Heterogeneousk-core versus bootstrap percolation on complex networks

Abstract

We introduce the heterogeneous-k-core, which generalizes the k-core, and contrast it with bootstrap percolation. Vertices have a threshold ki which may be different at each vertex. If a vertex has less than ki neighbors it is pruned from the network. The heterogeneous-k-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds ki are 1 with probability f or k ≥ 3 with probability (1 − f ), the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolation. We show that there are two types of transitions in this heterogeneous-k-core process: the giant heterogeneousk-core may appear with a continuous transition and there may be a second, discontinuous, hybrid transition. We compare critical phenomena, critical clusters and avalanches at the heterogeneous-kcore and bootstrap percolation transitions. We also show that network structure has a crucial effect on these processes, with the giant heterogeneous-k-core appearing immediately at a finite value for any f > 0 when the degree distribution tends to a power law P (q) ∼ q −γ with γ < 3.PACS numbers: 64.60.aq, 64.60.ah, 05.70.Fh Bootstrap percolation and the k-core are closely related concepts, and in fact it is easy to confuse the two. Both belong to a new class of systems with hybrid phase transitions, yet it can be clearly shown that the two processes do not map onto each other. Here we elucidate the relationship and differences between these two concepts by introducing a generalization of the k-core, the heterogeneous-k-core.The k-core is the maximal sub-graph whose vertices all have internal degree at least k [1]. It has proved a useful tool giving insight into the deep structure of complex networks [2][3][4][5][6] , and has found applications in diverse areas, from rigidity [7] and jamming [8] transitions to real neural networks [9,10] and evolution [11] . The kcore has been extensively studied on tree-like networks, starting with Bethe lattices [12,13] and Random graphs [14][15][16], before finally being extended to arbitrary degree distributions [5,[17][18][19]. Hyperbolic lattices have also been considered [20]. Other studies, mostly numerical, have considered the sizes of culling avalanches [21][22][23]. Results on non tree-like graphs have been largely numerical [24,25], although some analytic results incorporating clustering have recently been obtained [26,27]. At the same time, bootstrap percolation has emerged as a useful model for a variety of applications such as neuronal activity [28][29][30], jamming and rigidity transitions and glassy dynamics [31,32], and magnetic systems [33]. In bootstrap percolation, a set of seed vertices is initially activated, and other vertices become active if they have k active neighbors. This process has been investigated on two and three dimensional lattices (see [34][35][36][37] and references therein). Bootstrap percolation has been studied * gjbaxter@ua.pt on the random regular graph [38,39], on...

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