Graph Isomorphism in Fuzzy Cognitive Maps for ...

2014 International Conference on Intelligent Networking and Collaborative Systems

Graph Isomorphism in Fuzzy Cognitive Maps for Monitoring of Game-based Learning Holman Bolívar Baron Faculty of Engineering Catholic University of Colombia Bogotá, Colombia Email: [email protected]

Mario Martinez Rojas Faculty of Engineering Catholic University of Colombia Bogotá, Colombia Email: [email protected]

Johanna Trujillo Diaz Faculty of Engineering Catholic University of Colombia Bogotá, Colombia Email: [email protected]

Andres Velasquez Contreras Faculty of Engineering Catholic University of Colombia Bogotá, Colombia Email: [email protected]

academic learning processes are still few in number and rarely integrated with the activities of the school environment. [5]. Because the variables affecting school performance are different levels and cannot establish a direct measure given the complexity of the educational process on the dependency relationships between learning activity and its effectiveness, it must be tested inferences on academic process, validate causal hypotheses to identify mediating fundamental mechanisms [3]. To address the complexity of the evaluation process in interactive environments such as video games, we propose the use of structural equation modeling (SEM) [6] as methodology Multiple Criteria Decision, providing a general framework for dealing with decision problems without making assumptions about the independence between variables of different levels of hierarchy, unlike the Analytic Hierarchy Process (AHP) o Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) o ELimination Et Choix Traduisant la REalité (ELECTRE), that usually assume independence among the criteria [7]. In the construction process of the structural equations it is necessary to develop sequence diagrams. To develop sequence diagrams we will use Fuzzy Cognitive Maps (FCM). These maps are an effective way to model causal relationships, allowing to apply knowledge about the strength of the relationship. Quantifying the strength of the relationship between two concepts is repre-sented by nodes in the range normalized [−1, 1]. Where the value -1 represents full negative, + 1 total positive and 0 does not denote any causal effect. As a result, the FCM model is fully described by a set of concepts and cause-effect relationships, represented by weights between them. Besides the graphical representation, we will use an equivalent model defined by a square matrix, called connection matrix. The connection is very suitable for computational purposes, the matrix stores all the weight values for the corresponding edges between concepts represented by rows and columns. The system with n nodes

Abstract—Knowledge can be understood as the specification of classifications and causes, kinds and causes usually are fuzzy. Fuzzy Cognitive Maps has been applied in different areas to express the dynamic behavior of a set of related concepts. Its graphical structure allows systematic causal propagation forward and backward chaining. In the same way, nowadays individuals use a lot of different technological media: smart TV, Internet, video games, mobile phones, etc. The use of multimedia has become popular in the education field, these multimedia techniques can allow students to get more entertainment, immersion and interactivity. An interactive learning environment can provide useful information to analyze the student learning process. This research work presents a learning assessment system that uses Ullman algorithm to determine if there are graph isomorphism between the graph resulting from the interaction of a student with a gaming platform and the graph from causal relationships built by the teacher or tutor. In this evaluation, it was observed that most practical tasks obtain better results, confirming the Hebb rule on learning and cellular neurophysiological relationship. Keywords— Fuzzy cognitive map, Cognitive training, Game-based assessment, Game-based learning

I. INTRODUCTION Quality education is directly related to school effectiveness, which is evident in the percentage of compliance with the learning objectives [1] nevertheless, progressive assessment and monitoring of learning is one of the weakest links to achieve school effectiveness [2] [3]. Nowadays individuals use a lot of different technological media: tv, internet, video games, mobile phones, etc; which create new ways to play, express themselves, learn, communicate and explore texts, ideas and identities for which the use of animation and multimedia in learning and teaching has been extended allowing more immersion and interactivity [4]. However, while producers of the modern media have developed deeply entertaining products to engage our children, Examples of materials that support 978-1-4799-6387-4/14 $31.00 © 2014 IEEE DOI 10.1109/INCoS.2014.117

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can be represented by a connection matrix n × n [8]. The FCM models have been applied in different areas to express the dynamic behavior of a set of related concepts. These models are a convenient tool for the simulation and analysis of dynamic systems. Previously these models have been used successfully in many different domains, such as engineering, medicine, control and politics. Its popularity comes from simplicity and transparency of the underlying model. In most cases, the FCM requires the involvement of experts to develop the model, as the experts are humans they cannot handle complex networks [9]. The research presented, proposes, an approach to assessment of videogame-based learning using fuzzy cognitive maps, as from isomorphism graph generated in the causal relationships between different scenarios generated and the possibility of effective feedback from teachers, parents and counselors. For model validation was performed a quasiexperimental analytical empirical test [10] in which a design of nonequivalent control group was used and one experimental. Were randomly assigned children equivalently to each group, group 1 received cognitive training program in planning, while the control group, do not participate in the program, which consisted of a sequence of 6 scenarios with different activities such as routing bus and towers of Hanoi game, based on the Neuropsychological Evaluation of the Child. The structure of this paper is as follows: in section II, presents related works to the implementation of video games and virtual worlds in learning, followed by Section III, that describes the fuzzy cognitive maps, section IV, described the methodology developed in research applying graph isomorphism, followed by section V where the results obtained are presented and the related discussion, in Section VI, described the conclusions and future works.

intended to essentially help assessor's to conduct their work more effective and efficient. The proposed approach uses the semantic web technologies, such as RDFS ontologies and rules to establish correspondences between the skills and credentials of students. Reimann et al. [3], used Hasse diagrams to specify dependency relationships between tasks. Presents a assessment approach for a finite set and atomic of academic skills, defined of some kind of fitness, ability, knowledge or skill associated with a numerical scale to specify their level of achievement and the preceding requirements. It should be noted that the specification of a set of competencies is a learning domain. Aleven et al. [14], proposed a model based on an explicit representation of the components of the instruction targeted in knowledge model, which evaluates the actions troubleshooting of students comparing them with the actions that the model can does in the same situation, this process is called tracking model, additionally, with using a Bayesian algorithm, the tutor considers the probability that the student knows each component of the knowledge captured in its domain, on the historical basis of their successes and failures, and the results retrieved in each knowledge component from which proposes an individualized sequencing problems. Ibañez et al. [15], propose an intelligent tutor named Non Player Character (NPC), NPC is a character that is not controlled by any human being and is active in the virtual world, it is responsible for instructing the student through a prerecorded audio file using Artificial Intelligence Markup Language (AIML), which responds to linguistic patterns from the user actions. Chen and Chen [16], present an approach that combines: statistical correlation analysis, fuzzy clustering algorithms, gray relational analysis, K-means algorithm and the fuzzy inference to measure behaviors and learning outcomes. It begins with the procedure of factor analysis, responsible for identifying independent factors associated with learning objective, by means of a statistical correlation analysis which takes into account the time of reading, the concentration ratio, and the responses according to the item response theory proposed by Baker [17]. Different studies like the Lorenzo et al. [18], present analysis of relationships between instructors and learner on immersive environments, demonstrating that these platforms can be used for assessment through peer collaboration tasks and improve communication with the tutor, increasing patterns of interaction within a learning group. To test the concept of multi-user in 3D virtual environments as a means to teach, there are experiences such as that developed by Okutsu et al.[19] where through a software prototype called Aeroquest, made an experience with 135 college students, evidencing the potential of virtual worlds to get an exam score within a confidence interval of 95%. Kim et al. [20] provides a research model for the evaluation of virtual worlds by measuring the level of identification of users with avatars and virtual communities, concluded that communication improves efficiency and confidence, thereby facilitating sustained use in important theoretical and practical implications. Jarmon [21], explores the nature and process of learning in a graduate course, validating the

II. RELATED WORKS Instructional activities should allow applying the acquired knowledge and skills in everyday and in real situations [7], for which the video games and virtual worlds are a great alternative because they allow us to establish relations presence of place, social presence and co-presence [11], which influences the satisfaction of the student experience motivating learning [12]. Out of the motivational factor, video games facilitate the measurement of learning according to the sequence of instructional activities, learning objectives and achievement levels by the student. In the literature there are different models and techniques for formative evaluation, Chen y Tzeng [7] proposed a model for the evaluation of educational materials, from the integration of Decision Making Trial and Evaluation Laboratory (DEMATEL) with Analysis Network Process (ANP) to identify the hierarchical control from a network of criteria, sub-criteria relationships and interactions, and the network of influence between elements and groups through the specification of dependency relationships. Biletska et al. [13], propose an approach for building an expert system for the electronic assessment (eAssessment) academic credentials and competencies, 305

hypothesis that virtual worlds can be well suited for experiential learning, moreover, as a motivational factor, Freitas and Oliver [22] propose the use of computer games and simulations for teaching, reinforcement of formal and informal learning and cognitive development of higher order. The diegesis of the game is used to refer to: mode of presentation, interactivity, levels of immersion and fidelity of the game; framing the difference between being immersed in the game and the process of critical reflection that develops out of the game, can be used as a method to support the objectives learning. An example of using the computer as a mediator in the process of learning through play and social interaction, are the Massively Multiplayer Online Game (MMOG), which may be considered an educational platform because it allows players to learn together through the joint interaction along cooperative processes has been demonstrated that several unique characteristics as the virtual identity can trigger learning behaviors, which can be seen as a process resulting from the adoption and continued use of the game [23]. Fig 1. An example of causal relationships between scenarios

III. FUZZY COGNITIVE MAPS Fuzzy Cognitive Maps have been introduced in the fields of artificial intelligence [21], these structures are fuzzygraph to represent causal reasoning able to encode knowledge by using the fuzzy set. Its graphical structure allows systematic causal propagation forward and backward chaining. Causality is represented as a fuzzy relation between concepts and extends the logical implication, because it generates a different concept in a fuzzy subset to be a change in the degree of membership of a member in the whole variation is generated in the concept. A concept is defined by the disjunction of a set y [24].

As a digraph, the FCM can be represented by an adjacency matrix. The mathematical model of a FCM is as follows: (4) Where and are state values of causal concept and effect concept , respectively, is the weight of the causal relationship from concept to the concept and is the threshold function of concept . The status value can be a fuzzy value between [0, 1] representing the degree of a concept, or a bivalent logic between {0,1} representing the state of opening or closing of a concept. The weight of an edge indicates the degree of influence of the causal concept on effect concept , which can be a fuzzy value between [-1, 1] or a trivalent logic between {1, 0, 1}. If the weight is positive, the increase or decrease in the value of state of concept leads to the increase or decrease in the value of state of concept . If the weight is negative, the increase or decrease in the value of the concept state leads to the decrease or increase of the state value of concept [25]. The graphical representation makes it easy to generate the matrix labeled adjacency, where each time the student takes that decision join him 1 array is compared with the array of theoretical labeled adjacency, were the decisions that the teacher expects presented that students take the stage in every scene. (see Fig. 2).

(1) Where is the fuzzy complement of fuzzy arbitrary set , therefore, diffuse causality can be defined by the double implication of disjunction belonging between two fuzzy sets and their complements. (2) (3) The negative causation can be defined with the same quantifiers and fuzzy relations of positive causality [25]. The evaluation of learning in an interactive environment requires the specification of the causal relationships between the several activities of an instructional process. An example where each scene corresponds to an activity and the weight of each edge represents the proportion of the attainable level by the student (see Fig. 1).

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The best known algorithms for determining whether or not two isomorphic graphs are exponential in the worst case complexity. However, algorithms are designed in polynomial complexity to solve the problem average case. However, for many specific types of practical problems exist algorithms which respond efficiently. Ullmann [27] describes a recursive backward procedure to solve the problem of subgraph isomorphism. Cordella et al [28] demonstrated that runtime Ullman algorithm (n3), in the best case and (n!n) in the worst case. Memory usage is (n3) where n denotes the size of the graph. Eppstein [29] solves the problem of subgraph isomorphism in planar graphs in linear time. It uses a technique of partitioning planar graph and applies dynamic programming to each partition.

Fig 2. Adjacency matrices theoretical and real

Simple graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorphic if there is a bijective function f of V1 in V2 with the property that, for each pair of vertices u, v  V1, u and v are adjacent in G1 if and only if are adjacent in G2. It is said that this function f is an isomorphism. It is difficult to determine whether two simple graphs are isomorphic or not since there are n! possible bijections between the sets of vertices of two simple graphs n vertices. There is an isomorphism induced subgraph between a query graph G1 and a target graph G2, if G1 is isomorphic to an induced subgraph of G2, the query graph G1 is a subgraph target graph G2. The problem of finding an induced subgraph isomorphic is believed to be a problem, for which there is no efficient solution, belongs to the class of NP-complete problems [26]. Therefore, each subgraph isomorphism algorithm shows exponential execution time relative to the size of the input graph. To prove that subgraph isomorphism is NP-complete, it must be formulated as a decision problem. The input for the decision problem is a pair of graphs G and G’. The answer to the problem is positive if G’ is isomorphic to a subgraph of G and negative otherwise. They must prove that the graph isomorphism is in NP and it must be possible polynomially reduce any NP problem to that problem.

IV. DEVELOPMENT OF RESEARCH Given that school effectiveness is evidential in the percentage of compliance with the learning objectives [1], we associate each scene game to one goal related to a learning objective, establishing valid levels for each scenario and its dependence with other games scene. Each student interacts with videogames in a random sequence and directed depending on what they want teachers. With the information collected from the interaction of each student with the video game, the framework proceeds to perform relational analysis according to the dependency relationships established by teacher, this constitutes the assessment model, the information obtained is analyzed statistically according to the criteria of validity, reliability and fairness to calíbrate model and infractor estimates for identify theoretically inconsistent estimates or negative variances of error coefficients and standardized test standardized residuals (see Fig. 3).

Fig. 3 Activity Diagram of Assessment of Game-Based Learning

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For model validation was performed analytical empirical test quasi-experimental [10], in which a design of non-equivalent control group and an experimental group was used, each group presented a posttest but did not have pre-experimental sampling equivalence since groups are constituted entities formed naturally. Randomly children equivalently assigned to each group, group 1 received the digital cognitive training program in planning, while group 2 received a personalized way. A sequence of 6 scenarios with different activities such as routing bus routes up towers of Hanoi game type was applied, based on Pediatric Neuropsychological Evaluations (PNE). The

figure 4 shows a scene of the Video Game Bus station, where the child had to identify the best route to take each of the passengers in the shortest possible time, according to a map showing the routes and stops. All scenarios were developed for interactive cognitive training executive function of planning adopting the conceptual frameworks of Miyake et al. [30], The scenarios were tested by children aged 5 and 6 years (x = 5.31 and Sd = 0.48). The selection of children was not probabilistic and had the consent of the directives of the Lyceum of the Catholic University of Colombia, as well as parents and guardians.

Fig. 4 Bus Station Game

shows that there is always a positive correlation but low correlation (0.052 < r < 0.173) , which remains to analyze the correlation between the first environment is observed and recent environments (Bus_Station_Game , Movers_Game and Library_Hanoi_Game) the correlation is increased as more training is done , so much so that when analyzing the last two practice settings is a high correlation is observed over 0.7 . This suggests that there is a better more practical result , showing the Hebb learning rule on learning and cellular neurophysiological relationship.

V. RESULTS AND DISCUSSION According to the application of each environment, we sought to identify the correlation between the results obtained in the number of successes , attempts and average spent. Table I shows the results of bivariate correlation analysis, using the Pearson coefficient , used in place of the variance because it does not require the specification of a measurement scale. However, in the implementation of the first environments (Pyramid_Game, Backpack_Game and Catapult_Game) clearly

TABLE I CORRELATION BETWEEN ENVIRONMENTS INTERACTIVE

Pyramid_Game

Pyramid_Game

Backpack_Game

Catapult_Game

1

0,168

0,052

Backpack_Game

0,168

1

0,173

Catapult_Game

0,052

0,173

1

Bus_Station_Game

0,142

0,128

0,026

Movers_Game

0,192

0,134

0,168

0,18

0,192

0,119

Bus_Station_Game

Movers_Game

Library_Hanoi_Game

Pyramid_Game

0,142

0,192

0,18

Backpack_Game

0,128

0,134

0,192

Catapult_Game

0,026

0,168

0,119

1

0,459

0,508

Movers_Game

0,459

1

0,732

Library_Hanoi_Game

0,508

0,732

1

Library_Hanoi_Game

Bus_Station_Game

308

Clearly a relationship between the proposed and learning Hebbian model is observed, however you need to compare it with the personalized training you received the control group , for which the test of the Tower of London (TOL), test was used highly used for validation of the executive function of planning.

Figure 5 shows the relationship of the distribution of data against a normal distribution, as shown Training is closer to a normal distribution, compared to personal training, which can identify the degree of fairness of system proposed, regarding training led by a teacher, which may be subjective variables that affect student learning.

Fig. 5 Value of data against a normal distribution

Figure 6 shows the occurrence and variation of each of the tested relative to the workout process as a whole environment. The Bus_Station_Game environment is posing a greater causality in the workout process for the development of executive function in the plan-ning in children, according to the sequence applied from the diagram of causality established teacher in the moment of defining the strategy for achieving the learning objective sought.

VI. CONCLUSIONS AND FUTURE WORK Using an interactive environment as a teaching tool, plus motivational purposes, allowing the collection of many information. The teacher may use information collected to identify the level of progress in achieving the objectives of student learning, providing feedback in the educational process. The teacher can use learning assessment system to identify the influence and usefulness of a training exercise by the formalization of relations of dependence through the construction of cognitive map associated with teaching strategy that has been proposed. This will allow the teacher to use his time on the design of an instruction, validating constantly their effectiveness and at the same time allow that a good teaching strategy using interactive environments can be replicated by other teachers. One of the biggest difficulties is the development of the interactive environments, because most teachers do not know the technology for development, another difficulty is the generalization of association between a designed interactive environment and the purpose for which it was built, however, this will allow the discussion between teachers on convenience use and improving the quality of educational tools.

Fig. 6 Results of the environment applications

309

[17] F. Baker and S. Kim, "Item Response Theory: Parameter

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