PHYSICAL REVIEW E 89, 042812 (2014) Intergroup networks as random threshold graphs Sudipta Saha, Niloy Ganguly, and Animesh Mukherjee Department of Computer Science and Engineering, Indian Institute of Technology, Kharagpur 721302, India Tyll Krueger Department of Computer Science and Engineering, Technical University of Wroclaw, Poland (Received 9 February 2014; published 24 April 2014) Similar-minded people tend to form social groups. Due to pluralistic homophily as well as a sort of heterophily, people also participate in a wide variety of groups. Thus, these groups generally overlap with each other; an overlap between two groups can be characterized by the number of common members. These common members can play a crucial role in the transmission of information between the groups. As a step towards understanding the information dissemination, we perceive the system as a pruned intergroup network and show that it maps to a very basic graph theoretic concept known as a threshold graph. We analyze several structural properties of this network such as degree distribution, largest component size, edge density, and local clustering coefficient. We compare the theoretical predictions with the results obtained from several online social networks (LiveJournal, Flickr, YouTube) and find a good match. DOI: 10.1103/PhysRevE.89.042812 PACS number(s): 89.75.Fb, 89.90.+n I. INTRODUCTION Group formation [1–3] is a very common and popular feature among humans where a user (human) can participate in multiple groups [4]. Consequently, many social network- ing sites like LiveJournal, 1 Flickr, 2 YouTube, 3 etc., provide explicit facilities to form, maintain, and communicate within social groups [5]. These groups can be deemed as a medium of mass communication among its participating users [6–9]; there is, however, an interesting side effect to this communication. Common users belonging to multiple groups pass information of one group to another. Hence, analyzing the extent of connectivity among the groups can shed light into the amount of information propagated from one group to another. This connectivity structure can be best estimated by analyzing the properties of an evolving intergroup network, which is the primary focus of this paper. Intergroup networks can be modeled as the one-mode projection of evolving user-group bipartite networks. In the bipartite process, the user partition grows with time, while, if we consider only the popular groups, the group partition remains fixed. Such bipartite networks where one partition remains fixed is termed a alphabetic-bipartite network (α-BiN) [10]. A projection on the groups allows us to obtain the group-group network (i.e., the intergroup network) where two groups are neighbors if they have at least one common user. However, it can be safely assumed that two groups will have high mutual interaction, thus allowing more information to propagate if there are at least a critical number of common users (determined by a threshold value) [5,11]. Therefore, an interesting structure to study is the “pruned intergroup network” where two nodes (groups) are connected if they have more than a threshold number of common users. The first attempt to understand the “pruned intergroup network” was made in Ref. [5], where by mapping the 1 LiveJournal: www.livejournal.com 2 Flickr: www.flickr.com 3 YouTube: www.youtube.com underlying evolution dynamics to a Polya urn model, the degree distribution of the pruned intergroup network is derived. However, in order to gain a better insight into the connectivity structure, the more relevant structural properties such as largest component size, edge density, and local clustering coefficient (assuming a large number of groups) under various possible threshold values need to be analyzed as a first step. In this paper, we identify the mathematical relationship between the weight of an edge and the weights of the associated nodes (later these node weights are referred to as “attractiveness parameters”), derived in Ref. [5], as special importance. We heavily leverage on this relationship to derive the formula for the above mentioned structural properties of the “pruned intergroup network.” We also show that this class of networks can be appropriately modeled by a special variant of “random threshold graph”[12], termed a “multiplicative random threshold graph.” We compare the theoretical predictions with the same obtained from the available real datasets, and in most of the cases, we obtain a significantly accurate match. Hence, the main contributions of this paper are twofold: (a) we show how the intergroup relationships in social systems and in a more general sense the pruned one-mode projection of a preferentially grown α-BiN can be studied as a special kind of random threshold graph model, and (b) we show how the mathematical analysis of various structural aspects (degree distribution, largest component, edge density, as well as local clustering coefficient) of this specific kind of multiplicative random threshold graphs can be done in a more transparent way. The rest of the paper is organized as follows. In the next section, we precisely describe the basic model of α-BiN and the special property of the intergroup networks that allows us to map it to multiplicative random threshold graphs. In Sec. III we present a detailed description of the mathematical analysis of degree distribution, edge density, largest component, and the local clustering coefficient. Next, in Sec. IV we compare the mathematical findings with the observations made from the real dataset. Finally, in Sec. V we present a brief review of the state-of-the-art before drawing the conclusion. 1539-3755/2014/89(4)/042812(11) 042812-1 ©2014 American Physical Society