On measures of average degree for lattices
On measures of average degree for lattices Sven Erick Alm Department of Mathematics Uppsala University P.O. Box 480 SE-751 06 Uppsala, Sweden March 20, 2003
Abstract The usual definition of average degree for a non-regular lattice has the disadvantage that it takes the same value for many lattices with clearly different connectivity. We introduce an alternative definition of average degree, which better separates different lattices. These measures are compared on a class of lattices and are analyzed using a Markov chain describing a random walk on the lattice. Using the new measure, we conjecture the order of both the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.
AMS 2000 Classification: Primary: 05C50, Secondary: 60K35, 60G50. Key words: Lattice, average degree, random walk, percolation.
1
1
Introduction
By a lattice we will understand a connected periodic planar graph with a finite number,
, of
node classes, with degrees . The degree of a node is the number of edges from the node. We restrict interest to non-directed graphs. If , the lattice is regular and it is natural to define the degree of the lattice as . For lattices with , the average degree of the lattice is usually defined as a weighted average
(1)
are the proportions in which the node classes appear. This definition of average degree, or coordination number as it is called in the Physics
where the weights
literature, has some disadvantages, which are discussed in Section 2, and we will introduce an , and give several alternative definition of average degree, , in Section 4. We show that ! #" #" #" examples of pairs of graphs $% , where $& , but (' $% . We have found no examples where the opposite holds. A comparison of these measures of degree is performed on a class of lattices, the ALB lattices, which are defined in Section 3. Much of the analysis is based on a study of a Markov chain describing a random walk on the lattice, with the node classes as states. We conclude with a number of conjectures, including predicted order of the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.
2
Average degree
The measure of (1) is frequently used by physicists attempting to give universal formulas for critical probabilities in percolation; for a review, see Wierman [5]. These formulas are usually based on the dimension and the average degree of the lattice. As our lattices all have dimension 2, these formulas will predict the same critical probability for all graphs with the same average degree.
To compute , we need to know the weights
. These can be expressed by means * + )-,.0/13254 , describing a of the stationary distribution of the finite Markov chain, ) 2
random walk on the lattice, with the node classes as states. Let 687:9;7 be the transition matrix of this finite Markov chain. As the graph is connected, the Markov chain is irreducible, and has a unique stationary distribution =@?Aí?ú?úï whereas èû ê%ëìêÏí?î ïEðñxòEòuü
èû ê%ëýêÏí ñïEðï íïEð0òEòþó è û ê%ëìê Bow-tie òEò , ç ü ê úL÷ ò ÿJô "ó,ú0ÿJô »÷µð ÿJôöñ ÷í0ÿJô ó è3ç éê%ëýêÏíïEðï úïEð0òEò , and è3éê%ëìêÏíï úï íï úòEò½ó,í0ÿJôöñN÷%ú0ÿJôöñ'ó whereas èû ê%ëìêÏíï úï íï úòEò ó èû ê%ëýêÏíïEðï úïEð0òEò½ó .
There are non-trivial lattices for which è û ó Theorem 6. For a bipartite lattice with
ó ,
è ÿ ÷ è ñ ù
Proof. For a bipartite lattice with
so that
ê ÿ ï ÿ ò , è ó ñ ñ û ó
èç é .
ç è" û ó 3è é'ó
ó , ó
and
ó
÷èñ
ê%è ÿ ÷ è ñ ò ÿJôöñ and èç é'ó è ÿ
ó
è ñ
è ÿ
ï
ê%è ÿ ÷ è ñ ò ÿJôöñ .
A summary of average degrees for the ALB lattices is given in Table 1. In the table we also give some aliases for the lattices that are more or less common in the literature. From Table 1 we observe a number of properties that hold for the average degrees of the ALB lattices, and which we conjecture hold for general lattices. Conjecture 1. For general lattices
and
,
# !" è3ç éê ò% è3ç éê ò (i) "$ èç &ê ò'% èç &ê ò # !" èû ê ò% è û ê ò ç éê òþü è3ç éê ò 3 è (ii) "$ èç & ê ò'% èç &ê ò # !" èû ê òþü è û ê ò ç &ê òuü èç è & ê ò (iii) "$ è3ç éê òþü èç ( ê ò Conjecture 2. For general lattices and , èû ê òuü èû ê ò èû ê%ëýê òEò') èû ê%ëìê òEò (i) èç éê òuü è3ç éê ò èç éê%ëýê òEò') è3ç éê%ëìê òEò (ii) èû ê
òþü èû ê
ò
Remark 5. By the relation (2) between the average degree of a lattice and its dual, we get
& è ç Nê
& ò+* èç &ê%ëìê òEò', è-ç & ê%ëìê
òþü è ç Nê
òEò . 11
Table 1: Average degrees for the ALB lattices. Graph
Alias
46587:9;= ? Asanoha 465A@B9DC:9;F-? Triangular 5G587>=>9G@B9D7>=9G@H?I9-5A@J>?G? Bow-tie 587>=>9G@B9D7:9G@H? 587>K>9G@=-? 587 J 9DC? 46587:9DC:9D7:9DC? Dice 46587:9G@B9DC:9G@H? 4 (Ruby) 5A@J ? Square 587:9G@B9DC:9G@H? Ruby 587:9DC:9D7:9DC? Kagomé 46587J9DC? 46587>K>9G@=-? Pentagonal 46587>=>9G@B9D7:9G@H? 465G587>=>9G@B9D7>=9G@H?I9-5A@J?G? 4 (Bow-tie) 58C K ? Hexagonal 5A@B9DE>=? Bathroom tile, Briarwood 5A@B9DC:9;_yxMcdxMfR _{zf . However, this matrix does not p q kS |:D} ` k . correspond to an undirected lattice, as these must satisfy |~} ` would give
Remark 7. If we introduce the ordering relation
MBL NQPXR'` M:L NwR or M U W NQPSR`M U W NwR or Pb if MBL NwNQPSRGR' MBL Nw6Nw6RGR or MnU WNw6NQPXRGR'MnU WNwNwRGR r we get a complete ordering of the ALB lattices as shown in Table 1. This order is in full agreement with what is known of the partial order of these lattices that can be obtained by inclusions and non-inclusions, see Parviainen and Wierman [3]. It is also in agreement with the order of critical probabilities for bond percolation on these lattices, see Wierman and Parviainen [6], and the order of the connective constants of self-avoiding walks on these lattices, see Alm [1]. Conjecture 4. The critical probabilities for bond percolation on the ALB lattices are ordered, in increasing order, as in Table 1. Conjecture 5. The connective constants for self-avoiding walks on the ALB lattices are ordered, in decreasing order, as in Table 1. Let
| NQPXR
denote the critical probability for bond percolation and
constant for self-avoiding walks on the lattice degree for
| and
P
NQPSR
the connective
. It seems likely that the dependence on average
is more general than stated in Conjectures 4 and 5.
13
Conjecture 6. For general lattices,
and
O X'VO X' 6 , (ii) O X'VO ¢¡ S £¡ .
,
(i)
Acknowledgements: This work was supported by the Swedish Science Foundation (NFR). I would like to thank Svante Janson for stimulating discussions regarding the proof of Theorem 5 and Robert Parviainen for producing Figures 2–4.
References [1] Alm, Sven Erick. In preparation. [2] Grünbaum, Branko and Shephard, G.C. Patterns and Tilings. W.H. Freeman (1987). [3] Parviainen, Robert and Wierman, John C., The subgraph partial ordering of Archimedean and Laves lattices. U.U.D.M. Report 2002:13 (2002). [4] Wierman, John C., On the range of bond percolation thresholds for fully-triangulated graphs. J. Phys. A 35 (2001) 959-964. [5] Wierman, John C., Accuracy of universal formulas for percolation thresholds based on dimension and coordination number. Phys. Rev. E 66 (2002) 027105-1–4. [6] Wierman, John C. and Parviainen, Robert, Ordering bond percolation critical probabilities. U.U.D.M. Report 2002:44 (2002) .
14
Abstract The usual definition of average degree for a non-regular lattice has the disadvantage that it takes the same value for many lattices with clearly different connectivity. We introduce an alternative definition of average degree, which better separates different lattices. These measures are compared on a class of lattices and are analyzed using a Markov chain describing a random walk on the lattice. Using the new measure, we conjecture the order of both the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.
AMS 2000 Classification: Primary: 05C50, Secondary: 60K35, 60G50. Key words: Lattice, average degree, random walk, percolation.
1
1
Introduction
By a lattice we will understand a connected periodic planar graph with a finite number,
, of
node classes, with degrees . The degree of a node is the number of edges from the node. We restrict interest to non-directed graphs. If , the lattice is regular and it is natural to define the degree of the lattice as . For lattices with , the average degree of the lattice is usually defined as a weighted average
(1)
are the proportions in which the node classes appear. This definition of average degree, or coordination number as it is called in the Physics
where the weights
literature, has some disadvantages, which are discussed in Section 2, and we will introduce an , and give several alternative definition of average degree, , in Section 4. We show that ! #" #" #" examples of pairs of graphs $% , where $& , but (' $% . We have found no examples where the opposite holds. A comparison of these measures of degree is performed on a class of lattices, the ALB lattices, which are defined in Section 3. Much of the analysis is based on a study of a Markov chain describing a random walk on the lattice, with the node classes as states. We conclude with a number of conjectures, including predicted order of the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.
2
Average degree
The measure of (1) is frequently used by physicists attempting to give universal formulas for critical probabilities in percolation; for a review, see Wierman [5]. These formulas are usually based on the dimension and the average degree of the lattice. As our lattices all have dimension 2, these formulas will predict the same critical probability for all graphs with the same average degree.
To compute , we need to know the weights
. These can be expressed by means * + )-,.0/13254 , describing a of the stationary distribution of the finite Markov chain, ) 2
random walk on the lattice, with the node classes as states. Let 687:9;7 be the transition matrix of this finite Markov chain. As the graph is connected, the Markov chain is irreducible, and has a unique stationary distribution =@?Aí?ú?úï whereas èû ê%ëìêÏí?î ïEðñxòEòuü
èû ê%ëýêÏí ñïEðï íïEð0òEòþó è û ê%ëìê Bow-tie òEò , ç ü ê úL÷ ò ÿJô "ó,ú0ÿJô »÷µð ÿJôöñ ÷í0ÿJô ó è3ç éê%ëýêÏíïEðï úïEð0òEò , and è3éê%ëìêÏíï úï íï úòEò½ó,í0ÿJôöñN÷%ú0ÿJôöñ'ó whereas èû ê%ëìêÏíï úï íï úòEò ó èû ê%ëýêÏíïEðï úïEð0òEò½ó .
There are non-trivial lattices for which è û ó Theorem 6. For a bipartite lattice with
ó ,
è ÿ ÷ è ñ ù
Proof. For a bipartite lattice with
so that
ê ÿ ï ÿ ò , è ó ñ ñ û ó
èç é .
ç è" û ó 3è é'ó
ó , ó
and
ó
÷èñ
ê%è ÿ ÷ è ñ ò ÿJôöñ and èç é'ó è ÿ
ó
è ñ
è ÿ
ï
ê%è ÿ ÷ è ñ ò ÿJôöñ .
A summary of average degrees for the ALB lattices is given in Table 1. In the table we also give some aliases for the lattices that are more or less common in the literature. From Table 1 we observe a number of properties that hold for the average degrees of the ALB lattices, and which we conjecture hold for general lattices. Conjecture 1. For general lattices
and
,
# !" è3ç éê ò% è3ç éê ò (i) "$ èç &ê ò'% èç &ê ò # !" èû ê ò% è û ê ò ç éê òþü è3ç éê ò 3 è (ii) "$ èç & ê ò'% èç &ê ò # !" èû ê òþü è û ê ò ç &ê òuü èç è & ê ò (iii) "$ è3ç éê òþü èç ( ê ò Conjecture 2. For general lattices and , èû ê òuü èû ê ò èû ê%ëýê òEò') èû ê%ëìê òEò (i) èç éê òuü è3ç éê ò èç éê%ëýê òEò') è3ç éê%ëìê òEò (ii) èû ê
òþü èû ê
ò
Remark 5. By the relation (2) between the average degree of a lattice and its dual, we get
& è ç Nê
& ò+* èç &ê%ëìê òEò', è-ç & ê%ëìê
òþü è ç Nê
òEò . 11
Table 1: Average degrees for the ALB lattices. Graph
Alias
46587:9;= ? Asanoha 465A@B9DC:9;F-? Triangular 5G587>=>9G@B9D7>=9G@H?I9-5A@J>?G? Bow-tie 587>=>9G@B9D7:9G@H? 587>K>9G@=-? 587 J 9DC? 46587:9DC:9D7:9DC? Dice 46587:9G@B9DC:9G@H? 4 (Ruby) 5A@J ? Square 587:9G@B9DC:9G@H? Ruby 587:9DC:9D7:9DC? Kagomé 46587J9DC? 46587>K>9G@=-? Pentagonal 46587>=>9G@B9D7:9G@H? 465G587>=>9G@B9D7>=9G@H?I9-5A@J?G? 4 (Bow-tie) 58C K ? Hexagonal 5A@B9DE>=? Bathroom tile, Briarwood 5A@B9DC:9;_yxMcdxMfR _{zf . However, this matrix does not p q kS |:D} ` k . correspond to an undirected lattice, as these must satisfy |~} ` would give
Remark 7. If we introduce the ordering relation
MBL NQPXR'` M:L NwR or M U W NQPSR`M U W NwR or Pb if MBL NwNQPSRGR' MBL Nw6Nw6RGR or MnU WNw6NQPXRGR'MnU WNwNwRGR r we get a complete ordering of the ALB lattices as shown in Table 1. This order is in full agreement with what is known of the partial order of these lattices that can be obtained by inclusions and non-inclusions, see Parviainen and Wierman [3]. It is also in agreement with the order of critical probabilities for bond percolation on these lattices, see Wierman and Parviainen [6], and the order of the connective constants of self-avoiding walks on these lattices, see Alm [1]. Conjecture 4. The critical probabilities for bond percolation on the ALB lattices are ordered, in increasing order, as in Table 1. Conjecture 5. The connective constants for self-avoiding walks on the ALB lattices are ordered, in decreasing order, as in Table 1. Let
| NQPXR
denote the critical probability for bond percolation and
constant for self-avoiding walks on the lattice degree for
| and
P
NQPSR
the connective
. It seems likely that the dependence on average
is more general than stated in Conjectures 4 and 5.
13
Conjecture 6. For general lattices,
and
O X'VO X' 6 , (ii) O X'VO ¢¡ S £¡ .
,
(i)
Acknowledgements: This work was supported by the Swedish Science Foundation (NFR). I would like to thank Svante Janson for stimulating discussions regarding the proof of Theorem 5 and Robert Parviainen for producing Figures 2–4.
References [1] Alm, Sven Erick. In preparation. [2] Grünbaum, Branko and Shephard, G.C. Patterns and Tilings. W.H. Freeman (1987). [3] Parviainen, Robert and Wierman, John C., The subgraph partial ordering of Archimedean and Laves lattices. U.U.D.M. Report 2002:13 (2002). [4] Wierman, John C., On the range of bond percolation thresholds for fully-triangulated graphs. J. Phys. A 35 (2001) 959-964. [5] Wierman, John C., Accuracy of universal formulas for percolation thresholds based on dimension and coordination number. Phys. Rev. E 66 (2002) 027105-1–4. [6] Wierman, John C. and Parviainen, Robert, Ordering bond percolation critical probabilities. U.U.D.M. Report 2002:44 (2002) .
14
