Random Preferential Attachment Hypergraphs Chen Avin, Zvi Lotker, Yinon Nahum and David Peleg Abstract 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 prefer- ential 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 at random according to the preferential attachment rule, i.e., with probability propor- tional to their degree, where the degree of a vertex is the number of edges containing t−1 it. Denoting the total degree in the hyper graph by Dt = D0 +∑i=0 Yi, we allow Yt ≥ 1 E[Dt ]=t to be any integer-valued random variable satisfying (i) limt!¥ = G 2 (0;¥), E[Yt ]−p 2 2 (ii) E[j1=Dt − 1=E[Dt ]j] = o(1=t) and (iii) E Yt =Dt = o(1=t). Furthermore, if Yt 2 is either deterministic (i.e., not random) or satisfies E Yt = o(t), assumptions (ii) and (iii) can be omitted. 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−b , where b = 1 + G . This extends the special case of preferential attachment graphs, 2 where Yt = 2 for every t, yielding b = 1 + 2−p . Therefore, our results show that the exponent of the degree distribution is sensitive to whether one considers the struc- ture of a social network to be a hypergraph or a graph. We discuss, and provide examples for, the implications of these considerations. Chen Avin and Zvi Lotker Ben Gurion University of the Negev, Be’er-Sheva, Israel e-mail: favin,[email protected] Yinon Nahum and David Peleg Weizmann Institute of Science, Rehovot, Israel e-mail: fyinon.nahum,[email protected] 1 2 Chen Avin, Zvi Lotker, Yinon Nahum and David Peleg 1 Introduction Random structures have proved to be an extremely useful concept in many disci- plines, including mathematics, physics, economics, and communication systems. Examining the typical behavior of random instances of a structure allows us to un- derstand fundamental properties of the structure itself. In the context of graph structures, the foundations of random graph theory were laid in a seminal paper by Erdos˝ and Renyi´ in the late 1950’s [8]. Subsequently, several alternative models for random structures, suitable for different kinds of ap- plications, have been suggested. One of the most important alternative models is the preferential attachment (PA) model [14, 2], which was found to be particu- larly suitable for describing a variety of phenomena in nature, such as the “rich get richer” phenomenon, which cannot be adequately simulated within the original Erdos-R˝ enyi´ model. It has been shown that the preferential attachment model cap- tures some universal properties of real-world social networks and complex systems, such as heavy tail degree distribution and the “small world” phenomenon [12]. While graphs are extremely versatile and useful structures for representing in- terrelations among entities, one of their limitations is that they only capture dyadic (or binary) relations. In real-life, however, many natural, physical, and social phe- nomena involve k-ary relations for k > 2, or even relations of variable arity. For example, collaborations among researchers, as manifested through joint coauthor- ships of scientific papers, may be better represented using hyperedges instead of edges. Figure 1(a) depicts the hypergraph representation for coauthorship relations on four papers: paper 1 authored by fa;b;e; f g, paper 2 by fa;c;d;gg, paper 3 by fb;c;dg and paper 4 by fe; f g. Likewise, wireless communication networks [1] or social relations captured by photos which appear on social media also give rise to hyperedges in a natural way [17]. Affiliation models [11, 13], which are a popular model for social networks, are commonly interpreted as bipartite graphs, when, in fact, they may sometimes be represented more conveniently as hypergraphs. For ex- ample, Figure 1(b) presents the bipartite graph representation of the hypergraph H of Figure 1(a). While there have been attempts to reduce hypergraphs to graphs by converting every hyperedge to a clique, as this paper will show, this indirect analysis of hypergraphs can lead to inaccuracies. Sometimes, it is only possible (or conve- d c g b f 4 23 e c 3 e b b a a 1 g 1 g d 2 a d e 4 f c f (a) (b) (c) Fig. 1 (a) A hypergraph H with 7 nodes and 4 edges. (b) A bipartite graph representation of H. (c) The observed graph G(H): every hyperedge is replaced with a clique. Random Preferential Attachment Hypergraphs 3 nient) to access the observed graph G(H) of the original hypergraph H, namely, only the pairwise relation between players is visible or given (see Figure 1(c)). In some cases, this structure may be sufficient for the application at hand. However, in many cases (e.g., when studying degree distribution), direct analysis of the original hypergraph is needed for more accurate results. In order to deal with the latter cases, we tackle the more precise, albeit somewhat harder, analysis of hypergraphs. The study of hypergraphs, and in particular random hypergraph models, has its roots in a 1976 paper by Erdos˝ and Bollobas [3], which offers a model analogous to the Erdos-R˝ enyi´ random graph model [8]. Several interesting properties of the evolution of random hypergraphs in this model were recently studied [6, 7, 10]. A similar transition (from graphs to hypergraphs) for the preferential attachment model was first studied by Wang et al. in [15], which defined a basic evolving hyper- graph model with vertex arrival events and constant-size hyperedges. Specifically, at every step, m − 1 new nodes enter the network and a new edge of size m is added containing the m−1 new nodes and one existing node chosen by preferential attach- ment, i.e., with probability proportional to its degree. Analyzing the degree distribu- tion, it was shown that the resulting hypergraph follows a power law with exponent m+2. Note, however, that the model of [15] generates restricted hypergraphs. First, they are acyclic, since hyperedges cannot be added between existing nodes. Second, the hyperedges are uniform, i.e., of the same size. Third, every two nodes share at most one hyperedge. Hence the model results in a limited class of uniform hyper- graphs that is analogous to the class of trees within the family of graphs. Our work was motivated by the fact that most actual aplications where hyper- graph structures come into play do not fall in the limited class of [15]. Aiming to model a larger set of networks, we extend the G(p) evolving graph model of Chung and Lu [4], which allows edge arrivals, to include hyperedges of random hyper- edge size. Thus, we allow more complex structures for the resulting hypergraphs, including cycles and nonuniformity. The main technical contribution of this paper is an analysis of the degree dis- tribution of random preferential attachment hypergraphs, showing that they possess heavy tail degree distribution properties, similar to those of random preferential at- tachment graphs. However, our results also show that the exponent of the degree distribution of a random preferential attachment model is sensitive to whether one considers the structure to be a hypergraph or a graph. In fact, in the setting of hy- peredges which grow in size, namely Yt ! ¥, the exponent can drop below 2. The model proposed here extends Chung and Lu’s model [4] by supporting hy- pergraphs, and moreover, allowing hyperedges of random size. The process starts with an initial hypergraph, and at each time step t, a new hyperedge of random size joins the network. With probability p, this new hyperedge includes a new node and possibly other existing nodes, i.e., nodes which have arrived before time t, and with probability 1 − p, all the nodes of the new arriving hyperedge are existing nodes. Our model allows the hyperedge sizes to be random (with some restrictions), and the existing nodes of each hyperedge are selected randomly according to the prefer- ential attachment rule, namely, with probability proportional to their degree. 4 Chen Avin, Zvi Lotker, Yinon Nahum and David Peleg Fig. 2 The exponent b of a preferential attachment (PA) graph and a 3-uniform hypergraph as a function of p (the probability of a new vertex arrival event). In graphs, b is between 2 and 3, whereas in 3-hypergraphs b is between 2 and 2.5. We show that the degree distribution of the resulting hypergraph follows a power law, i.e., the fraction of nodes with degree k is proportional to k−b , with exponent [D ]=t b = 1 +G ; where G = lim E t 2 (0;¥) : t!¥ E[Yt ] − p This naturally extends the known result for the Chung and Lu’s model for graphs.