Sharp Thresholds for the Existence of Maximum δ-Temporal Cliques in Random Simple Temporal Graphs

Khái niệm cốt lõi

The size of a maximum δ-temporal clique in a random simple temporal graph on n vertices is approximately 2 log n / log (1/δ) with high probability.

Tóm tắt

The paper studies random simple temporal graphs, where the underlying graph is the complete graph on n vertices and the edge labels (timestamps) are chosen uniformly at random from the interval [0, 1]. The main result is a sharp threshold on the size of any maximum δ-temporal clique, which is a clique where the time difference between any pair of edges is at most δ.
The key insights are:

Using the probabilistic method, the authors prove that the size of a maximum δ-temporal clique is approximately 2 log n / log (1/δ) with high probability.
This is surprising because the random simple temporal graph contains Θ(n^2) overlapping δ-windows, each of which corresponds to a different random instance of the Erdős-Rényi random graph model Gn,δ. Yet, the size of the maximum δ-temporal clique and the maximum clique size of Gn,δ are approximately the same.
The authors also show that the minimum interval containing a δ-temporal clique is δ - o(δ) with high probability, and use this result to argue that any polynomial-time algorithm for finding a maximum δ-temporal clique is unlikely to have a very large probability of success.

Thống kê

None

Trích dẫn

None

Thông tin chi tiết chính được chắt lọc từ

On the existence of $δ$-temporal cliques in random simple temporal graphs

by George B. Me... lúc arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.07147.pdf

Yêu cầu sâu hơn

What are the implications of the connection between the maximum δ-temporal clique size in random simple temporal graphs and the maximum clique size in Erdős-Rényi random graphs Gn,δ

The connection between the maximum δ-temporal clique size in random simple temporal graphs and the maximum clique size in Erdős-Rényi random graphs Gn,δ has significant implications. This connection reveals that even though the random simple temporal graph contains multiple instances of Gn,δ within overlapping δ-windows, the size of the maximum δ-clique and the maximum clique size of Gn,δ are approximately the same. This surprising result suggests a strong correlation between the structures of these two types of graphs, despite their different temporal dynamics.
Algorithmically, this connection can be leveraged in various ways. For instance, if there is a polynomial time algorithm that can find a clique of size close to the maximum in Gn,δ, this algorithm can be adapted to find a similarly sized δ-temporal clique in random simple temporal graphs with high probability. This implies that insights and algorithms developed for static graphs can be extended to temporal graphs, opening up new possibilities for solving temporal graph problems efficiently.

Can this connection be leveraged algorithmically

If the underlying graph is not the complete graph but a sparser random graph, the results may change in terms of the threshold for the size of maximum δ-cliques. In a sparser random graph, the density of edges is lower, leading to fewer potential cliques and different structural properties. This could affect the sharp threshold on the size of maximum δ-cliques in random instances of the model. The relationship between the maximum clique size in the sparser random graph and the maximum δ-clique size in the corresponding temporal graph may also vary, depending on the specific characteristics of the sparser graph.
The sparsity of the underlying graph can influence the formation and size of cliques in the temporal graph, potentially impacting the complexity and distribution of δ-temporal cliques. Therefore, studying the behavior of δ-temporal cliques in sparser random graphs can provide insights into how graph density affects temporal clique formation and properties.

How might the results change if the underlying graph is not the complete graph, but a sparser random graph

There are several other non-path-related temporal graph problems where similar surprising connections between the temporal and static versions can be established. One such problem is the temporal vertex cover, where the goal is to find a set of vertices that covers all edges at different time points. The static version of vertex cover involves finding a set of vertices that covers all edges in the graph. By introducing time as a factor, the temporal vertex cover problem adds a temporal dimension to the traditional vertex cover problem.
Similarly, problems like temporal graph coloring, temporally transitive orientations of temporal graphs, and cluster editing in temporal graphs also exhibit interesting connections between their temporal and static versions. These problems involve optimizing graph structures or properties over time, introducing new challenges and complexities compared to their static counterparts. By exploring these connections, researchers can gain a deeper understanding of how temporal dynamics impact graph optimization and analysis.