Degree-preserving graph dynamics -- a versatile process to construct random networks

TL;DR

Degree-preserving network growth (DPG) provides a rigorous framework wherein incoming vertices attach while preserving existing vertex degrees, formalized via p-degrees, generalized lifting, and forward/backward DP steps. The work shows DPG can reproduce outputs of several real-world growth models and proves that DPG-feasibility is NP-complete, underscoring computational hardness of reconstructing kernels. It develops Linear DPG and Scale-free DPG, establishing degree-growth bounds, core-periphery structure, and scale-free behavior (for $oldsymbol{b7}>1$), while discussing sampling implications via Markov chains and P-stability. The paper also explores regular-graph variants and proves NP-hardness for DP-removal of many vertices, highlighting a tension between theoretical hardness and empirical success of DPG in modeling real networks. Together, these results lay a foundation for a formal theory of DPG dynamics and pose open questions about the structural properties that make real networks DPG-feasible.

Abstract

Real-world networks evolve over time via additions or removals of vertices and edges. In current network evolution models, vertex degree varies or grows arbitrarily. A recently introduced degree-preserving network growth (DPG) family of models preserves vertex degree, resulting in structures significantly different from and more diverse than previous models ([Nature Physics 2021, DOI: 10.1038/s41567-021-01417-7]). Despite its degree preserving property, the DPG model is able to replicate the output of several well-known real-world network growth models. Simulations showed that many well-studied real-world networks can be constructed from small seed graphs. Here we start the development of a rigorous mathematical theory underlying the DPG family of network growth models. We prove that the degree sequence of the output of some of the well-known, real-world network growth models can be reconstructed via the DPG process, using proper parametrization. We also show that the general problem of deciding whether a simple graph can be obtained via the DPG process from a small seed (DPG feasibility) is, as expected, NP-complete. It is an important open problem to uncover whether there is a structural reason behind the DPG-constructibility of real-world networks.

Figures (5)

• Figure 1: 2 DP-steps operations followed by 6 DP-removals transforms $2K_4$ into $K_4$. Red edges: to be removed by DP steps; blue edges: new edges created by inverse DP steps. Dash-dotted vertices and edges represent the stub vertex and the stub edge.
• Figure 2: A graph which is irreducible.
• Figure 3: Making a vertex $x\in V(G)$ non-DP-removable by joining it to every vertex of a unique copy of a large enough clique.
• Figure 4: The variable gadget $H_i$. Dashed lines represent non-edges.
• Figure 5: The clause gadget (a portion of the graph $G$) associated to $c_\ell=x_i\vee \neg x_k\vee x_r$.

Theorems & Definitions (28)

• Definition 2.1: Irreducibility
• Example 2.2
• Example 2.3
• Theorem 3.1: Vizing vizing
• Theorem 3.2: Pósa, 1962 posa
• Definition 3.3
• Theorem 3.4: DPG21
• Corollary 3.5
• proof
• Theorem 3.6: DPG21