Theory and algorithms for clusters of cycles in graphs for material networks

TL;DR

Addressing the challenge of cycle-rich topology in material networks, this work tackles non-uniqueness and potential exponential growth of cycles by proposing uniform random sampling of minimum cycle bases (MCB) and introducing two partitions, polyhedron-interchangeability ($\pi$) and short loop-interchangeability ($\text{sli}$), to organize the relevant cycles. A postprocessing step enforces pairwise cycle intersections to a single path, enabling a dual graph representation of cycle clusters and facilitating interpretable ring structures. The authors develop practical algorithms (modified Vismara $V'$-families, MCB construction with witness vectors, and basis-change procedures) and demonstrate that $\pi$- and $\text{sli}$-classes can be computed efficiently, even for large graphs arising in hydrocarbon pyrolysis simulations. Applying the framework to ReaxFF-based hydrocarbon networks reveals that the number of relevant cycles can grow rapidly with system size and identifies polyhedral structures reminiscent of diamond, suggesting implications for crystallography and materials design.

Abstract

Analysis of complex networks, particularly material networks such as the carbon skeleton of hydrocarbons generated in hydrocarbon pyrolysis in carbon-rich systems, is essential for effectively describing, modeling, and predicting their features. An important and the most challenging part of this analysis is the extraction and effective description of cycles, when many of them coalesce into complex clusters. A deterministic minimum cycle basis (MCB) is generally non-unique and biased to the vertex enumeration. The union of all MCBs, called the set of relevant cycles, is unique, but may grow exponentially with the graph size. To resolve these issues, we propose a method to sample an MCB uniformly at random. The output MCB is statistically well-defined, and its size is proportional to the number of edges. We review and advance the theory of graph cycles from previous works of Vismara, Gleiss et al., and Kolodzik et al. In particular, we utilize the polyhedron-interchangeability (pi) and short loop-interchangeability (sli) classes to partition the relevant cycles. We introduce a postprocessing step forcing pairwise intersections of relevant cycles to consist of a single path. This permits the definition of a dual graph whose nodes are cycles and edges connect pairs of intersecting cycles. The pi classes identify building blocks for crystalline structures. The sli classes group together sets of large redundant cycles. We present the application to an amorphous hydrocarbon network, where we (i) theorize how the number of relevant cycles may explode with system size and (ii) observe small polyhedral structures related to diamond.

Figures (37)

• Figure 1: (a) Snapshot of the giant biconnected component in a simulation initialized to adamantane ($\rm C_{10}H_{16}$) at 4000K and 40.5GPa. We highlight (i) essentialgleissInterchangeability2000 pentagonal and hexagonal rings belonging to every MCB in blue, (ii) a sli class composed of two 11-membered rings separated by a hexagonal ring in orange, and (iii) two pi classes with the same shape as barallene ($\rm C_8H_8$) in green. (b) A graph made of diamond-shaped cycles chained into a loop like a bracelet. An MCB of this graph includes all of the small diamond-shaped loops and one large loop. (c) Carbon skeleton of adamantane, the smallest diamondoid, i.e., diamond is formed by repeatedly tiling adamantane. An MCB of adamantane includes any 3 of the 4 hexagonal faces. (d) Depiction of the process of modifying a cycle $C_2\to C_2'$ such that the cycles $C_1$ and $C_2$ overlap over a single path.
• Figure 2: A graph with four face cycles $B_1$, $B_2$, $B_3$, and $B_4$ forming a basis. Three additional cycles $C_1$, $C_2$, and $C_3$ are shown. $C_1$ is a single loop while $C_2$ and $C_3$ are not.
• Figure 3: (a) A graph $G$ with edges colored according to whether they are inside (blue) or outside (red) the spanning tree $T$. The face cycles $B_1$ and $B_2$ form a fundamental cycle basis $\mathcal{F}$, where they are composed of an edge in $E(G){\setminus} E(T)$ joined by a path in $T$. (b) A depiction of the cycles $B_1$ and $B_1\,{\oplus}\, B_2$ alongside their binary representation as fundamental cycle basis $\mathcal{F}$ vectors. (c) Another spanning tree $T'$ resulting in a different fundamental cycle basis that contains $B_2'\,{=}\,B_1\,{\oplus}\, B_2$.
• Figure 4: A graph $G$ that does not have a unique MCB. Ignoring double bonds, $G$ is the carbon skeleton of barallene with formula $\rm C_8 H_8$. (a) 3D printed visualization of $G$. (b) Two-dimensional representation of $G$. The three hexagonal relevant cycles $C_1$, $C_2$, and $C_3$ are shown separately. Any pair of these cycles forms an MCB, where the third cycle is the sum of the other two, e.g., $\mathcal{M}_1=\{C_1,C_2\}$ is an MCB and $C_3=C_1\oplus C_2$.
• Figure 5: Depiction of the descriptors used to organize the relevant cycles into V-families as in Ref. vismaraUnion1997. For an odd length cycle (left), we fix its largest index node $r$ and the edge $(p,q)$ at the opposite end of the cycle. For an even length cycle (right), we fix its largest index node $r$ and the triple $(p,x,q)$ at the opposite end of the cycle. In the left, we also draw the exchange of the path $P_1$ for $P_2$ by adding the cycle $C_e=P_1\oplus P_2$ to $C_1$ to obtain $C_2=C_1\oplus C_e$.

Theorems & Definitions (53)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5: Vismara cycle family
• Lemma 1: Ref. vismaraUnion1997, Lemma 2
• proof
• Lemma 2
• proof
• Definition 6