Minimal Cover of Implication Rules to Represent Two Mode Networks Sebasti˜ ao M. Neto, Luis E. Z´ arate and Mark A. J. Song UNA University Center Pontifical Catholic University Belo Horizonte – MG – Brazil Email: sebastiaomendesneto@gmail.com, zarate@pucminas.br, mark@pucminas.br S´ ergio M. Dias Federal Service of Data Processing - SERPRO Federal University of Minas Gerais - UFMG Belo Horizonte – MG – Brazil Email: sergio.dias@serpro.gov.br Abstract—In a world full of connections between people and objects, new needs arise requiring multidisciplinary analysis of these new networks. This work presents a approach to analyze an Internet Service Provider (ISP) database using a minimal cover of implications extracted from formal concept analysis and complex network techniques. Our goal is to analyze access to the 25 most visited websites to find access patterns and its dependencies using a minimal cover of implications. 1,470,450 accesses carried out over a month were selected. The data were converted to a clarified formal context and the NextClosure algorithm was used to extract a minimal cover of implications. Implication rules were used to explore hidden substructures from the two mode networks. As a result, we found a set of 216 implications that explain every user access behavior. With a set of axioms we present the possibility of obtaining more specific rules by derivation. These implications represent access patterns that guarantee that whenever a set of websites are accessed, so is another set of websites. These results can aid in creating security policies and network configurations to help predict future accesses. Furthermore, with the implications we save a total of 99,90% of storage space needed to represent access behavior. Finally, we show that with the construction of the network based on implication rules, the relationships between the events (websites) of a two mode networks can be made. I. I NTRODUCTION In the last few years, attention has turned towards the increasing complexity of the connected world, as noted by Easley and Kleinberg [1]. This connectivity is propelled by a variety of factors, such as the Internet itself, telephone networks and the speed with which information travels around the globe. These factors enable the genesis of social networks formed by relationships between people, defining a behavior which connects the aforementioned people. Motivated by the interconnected world, research has surfaced and disciplines interlink to contribute with techniques and new perspectives for the analysis of complex networks composed by these connections. For some time now, social network analysis is focused on the discovery of social relationship patterns which unite social actors. These relationships can occur between subjects, events, or subjects and events. They can pinpoint instance properties such as importance, ranking and category. According to Getoor and Diehl [2], in some cases some relationships are not observed, thus it can be relevant to discover the existence of potential relationships between instances. It is of general interest to unveil hidden substructures as possible and potential communities. The discovery of these substructures is a field of data mining related to relationship mining in social networks. The field seeks to find common substructures, which can be useful in the classification of networks and their structures. However, the identification of substructures demands work towards perfecting methods to clarify network visualization and extract representations and important information from them [3-4]. Presently, social networks are undergoing a dra- matic growth spur which makes it harder and harder to foresee patterns in relationships. Along with that, it is also important to think of new computational models that enable representing, characterizing and analyzing social networks. Computational models based on implication rules can be very useful if the rule set allows for the extraction of new knowledge. The extraction, on the other hand, can be done via Formal Concept Analysis (FCA) [5], which is a mathematical research field introduced in the early 80s by Rudolf Wille and has found use in different fields of academia. FCA is a data analysis theory which identifies concept structures within a data set. It applies concept lattice theory to hierarchically organize data from a formal context made of objects, attributes and their incidences (or relationships). These sets can contain ordinal, categorical, nominal, or interval data. Therefore, due to its potential in knowledge representation, FCA has already benefited complex network analysis, as discussed by Freeman and White [6], Sn´ aˇ sel [7-8], Freeman [9], Poelmans et al [10], Rome and Haralick [11] and Ja-schke et al [12]. In this work, based on FCA theory, we propose an approach to build computational models based on implication rules to represent two mode networks. The NextClosure [5] algorithm, which is based on closed sets, was used to extract a minimal set of implications. The goal is to increase our knowledge on these networks finding a minimal set of user access patterns capable of representing hidden substructures (relationships between different websites) in the original network. An access pattern is considered to be the set of websites