--- title: Graph Bootstrap Percolation url: https://www.emergentmind.com/topics/graph-bootstrap-percolation type: topic --- # Graph Bootstrap Percolation Searching arXiv for recent and foundational papers on graph bootstrap percolation, edge-bootstrap, vertex-bootstrap, thresholds, and running time. Graph bootstrap percolation is a family of deterministic cellular automata on graphs in which an initial random or prescribed set of infected vertices or edges evolves by a local completion rule. Two formulations dominate the literature represented here. In the edge-based formulation, usually studied on the complete graph \(K_n\), an edge becomes active when it is the unique missing edge of a copy of a fixed graph \(H\). In the vertex-based formulation, a vertex becomes infected when it has at least \(r\) infected neighbours. In both cases the process is monotone in its classical form, and the central questions are threshold location, final infected size, structural characterization of witnesses, and the maximal time before stabilization [1107.1381][1111.1339]. ## 1. Two canonical formulations The literature uses closely related but technically distinct notions of graph bootstrap percolation. | Formulation | State space | Update rule | |---|---|---| | \(H\)-bootstrap percolation | Edges of \(K_n\) | add \(e\) when it is the unique inactive edge in a copy of \(H\) | | \(r\)-bootstrap percolation | Vertices of a graph \(G\) | infect \(v\) when \(|N(v)\cap I_{t-1}|\ge r\) | In the edge-based model, one starts from \(G_0\subseteq K_n\) and iterates \[ G_{t+1}=G_t\cup\{e\in E(K_n)\setminus E(G_t): \exists\,H'\cong H,\ E(H')\setminus\{e\}\subseteq E(G_t)\}. \] The closure is \(\langle G\rangle_H=\bigcup_{t\ge 0}G_t\), and \(H\)-percolation means \(\langle G\rangle_H=K_n\). In the random setting the initial graph is typically \(G(n,p)\), and the critical probability is \[ p_c(n,H)=\inf\{p>0:\mathbb P(\langle G\rangle_H=K_n)\ge 1/2\}. \] This is the formal framework developed in the clique and general-\(H\) threshold literature [2605.15066]. In the vertex-based model, with threshold \(r\ge 2\), one starts from \(I_0\subseteq V(G)\) and evolves by \[ v\in I_t \quad \text{if and only if} \quad |N(v)\cap I_{t-1}|\ge r, \] with \(I_\infty=\bigcup_{t\ge 0}I_t\). Depending on the host graph, the key observable is either complete infection, infection of a linear fraction of the graph, or the asymptotic final density at a typical site [1012.3535]. ## 2. Historical development and relation to weak saturation The edge process was introduced implicitly by Bollobás in 1968 in the study of weak saturation, and was later named and developed as a percolation model by Balogh, Bollobás, and Morris. This historical route explains why graph bootstrap percolation sits at the intersection of extremal combinatorics, probabilistic combinatorics, and cellular automata [2602.12736]. Weak saturation remains structurally central. For a graph \(H\), the weak saturation number \(wsat(n,H)\) is the minimum number of edges in an \(H\)-percolating graph on \(n\) vertices, while \(sat(n,H)\) is the minimum number of edges in an \(H\)-saturated graph, and \(ex(n,H)\) is the Turán number. For cliques \(K_k\), Kalai and Alon showed that \(sat(n,K_k)=wsat(n,K_k)\) for all \(k\), and the extremal construction is the unique join \(K_{k-2}\vee \overline{K}_{n-k+2}\) [2602.12736]. This weak-saturation origin also clarifies a frequent point of confusion. Ordinary bond or site percolation studies random connected components in a static graph. Graph bootstrap percolation instead studies deterministic closure after random initialization. The random object is the initial condition; the subsequent dynamics are rule-driven and closure-based. ## 3. Threshold theory for edge-bootstrap percolation The threshold problem asks for the scale of \(p_c(n,H)\) when the initial active edges are sampled from \(G(n,p)\). For clique rules, a foundational result showed that for \(r\ge 4\), \[ c(r)\,\frac{n^{-1/X(r)}}{\log n}\le p_c(n,K_r)\le C(r)\,n^{-1/X(r)}\Big(\frac{\log n}{\log\log n}\Big)^{2/X(r)}, \] where \[ X(r)=\frac{\binom{r}{2}-2}{r-2}. \] For \(K_4\), the threshold is \(\Theta((n\log n)^{-1/2})\), and for \(K_3\) the process percolates if and only if the initial graph is connected, so \(p_c(n,K_3)\sim (\log n)/n\) [1107.1381]. A general theory is now available for every fixed rule graph \(H\). The key parameter is the critical activation density \[ \rho(H)=\inf_{(e,A)\in\mathcal A(H)}\max_{e\subset F\subseteq A\cup e}\frac{e(F)-1}{v(F)-2}, \] where \(\mathcal A(H)\) consists of witness pairs \((e,A)\) in which the edge \(e\) is eventually activated starting from \(A\). The main theorem identifies the threshold exponent by \[ \ell(H):=\lim_{n\to\infty}\frac{\log p_c(n,H)}{\log n}=-\frac{1}{\rho(H)}. \] When \(\rho(H)>1\), \[ \Omega(1/\log n)=n^{1/\rho(H)}p_c(n,H)=O(\log^2 n), \] and, more precisely, \[ n^{1/\rho(H)}p_c(n,H)=O\big(\log^{\,2/\rho(H)+2/a}n\big), \] where \(a/b\) is \(\rho(H)\) in lowest terms [2605.15066]. Balanced graphs form the classical tractable subclass. If \[ \frac{e(F)-1}{v(F)-2}\le \lambda(H):=\frac{e(H)-2}{v(H)-2} \] for every proper subgraph \(F\subset H\) with \(3\le v(F)3k/4\), then \(M_H(n)=\Omega(n^2)\). If \(\delta(H)\ge k/2+1\), then \[ M_H(n)\ge n^{2-O(1/\sqrt{\log n})}=n^{2-o(1)}. \] For random rules \(H=G(k,p)\), there is a sharp transition near \(p\approx (\log k)/k\): below \((1/2-o(1))(\log k)/k\), one has \(M_H(n)=O(1)\), whereas \(p=\omega((\log k)/k)\) yields \(M_H(n)=\Omega(n^2)\) with high probability [2602.12736]. Bipartite rules exhibit a different spectrum. A general upper bound is \[ M_H(n)\le 2\,ex(n,H). \] For \(K_{r,s}\), \[ M_{K_{r,s}}(n)\ge n^{2-1/r-1/(s-1)-o(1)}, \] and for \(Q_3\), \[ \Omega(n^{3/2})\le M_{Q_3}(n)\le O(n^{8/5}). \] Thus the model realizes infinitely many polynomial exponents strictly between \(3/2\) and \(2\) [2602.12736]. Time-constrained thresholds interpolate between eventual percolation and fast percolation. For clique rules and \(1\le t\le (\log\log n)/(3\log \tau)\), where \(\tau=\binom{r}{2}-1\), the threshold for percolation by time \(t\) satisfies \[ n^{-(v_t-2)/e_t}/\omega(n)\le p_c(n,K_r,t)\le n^{-(v_t-2)/e_t}\log n, \] with \(e_t=\tau^t\), \[ v_t=(r-2)\frac{\tau^t-1}{\tau-1}+2, \qquad \frac{v_t-2}{e_t}=\frac{1}{\lambda(K_r)}-\frac{1}{\lambda(K_r)\tau^t}. \] This identifies the witness exponent for percolation within a prescribed number of rounds [1503.01454]. ## 5. Vertex-bootstrap percolation on random, inhomogeneous, and heavy-tailed graphs For vertex-based \(r\)-bootstrap percolation on \(G_{n,p}\), the classical sparse random-graph theory is centered on \[ t_c=\Big(\frac{(r-1)!}{n p^r}\Big)^{1/(r-1)}, \qquad a_c=\Big(1-\frac1r\Big)t_c. \] In the regime \(n^{-1}\ll p\ll n^{-1/r}\), if \(a/a_c\to \alpha<1\), then \[ A^*=(\theta(\alpha)+o_p(1))\,t_c, \] where \(\theta(\alpha)\) is the unique solution of \[ r\theta-\theta^r=(r-1)\alpha. \] If \(a/a_c\ge 1+\delta\), then \[ A^*=n-o(n) \] with high probability. Complete percolation is then controlled by the endgame condition \[ np-\big(\log n+(r-1)\log\log n\big)\to +\infty. \] Around the refined threshold \(\hat a\), the critical window has width \(\Theta(\sqrt a)\), and within that window the transition probabilities are asymptotically Gaussian [1012.3535]. Heavy-tailed inhomogeneous graphs behave differently. In Chung–Lu random graphs with power-law weights and exponent \(2<\beta<3\), there is a sublinear critical function \(a_c(n)=o(n)\) such that \[ a(n)\ll a_c(n)\Rightarrow I_\infty=I_0 \text{ whp}, \qquad a(n)\gg a_c(n)\Rightarrow |I_\infty|\ge \varepsilon n \text{ whp} \] for some \(\varepsilon>0\). In the extremal maximum-degree regime, \[ a_c(n)=n^{\frac{\beta-2}{\beta-1}}, \] independent of \(r\). In the more general regime \[ \frac{r-1}{2r-\beta+1}<\zeta\le \frac{1}{\beta-1}, \] with maximum degree \(n^\zeta\), \[ a_c(n)=n^{\frac{r(1-\zeta)+\zeta(\beta-1)-1}{r}}. \] The mechanism is a dense hub kernel: infection first saturates a high-weight core and then descends across degree layers \(L_j\) defined by the recursion \[ f_{j+1}(n)=C\,f_j(n)^{\beta-2}. \] This produces linear cascades from sublinear seeds, in stark contrast to Erdős–Rényi and regular graphs [1111.1339]. Directed inhomogeneous random graphs with heterogeneous thresholds admit an explicit fixed-point theory. If vertices carry in- and out-weights \(W^-,W^+\) and threshold \(C\), the final infected fraction is determined by the smallest positive solution \(\hat z\) of \[ \mathbb E[W^+\psi_C(W^- z)]-z=0, \qquad \psi_r(x)=\mathbb P(\mathrm{Poi}(x)\ge r). \] Under a local contraction condition, \[ \frac{|\mathcal A_n|}{n}\xrightarrow{p}\mathbb E[\psi_C(W^-\hat z)]. \] The derivative criterion is \[ f'(z)=\mathbb E\!\left[W^+W^-\,\mathbf 1\{\mathrm{Poi}(zW^-)=C-1,\ C\ge 1\}\right]-1. \] In particular, heavy-tailed weights can yield large cascades even when \(\mathbb P(C=1)=0\) [1511.07993]. Geometry adds another control parameter. On geometric inhomogeneous random graphs with weight exponent \(\beta\in(2,3)\), a localized seed in a ball of expected size \(\nu=n|B_0|\) has metastability threshold \[ \rho_c(\nu)=\nu^{-1/(\beta-1)}. \] If \(\rho\gg \rho_c(\nu)\), then \(|A^{\le \infty}|=\Theta(n)\) with high probability; if \(\rho\ll \rho_c(\nu)\), the infection stays localized. In the supercritical regime the macroscopic outbreak time is doubly logarithmic: \[ |A^{\le (1+\delta)i_\infty}|=\Theta(n), \qquad i_\infty=\frac{\log \log_\nu n+\log\log n}{|\log(\beta-2)|}, \] up to lower-order terms. For \(\alpha>\beta-1\), infection times of individual vertices are determined up to lower-order terms by their positions and weights, and the outbreak can be contained by removing relatively few boundary edges [1603.02057]. ## 6. Variants, geometry, non-monotone models, and open problems A broad later literature studies how graph structure modifies classical threshold behavior. On the hybrid graph \(G_{n,p}^1\), obtained by adding a ring to \(G_{n,p}\), the critical first-order scale for \(r=2\) remains \[ t_c=a_c=\frac{2}{n p^2}, \] but the local ring edges narrow the critical window. In the regime \(n^{-1}1/\ell\), the transition is sharp, while when \(\gamma<1/\ell\), it is gradual. Even thresholds \(\theta=2\ell\) are sharp for all \(\gamma\). At the boundary \(\gamma=1/\ell\), a mixed transition appears [1505.03204]. Non-monotone variants change the theory qualitatively. In bootstrap percolation with recovery on \([n]^2\), healthy vertices with at least two infected neighbours become infected, but infected vertices with no infected neighbours become healthy. The critical probability satisfies \[ p_c([n]^2,R)=\sqrt{\frac{A_R}{\log n}}+o\!\left(\frac{1}{\sqrt{\log n}}\right), \] where \[ A_R=\int_0^\infty g(x)\,dx\approx 0.0779. \] This replaces the monotone \((\log n)^{-1}\) scale by \((\log n)^{-1/2}\), reflecting the fact that stable local seeds are effectively 2-site structures [1505.08030]. Inhibitory and majority rules reveal further departures from monotone fixed-threshold behavior. With inhibition on sparse Erdős–Rényi graphs, the synchronous discrete-time model is non-monotone in the size of the starting set and can realize essentially any final size between \((\log n)/p\) and \(n\) in the regime \(\tau>1/(1+\gamma)\), whereas the asynchronous model with exponential transmission times stabilizes to a deterministic linear fraction and does so in \(O(1)\) time [1410.3291]. For majority bootstrap percolation on high-dimensional geometric graphs in the class \(\mathcal H(K)\), the critical window is universally located below \(1/2\); for \(d\)-regular graphs in this class, \[ p_c(G)=\frac12-\frac12\sqrt{\frac{\log d}{d}}+o\!\left(\sqrt{\frac{\log d}{d}}\right), \] extending the hypercube picture to grids, tori, Hamming graphs, middle-layer graphs, odd graphs, and folded hypercubes [2406.17486]. Several structural problems remain open. For edge-bootstrap percolation, the computability and rationality of \(\rho(H)\) are unresolved, as is the conjectured uniform upper bound \(p_c(n,H)=O(n^{-1/\rho(H)})\) for all \(H\) with \(\rho(H)>1\). The proposed sharpness criterion via attainment of \(\rho(H)\) is also open beyond the families already treated [2605.15066]. On the extremal-time side, the principal unsolved case is whether \(M_{K_5}(n)=o(n^2)\), along with broader questions about the minimum-degree threshold for quadratic running time and the existence of a universal exponent \(\lim_{n\to\infty}\log M_H(n)/\log n\) for every rule graph \(H\) [2602.12736]. Source: https://www.emergentmind.com/topics/graph-bootstrap-percolation