TY - GEN
T1 - Random preferential attachment hypergraph
AU - Avin, Chen
AU - Lotker, Zvi
AU - Nahum, Yinon
AU - Peleg, David
N1 - Publisher Copyright:
© 2019 Association for Computing Machinery.
PY - 2019/8/27
Y1 - 2019/8/27
N2 - In the future, analysis of social networks will conceivably move from graphs to hypergraphs. However, theory has not yet caught up with this type of data organizational structure. By introducing and analyzing a general model of preferential attachment hypergraphs, this paper makes a step towards narrowing this gap. We consider a random preferential attachment model H(p,Y) for network evolution that allows arrivals of both nodes and hyperedges of random size. At each time step t, two possible events may occur: (1) [vertex arrival event:] with probability p > 0 a new vertex arrives and a new hyperedge of size Yt, containing the new vertex and Yt − 1 existing vertices, is added to the hypergraph; or (2) [hyperedge arrival event:] with probability 1−p, a new hyperedge of size Yt, containing Yt existing vertices, is added to the hypergraph. In both cases, the involved existing vertices are chosen independently at random according to the preferential attachment rule, i.e., with probability proportional to their degree, where the degree of a vertex is the number of edges containing it. Assuming general restrictions on the distribution of Yt, we prove that the H(p,Y) model generates power law networks, i.e., the expected fraction of nodes with degree k is proportional to k−1−Γ, where Γ = limt→∞ Pt(Et i=0 −[Y1tE]−[Ypi)] ∈ (0,∞). This extends the special case of preferential attachment graphs, where Yt = 2 for every t, yielding Γ = 2/(2 − p). Therefore, our results show that the exponent of the degree distribution is sensitive to whether one considers the structure of a social network to be a hypergraph or a graph. We discuss, and provide examples for, the implications of these considerations.
AB - In the future, analysis of social networks will conceivably move from graphs to hypergraphs. However, theory has not yet caught up with this type of data organizational structure. By introducing and analyzing a general model of preferential attachment hypergraphs, this paper makes a step towards narrowing this gap. We consider a random preferential attachment model H(p,Y) for network evolution that allows arrivals of both nodes and hyperedges of random size. At each time step t, two possible events may occur: (1) [vertex arrival event:] with probability p > 0 a new vertex arrives and a new hyperedge of size Yt, containing the new vertex and Yt − 1 existing vertices, is added to the hypergraph; or (2) [hyperedge arrival event:] with probability 1−p, a new hyperedge of size Yt, containing Yt existing vertices, is added to the hypergraph. In both cases, the involved existing vertices are chosen independently at random according to the preferential attachment rule, i.e., with probability proportional to their degree, where the degree of a vertex is the number of edges containing it. Assuming general restrictions on the distribution of Yt, we prove that the H(p,Y) model generates power law networks, i.e., the expected fraction of nodes with degree k is proportional to k−1−Γ, where Γ = limt→∞ Pt(Et i=0 −[Y1tE]−[Ypi)] ∈ (0,∞). This extends the special case of preferential attachment graphs, where Yt = 2 for every t, yielding Γ = 2/(2 − p). Therefore, our results show that the exponent of the degree distribution is sensitive to whether one considers the structure of a social network to be a hypergraph or a graph. We discuss, and provide examples for, the implications of these considerations.
KW - Degree distribution
KW - Preferential attachment
KW - Random hypergraphs
KW - Social networks
UR - http://www.scopus.com/inward/record.url?scp=85078843134&partnerID=8YFLogxK
U2 - 10.1145/3341161.3342867
DO - 10.1145/3341161.3342867
M3 - Conference contribution
AN - SCOPUS:85078843134
T3 - Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2019
SP - 398
EP - 405
BT - Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2019
A2 - Spezzano, Francesca
A2 - Chen, Wei
A2 - Xiao, Xiaokui
PB - Association for Computing Machinery, Inc
T2 - 11th IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2019
Y2 - 27 August 2019 through 30 August 2019
ER -