Hypergraph Theory in Wireless Communication Networks by Hongliang Zhang Lingyang Song Zhu Han & Yingjun Zhang

Author:Hongliang Zhang, Lingyang Song, Zhu Han & Yingjun Zhang
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proposition 2.2

We divide the vertex set X of hypergraph H into cellular set X c and D2D set X d . When the number of cellular UEs increases by 1, the cellular UEs and D2D pairs form a new hypergraph H ′ . If , then M(H ′ ) = M(H) + 1; Otherwise, M(H) ≤ M(H ′ ) ≤ M(H) + 1.

Proof

In hypergraph construction, if the number of vertices increases by 1, the monodegree of the other vertices will increase by at most 1. The reason is that once two vertices form an edge, one vertex will not be the cumulative interferer of the other, and they cannot form a hyperedge. In addition, any two cellular UEs are bound to form an edge, and thus, if the number of cellular UEs increases by 1, the monodegree of each cellular UE will increase by 1 as well.

Under the assumption , cellular UE x is the vertex which has the maximum value of the minimum monodegree. According to the aforementioned analysis, if the monodegree of cellular UE x increases by 1, then the monodegree of the other vertices will increase by at most 1. Thus, cellular UE x is still the vertex which has the maximum value of the minimum monodegree, and M(H ′) = M(H) + 1. Otherwise, a D2D pair x is the vertex which has the maximum value of the minimum monodegree. If the mutual interference between D2D pair x and the new cellular UE cannot form an edge nor a hyperedge, M(H ′) = M(H). Therefore, if the vertex is not a cellular UE, M(H) ≤ M(H ′) ≤ M(H) + 1. □

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