Irreversible aggregation and network renormalization

Abstract

Irreversible aggregation is revisited in view of recent work on renormalization of complex networks. Its scaling laws and phase transitions are related to percolation transitions seen in the latter. We illustrate our points by giving the complete solution for the probability to find any given state in an aggregation process (k + 1)X → X, given a fixed number of unit mass particles in the initial state. Exactly the same probability distributions and scaling are found in one dimensional systems (a trivial network) and well-mixed solutions. This reveals that scaling laws found in renormalization of complex networks do not prove that they are self-similar. PACS numbers: 89.75.Hc, 02.10.Ox, 05.70.Ln Droplets beget rain, goblets coagulate to make butter or cream, and dust particles stick together to form aggregates that can eventually coalesce into planets. At the microscopic level, irreversible aggregation of atoms and molecules creates many familiar forms of matter such as aerosols, colloids, gels, suspensions, clusters and solids [1]. Almost a century ago, Smoluchowski proposed a theory based on rate equations to describe processes governed by diffusion, collision and irreversible merging of aggregates [2]. The theory predicts how many small and large clusters exist at any given time and yields a mass distribution that depends on certain details such as the initial conditions, reactions present, relative rates, the presence or absence of spatial structure, etc. A key interest to physicists has been to derive scaling laws that characterize different universality classes [3, and references therein].By contrast, wide interest in complex networks [4-7] has emerged recently. Vast applications to physics, computer science, biology, and sociology [8-10, and references therein] continue to be vigorously investigated. An important question is whether or not complex networks exhibit self-similarity at different length scales and if they can be grouped into universality classes on that basis. Renormalization schemes for networks were proposed [11][12][13][14] to address this question. Scaling of the mass or degree distribution of the renormalized nodes was used to argue that many complex networks are selfsimilar. The semi-sequential renormalization group (RG) flow underlying the box covering of [11][12][13][14] was studied carefully in [15,16], where it was found that scaling laws may be related to an "RG fixed point" which was observed for a wide variety of networks. A convenient, fully sequential scheme called random sequential renormalization (RSR) was introduced [17]. At each RSR step, one node is selected at random, and all nodes within a fixed distance ℓ of it are replaced by a single super-node.We point out a simple mapping between RSR and irreversible aggregation on any graph. Hence any conclusion drawn for one process holds also for the other. Indeed, a local coarse-graining step to produce a new super-node represents one aggregation event, where a 'molecule' aggregates with all its neighbors within distance ℓ to p...

Graph states as ground states of two-body frustration-free Hamiltonians

“…( ) ( ) that appears in equation (9). If we imagine flipping a spin in σ, we see that σ L ( ) will increase by 1 if and only if a loop of like outcomes is created, and will decrease by 1 if and only if a loop of like outcomes is lost.…”

“…We will call a connected region in the parameter space of a Hamiltonian with ground states universal for MBQC a computational phase. Previous work on computational phases has investigated the cluster model in external fields (also at non-zero temperature) [7,8], with competing Ising interactions [9], in three dimensions at non-zero temperature [10], with errorsuppressing interacting cluster terms [11], and under general symmetry preserving perturbations [12,13]. For two-body models, studies have looked at the Haldane phase in one dimension [14], an anisotropic AKLT model on a honeycomb lattice [15] and versions of the Hamiltonians in [6], which can also be universal at non-zero temperature [16,17].…”

Agglomerative percolation on bipartite networks: Nonuniversal behavior due to spontaneous symmetry breaking at the percolation threshold

“…Recent discoveries that widened enormously the scope of different behaviors at the percolation threshold include infinite order transitions in growing networks [1], supposedly first order transitions in Achlioptas processes [2] (that are actually continuous [3,4] but show very unusual finite size behavior [5]), and real first order transitions in interdependent networks [6][7][8][9]. Another class of "non-classical" percolation models, inspired by attempts to formulate a renormalization group for networks [10,11], was introduced in [12][13][14][15][16] and is called 'agglomerative percolation' (AP).…”

“…The new combined cluster is then linked to all neighbors of its constituents. AP can be solved rigorously on 1-d linear chains [14,15], where it is found to be in a different universality class from OP. Although a similarly complete mathematical analysis is not possible on random graphs, both numerics and nonrigorous analytical arguments show that the same is true for 'critical' trees [12] and Erdös-Rényi graphs [16].…”

Agglomerative percolation on the Bethe lattice and the triangular cactus

“…In contrast, the universality of the transition of AP on the triangular lattice, which is not bipartite, is the same as that of the random percolation [12]. Using analytical methods and numerical simulations, APs on the one-dimensional ring [8], the two-dimensional square lattice and triangular lattice [9], critical tree [10], and complex network [11] were studied.…”