Hierarchical cycle-tree packing model for $K$-core attack problem

TL;DR

This work addresses the problem of selecting a minimal initial seed set within the $K$-core to trigger complete collapse under irreversible pruning. It advances a hierarchical cycle-tree packing framework, solved with the replica-symmetric cavity method and coarse-grained belief propagation, yielding the hCTGA algorithm that constructs near-optimal attack sets on sparse random graphs. Theoretical predictions show the minimum attack density $\rho_{\min}$ decreases and converges rapidly as the layer count $H$ grows, while empirical results on regular random and Erdős–Rényi graphs demonstrate that hCTGA outperforms prior heuristics and approaches RS-based minima. The approach provides a scalable, physics-inspired tool for optimizing irreversible dynamics on graphs and offers insights into potential hierarchical organization in complex networks.

Abstract

The $K$-core of a graph is the unique maximum subgraph within which each vertex connects to $K$ or more other vertices. The optimal $K$-core attack problem asks to delete the minimum number of vertices from the $K$-core to induce its complete collapse. A hierarchical cycle-tree packing model is introduced here for this challenging combinatorial optimization problem. We convert the temporally long-range correlated $K$-core pruning dynamics into locally tree-like static patterns and analyze this model through the replica-symmetric cavity method of statistical physics. A set of coarse-grained belief propagation equations are derived to predict single vertex marginal probabilities efficiently. The associated hierarchical cycle-tree guided attack ({\tt hCTGA}) algorithm is able to construct nearly optimal attack solutions for regular random graphs and Erdös-Rényi random graphs. Our cycle-tree packing model may also be helpful for constructing optimal initial conditions for other irreversible dynamical processes on sparse random graphs.

Figures (4)

• Figure 1: Example of hierarchical cycle-tree packing with $H=2$ and $K=3$. The three seed vertices at the lowest layer $h=0$ form the attack set. The edges attached to these seed vertices are drawn as dashed lines. Each normal (non-seed) vertex chooses one of its normal nearest neighbors or itself as the target and this is indicated by an arrow, leading to the formation of a directed tree at layer $h=1$ and a directed cycle-tree at layer $h=2$.
• Figure 2: Theoretical predictions for the regular random graph ensemble of degree $D=7$ at different values of layer number $H$. The $K$-core threshold is $K = 3$. (a) Energy density $\rho$ versus inverse temperature $\beta$. (b) Entropy density $s$ versus $\beta$. (c) Entropy density $s$ versus energy density $\rho$. (d) Minimum energy density $\rho_{\textrm{min}}$ versus the value of $H$. As a comparison, the gray points are the results obtained by the model of Ref. Guggiola-2015 using the maximum time step $T$ as the hyperparameter.
• Figure 3: Sensitivity of the results of the hCTGA algorithm on the inverse temperature $\beta$ and on the number $N$ of vertices in the graph. The mean values averaged over $12$ regular random graph instances of degree $D$ and size $N$ are shown, and the error bars indicate standard error of the mean. The maximum number $H$ of layers is fixed to $H = 3$. (a) Relative size $\rho$ of constructed attack set versus $\beta$ on graphs of size $N = 10,000$ and degree $D=6$ with threshold value $K = 5$. (b) Relative size $\rho$ versus graph size $N$, obtained on graphs of degree $D = 4$ with threshold value $K = 3$ and $\beta = 23$. The algorithmic results obtained by the WN algorithm Schmidt-2019 are also shown as a comparison.
• Figure 4: Factor-graph representation for deriving the mean field equations. (top) Vertices and edges in the original graph $G$, with $\partial i = \{j, k, l\}$ and $\partial j = \{ i, m, n\}$. (bottom) Variable nodes and factor nodes in the factor graph. Each variable node corresponds to an edge of graph $G$, and each factor node corresponds to the local constraint induced by a vertex of graph $G$. Notice that each variable node is linked to exactly two factor nodes.