- Email: [email protected]
Effective dimensions in networks with long-range connections
ARTICLE IN PRESS
Physica A 356 (2005) 157–161 www.elsevier.com/locate/physa
Effective dimensions in networks with long-range connections Cristian F. Moukarzel CINVESTAV del IPN, Depto. de Fı´sica Aplicada, 97310 Me´rida, Yucata´n, Me´xico Available online 9 June 2005
Abstract One- and two-dimensional lattices of points are connected with long-range links, whose lengths are distributed according to PðrÞrm . By changing the decay exponent m one can go from d-dimensional short-range networks to 1-dimensional networks topologically similar to random graphs. An effective dimension d chem ðmÞ can be defined in terms of the shortest-path properties of these networks. These effective dimensions d chem are calculated here in one and two dimensions, for system sizes of up to 107 points. r 2005 Elsevier B.V. All rights reserved. Keywords: Long-range interactions; Power-law; Effective dimension; Chemical dimension; Small-world; Networks; Phase transitions; Graphs
1. Introduction Long-range interactions (LRI) whose intensity decays with distance r as JðRÞrm can radically change the critical behavior of a spin system, if m is small enough. Ising [1–10] and Potts [11–14] models with LRI have been intensely studied recently in order to better understand these effects. Renormalization group arguments [3] suggest the following picture for ferromagnetic d-dimensional n-component spin systems: If mo 32 d, the system behaves in a mean field manner. If 32 domod þ 2, its critical indices depend on m, while if m4d þ 2 the behavior is short-range like. It can be said that the addition of LRI may endow the system with an ‘‘effective E-mail address: [email protected]. 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.05.029
ARTICLE IN PRESS C.F. Moukarzel / Physica A 356 (2005) 157–161
158
dimension’’ d eff that differs from the embedding Euclidean dimension d. This idea has been exploited to study the scaling behavior of critical systems above their effective upper critical dimension d u , while still working on lattices of low Euclidean dimension [7]. All these results concern systems where all pairs of sites interact through a coupling whose intensity decays with distance. An alternative way to introduce LRIs is to let a unit-strength LR coupling be present with a probability PðrÞ that decays with distance [15]. This second type of LRI is the focus of this work. A network can be constructed to represent these systems, in which a link between two points represents a unit coupling. Because all links are now equally strong, effective dimensions can be defined, that only depend on the topology or connectivity, of the resulting network or graph.
2. Shortest paths and effective dimensions For a connected but otherwise arbitrary graph, we define the shortest path ‘ij between two nodes i and j to be the minimum number of links in a path joining i and j. For undirected graphs, ‘ij ¼ ‘ji . The shortest path ‘ is also called chemical distance between i and j. The volume V i ð‘Þ within a distance ‘ around a given point i is the number of nodes that can be reached from i in at most ‘ steps. On a d-dimensional lattice with only nearest-neighbor links, V ð‘Þ grows asymptotically as ‘d . This suggests the possibility to define, for arbitrary graphs, an effective (or chemical) dimension as d chem ¼ lim
‘!1
loghV i ð‘Þi , log ‘
(1)
where the average is taken over all nodes pertaining to the graph or realizations of graphs under consideration. However in many cases, definition (1) is not practical for numerical measurements, because often in a few steps from a site the whole of the graph can be visited. This has the consequence that only small values of ‘ are available. For graphs whose nodes are located in d-dimensional Euclidean space, we have proposed an alternative definition that makes use of Euclidean distances in order to measure d chem [15]. Since the lattice is embedded in d dimensions, clearly V ðrÞrd , where V ðrÞ is the average number of nodes within a sphere of radius r around a point. Let ‘ðrÞ be the average chemical distance for pairs of nodes separated by an Euclidean distance r. Eq. (1) means that V ð‘Þ‘d chem so that, asymptotically log ‘ðrÞ 1 . (2) ‘ðrÞrd=d chem ) d chem ¼ d lim r!1 log r This last definition is based on (1) but is only applicable for networks for which an Euclidean distance exists. The advantage of (2) resides in that since often ‘ðrÞ5r, this last definition suffers from less severe finite-size corrections, in many practical cases. In this work we use definition (2) in order to measure chemical dimensions of
ARTICLE IN PRESS C.F. Moukarzel / Physica A 356 (2005) 157–161
159
one- and two-dimensional arrays of points connected with power-law distributed LR links, as described in the following.
3. Networks and results We consider d-dimensional hypercubic lattices of points and connect them with links in the following way: Z links stem from each point i, onto each of the d-axis directions. For each link we choose the neighbor j at the other end, subjected to the condition that ~ rij be parallel to one of the principal directions of the lattice (see Fig. 1). The link ‘‘length’’ is distributed according to Pðrij Þ ¼ C
1 , rmij
(3)
where C is a normalization constant and rij is the Euclidean distance between points i and j on the d-dimensional array. The resulting network has dZ links per site. The link distribution is not rotationally invariant. In two dimensions, for example, only horizontal or vertical links exist. As compared with a procedure in which each link was chosen according to (3) disregarding its orientation, the chosen method is considerably simpler to implement, but has d ‘‘preferred directions’’. Calculations are under way with a more general, asymptotically rotationally invariant, distribution. However, we do not expect the effective dimensions (see (2)) to be strongly modified by this source of anisotropy. An important difference with Ref. [15] is that in this case, LR links are not added to a d-dimensional network. All links are random and their length distribution is given by (3), in this work. Once the network is constructed, shortest paths are calculated by breadth-first-search as described somewhere else [15–17].
Fig. 1. For each of the points in a square array, Z LR links connect it to a neighbor that lays on one of the axis directions, and is located at a distance r chosen at random with PðrÞrm . The resulting network has thus exactly dZ links per site. In this example, Z ¼ 3 and d ¼ 2.
ARTICLE IN PRESS C.F. Moukarzel / Physica A 356 (2005) 157–161
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
10
L = 107
dchem
L = 107
4
L = 106
2
L = 105
d=1
0 0.5 1
1.5 2 µ
(a)
d/dchem
L = 106
6
0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
L = 105
8
d=1
2.5 3
3.5 4
0
0.5
1
1.5
(b)
2 µ
2.5
3
3.5
4
10
L = 102 L = 3x102 L = 103 L = 3x103
0
1
2
(c)
L = 102 L = 3x102 L = 103 L = 3x103
8
d=2 dchem
d/dchem
160
3 µ
4
5
6 4 2 0
6 (d)
d=2 0
1
2
3 µ
4
5
6
Fig. 2. Networks comprised of one- (a,b) and two-dimensional (c,d) arrays of points were connected by LR links whose distribution is given by Eq. (3). Shortest-path distances ‘ðrÞ are averaged as a function of Euclidean distance r between points. The asymptotic behavior (Eq. (2)) of ‘ðrÞ allows us to calculate d=d chem (a,c) and from it the effective dimensions d chem (b,d) as a function of the decay exponent m. The largest systems studied were comprised of 107 points, both in one and two dimensions. 103 networks with Z ¼ 2 links per site and per principal direction were generated for each data point on these plots.
These are averaged as a function of Euclidean distance r, to obtain ‘ðrÞ, which is the average number of links needed to join two points separated by Euclidean distance r. For large m, the network turns out to be a short-range d-dimensional network and therefore ‘ðrÞr, i.e. d=d chem ! 1 and the effective dimension d chem coincides with the Euclidean dimension d (Fig. 2). For intermediate values of m, the effective dimension depends on m, growing as m decreases. When m is very small, the network is essentially randomly connected (however, only along the d-principal directions) and ‘ðrÞ is asymptotically constant, indicating that d=d chem ¼ 0, or that d chem is divergent in this limit. d chem appears to be divergent for mp1 in one dimension and for mp 12 in two dimensions. The upper value of m beyond which the network is shortd¼2 range seems to be md¼1 SR ¼ 2 in one dimension and mSR ¼ 4 in two dimensions.
Acknowledgements This work was partially supported by CONACYT, Me´xico, through research project 36256-E.
ARTICLE IN PRESS C.F. Moukarzel / Physica A 356 (2005) 157–161
161
References [1] D. Thouless, Long-range order in one-dimensional ising systems, Phys. Rev. 187 (2) (1969) 732. [2] F.J. Dyson, Ising ferromagnet with discontinuous long-range order, Commun. Math. Phys. 21 (4) (1971) 269. [3] M. Fisher, B. Nickel, S. Ma, Critical exponents for long-range interactions, Phys. Rev. Lett. 29 (4) (1972) 917. [4] J.M. Kosterlitz, Phase-transitions in long-range ferromagnetic chains, Phys. Rev. Lett. 37 (23) (1976) 1577. [5] J.L. Monroe, Bethe lattice approximation of long-range interaction Ising-models, Phys. Lett. A 171 (5–6) (1992) 427. [6] S.A. Cannas, One-dimensional Ising-model with long-range interactions—a renormalization-group treatment, Phys. Rev. B-Condens. Matter 52 (5) (1995) 3034. [7] E. Luijten, H. Blote, Finite-size scaling and universality above the upper critical dimensionality, Phys. Rev. Lett. 76 (19) (1996) 1557 (see errata on p. 3662). [8] E. Luijten, H. Blote, Classical critical behavior of spin models with long-range interactions, Phys. Rev. B-Condens. Matter 56 (14) (1997) 8945. [9] J.L. Monroe, One-dimensional Ising models with long-range interactions, J. Phys. A-Math. Gen. 31 (49) (1998) 9809. [10] E. Bayong, H. Diep, Effect of long-range interactions on the critical behavior of the continuous Ising model, Phys. Rev. B-Condens. Matter 59 (18) (1999) 11919. [11] Z. Glumac, K. Uzelac, Critical-behavior of the 1-d q-state Potts-model with long-range interactions, J. Phys. A-Math. Gen. 26 (20) (1993) 5267. [12] S. Cannas, A. de Magalhaes, The one-dimensional Potts model with long-range interactions: a renormalization group approach, J. Phys. A-Math. Gen. 30 (10) (1997) 3345. [13] J.L. Monroe, Long-range one-dimensional Potts models: a cluster mean-field and extrapolation approach, J. Phys. A-Math. Gen. 32 (41) (1999) 7083. [14] E. Bayong, H. Diep, V. Dotsenko, Potts model with long-range interactions in one dimension, Phys. Rev. Lett. 83 (1) (1999) 14. [15] C. Moukarzel, M. de Menezes, Shortest paths on systems with power-law distributed long-range connections, Phys. Rev. E 65 (5) (2002) 56709. [16] M. de Menezes, C. Moukarzel, T. Penna, Geometric phase-transition on systems with sparse longrange connections, Physica A 295 (1–2) (2001) 132. [17] M. De Menezes, C. Moukarzel, T. Penna, First-order transition in small-world networks, Europhys. Lett. 50 (5) (2000) 574.
Physica A 356 (2005) 157–161 www.elsevier.com/locate/physa
Effective dimensions in networks with long-range connections Cristian F. Moukarzel CINVESTAV del IPN, Depto. de Fı´sica Aplicada, 97310 Me´rida, Yucata´n, Me´xico Available online 9 June 2005
Abstract One- and two-dimensional lattices of points are connected with long-range links, whose lengths are distributed according to PðrÞrm . By changing the decay exponent m one can go from d-dimensional short-range networks to 1-dimensional networks topologically similar to random graphs. An effective dimension d chem ðmÞ can be defined in terms of the shortest-path properties of these networks. These effective dimensions d chem are calculated here in one and two dimensions, for system sizes of up to 107 points. r 2005 Elsevier B.V. All rights reserved. Keywords: Long-range interactions; Power-law; Effective dimension; Chemical dimension; Small-world; Networks; Phase transitions; Graphs
1. Introduction Long-range interactions (LRI) whose intensity decays with distance r as JðRÞrm can radically change the critical behavior of a spin system, if m is small enough. Ising [1–10] and Potts [11–14] models with LRI have been intensely studied recently in order to better understand these effects. Renormalization group arguments [3] suggest the following picture for ferromagnetic d-dimensional n-component spin systems: If mo 32 d, the system behaves in a mean field manner. If 32 domod þ 2, its critical indices depend on m, while if m4d þ 2 the behavior is short-range like. It can be said that the addition of LRI may endow the system with an ‘‘effective E-mail address: [email protected]. 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.05.029
ARTICLE IN PRESS C.F. Moukarzel / Physica A 356 (2005) 157–161
158
dimension’’ d eff that differs from the embedding Euclidean dimension d. This idea has been exploited to study the scaling behavior of critical systems above their effective upper critical dimension d u , while still working on lattices of low Euclidean dimension [7]. All these results concern systems where all pairs of sites interact through a coupling whose intensity decays with distance. An alternative way to introduce LRIs is to let a unit-strength LR coupling be present with a probability PðrÞ that decays with distance [15]. This second type of LRI is the focus of this work. A network can be constructed to represent these systems, in which a link between two points represents a unit coupling. Because all links are now equally strong, effective dimensions can be defined, that only depend on the topology or connectivity, of the resulting network or graph.
2. Shortest paths and effective dimensions For a connected but otherwise arbitrary graph, we define the shortest path ‘ij between two nodes i and j to be the minimum number of links in a path joining i and j. For undirected graphs, ‘ij ¼ ‘ji . The shortest path ‘ is also called chemical distance between i and j. The volume V i ð‘Þ within a distance ‘ around a given point i is the number of nodes that can be reached from i in at most ‘ steps. On a d-dimensional lattice with only nearest-neighbor links, V ð‘Þ grows asymptotically as ‘d . This suggests the possibility to define, for arbitrary graphs, an effective (or chemical) dimension as d chem ¼ lim
‘!1
loghV i ð‘Þi , log ‘
(1)
where the average is taken over all nodes pertaining to the graph or realizations of graphs under consideration. However in many cases, definition (1) is not practical for numerical measurements, because often in a few steps from a site the whole of the graph can be visited. This has the consequence that only small values of ‘ are available. For graphs whose nodes are located in d-dimensional Euclidean space, we have proposed an alternative definition that makes use of Euclidean distances in order to measure d chem [15]. Since the lattice is embedded in d dimensions, clearly V ðrÞrd , where V ðrÞ is the average number of nodes within a sphere of radius r around a point. Let ‘ðrÞ be the average chemical distance for pairs of nodes separated by an Euclidean distance r. Eq. (1) means that V ð‘Þ‘d chem so that, asymptotically log ‘ðrÞ 1 . (2) ‘ðrÞrd=d chem ) d chem ¼ d lim r!1 log r This last definition is based on (1) but is only applicable for networks for which an Euclidean distance exists. The advantage of (2) resides in that since often ‘ðrÞ5r, this last definition suffers from less severe finite-size corrections, in many practical cases. In this work we use definition (2) in order to measure chemical dimensions of
ARTICLE IN PRESS C.F. Moukarzel / Physica A 356 (2005) 157–161
159
one- and two-dimensional arrays of points connected with power-law distributed LR links, as described in the following.
3. Networks and results We consider d-dimensional hypercubic lattices of points and connect them with links in the following way: Z links stem from each point i, onto each of the d-axis directions. For each link we choose the neighbor j at the other end, subjected to the condition that ~ rij be parallel to one of the principal directions of the lattice (see Fig. 1). The link ‘‘length’’ is distributed according to Pðrij Þ ¼ C
1 , rmij
(3)
where C is a normalization constant and rij is the Euclidean distance between points i and j on the d-dimensional array. The resulting network has dZ links per site. The link distribution is not rotationally invariant. In two dimensions, for example, only horizontal or vertical links exist. As compared with a procedure in which each link was chosen according to (3) disregarding its orientation, the chosen method is considerably simpler to implement, but has d ‘‘preferred directions’’. Calculations are under way with a more general, asymptotically rotationally invariant, distribution. However, we do not expect the effective dimensions (see (2)) to be strongly modified by this source of anisotropy. An important difference with Ref. [15] is that in this case, LR links are not added to a d-dimensional network. All links are random and their length distribution is given by (3), in this work. Once the network is constructed, shortest paths are calculated by breadth-first-search as described somewhere else [15–17].
Fig. 1. For each of the points in a square array, Z LR links connect it to a neighbor that lays on one of the axis directions, and is located at a distance r chosen at random with PðrÞrm . The resulting network has thus exactly dZ links per site. In this example, Z ¼ 3 and d ¼ 2.
ARTICLE IN PRESS C.F. Moukarzel / Physica A 356 (2005) 157–161
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
10
L = 107
dchem
L = 107
4
L = 106
2
L = 105
d=1
0 0.5 1
1.5 2 µ
(a)
d/dchem
L = 106
6
0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
L = 105
8
d=1
2.5 3
3.5 4
0
0.5
1
1.5
(b)
2 µ
2.5
3
3.5
4
10
L = 102 L = 3x102 L = 103 L = 3x103
0
1
2
(c)
L = 102 L = 3x102 L = 103 L = 3x103
8
d=2 dchem
d/dchem
160
3 µ
4
5
6 4 2 0
6 (d)
d=2 0
1
2
3 µ
4
5
6
Fig. 2. Networks comprised of one- (a,b) and two-dimensional (c,d) arrays of points were connected by LR links whose distribution is given by Eq. (3). Shortest-path distances ‘ðrÞ are averaged as a function of Euclidean distance r between points. The asymptotic behavior (Eq. (2)) of ‘ðrÞ allows us to calculate d=d chem (a,c) and from it the effective dimensions d chem (b,d) as a function of the decay exponent m. The largest systems studied were comprised of 107 points, both in one and two dimensions. 103 networks with Z ¼ 2 links per site and per principal direction were generated for each data point on these plots.
These are averaged as a function of Euclidean distance r, to obtain ‘ðrÞ, which is the average number of links needed to join two points separated by Euclidean distance r. For large m, the network turns out to be a short-range d-dimensional network and therefore ‘ðrÞr, i.e. d=d chem ! 1 and the effective dimension d chem coincides with the Euclidean dimension d (Fig. 2). For intermediate values of m, the effective dimension depends on m, growing as m decreases. When m is very small, the network is essentially randomly connected (however, only along the d-principal directions) and ‘ðrÞ is asymptotically constant, indicating that d=d chem ¼ 0, or that d chem is divergent in this limit. d chem appears to be divergent for mp1 in one dimension and for mp 12 in two dimensions. The upper value of m beyond which the network is shortd¼2 range seems to be md¼1 SR ¼ 2 in one dimension and mSR ¼ 4 in two dimensions.
Acknowledgements This work was partially supported by CONACYT, Me´xico, through research project 36256-E.
ARTICLE IN PRESS C.F. Moukarzel / Physica A 356 (2005) 157–161
161
References [1] D. Thouless, Long-range order in one-dimensional ising systems, Phys. Rev. 187 (2) (1969) 732. [2] F.J. Dyson, Ising ferromagnet with discontinuous long-range order, Commun. Math. Phys. 21 (4) (1971) 269. [3] M. Fisher, B. Nickel, S. Ma, Critical exponents for long-range interactions, Phys. Rev. Lett. 29 (4) (1972) 917. [4] J.M. Kosterlitz, Phase-transitions in long-range ferromagnetic chains, Phys. Rev. Lett. 37 (23) (1976) 1577. [5] J.L. Monroe, Bethe lattice approximation of long-range interaction Ising-models, Phys. Lett. A 171 (5–6) (1992) 427. [6] S.A. Cannas, One-dimensional Ising-model with long-range interactions—a renormalization-group treatment, Phys. Rev. B-Condens. Matter 52 (5) (1995) 3034. [7] E. Luijten, H. Blote, Finite-size scaling and universality above the upper critical dimensionality, Phys. Rev. Lett. 76 (19) (1996) 1557 (see errata on p. 3662). [8] E. Luijten, H. Blote, Classical critical behavior of spin models with long-range interactions, Phys. Rev. B-Condens. Matter 56 (14) (1997) 8945. [9] J.L. Monroe, One-dimensional Ising models with long-range interactions, J. Phys. A-Math. Gen. 31 (49) (1998) 9809. [10] E. Bayong, H. Diep, Effect of long-range interactions on the critical behavior of the continuous Ising model, Phys. Rev. B-Condens. Matter 59 (18) (1999) 11919. [11] Z. Glumac, K. Uzelac, Critical-behavior of the 1-d q-state Potts-model with long-range interactions, J. Phys. A-Math. Gen. 26 (20) (1993) 5267. [12] S. Cannas, A. de Magalhaes, The one-dimensional Potts model with long-range interactions: a renormalization group approach, J. Phys. A-Math. Gen. 30 (10) (1997) 3345. [13] J.L. Monroe, Long-range one-dimensional Potts models: a cluster mean-field and extrapolation approach, J. Phys. A-Math. Gen. 32 (41) (1999) 7083. [14] E. Bayong, H. Diep, V. Dotsenko, Potts model with long-range interactions in one dimension, Phys. Rev. Lett. 83 (1) (1999) 14. [15] C. Moukarzel, M. de Menezes, Shortest paths on systems with power-law distributed long-range connections, Phys. Rev. E 65 (5) (2002) 56709. [16] M. de Menezes, C. Moukarzel, T. Penna, Geometric phase-transition on systems with sparse longrange connections, Physica A 295 (1–2) (2001) 132. [17] M. De Menezes, C. Moukarzel, T. Penna, First-order transition in small-world networks, Europhys. Lett. 50 (5) (2000) 574.