From explosive to infinite-order transitions on a hyperbolic network

Abstract

We analyze the phase transitions that emerge from the recursive design of certain hyperbolic networks that includes, for instance, a discontinuous ("explosive") transition in ordinary percolation. To this end, we solve the q-state Potts model in the analytic continuation for non-integer q with the real-space renormalization group. We find exact expressions for this one-parameter family of models that describe the dramatic transformation of the transition. In particular, this variation in q shows that the discontinuous transition is generic in the regime q < 2 that includes percolation. A continuous ferromagnetic transition is recovered in a singular manner only for the Ising model, q = 2. For q > 2 the transition immediately transforms into an infinitely smooth order parameter of the Berezinskii-Kosterlitz-Thouless (BKT) type. PACS numbers: 64.60.ah, 64.60.ae, 64.60.aq Real-world networks [1-3] exhibit dramatically distinct phenomenology, with a profound imprint of their geometry on the dynamics, when compared with lattices or mean-field systems. What we now call complex networks, aside from being random, possess geometries dominated by small-world bonds and scale-free degree distributions [4,5]. These lead to novel, and often nonuniversal, scaling behaviors unknown for lattices. For example, they exhibit discontinuous transition even in ordinary percolation [6] as exactly solvable realizations of the so-called "explosive" percolation transitions [7][8][9][10][11][12][13]. Unlike the Achlioptas process, it is easy to prove that this discontinuity exists [6,14], however, its origin results from the network structure, not from a correlated process [7].Hyperbolic networks combine a lattice geometry with a hierarchy of small-world connections. They were originally proposed as solvable models for complex networks [15][16][17][18][19]. These recursively defined structures provide deeper insights into small-world effects compared to random networks that otherwise require approximate or numerical methods. Work on percolation [6,17,[20][21][22], the Ising model [15,[23][24][25], and the q-state Potts model [19,26] have shown that critical behavior, once thought to be exotic and model-specific [3], can be universally categorized near the transition point [26,27] for a large class of hyperbolic networks, such as those discontinuous percolation transitions described in Refs. [6,14]. They exemplify a non-trivial category of the explosive cluster-growth mechanisms leading to the discontinuous transition [28]. Yet, it has been noted that this discontinuity arises only as a non-generic case of the general theory [27], making its prevalence in percolation on hyperbolic networks somewhat surprising.Here, we illuminate the origin of the explosive percolation transition by investigating the q-state Potts model on a simple hyperbolic network, MK1 [6]. The analytic continuation in q [29] reveals that the discontinuity that characterizes the explosive percolation transition related to the limit q → 1, rather than being a special ca...

Classification of critical phenomena in hierarchical small-world networks

“…[4], or in field theory, where it signals the loss of conformality [39]. However, it is most closely related to the recent observation of discontinuous ("explosive") transitions in ordinary percolation on hierarchical networks [18,20,28,40]. Our study shows that criticality in these models is generally non-universal but falls into three generic regimes.…”

“…In particular, the iterative structure of hierarchical networks may facilitate their realization in engineered devices to unlock and control their unconven-tional behaviors. Work on percolation [15][16][17][18][19][20], the Ising model [21][22][23][24][25], and the q-state Potts model [26][27][28] have shown that critical behavior, once thought to be exotic and model-specific [4], can be categorized with the renormalization group [26] for a large class of hierarchical networks with a hyperbolic structure.…”

Griffiths phase on hierarchical modular networks with small-world edges

“…Certain hierarchical networks, with a self-similar structure and small-world connections, have shown to exhibit novel dynamics [16][17][18][19][20][21]. Here, we design hierarchical models based on one such example, the Hanoi networks [16,18,22,23].…”

Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes

“…These networks are well suited to perform exact real-space renormalization group (RG) calculations. Using RG theory there has been very important progress in characterizing the critical properties of percolation [14][15][16][31][32][33][34][35][36], spin (Ising and Potts) models [37][38][39][40], and Gaussian models [25,26] in these structures. In particular in Refs.…”