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 N i x n−1 i y 4/n−1 i is independent of time ∀ n, where xi and yi are the length and width of the ith 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) ∼ k −γ where γ = 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 ith block is populated with pi ∼ x 3 i then its distribution in the WPSL exhibits multifractality.

In this article, we show that if each node of the Barabási-Albert (BA) network is characterized by the generalized degree q, i.e. the product of their degree k and the square root of their respective birth time, then the distribution function F (q, t) exhibits dynamic scaling F (q, t → ∞) ∼ t −1/2 φ(q/t 1/2 ) where φ(x) is the scaling function. We verified it by showing that a series of distinct F (q, t) vs q curves for different network sizes N collapse onto a single universal curve if we plot t 1/2 F (q, t) vs q/t 1/2 instead. Finally, we show that the BA network falls into two universality classes depending on whether new nodes arrive with single edge (m = 1) or with multiple edges (m > 1).PACS numbers: 61.43. Hv, 64.60.Ht, 68.03.Fg, 89.75.Da Many complex systems can be described as an interwoven web of a large network if the constituents are regarded as nodes or vertices and the interactions between constituents as links or edges. For example, cells of living systems are networks of molecules linked by chemical interaction [1][2][3], the Internet is a network of routers and computers linked by cables or wireless connections [4], the power-grid is a network of substations linked by transmission lines [5], and social networks are networks of individuals linked by friendships, professional ties, etc [6][7][8][9][10]. The first theoretical attempt to guide our understanding about complex network topology began with the seminal work of Paul Erdös and Alfréd Rényi in 1959 [11]. The main result of the Erdös-Rényi (ER) model is that the degree distribution P (k), defined as the probability that a randomly chosen node is connected to k other nodes by one edge, is Poissonian revealing that it is almost impossible to find nodes that have significantly higher or fewer links than the average degree. However, real networks are neither completely regular where all the nodes have the same degree k nor completely random where P (k) is Poissonian instead they are scale-free in character where P (k) obeys power-law.Just over a decade ago Barabási and Albert revolutionized the notion of the network theory by recognizing the fact that natural and man-made networks are not static rather they grow by continuous addition of new nodes. They further argued that the new nodes establish links to the well-connected existing ones preferentially rather than randomly known as the preferential attachment (PA) rule. It essentially embodies the intuitive idea of the rich get richer principle of the Matthew effect in sociology [12]. BA then presented a simple theoretical model incorporating both the ingredients and showed that the resulting network can reproduce the power-law degree distribution which most real life networks exhibit [13,14]. Recently, we have shown that random sequential partition of a square into contiguous and non-overlapping blocks can be described as a network with power-law degree distribution if blocks are regarded as nodes and common border between blocks as links [15].A power-law distribution function is considered ...

Firstly, we propose and investigate a dyadic Cantor set (DCS) and its kinetic counterpart where a generator divides an interval into two equal parts and removes one with probability $(1-p)$. The generator is then applied at each step to all the existing intervals in the case of DCS and to only one interval, picked with probability according to interval size, in the case of kinetic DCS. Secondly, we propose a stochastic DCS in which, unlike the kinetic DCS, the generator divides an interval randomly instead of equally into two parts. Finally, the models are solved analytically; an exact expression for fractal dimension in each case is presented and the relationship between fractal dimension and the corresponding conserved quantity is pointed out. Besides, we show that the interval size distribution function in both variants of DCS exhibits dynamic scaling and we verify it numerically using the idea of data-collapse.Comment: 8 pages, 6 figures, To appear in Chaos, Solitons & Fractal

The square lattice is perhaps the simplest cellular structure. In this work, however, we investigate the various structural and topological properties of the kinetic and stochastic counterpart of the square lattice and termed them as kinetic square lattice (KSL) and weighted planar stochastic lattice (WPSL) respectively. We find that WPSL evolves following several non-trivial conservation laws,where xi and yi are the length and width of the ith block. The KSL, on the other hand, evolves following only one conservation law, namely the total area, although one find three apparently different conserved integrals which effectively the total area. We show that one of the conserved quantity of the WPSL obtained either by setting n = 1 or n = 4 can be used to perform multifractal analysis. For instance, we show that if the ith block is populated with either pi ∼ x 3 i or pi ∼ y 3 i then the resulting distribution in the WPSL exhibits multifractality. Furthermore, we show that the dual of the WPSL, 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 scale-free network since its degree distribution exhibits power-law P (k) ∼ k −γ with exponent γ = 5.66. It implies that the coordination number distribution of the WPSL is scale-free in character as we find that P (k) also describes the fraction of blocks having k neighbours.