Scale-free network topology and multifractality in weighted planar stochastic lattice

We propose a weighted planar stochastic lattice (WPSL) formed by the random sequential partition of a plane into contiguous and non-overlapping blocks and find that it evolves following several non-trivial conservation laws, namely $\sum_i^N x_i^{n-1} y_i^{4/n-1}$ is independent of time $\forall \ n$, where $x_i$ and $y_i$ are the length and width of the $i$th block. Its dual on the other hand, obtained by replacing each block with a node at its center and common border between blocks with an edge joining the two vertices, emerges as a network with a power-law degree distribution $P(k)\sim k^{-\gamma}$ where $\gamma=5.66$ revealing scale-free coordination number disorder since $P(k)$ also describes the fraction of blocks having $k$ neighbours. To quantify the size disorder, we show that if the $i$th block is populated with $p_i\sim x_i^3$ then its distribution in the WPSL exhibits multifractality.


I. INTRODUCTION
Planar cellular structures formed by tessellation, tiling, or subdivision of plane into contiguous and nonoverlapping cells have always generated interest among scientists in general and physicist in particular because cellular structures are ubiquitous in nature. Examples include acicular texture in martensite growth, tessellated pavement on ocean shores, agricultural land division according to ownership, grain texture in polycrystals, cell texture in biology, soap froths and so on [1][2][3][4]. For instance, Voronoi lattice formed by partitioning of a plane into convex polygons or Apollonian packing generated by tiling a plane into contiguous and non-overlapping disks has found widespread applications [5,6]. There are some theoretical models, both random and deterministic, developed either to directly mimic these structures or to serve as a tool on which one can study various physical problems [1]. However, cells in most of the existing structures do not have sides which share with more than one side of other cells. In reality, the sides of a cell can share a part or a whole side of another cell. As a result, the number of neighbours of a cell can be higher than the number of sides it has. Moreover, cellular structure may emerge through evolution where cells can be of different sizes and have different number of neighbours since nature favours these properties as a matter of rule rather than exception. A lattice with such properties can be of great interest as it can mimic disordered medium on which one can study percolation or random walk like problems.
In this article, we propose a weighted planar stochastic lattice (WPSL) as a space-filling cellular structure where annealed coordination number disorder and size disorder are introduced in a natural way. The definition of the model is trivially simple. It starts with an initiator, say a square of unit area, and a generator that divides it randomly into four blocks. The generator thereafter is sequentially applied over and over again to only one of the available blocks picked preferentially with respect to their areas. It results in the partitioning of the square into ever smaller mutually exclusive rectangular blocks. A snapshot of the WPSL at late stage (figure 1) provides an awe-inspiring perspective on the emergence of an intriguing and rich pattern of blocks. We intend to investigate its topological and geometrical properties in an attempt to find some order in this seemingly disordered lattice. If the blocks of the WPSL are regarded as isolated fragments then the model can also describe the fragmentation of a 2D object by random sequential nucleation of seeds from which two orthogonal cracks parallel to the sides of the parent object are grown until intercepted by existing cracks [7,8]. In reality, fragments produced in fracture of solids by propagation of interacting cracks is a formidable mathematical problem. The model in question can, however, be considered as the minimum model which should be capable of capturing the essential fea-tures of the underlying mechanism. The WPSL can also describe the martensite formation as we find its definition remarkably similar to the model proposed by Rao et al. which is also reflected in the similarity between the figure 1 and figure 2 of [1,9]. Yet another application, perhaps a little exotic, is the random search tree problem in computer science [10].
Searching for an order in the disorder is always an attractive proposition in physics. To this end, we invoke the concept of complex network topology to quantify the coordination number disorder and the idea of multifractality to quantify the size disorder of the blocks in WPSL. It is interesting to note that the dual of the WPSL (DW-PSL) obtained by replacing each block with a node at its center and common border between blocks with an edge joining the two corresponding vertices emerges as a network. The area of the respective blocks is assigned to the corresponding nodes to characterize them as their fitness parameter. Nodes in the DWPSL, therefore, are characterized by their respective fitness parameter and the corresponding degree k defined as the number of links a node has. For a decade, there has been a surge of interest in finding the degree distribution P (k) triggered by the work of A.-L. Barabasi and his co-workers who have revolutionized the notion of the network theory by recognizing the fact that real networks are not static rather grow by addition of new nodes establishing links preferentially, known as the preferential attachment (PA) rule, to the nodes that are already well connected [11]. Incorporating both the ingredients, growth and the PA rule, Barabasi and Albert (BA) presented a simple theoretical model and showed that such network self-organizes into a power-law degree distribution P (k) ∼ k −γ with γ = 3 [12]. The phenomenal success of the BA model lies in the fact that it can capture, at least qualitatively, the key features of many real life networks. Interestingly, we find that the DWPSL has all the ingredients of the BA model and its degree distribution P (k) follows heavytailed power-law but with exponent γ = 5.66 revealing that the coordination number of the WPSL is scale-free in character.
In addition to characterizing the blocks of the WPSL by the coordination number k, they can also be characterized by their respective length x and width y. We then find that the dynamics of the WPSL is governed by infinitely many conservation laws, namely the quantity  i is a non-trivial conserved quantity obtained by setting n = 1 (or n = 4). Interestingly, we find that when the ith block is populated with the fraction of the measure p i ∼ x 3 i then the distribution of the population in the WPSL emerges as multifractal indicating further development towards gaining deeper insight into the complex nature of the WPSL we proposed. Multifractal analysis was initially proposed to treat turbulence but later successfully applied in a wide range of exciting field of research [13].
Recently though it has received a renewed interest as it has been found that the wild fluctuations of the wave functions in the vicinity of the Anderson and the quantum Hall transition can be best quantified by using a multifractal analysis [14].
The organization of this paper is as follows. In section 2, we give its exact algorithm of the model. In section 3 various structural topological properties of the WPSL and its dual are discussed in order to quantify the annealed coordination number disorder. In section 4 we discuss the geometric properties of the WPSL in an attempt to quantify the annealed size disorder. Finally, section 5 gives a short summary of our results.

II. ALGORITHM OF THE MODEL
Perhaps an exact algorithm can provide a better description of the model than the mere definition. In step one, the generator divides the initiator, say a square of unit area, randomly into four smaller blocks. We then label the four newly created blocks by their respective areas a 1 , a 2 , a 3 and a 4 in a clockwise fashion starting from the upper left block (see figure 2). In each step thereafter only one block is picked preferentially with respect to their respective area (which we also refer as the fitness parameter) and then it is divided randomly into four blocks. In general, the jth step of the algorithm can be described as follows. 1] each of which represents the blocks labelled by their areas a 1 , a 2 , ..., a (3j−2) respectively. (ii) Generate a random number R from the interval [0, 1] and find which of the (3i − 2) sub-interval contains this R. The corresponding block it represents, say the pth block of area a p , is picked. (iii) Calculate the length x p and the width y p of this block and keep note of the coordinate of the lower-left corner of the pth block, say it is (x low , y low ). (iv) Generate two random numbers x R and y R from [0, x p ] and [0, y p ] respectively and hence the point (x R + x low , y R + y low ) mimics a random point chosen in the block p.
(v) Draw two perpendicular lines through the point (x R + x low , y R + y low ) parallel to the sides of the pth block in order to divide it into four smaller blocks. The label a p is now redundant and hence it can be reused. (vi) Label the four newly created blocks according to their areas a p , a (3j−1) , a 3j and a (3j+1) respectively in a clockwise fashion starting from the upper left corner.
(vii) Increase time by one unit and repeat the steps (i) -(vi) ad infinitum.

III. TOPOLOGICAL PROPERTIES OF WPSL
We first focus our analysis on the blocks of the WPSL and their coordination numbers. Note that for square lattice, the deterministic counterpart of the WPSL, the coordination number is a constant equal to 4. However, the coordination number in the WPSL is neither a constant nor has a typical mean value, rather the coordination number that each block assumes in the WPSL is random. Moreover, it is allowed to evolve with time and hence the coordination number disorder in the WPSL can be regarded as of annealed type. Defining each step of the algorithm as one time unit and imposing periodic boundary condition in the simulation, we find that the number of blocks N k (t) which have coordination number k (or k neighbours) continue to grow linearly with time N k (t) = m k t (see figure 3). On the other hand, the number of total blocks N (t) = k N k (t) at time t in the lattice also grow linearly with time N (t) = 1 + 3t which we can write N (t) ∼ 3t in the asymptotic regime. The ratio of the two quantities ρ k (t) = N k (t)/N (t) that describes the fraction of the total blocks which have coordination number k is ρ k (t) = m k /3. It implies that ρ k (t) becomes a global property since we find it independent of time and size of the lattice. We now take WPSL of fixed size or time and look into its structural properties. For instance, we want to find out what fraction of the total blocks of the WPSL of a given size has coor- dination number k. For this we collect data for ρ t as a function of k and obtain the coordination number distribution function ρ t (k) where subscript t indicates fixed time. Interestingly, the same WPSL can be interpreted as a network if the blocks of the lattice are regarded as nodes and the common borders between blocks as their links which is topologically identical to the dual of the WPSL (DWPSL). In fact, the data for the degree distribution P (k) of the DWPSL network is exactly the same as the coordination number distribution function ρ t (k) of the WPSL i.e., ρ t (k) ≡ P (k). The plot of ln P (k) vs ln(k) in figure 4 using data obtained after ensemble average over 50 independent realizations clearly suggests that fitting a straight line to data is possible. This implies that the degree distribution decays obeying power-law However, note that figure 4 has heavy or fat-tail, bench-mark of the scale-free network, which represents highly connected hub nodes. The presence of messy tail-end complicates the process of fitting the data into powerlaw forms, estimating its exponent γ, and identifying the range over which power-law holds. One way of reducing the noise at the tail-end of the degree distribution is to plot cumulative distribution P (k ′ > k). We therefore plot ln(P (k ′ > k)) vs ln(k) in figure 5 using the same data of figure 4 and find that the heavy tail smooths out naturally where no data is obscured. The straight line fit of figure 5 has a slope γ − 1 = 4.66 which indicates that the degree distribution (figure 4) decays following powerlaw with exponent γ = 5.66. We find it worthwhile to mention as a passing note that the mean coordination number k kP (k) reaches asymptotically to a constant value equal to 5.333. We thus find that in the large-size limit the WPSL develops some order in the sense that its annealed coordination number disorder or the degree distribution of its dual is scale-free in character. This is in sharp contrast to the quenched coordination number disorder found in the Voronoi lattice where it is almost impossible to find cells which have significantly higher or fewer neighbours than the mean value k = 6 [16]. In fact, it has been shown that the degree distribution of the dual of the Voronoi lattice is Gaussian. The square lattice, on the other hand, is self-dual and its degree distribution is P (k) = δ(k − 4). Further, it is interesting to point out that the exponent γ = 5.66 is significantly higher than usually found in most real-life network which is typically 2 < γ ≤ 3. This suggests that in addition to the PA rule the network in question has to obey some constraints. For instance, nodes in the WPSL are spatially embedded in Euclidean space, links gained by the incoming nodes are constrained by the spatial location and the fitting parameter of the nodes. Owing to its unique dynamics this was not unexpected. Perhaps, it is noteworthy to mention that the the degree distribution of the electric power grid, whose nodes like WPSL are also embedded in the spatial position, is shown to exhibit power-law but with exponent γ power = 4 [11].
The power-law degree distribution P (k) has been found in many seemingly unrelated real life networks. It implies that there must exists some common underlying mechanisms for which disparate systems behave in such a remarkably similar fashion [12]. Barabasi and Albert argued that the growth and the PA rule are the main essence behind the emergence of such power-law. Indeed, the DWPSL network too grows with time but in sharp contrast to the BA model where network grows by addition of one single node with m edges per unit time, the DWPSL network grows by addition of a group of three nodes which are already linked by two edges. It also differs in the way incoming nodes establish links with the existing nodes.
To understand the growth mechanism of the DWPSL network let us look into the jth step of the algorithm. First a node, say it is labeled as a p , is picked from the (3j − 2) nodes preferentially with respect to the fitness parameter of a node (i.e., according to their respective areas). Secondly, connect the node a p with two new nodes (3j − 1) and (3j + 1) in order to establish their links with the existing network. At the same time, at least two or more links of a p with other nodes are removed (though the exact number depends on the number of neighbours a p already has) in favour of linking them among the three incoming nodes in a self-organised fashion. In the process, the degree kj p of the node a p will either decrease (may turn into a node with marginally low degree in the case it is highly connected node) or at best remain the same but will never increase. It, therefore, may appear that the PA rule is not followed here. A closer look into the dynamics, however, reveals otherwise. It is interesting to note that an existing nodes during the process gain links only if one of its neighbour is picked not itself. It implies that the higher the links (or degree) a node has, the higher its chance of gaining more links since they can be reached in a larger number of ways. It essentially embodies the intuitive idea of PA rule. Therefore, the DWPSL network can be seen to follow preferential attachment rule but in disguise.

IV. GEOMETRIC PROPERTIES OF WPSL
We again focus on the blocks of the WPSL but this time we characterize them by their length x and width y instead of the number of neighbours they have. Then the evolution of their distribution function C(x, y; t) can be described by the following kinetic equation [7,8] ∂C(x, y; t) ∂t = −xyC(x, y; t) + 4 ∞ x ∞ y C(x 1 , y 1 ; t)dx 1 dy 1 .
Iterating it to get all the derivatives of M (m, n; t) and then substituting them into the Taylor series expansion of M (m, n; t) about t = 0 one can write its solution in terms of generalized hypergeometric function [17] M (m, n; t) = 2 F 2 a + , a − ; m, n; −t , where M (m, n; t) = M (n, m; t) for symmetry reason and vs N for n = 3, 4, 5 are drawn using data collected from one realization.
One can immediately see that (i) M (1, 1; t) = 1 + 3t is the total number of blocks N (t) and (ii) M (2, 2; t) = 1 is the total area of all the blocks which is obviously a conserved quantity. The behaviour of M (m, n; t) in the long time limit is It implies that the system is in fact governed by several conservation laws namely, M (n, 4/n; t) = are independent of time for all n which has been confirmed by numerical simulation (see figure 6). We now focus on the distribution function n(x, t) = ∞ 0 C(x, y, t)dy that describes the concentration of blocks of length x at time t regardless of the size of their widths y. Then the qth moment of n(x, t) is defined as Appreciating the fact that M q (t) = M (q + 1, 1; t) and using equation (7) one can immediately write that Note that had we focus on n(y, t) instead of n(x, t) we would have exactly the same result since their qth moments are identical, M (q + 1, 1; t) = M (1, q + 1; t), due to symmetry reason. We find that the quantity M 3 (t) and hence i is a conserved quantity. However, yet another interesting fact is that although M 3 (t) remains a constant against time in every independent realization, their exact numerical value fluctuates sample to sample (see figure 7). It clearly indicates lack of selfaveraging or wild fluctuation. Nonetheless, during each realization we can use M 3 (t) as a measure to populate the ith block with the fraction of the total population p i = x 3 i / i x 3 i . The corresponding "partition function" then is whose solution can be written immediately from equation (9) to give We find it instructive to express Z q in terms of square root of the mean area δ(t) = M (2, 2; t)/M (1, 1; t) ∼ t −1/2 as it gives the weighted number of squares N (q, δ) needed to cover the measure which scales as where the mass exponent τ (q) = 9q 2 + 16 − (3q + 2).
The non-linear nature of τ (q) suggests that an infinite hierarchy of exponents is required to specify how the moments of the probabilities {p}s scales with δ. Note that τ (0) = 2 is the Hausdorff-Besicovitch (H-D) dimension of the WPSL since the bare number N (q = 0, δ) ∼ δ −τ (0) and τ (1) = 0 follows from the conservation laws (or normalization of the probabilities p i = 1). We now perform the Legendre transformation of τ (q) by using the Lipschitz-Hölder exponent as an independent variable to obtain the new function Substituting it in equation (12) we find that since it is the dominant value of the integral obtained by the extremal conditions. We can infer from this that the number of squares dN (α, δ) needed to cover the PSL with α in the range α to α + dα scales as where ρ(α)dα is the number of times the subdivided regions indexed by α [18]. It implies that a spectrum of spatially intertwined fractal dimensions f (α(q)) = 16 are needed to characterize the measure which is always concave in character (see figure 8). It implies that the size disorder of the blocks are multifractal in character since the measure {p α } is related to size of the blocks. That is, the distribution of {p α } in WPSL can be subdivided into a union of fractal subsets each with fractal dimension f (α) ≤ 2 in which the measure p α scales as δ α . Note that f (α) is always concave in character (see figure 8) with a single maximum at q = 0 correspond to the dimension of the WPSL with empty blocks. On the other hand, we find that the entropy S(δ) = − i p i ln p i associated with the partition of the measure on the support (WPSL) by using the relation i p q i ∼ δ −τ (q) in the definition of S(δ). Then a few steps of algebraic manipulation reveals that S(δ) exhibits scaling where the exponent α 1 = 6 5 obtained from It is interesting to note that α 1 is related to the generalized dimension D q , also related to the Rényi entropy H q (p) = 1 q−1 ln i p q i in the information theory, given by which is often used in the multifractal formalism as it can also provide insightful interpretation. For instance, D 0 = τ (0) is the dimension of the support, D 1 = α 1 is the Rényi information dimension and D 2 is known as the correlation dimension [19,20].

V. SUMMARY
We have proposed and studied a weighted planar stochastic lattice (WPSL) which has annealed coordination number and the block size disorder. We have shown that the coordination number disorder is scalefree in character since the degree distribution of its dual (DWPSL), which is topologically identical to the network obtained by considering blocks of the WPSL as nodes and the common border between blocks as links, exhibits power-law. However, the novelty of this network is that it grows by addition of a group of already linked nodes which then establish links with the existing nodes following PA rule though in disguise in the sense that existing nodes gain links only if one of their neighbour is picked not itself. Besides, we have shown that if the blocks of the WPSL are characterized by their respective length x and width y then we find N i x n−1 i y 4/n−1 i remains a constant regardless of the size of the lattice. However, the numerical values of the conserved quantities except n = 2 varies from sample to sample revealing absence of self-averaging -an indication of wild fluctuation. We have shown that if the blocks are occupied with a fraction of the measure equal to cubic power of their respective length or width then its distribution on the WPSL is multifractal nature. Such multifractal lattice with scale-free coordination disorder can be of great interest as it has the potential to mimic disordered medium on which one can study various physical phenomena like percolation and random walk problems etc. NIP acknowledges support from the Bose Centre for Advanced Study and Research in Natural Sciences.