Renormalization Group Analysis of a Small-world Network Model

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1 Renormalization Group Analysis of a Small-world Network Model Thomas Payne December 4, 2009 A small-world network is a network where most of the nodes aren t directly connected to each other, but any node can reach any other node in a relatively few number of steps. Small-world behavior has been discovered in a wide range of networks, from the way neurons communicate in the brain to how the internet is connected. In 1998, Strogatz and Watts[1] presented a computational model of a network that displays small-world network behavior. A year later, Newman and Watts[2] presented a renormalization group analysis of a similar model, where they detected a phase transition and defined a critical exponent τ that governs the onset of small-world network behavior. This is a simplified presentation of their analysis, with use of their figures. The model presented by Newman and Watts is a onedimensional with periodic boundary conditions, and so is modeled by a ring (Figure 1). The ring has L nodes, connected to each other in a regular lattice with range k. That is, when k = 1, each node is connected to all other nodes one space away. When k = 3, each node is connected to all nodes three spaces away, and so on. The number of regular connections in the model is kl. Finally, Figure 1: The RG transformations used in the calculations described in the text: (a) the transformation used for the k = 1 system; (b) the transformation used for the k = 3 system. shortcut connections between random nodes are added with probability p. p is the probability per regular connection that a new shortcut connection will be added to the model. The number of shortcut connections present is therefore pkl. The independent parameters of this model are p, k, and L. Our observable is l, the average path length. The prescence of shortcut connections is what allows for small-world behavior. In a regularly-spaced lattice, l L, because to get between any two nodes, every node in-between must be traversed. But when sufficient shortcut connections are present, l log L. The conditions that represent this are 1

2 pkl 1 l L pkl 1 l log L For a constant p and k, then, there must be some turning point L = ξ that signals a crossover from regular lattice behavior to small-world behavior. Judging from the above conditions, this is the point pkξ 1. For right now, let s set k = 1. In that case, ξ 1 p (1) ξ was chosen because this quantity is analogous to a correlation length. When p is large, ξ becomes smaller, indicating a shorter path length between nodes. But as p 0, the correlation length diverges, and the only distance remaining is the lattice spacing. From this, Newman and Watts conclude that there is a one-sided phase transition at p = 0, similar to that of the 1D Ising model. Because we want to relate the small-world network behavior to the number of shortcuts, we want to relate ξ to p. We will make the assumption that near the critical point (ξ 1), ξ p τ (2) Our goal is to find the critical exponent τ. To do this, we will investigate the model under a coarse-graining operation. For right now, we will work under conditions of large L and k = 1, and show later that greater values of k are equivalent. To coarse-grain, we will combine pairs of nodes, as shown in Figure 1. Notice that although the number of nodes and regular connections has decreased by half, the number of shortcut connections is preserved. This results in the parameters rescaling as follows: L = 1 2 L k = k p = 2p (3) Since p is the probability per regular connection, and the number of shortcuts is preserved, p must double since kl is halved. As a result, the new path length is l = 1 2 l (4) Although the number of shortcuts after the coarse-graining is larger compared to the number of regular connections, a given path through the network is composed largely of steps through the regular lattice. In the limit of large L, l scales as L. 2

3 Now we make the assumption that near the critical point, l has a scaling relation l = Lf ( ) L ξ { const x 1 where f(x) = log x x x 1 (5) where f(x) is a scaling function. When L/ξ is small, we are in the regular network regime, and l scales linearly. When L/ξ is large, we observe the logarithmic dependence on L that characterizes a small-world network. Using this scaling relation, we plug in from Eqn. (2) to obtain l = Lf (p τ L) (6) After coarse-graining, this will need to be rescaled, so by plugging Eqn. (6) into Eqn. (4) we get L f ( p τ L ) = 1 2 Lf (pτ L) (7) Because L = 1 2L, we re left with the scaling function being equivalent to itself after coarse-graining. As a result, the arguments must be equal. Substituting the rescaling relation in for p, we find p τ L = p τ L (8) (2p) τ 1 2 L = pτ L (9) 2 τ 1 = 1 (10) Naturally, the only value of τ that satisfies this is τ = 1. We have just shown that in the limit of large L and small p for k = 1, this model will demonstrate small-world behavior at characteristic size ξ 1 p. What about for k > 1? While there is a mathematical argument to support this, the easiest thing to do is to look at what happens in the k = 3 case under coarse-graining in Figure (1). By combining nodes in groups of size k, we see that after one round of coarse-graining, the model has collapsed into the k = 1 case, and the previous analysis applies. The model behaves the same for all values of k that are small compared to L. If you need further convincing, Figure (2) shows the average path length vs. number of shortcuts for different values of k. Finally, we can generalize this analysis to higher dimensions. Imagine a two-dimensional grid, or a three-dimensional lattice, with sides of length L. All of the nodes are regularly spaced and connected by degree k. The average path length in these systems scales similarly to the onedimensional model. Now coarse-grain it as we did the first model, by combining pairs of nodes 3

4 Figure 2: Linear-log plot of dependence of l on the number of shortcuts for two values of k. Ignore the inset. Figure 3: Linear-log plot of dependence of l on p for a 2D model. 4

5 in each dimension. So for a 2D model, four nodes in a square become one new node. In a cubic lattice, eight nodes become one node. This results in a slightly different scaling behavior where L = 1 2 L k = 1 p = 2 d l = 1 2 l (11) where d is the number of dimensions. Since the number of connections has gone down by a factor of 2 in each dimension, p must increase by 2 in each dimension. Following through with the analysis starting in Eqn. (8), we discover p τ L = p τ L (12) (2 d p) τ 1 2 L = pτ L (13) (2 d ) τ = 2 (14) and so of course τ = 1 d. Figure (3) shows the results of a simulation in 2D and confirms that l goes as p 1 2. The analysis by Newman and Watts demonstrates that in this simple model, the onset of smallworld network behavior can be explained in terms of a simple dependance on a single parameter, and that this relationship can be generalized rather neatly to larger dimensions and connectivities. References [1] D.J.Watts, S.H. Strogatz, Nature 393 (1998) 440 [2] M.E.J Newman, D.J. Watts, Physics Letters A 263 (1999) 341 5