistance between any two nodes in a resistor network in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. However, it is not always explicit formulas. Afterwards, together with Wu, Izmailian and Kenna [15] improved Wu’s results and got a simpler expression for it. And they used the new formulas to get the resistance between two nodes in the cobweb resistor network and solved the spanning tree problem on this network. In [16], using the intimate relations between random walks and electric networks, Chen established the relation between effective resistance and conductance in resistor networks. Researchers considered resistance distance in kinds of specific resistor networks or graphs. For instance, based on an elementary triangular circuit, Saggese and Luca [17] constructed a fractal-like network and calculated the equivalent resistance of finite approximations of the network. Shangguan and Chen [18] gave a recursive algorithm to compute resistance between two nodes of flower networks. Researchers have also studied resistance distances in the complete n-partite graph Km1 ,m2 ,··· ,mn [19], Apollonian network [20], corona and neighborhood corona networks [21] and many other graphs ([22]-[26]). In this paper, we will study the potting network and obtain the resistance distance between any pair of nodes of the network. 2000 Mathematics Subject Classification. Primary 28A80. Key words and phrases. fractal network, resistance distance, potting network. Qin Wang is the corresponding author. 1
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
2. Constructions and properties of potting networks
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In this section, we will first introduce the model of potting networks Gt (n, m) = (Vt , Et ) (t ≥ 0) which are built in an iterative way and controlled by a pair of positive-integer parameters (n, m). Initially, as we can see in Fig. 1, when t = 0, we suppose G0 (n, m) is an edge which connect two nodes, x∗ and y ∗ , called the initial nodes. Then fix x∗ and y ∗ , there are two parallel paths connecting x∗ and y ∗ with length m and 2 respectively, and n edges at the midpoint of the path with length of 2 in G1 (n, m). Vividly, we call those n edges grass. It is obvious that ♯V0 = 2, ♯E0 = 1 and ♯E1 = ♯V1 = n + m + 2. For t ≥ 2, by replacing every edge in Gt−1 with a copy of the initial graph G1 , identifying x∗ , y ∗ with two nodes of edge in Gt−1 , we can obtain Gt . Now we have {Gt = (Vt , Et )}t with ♯Et = (♯E1 )t and t 1 ) −1 ♯Vt = 2 + (♯V1 − 2) (♯E ♯E1 −1 . We define it as a potting network, because of the shape characteristics of G1 . Fig. 2 illustrates the construction process of a particular potting network in the case of m = 3 and n = 3.
x*
y*
x*
A1
A2
A n- 1 An
B1 B2
G0
y*
A0 Bm-2 Bm-1
G1
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Figure 1. Iterated structure of (n, m)-potting network.
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x*
G0
y*
x*
G1
y*
x*
G2
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Figure 2. Iterated structure of (3, 3)-potting network 3. Main Results
In this section, our purpose is to establish a theorem to compute resistances in (n, m)-potting networks. Suppose that N = (V (N ), E(N ), w) is a resistor network with underlying graph G. Here V (N ) is the set of nodes, E(N ) is the set of edges and w : E(N ) → R is the conductance function on edge set E(N ). For e = (i, j) ∈ E(N ), we use wij or w(e) to denote the conductance of e. As we know in physics, 1/w(e) is the resistance of e. As a convention, we call a resistor network simple [16], if for any e ∈ E(N ), w(e) = 1. For u, v ∈ V (N ), we use ruv = ruv (N ) to denote the effective resistance of the resistor network N .
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RESISTANCE DISTANCE IN POTTING NETWORKS
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We have the following lemma and two useful principles.
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Lemma 1. [16] Let N = (V (N ), E(N ), w) be a resistor network, then for any two distinct nodes i and j, X wik (rik − rjk ) = 2, wi rij + k∈Γ(i)
P
k∈Γ(i)
p ro
where Γ(i) denotes the set of neighbors of node i and wi =
wik .
Principle 1 (Substitution Principle). [24] If H is a subnetwork of N and H ∗ is a network such that V (H) ⊂ V (H ∗ ) and rh1 h2 (H) = rh1 h2 (H ∗ ) for all h1 , h2 ∈ V (H), then the network N ∗ obtained from N by replacing H with H ∗ satisfies ruv (N ) = ruv (N ∗ ) for all u, v ∈ V (N ).
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In a connected graph G, we say that a vertex is a cut-vertex of G if its removal from G disconnects G. A maximal connected subgraph of G is a block of G if it does not contain a cut-vertex of itself. Principle 2 (Elimination Principle). [23] Let N be a resistor network with underlying graph G which is connected. Let B be a block of G containing exactly one cut-vertex x of G. If N ′ is the network obtained from N by deleting all the vertices of B − x, then for all u, v ∈ V (N ′ ), ruv (N ) = ruv (N ′ ).
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e e
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1
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b
em-2 em-1
2
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n-1
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f0
d
f m -2 f m -1
f1 f2
G t-1
n-1 n
0
e1 e2
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c
e e1
Gt
Figure 3. New nodes generated by e = (a, b) and f = (c, d) of Gt−1 .
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Let us return to the potting network. Fix (n, m) and let Nt = (Gt , w) be a simple resistor network. As is shown in Fig. 3, we label the new nodes in Gt , which are generated by the edge e = (a, b) of Gt−1 , with e0 , e1 , e2 , · · · , en , e1 , e2 , · · · , em−1 . For convenience, when u, v ∈ Vt , we use ruv (t) to denote ruv (Nt ). The following theorem gives a recursive algorithm for computing the resistance distance between any two nodes in (n, m)-potting networks Nt . Theorem 1. For any two distinct nodes u, v ∈ Vt , we have three cases. (1) If u, v ∈ Vt−1 , then ruv (t) =
2m ruv (t − 1). m+2
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
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(2) If u ∈ Vt−1 , v ∈ {e0 , e1 , e2 , · · · , en , e1 , e2 , · · · , em−1 } with e = (a, b), then m θ1 , if v = e0 , η1 + m + 2 m if v = ei , ruv (t) = η2 + m + 2 θ2 , 2 η3 + θ3 , if v = ej , m+2 where 1 3 j(m − j) η1 = , η2 = , η3 = , 2 2 m 1 θ1 = θ2 = rua (t − 1) + rub (t − 1) − rab (t − 1), 2 j(m − j) rab (t − 1). θ3 = (m − j)rua (t − 1) + jrub (t − 1) − m (3) If u, v ∈ {e0 , e1 , · · · , en , e1 , · · · , em−1 } ∪ {f 0 , f 1 , · · · , f n , f1 , · · · , fm−1 } with e = (a, b) and f = (c, d), then if u = ei , v = e0 , α1 , α2 , if u = ei , v = ej (i 6= j), 2 β3 , if u = ei , v = ej (i < j), α3 + m(m + 2) m β4 , if u = e0 , v = f 0 , α4 + 2(m + 2) m α5 + β5 , if u = e0 , v = f i , 2(m + 2) ruv (t) = m β6 , if u = ei , v = f j , α6 + 2(m + 2) α7 + 1 β7 , if u = e0 , v = fi , m+2 1 α8 + β8 , if u = ei , v = fj , m+2 2 α9 + β9 , if u = ei , v = fj , m(m + 2) where α1 = α4 = 1, α2 = α5 = 2, α6 = 3, (m + i − j)(j − i) , m 3 j(m − j) , α8 = + 2 m β3 = (j − i)2 rab (t − 1),
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α3 =
1 i(m − i) + , 2 m i(m − i) + j(m − j) α9 = , m
α7 =
β4 = β5 = β6 = rac (t − 1) + rad (t − 1) + rbc (t − 1)
+ rbd (t − 1) − rab (t − 1) − rcd (t − 1),
β7 = (m − i)[rac (t − 1) + rad (t − 1)] + i[rbc (t − 1) + rbd (t − 1)] −
m 2i(m − i) rab (t − 1) − rcd (t − 1), 2 m
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RESISTANCE DISTANCE IN POTTING NETWORKS
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β8 = (m − j)[rac (t − 1) + rbc (t − 1)]
2j(m − j) m rab (t − 1) − rcd (t − 1), 2 m β9 = (m − i)(m − j)rac (t − 1) + j(m − i)rad (t − 1)
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+ j[rad (t − 1) + rbd (t − 1)] −
+ i(m − j)rbc (t − 1) + ijrbd (t − 1) − i(m − i)rab (t − 1) − j(m − j)rcd (t − 1).
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Proof. Let us consider the resistance rx∗ y∗ (N1 ). Using Elimination Principle, we can delete all the grass in N1 , then 2m rx∗ y∗ (N1 ) = (3.1) m+2 due to series and parallel principles. Case 1. Suppose u, v ∈ Vt−1 . Utilizing (3.1) and Substitution Principle on Nt , we directly obtain that 2m ruv (t − 1). (3.2) ruv (t) = m+2 Case 2. Suppose u ∈ Vt−1 and v ∈ {e0 , e1 , e2 , · · · , en , e1 , e2 , · · · , em−1 }. Subcase 2.1. When v = e0 , by Substitution Principle and Elimination Principle, we can delete the grass and find an equivalent network shown in Fig. 4, where the weight of edge is the corresponding conductance.
u
a
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1
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Figure 4. u ∈ Vt−1 and v = e0 .
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Making use of Lemma 1, we have 2re0 a (t) + (re0 a (t) − 0) + (re0 b (t) − rab (t)) = 2, 2re0 b (t) + (re0 a (t) − rba (t)) + (re0 b (t) − 0) = 2.
Solving it, we can obtain
re0 a (t) = re0 b (t) =
1 1 rab (t) + . 4 2
(3.3)
Using Lemma 1 again, we have 2re0 u (t) + (re0 a (t) − rua (t)) + (re0 b (t) − rub (t)) = 2,
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applying (3.2) and (3.3) to the equation (3.4), it follows that 1 m 1 re0 u (t) = + rua (t − 1) + rub (t − 1) − rab (t − 1) . 2 m+2 2
(3.4)
(3.5)
Subcase 2.2. For v ∈ {e1 , e2 , · · · , en }, according to Elimination Principle, we can delete other grass and find an equivalent network shown in Fig. 5. Let v = ei . By Lemma 1, we have 3re0 a (t) + (re0 a (t) − 0) + (re0 b (t) − rab (t)) + (re0 ei (t) − raei (t)) = 2, 3r 0 (t) + (re0 a (t) − rba (t)) + (re0 b (t) − 0) + (re0 ei (t) − rbei (t)) = 2, (3.6) eb re0 ei (t) + (re0 ei (t) − 0) = 2.
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
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Figure 5. u ∈ Vt−1 and v = ei .
Applying (3.1) and (3.2) to equation set (3.6) and solving it, we obtain that re0 ei (t) = 1, and raei (t) = rbei (t) = Using Lemma 1 again, we obtain
1 3 rab (t) + . 4 2
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rvu (t) + (rve0 (t) − rue0 (t)) = 2,
together with (3.5), we can get m 3 ruei (t) = + 2 m+2
1 rua (t − 1) + rub (t − 1) − rab (t − 1) . 2
Subcase 2.3. For v ∈ {e1 , e2 , · · · , em−1 }, in the same way, the equivalent network is shown in Fig. 6.
u
a
b
½
1/(m - j)
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ej
Figure 6. u ∈ Vt−1 and v = ej .
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Let v = ej . In the same way, we have 1 1 1 1 )rva (t) + (rva (t) − 0) + (rvb (t) − rab (t)) = 2, ( + j m−j j m−j 1 1 1 1 ( + )rvb (t) + (rva (t) − rab (t)) + (rvb (t) − 0) = 2. j m−j j m−j Solving it, we obtain that j2 j(m − j) rva (t) = + 2 rab (t), m m (m − j)2 j(m − j) rvb (t) = rab (t). + m m2 Utilizing Lemma 1 again, we have 1 1 1 1 ( + )re u (t) + (rej a (t) − rua (t)) + (re b (t) − rub (t)) = 2. j m−j j j m−j j Combining (3.2), (3.7) and (3.8), we can obtain ruej (t) =
2 j(m − j) + [(m − j)rua (t − 1) m m+2 j(m − j) + jrub (t − 1) − rab (t − 1)]. m
(3.7)
(3.8)
(3.9)
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Case 3. Suppose u, v ∈ Ve ∪ Vf , where Ve = {e0 , e1 , · · · , en , e1 , · · · , em−1 } and Vf = {f 0 , f 1 , · · · , f n , f1 , · · · , fm−1 }. Subcase 3.1. Assume u ∈ {e1 , e2 , · · · , en } and v ∈ {e0 }. In this case, the equivalent network is shown in Fig. 7.
e0 1/m
Figure 7. u = ei and v = e0 . Let u = ei . Making use of Lemma 1, we have
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rei e0 (t) + (rei e0 (t) − 0) = 2.
Solving this equation, we can obtain
rei e0 (t) = 1.
1
Subcase 3.2. Assume u, v ∈ {e , e2 , · · · , en }. In this case, the equivalent network is shown in Fig. 8.
ej
ei
a
1
1
1
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e0
b
1/m
Figure 8. u = ei and v = ej (i 6= j).
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Let u = ei and v = ej (i 6= j). In the same way, we have the following equation, Hence
rei ej (t) + (rei e0 (t) − rej e0 (t)) = 2.
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rei ej (t) = 2. Subcase 3.3. Assume u, v ∈ {e1 , e2 , · · · , em−1 }. Fig. 9 is the equivalent network when u = ei , v = ej (i < j).
a
b
1/2
1/( m - j )
1/ i
ei
1/( j - i )
ej
Figure 9. u = ei and v = ej (i < j).
Using this equivalent network, we can obtain 1 1 1 1 ( + )rei ej (t) + (rei a (t) − rej a (t)) + (re e (t) − 0) = 2, i j−i i j−i i j
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
with (3.9), we have
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2(j − i)2 (m + i − j)(j − i) + rab (t − 1). m m(m + 2)
rei ej (t) =
Subcase 3.4. Assume u = e0 and v = f 0 . In this case, the equivalent network is shown in Fig. 10. 1
1
e0
b
c
1/m
1
1
d
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a
f0
1/m
Figure 10. u = e0 and v = f 0 . In the same way, by Lemma 1, we have therefore,
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2re0 f 0 (t) + (re0 a (t) − rf 0 a (t)) + (re0 b (t) − rf 0 b (t)) = 2, m [rac (t − 1) + rad (t − 1) + rbc (t − 1) 2(m + 2) + rbd (t − 1) − rab (t − 1) − rcd (t − 1)].
re0 f 0 (t) =1 +
Subcase 3.5. Assume u = e0 and v ∈ {f 1 , f 2 , · · · , f n }. In this case, the equivalent network is shown in Fig. 11.
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1
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0
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1/m
f
i
f
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1 1
1
d
1/m
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Figure 11. u = e0 and v = f i .
Let v = f i . In terms of Lemma 1, we have that is
2re0 f i (t) + (re0 a (t) − rf i a (t)) + (re0 b (t) − rf i b (t)) = 2, m [rac (t − 1) + rad (t − 1) 2(m + 2) + rbc (t − 1) + rbd (t − 1) − rab (t − 1) − rcd (t − 1)].
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re0 f i (t) = 2 +
Subcase 3.6. Assume u ∈ {e1 , e2 , · · · , en } and v ∈ {f 1 , f 2 , · · · , f n }. In this case, the equivalent network is shown in Fig. 12. Let u = ei , v = f j . In terms of Lemma 1, we have
thus,
rei f j (t) + (rei e0 (t) − rf j e0 (t)) = 2, m [rac (t − 1) + rad (t − 1) + rbc (t − 1) 2(m + 2) + rbd (t − 1) − rab (t − 1) − rcd (t − 1)].
rei f j (t) = 3 +
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RESISTANCE DISTANCE IN POTTING NETWORKS
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j
f 1
1
e0
1
b c
1
1
f0
1/m
d
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1
a
9
1/m
p ro
Figure 12. u = ei and v = f j .
Subcase 3.7. Assume u = e0 and v ∈ {f1 , f2 , · · · , fm−1 }. In this case, the equivalent network is shown in Fig. 13.
a
1
1
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b
c
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1/ i
1/m
d
½
1/(m - i)
fi
Figure 13. u = e0 and v = fi . Let v = fi . Making use of Lemma 1, we have
2re0 fi (t) + (re0 a (t) − rfi a (t)) + (re0 b (t) − rfi b (t)) = 2.
(3.10)
Applying (3.3) and (3.9) to equation (3.10), we can obtain
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1 1 i(m − i) + [(m − i)(rac (t − 1) + rbc (t − 1)) re0 fi (t) = + 2 m m+2 m 2i(m − i) + i(rad (t − 1) + rbd (t − 1)) − rab (t − 1) − rcd (t − 1)]. 2 m Subcase 3.8. Assume u ∈ {e1 , e2 , · · · , en } and v ∈ {f1 , f2 , · · · , fm−1 }. ei
a
1
1
e
0
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1/m
1
b c
d 1/(m - j)
1/ j
fj
Figure 14. u = ei and v = fj .
Let u = ei , v = fj . Observing the equivalent network in Fig. 14, by Lemma 1, we can obtain rei fj (t) + (rei e0 (t) − rfj e0 (t)) = 2, thus, 3 j(m − j) 1 rei fj (t) = + + [(m − j)(rac (t − 1) + rbc (t − 1)) 2 m m+2 m 2j(m − j) + j(rad (t − 1) + rbd (t − 1)) − rab (t − 1) − rcd (t − 1)]. 2 m
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
a
b c
½
1/(m - i)
1/ i
d
½ 1/ j
ei
1/(m - j)
fj
of
10
p ro
Figure 15. u = ei and v = fj .
Subcase 3.9. Assume u ∈ {e1 , e2 , · · · , em−1 } and v ∈ {f1 , f2 , · · · , fm−1 }. Let u = ei , v = fj . In the same way, observing the equivalent network in Fig. 15, by Lemma 1, we can obtain 1 1 1 1 )ruv (t) + (rua (t) − rva (t)) + (rub (t) − rvb (t)) = 2, ( + i m−i i m−i therefore,
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j(m − i) i(m − i) + j(m − j) (m − i)(m − j) rac (t) + rad (t) + 2 m m m2 ij i(m − i) j(m − j) i(m − j) rbc (t) + 2 rbd (t) − rab (t) − rcd (t) + 2 2 m m m m2 i(m − i) + j(m − j) 2 = + [(m − i)(m − j)rac (t − 1) m m(m + 2) + j(m − i)rad (t − 1) + i(m − j)rbc (t − 1) + ijrbd (t − 1)
rei fj (t) =
− i(m − i)rab (t − 1) − j(m − j)rcd (t − 1)].
The proof is complete.
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4. Conclusion
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In this paper, we focus on resistance distance in potting network. In fact, The initial graph in potting networks has a hybrid pattern which is quite different from that of flower networks in [4]. What we are interested is the hybrid fractal networks, such as their eigentime identities and resistance distances. Both of them are related to the Laplacian and random walk on evolving networks. In our future work, based on the resistance distance, we will investigate eigentime identities of fractal networks using the formula ([27]) Fij + Fji = 2(♯E)rij (N ) where Fij is the mean-first passage time from node i ∈ V to node j ∈ V in the network N = (V, E, w) with w ≡ 1. References
[1] D. J. Watts, S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393 (1998) 440–442. [2] A. L. Barab´ asi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509–512. [3] M. E. Newman, The structure and function of complex networks, SIAM Rev. 45(2) (2003) 167–256. [4] H. D. Rozenfeld, S. Havlin, D. Ben-Avraham, Fractal and transfractal recursive scale-free nets, New J. Phys. 9(6) (2007) 175. [5] H. Cui, Y. H. Mao, Eigentime identity for asymmetric finite Markov chains, Front. Math. China 5 (2010) 623–634.
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[6] Z. Z. Zhang, Z. Y. Hu, Y. B. Sheng, G. R. Chen, Exact eigenvalue spectrum of a class of fractal scale-free networks, Europhys. Lett. 99(1) (2012) 10007. [7] F. Zhu, M. F. Dai, Y. J. Dong, J. Liu, Random walk and first passage time on a weighted hierarchical network, Internat. J. Modern Phys. C 25(9) (2014) 1450037. [8] J. Chen, A. B. Le, Q. Wang, L. F. Xi, A small-world scale-free network generated by Sierpinski Pentagon, Phys. A 449 (2016) 126–135. [9] S. Wang, L. F. Xi, H. Xu, L. H. Wang, Scale-free and small-world properties of Sierpinski networks, Phys. A 465 (2017) 690–700. [10] Q. Q. Ye, J. W. Gu, L. F. Xi, Eigentime identity of fractal flower networks, Fractals 27 (2018) 1950008. [11] L. F. Xi, Q. Q. Ye, Eigentime identities of potting networks, Phys. A 526 (2019) 120934. [12] B. B. Sun, J. L. Yao, L. F. Xi, Eigentime identities of fractal sailboat networks, Phys. A 520 (2019) 338–349. [13] G. Kirchhoff, Ueber die Aufl¨ osung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Str¨ ome gef¨ uhrt wird, Ann. Phys. 148(12) (1847) 497–508. [14] F. Y. Wu, Theory of resistor networks: the two-point resistance, J. Phys. A 37 (2004) 6653– 6673. [15] N. S. Izmailian, R. Kenna, F. Y. Wu, The two-point resistance of a resistor network: a new formulation and application to the cobweb network, J. Phys. A 47 (2013) 035003. [16] H. Y. Chen, Random walks and the effective resistance sum rules, Discrete Appl. Math. 158 (2010) 1691–1700. [17] A. Saggese, R. D. Luca, A fractal-like resistive network, European J. Phys. 35 (2014) 065006. [18] Y. M. Shangguan, H. Y. Chen, Two-point resistances in a family of self-similar (x, y)-flower networks, Phys. A 523 (2019) 382–391. [19] S. V. Gervacio, Resistance distance in complete n-partite graphs, Discrete Appl. Math. 203 (2016) 53–61. [20] Y. M. Shangguan, H. Y. Chen, Two-point resistances in an Apollonian network, Phys. Rev. E 96 (2017) 062140. [21] J. D. Cao, J. B. Liu, S. H. Wang, Resistance distances in corona and neighborhood corona networks based on Laplacian generalized inverse approach, J. Algebra Appl. 18 (2019) 1950053. [22] M. Vaskouski, A. Zadorozhnyuk, Resistance distances in Cayley graphs on symmetric groups, Discrete Appl. Math. 227 (2017) 121–135. [23] Z. Z. Jiang, W. G. Yan, Resistance between two nodes of a ring network, Phys. A 484 (2017) 21–26. [24] Z. Z. Jiang, W. G. Yan, Some two-point resistances of the Sierpinski gasket network, J. Stat. Phys. 172 (2018) 824–832. [25] L. Zhang, J. Zhao, J. B. Liu, M. Arockiaraj, Resistance distance in H-Join of graphs G1 , G2 , · · · , Gk , Mathematics 6(12) (2018) 283. [26] L. Zhang, J. Zhao, J. B. Liu, S. N. Daoud, Resistance distance in the double corona based on R-graph, Mathematics 7(1) (2019) 92. [27] P. G. Doyle, J. L. Snell, Random Walks and Electical Networks, Math. Assoc. America: Washington,DC (1984). Department of Mathematics, Ningbo University, Ningbo 315211, P. R. China E-mail address: [email protected]
Jo
Department of Mathematics, Ningbo University, Ningbo 315211, P. R. China E-mail address: [email protected] Department of Mathematics, Ningbo University, Ningbo 315211, P. R. China E-mail address: [email protected]
Department of software engineering, Zhejiang Wanli University, Ningbo 315100, P. R. China E-mail address: [email protected]
PII: DOI: Reference:
S0378-4371(19)31725-X https://doi.org/10.1016/j.physa.2019.123053 PHYSA 123053
To appear in:
Physica A
Received date : 29 August 2019 Please cite this article as: J. Fan, J. Zhu, L. Tian et al., Resistance Distance in Potting Networks, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123053. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights (for review)
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Highlights
1. A class of self-similar and symmetric networks named potting networks are
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constructed. 2. The elimination principle of electric circuits are used to calculate the resistance
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distance.
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3. The recursion formula of two-point resistance on potting networks is established.
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RESISTANCE DISTANCE IN POTTING NETWORKS
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
Abstract. On a resistor network, the resistance distance is the effective resistance ruv between two nodes u, v of the network. In this paper, we study a family of self-similar and symmetric networks named potting networks and obtain a recursion formula to calculate resistance distances.
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1. Introduction
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For decades the study of complex networks has received the attentions of researchers from different scientific fields. For example Watts and Strogatz [1] studied small-world network model, Barab´ asi and Albert [2] investigated scale-free network model, Newman [3] studied the structure and function of complex networks and many researchers discussed various fractal networks ([4]-[12]). As a class of complex networks, the resistor network is also of great importance and attracts increasing attention from researchers. In 1847, Kirchhoff [13] formulated the study of electric networks as an instance of a linear analysis. In electric circuit theory and graph theory, the computation of two-point resistances in resistor networks is a classical problem. Many scholars are devoted to the study of it. In [14], Wu obtained the resistance between any two nodes in a resistor network in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. However, it is not always explicit formulas. Afterwards, together with Wu, Izmailian and Kenna [15] improved Wu’s results and got a simpler expression for it. And they used the new formulas to get the resistance between two nodes in the cobweb resistor network and solved the spanning tree problem on this network. In [16], using the intimate relations between random walks and electric networks, Chen established the relation between effective resistance and conductance in resistor networks. Researchers considered resistance distance in kinds of specific resistor networks or graphs. For instance, based on an elementary triangular circuit, Saggese and Luca [17] constructed a fractal-like network and calculated the equivalent resistance of finite approximations of the network. Shangguan and Chen [18] gave a recursive algorithm to compute resistance between two nodes of flower networks. Researchers have also studied resistance distances in the complete n-partite graph Km1 ,m2 ,··· ,mn [19], Apollonian network [20], corona and neighborhood corona networks [21] and many other graphs ([22]-[26]). In this paper, we will study the potting network and obtain the resistance distance between any pair of nodes of the network. 2000 Mathematics Subject Classification. Primary 28A80. Key words and phrases. fractal network, resistance distance, potting network. Qin Wang is the corresponding author. 1
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
2. Constructions and properties of potting networks
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In this section, we will first introduce the model of potting networks Gt (n, m) = (Vt , Et ) (t ≥ 0) which are built in an iterative way and controlled by a pair of positive-integer parameters (n, m). Initially, as we can see in Fig. 1, when t = 0, we suppose G0 (n, m) is an edge which connect two nodes, x∗ and y ∗ , called the initial nodes. Then fix x∗ and y ∗ , there are two parallel paths connecting x∗ and y ∗ with length m and 2 respectively, and n edges at the midpoint of the path with length of 2 in G1 (n, m). Vividly, we call those n edges grass. It is obvious that ♯V0 = 2, ♯E0 = 1 and ♯E1 = ♯V1 = n + m + 2. For t ≥ 2, by replacing every edge in Gt−1 with a copy of the initial graph G1 , identifying x∗ , y ∗ with two nodes of edge in Gt−1 , we can obtain Gt . Now we have {Gt = (Vt , Et )}t with ♯Et = (♯E1 )t and t 1 ) −1 ♯Vt = 2 + (♯V1 − 2) (♯E ♯E1 −1 . We define it as a potting network, because of the shape characteristics of G1 . Fig. 2 illustrates the construction process of a particular potting network in the case of m = 3 and n = 3.
x*
y*
x*
A1
A2
A n- 1 An
B1 B2
G0
y*
A0 Bm-2 Bm-1
G1
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Figure 1. Iterated structure of (n, m)-potting network.
y*
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x*
G0
y*
x*
G1
y*
x*
G2
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Figure 2. Iterated structure of (3, 3)-potting network 3. Main Results
In this section, our purpose is to establish a theorem to compute resistances in (n, m)-potting networks. Suppose that N = (V (N ), E(N ), w) is a resistor network with underlying graph G. Here V (N ) is the set of nodes, E(N ) is the set of edges and w : E(N ) → R is the conductance function on edge set E(N ). For e = (i, j) ∈ E(N ), we use wij or w(e) to denote the conductance of e. As we know in physics, 1/w(e) is the resistance of e. As a convention, we call a resistor network simple [16], if for any e ∈ E(N ), w(e) = 1. For u, v ∈ V (N ), we use ruv = ruv (N ) to denote the effective resistance of the resistor network N .
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RESISTANCE DISTANCE IN POTTING NETWORKS
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We have the following lemma and two useful principles.
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Lemma 1. [16] Let N = (V (N ), E(N ), w) be a resistor network, then for any two distinct nodes i and j, X wik (rik − rjk ) = 2, wi rij + k∈Γ(i)
P
k∈Γ(i)
p ro
where Γ(i) denotes the set of neighbors of node i and wi =
wik .
Principle 1 (Substitution Principle). [24] If H is a subnetwork of N and H ∗ is a network such that V (H) ⊂ V (H ∗ ) and rh1 h2 (H) = rh1 h2 (H ∗ ) for all h1 , h2 ∈ V (H), then the network N ∗ obtained from N by replacing H with H ∗ satisfies ruv (N ) = ruv (N ∗ ) for all u, v ∈ V (N ).
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In a connected graph G, we say that a vertex is a cut-vertex of G if its removal from G disconnects G. A maximal connected subgraph of G is a block of G if it does not contain a cut-vertex of itself. Principle 2 (Elimination Principle). [23] Let N be a resistor network with underlying graph G which is connected. Let B be a block of G containing exactly one cut-vertex x of G. If N ′ is the network obtained from N by deleting all the vertices of B − x, then for all u, v ∈ V (N ′ ), ruv (N ) = ruv (N ′ ).
b
al
a
a
d
2
e e
c
f
1
f
e
b
em-2 em-1
2
f
n-1
f
n
f0
d
f m -2 f m -1
f1 f2
G t-1
n-1 n
0
e1 e2
urn
c
e e1
Gt
Figure 3. New nodes generated by e = (a, b) and f = (c, d) of Gt−1 .
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Let us return to the potting network. Fix (n, m) and let Nt = (Gt , w) be a simple resistor network. As is shown in Fig. 3, we label the new nodes in Gt , which are generated by the edge e = (a, b) of Gt−1 , with e0 , e1 , e2 , · · · , en , e1 , e2 , · · · , em−1 . For convenience, when u, v ∈ Vt , we use ruv (t) to denote ruv (Nt ). The following theorem gives a recursive algorithm for computing the resistance distance between any two nodes in (n, m)-potting networks Nt . Theorem 1. For any two distinct nodes u, v ∈ Vt , we have three cases. (1) If u, v ∈ Vt−1 , then ruv (t) =
2m ruv (t − 1). m+2
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
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(2) If u ∈ Vt−1 , v ∈ {e0 , e1 , e2 , · · · , en , e1 , e2 , · · · , em−1 } with e = (a, b), then m θ1 , if v = e0 , η1 + m + 2 m if v = ei , ruv (t) = η2 + m + 2 θ2 , 2 η3 + θ3 , if v = ej , m+2 where 1 3 j(m − j) η1 = , η2 = , η3 = , 2 2 m 1 θ1 = θ2 = rua (t − 1) + rub (t − 1) − rab (t − 1), 2 j(m − j) rab (t − 1). θ3 = (m − j)rua (t − 1) + jrub (t − 1) − m (3) If u, v ∈ {e0 , e1 , · · · , en , e1 , · · · , em−1 } ∪ {f 0 , f 1 , · · · , f n , f1 , · · · , fm−1 } with e = (a, b) and f = (c, d), then if u = ei , v = e0 , α1 , α2 , if u = ei , v = ej (i 6= j), 2 β3 , if u = ei , v = ej (i < j), α3 + m(m + 2) m β4 , if u = e0 , v = f 0 , α4 + 2(m + 2) m α5 + β5 , if u = e0 , v = f i , 2(m + 2) ruv (t) = m β6 , if u = ei , v = f j , α6 + 2(m + 2) α7 + 1 β7 , if u = e0 , v = fi , m+2 1 α8 + β8 , if u = ei , v = fj , m+2 2 α9 + β9 , if u = ei , v = fj , m(m + 2) where α1 = α4 = 1, α2 = α5 = 2, α6 = 3, (m + i − j)(j − i) , m 3 j(m − j) , α8 = + 2 m β3 = (j − i)2 rab (t − 1),
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α3 =
1 i(m − i) + , 2 m i(m − i) + j(m − j) α9 = , m
α7 =
β4 = β5 = β6 = rac (t − 1) + rad (t − 1) + rbc (t − 1)
+ rbd (t − 1) − rab (t − 1) − rcd (t − 1),
β7 = (m − i)[rac (t − 1) + rad (t − 1)] + i[rbc (t − 1) + rbd (t − 1)] −
m 2i(m − i) rab (t − 1) − rcd (t − 1), 2 m
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RESISTANCE DISTANCE IN POTTING NETWORKS
5
β8 = (m − j)[rac (t − 1) + rbc (t − 1)]
2j(m − j) m rab (t − 1) − rcd (t − 1), 2 m β9 = (m − i)(m − j)rac (t − 1) + j(m − i)rad (t − 1)
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+ j[rad (t − 1) + rbd (t − 1)] −
+ i(m − j)rbc (t − 1) + ijrbd (t − 1) − i(m − i)rab (t − 1) − j(m − j)rcd (t − 1).
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Proof. Let us consider the resistance rx∗ y∗ (N1 ). Using Elimination Principle, we can delete all the grass in N1 , then 2m rx∗ y∗ (N1 ) = (3.1) m+2 due to series and parallel principles. Case 1. Suppose u, v ∈ Vt−1 . Utilizing (3.1) and Substitution Principle on Nt , we directly obtain that 2m ruv (t − 1). (3.2) ruv (t) = m+2 Case 2. Suppose u ∈ Vt−1 and v ∈ {e0 , e1 , e2 , · · · , en , e1 , e2 , · · · , em−1 }. Subcase 2.1. When v = e0 , by Substitution Principle and Elimination Principle, we can delete the grass and find an equivalent network shown in Fig. 4, where the weight of edge is the corresponding conductance.
u
a
1
1
e0
b
1/m
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Figure 4. u ∈ Vt−1 and v = e0 .
urn
Making use of Lemma 1, we have 2re0 a (t) + (re0 a (t) − 0) + (re0 b (t) − rab (t)) = 2, 2re0 b (t) + (re0 a (t) − rba (t)) + (re0 b (t) − 0) = 2.
Solving it, we can obtain
re0 a (t) = re0 b (t) =
1 1 rab (t) + . 4 2
(3.3)
Using Lemma 1 again, we have 2re0 u (t) + (re0 a (t) − rua (t)) + (re0 b (t) − rub (t)) = 2,
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applying (3.2) and (3.3) to the equation (3.4), it follows that 1 m 1 re0 u (t) = + rua (t − 1) + rub (t − 1) − rab (t − 1) . 2 m+2 2
(3.4)
(3.5)
Subcase 2.2. For v ∈ {e1 , e2 , · · · , en }, according to Elimination Principle, we can delete other grass and find an equivalent network shown in Fig. 5. Let v = ei . By Lemma 1, we have 3re0 a (t) + (re0 a (t) − 0) + (re0 b (t) − rab (t)) + (re0 ei (t) − raei (t)) = 2, 3r 0 (t) + (re0 a (t) − rba (t)) + (re0 b (t) − 0) + (re0 ei (t) − rbei (t)) = 2, (3.6) eb re0 ei (t) + (re0 ei (t) − 0) = 2.
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
ei
a
e
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b
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u
1
1
1/m
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Figure 5. u ∈ Vt−1 and v = ei .
Applying (3.1) and (3.2) to equation set (3.6) and solving it, we obtain that re0 ei (t) = 1, and raei (t) = rbei (t) = Using Lemma 1 again, we obtain
1 3 rab (t) + . 4 2
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rvu (t) + (rve0 (t) − rue0 (t)) = 2,
together with (3.5), we can get m 3 ruei (t) = + 2 m+2
1 rua (t − 1) + rub (t − 1) − rab (t − 1) . 2
Subcase 2.3. For v ∈ {e1 , e2 , · · · , em−1 }, in the same way, the equivalent network is shown in Fig. 6.
u
a
b
½
1/(m - j)
1/ j
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ej
Figure 6. u ∈ Vt−1 and v = ej .
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Let v = ej . In the same way, we have 1 1 1 1 )rva (t) + (rva (t) − 0) + (rvb (t) − rab (t)) = 2, ( + j m−j j m−j 1 1 1 1 ( + )rvb (t) + (rva (t) − rab (t)) + (rvb (t) − 0) = 2. j m−j j m−j Solving it, we obtain that j2 j(m − j) rva (t) = + 2 rab (t), m m (m − j)2 j(m − j) rvb (t) = rab (t). + m m2 Utilizing Lemma 1 again, we have 1 1 1 1 ( + )re u (t) + (rej a (t) − rua (t)) + (re b (t) − rub (t)) = 2. j m−j j j m−j j Combining (3.2), (3.7) and (3.8), we can obtain ruej (t) =
2 j(m − j) + [(m − j)rua (t − 1) m m+2 j(m − j) + jrub (t − 1) − rab (t − 1)]. m
(3.7)
(3.8)
(3.9)
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RESISTANCE DISTANCE IN POTTING NETWORKS
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ei 1
1
1
b
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a
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Case 3. Suppose u, v ∈ Ve ∪ Vf , where Ve = {e0 , e1 , · · · , en , e1 , · · · , em−1 } and Vf = {f 0 , f 1 , · · · , f n , f1 , · · · , fm−1 }. Subcase 3.1. Assume u ∈ {e1 , e2 , · · · , en } and v ∈ {e0 }. In this case, the equivalent network is shown in Fig. 7.
e0 1/m
Figure 7. u = ei and v = e0 . Let u = ei . Making use of Lemma 1, we have
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rei e0 (t) + (rei e0 (t) − 0) = 2.
Solving this equation, we can obtain
rei e0 (t) = 1.
1
Subcase 3.2. Assume u, v ∈ {e , e2 , · · · , en }. In this case, the equivalent network is shown in Fig. 8.
ej
ei
a
1
1
1
1
al
e0
b
1/m
Figure 8. u = ei and v = ej (i 6= j).
urn
Let u = ei and v = ej (i 6= j). In the same way, we have the following equation, Hence
rei ej (t) + (rei e0 (t) − rej e0 (t)) = 2.
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rei ej (t) = 2. Subcase 3.3. Assume u, v ∈ {e1 , e2 , · · · , em−1 }. Fig. 9 is the equivalent network when u = ei , v = ej (i < j).
a
b
1/2
1/( m - j )
1/ i
ei
1/( j - i )
ej
Figure 9. u = ei and v = ej (i < j).
Using this equivalent network, we can obtain 1 1 1 1 ( + )rei ej (t) + (rei a (t) − rej a (t)) + (re e (t) − 0) = 2, i j−i i j−i i j
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
with (3.9), we have
of
2(j − i)2 (m + i − j)(j − i) + rab (t − 1). m m(m + 2)
rei ej (t) =
Subcase 3.4. Assume u = e0 and v = f 0 . In this case, the equivalent network is shown in Fig. 10. 1
1
e0
b
c
1/m
1
1
d
p ro
a
f0
1/m
Figure 10. u = e0 and v = f 0 . In the same way, by Lemma 1, we have therefore,
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2re0 f 0 (t) + (re0 a (t) − rf 0 a (t)) + (re0 b (t) − rf 0 b (t)) = 2, m [rac (t − 1) + rad (t − 1) + rbc (t − 1) 2(m + 2) + rbd (t − 1) − rab (t − 1) − rcd (t − 1)].
re0 f 0 (t) =1 +
Subcase 3.5. Assume u = e0 and v ∈ {f 1 , f 2 , · · · , f n }. In this case, the equivalent network is shown in Fig. 11.
a
1
b
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1
e
0
c
1/m
f
i
f
0
1 1
1
d
1/m
urn
Figure 11. u = e0 and v = f i .
Let v = f i . In terms of Lemma 1, we have that is
2re0 f i (t) + (re0 a (t) − rf i a (t)) + (re0 b (t) − rf i b (t)) = 2, m [rac (t − 1) + rad (t − 1) 2(m + 2) + rbc (t − 1) + rbd (t − 1) − rab (t − 1) − rcd (t − 1)].
Jo
re0 f i (t) = 2 +
Subcase 3.6. Assume u ∈ {e1 , e2 , · · · , en } and v ∈ {f 1 , f 2 , · · · , f n }. In this case, the equivalent network is shown in Fig. 12. Let u = ei , v = f j . In terms of Lemma 1, we have
thus,
rei f j (t) + (rei e0 (t) − rf j e0 (t)) = 2, m [rac (t − 1) + rad (t − 1) + rbc (t − 1) 2(m + 2) + rbd (t − 1) − rab (t − 1) − rcd (t − 1)].
rei f j (t) = 3 +
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RESISTANCE DISTANCE IN POTTING NETWORKS
ei
j
f 1
1
e0
1
b c
1
1
f0
1/m
d
of
1
a
9
1/m
p ro
Figure 12. u = ei and v = f j .
Subcase 3.7. Assume u = e0 and v ∈ {f1 , f2 , · · · , fm−1 }. In this case, the equivalent network is shown in Fig. 13.
a
1
1
e0
b
c
Pr e-
1/ i
1/m
d
½
1/(m - i)
fi
Figure 13. u = e0 and v = fi . Let v = fi . Making use of Lemma 1, we have
2re0 fi (t) + (re0 a (t) − rfi a (t)) + (re0 b (t) − rfi b (t)) = 2.
(3.10)
Applying (3.3) and (3.9) to equation (3.10), we can obtain
urn
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1 1 i(m − i) + [(m − i)(rac (t − 1) + rbc (t − 1)) re0 fi (t) = + 2 m m+2 m 2i(m − i) + i(rad (t − 1) + rbd (t − 1)) − rab (t − 1) − rcd (t − 1)]. 2 m Subcase 3.8. Assume u ∈ {e1 , e2 , · · · , en } and v ∈ {f1 , f2 , · · · , fm−1 }. ei
a
1
1
e
0
Jo
1/m
1
b c
d 1/(m - j)
1/ j
fj
Figure 14. u = ei and v = fj .
Let u = ei , v = fj . Observing the equivalent network in Fig. 14, by Lemma 1, we can obtain rei fj (t) + (rei e0 (t) − rfj e0 (t)) = 2, thus, 3 j(m − j) 1 rei fj (t) = + + [(m − j)(rac (t − 1) + rbc (t − 1)) 2 m m+2 m 2j(m − j) + j(rad (t − 1) + rbd (t − 1)) − rab (t − 1) − rcd (t − 1)]. 2 m
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JIAQI FAN, JIALI ZHU, LI TIAN, AND QIN WANG
a
b c
½
1/(m - i)
1/ i
d
½ 1/ j
ei
1/(m - j)
fj
of
10
p ro
Figure 15. u = ei and v = fj .
Subcase 3.9. Assume u ∈ {e1 , e2 , · · · , em−1 } and v ∈ {f1 , f2 , · · · , fm−1 }. Let u = ei , v = fj . In the same way, observing the equivalent network in Fig. 15, by Lemma 1, we can obtain 1 1 1 1 )ruv (t) + (rua (t) − rva (t)) + (rub (t) − rvb (t)) = 2, ( + i m−i i m−i therefore,
Pr e-
j(m − i) i(m − i) + j(m − j) (m − i)(m − j) rac (t) + rad (t) + 2 m m m2 ij i(m − i) j(m − j) i(m − j) rbc (t) + 2 rbd (t) − rab (t) − rcd (t) + 2 2 m m m m2 i(m − i) + j(m − j) 2 = + [(m − i)(m − j)rac (t − 1) m m(m + 2) + j(m − i)rad (t − 1) + i(m − j)rbc (t − 1) + ijrbd (t − 1)
rei fj (t) =
− i(m − i)rab (t − 1) − j(m − j)rcd (t − 1)].
The proof is complete.
al
4. Conclusion
Jo
urn
In this paper, we focus on resistance distance in potting network. In fact, The initial graph in potting networks has a hybrid pattern which is quite different from that of flower networks in [4]. What we are interested is the hybrid fractal networks, such as their eigentime identities and resistance distances. Both of them are related to the Laplacian and random walk on evolving networks. In our future work, based on the resistance distance, we will investigate eigentime identities of fractal networks using the formula ([27]) Fij + Fji = 2(♯E)rij (N ) where Fij is the mean-first passage time from node i ∈ V to node j ∈ V in the network N = (V, E, w) with w ≡ 1. References
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Journal Pre-proof
RESISTANCE DISTANCE IN POTTING NETWORKS
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Pr e-
p ro
of
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Jo
Department of Mathematics, Ningbo University, Ningbo 315211, P. R. China E-mail address: [email protected] Department of Mathematics, Ningbo University, Ningbo 315211, P. R. China E-mail address: [email protected]
Department of software engineering, Zhejiang Wanli University, Ningbo 315100, P. R. China E-mail address: [email protected]