Clustering Social Networks to Remove Neutral Nodes

Fatemeh Hendijani FardDepartment of Electrical and Computer Engineering

University of CalgaryCalgary, AB, Canadafhendija@ucalgary.ca

Behrouz H. FarDepartment of Electrical and Computer Engineering

University of CalgaryCalgary, AB, Canada

far@ucalgary.ca

Abstract—Multi agent systems with autonomous interaction, negotiation and learning capabilities can efficiently model social behavior of individuals participating in a social network. A central problem in a social network is to identify the nodes that actively participate in the expansion of the net both physically and functionally. Several metrics have already been proposed to identify those hot spots. The algorithms to identify hot spots are either heuristic based or computationally expensive. In this paper we use an agent model of the social net and propose a method that can identify the neutral nodes, i.e. the nodes that can never be considered as hot spot nodes given the network topology and rules of negotiation among nodes. Therefore these nodes can be eliminated from the net. A direct advantage of this method is reducing the computational complexity for the configuration and identification of hot spots. Through a case study we have shown that the proposed method can lead to 33% reduction of computation regarding the number of agent types in the example.

Keywords- Agent-based modeling and analysis; clustering; social networks; neutral agents

I. INTRODUCTION AND RELATED WORKS

Agent based modeling and simulation models systems with many constituent elements interacting with each other using defined interaction protocols [1]. This approach is needed because of increasing grow of size and complexity of systems[1,2]. Agent based modeling is particularly useful when modeling systems without a centralized controller. In this case, one can consider the entities participating in a system as agents.There are usually rules for the behavior of each agent and also rules for negotiating among agents. The agent model can helpinvestigate characteristics of the system, observe the changes,and detect emergent behaviors from the interactions among the elements. By emergent behavior in this paper we mean the unexpected behaviors of the entities which is also mentioned in the studying of emergent behavior in [3].

There has been lots of researches using agent based modeling in different applications areas. A categorization of different applications of agent based modeling, such as economics and financial markets, crowds, transportation, society and social sciences, biology, etc., is given in [10].Modeling systems to find the emergent behaviors in the networks regarding timing mechanisms [4], modeling social networks and web service selections in recommender systems[5], modeling migrations in social networks [6], modeling the spread of diseases [7,8] and mining different measures on groups and teams in social networks [9] are among applications of agent based modeling and analysis.

This modeling in any social network tries to analyze the behavior of the related network or simulates the agents’ behavior in a certain situation. For example, in e-commerce applications, this modeling tries to find the emergent behavior of involved agents and detect frauds, or finds reputations of participating agents and etc. In this analysis the modeled agents play role of each component of social network, includingpeople, physical locations, etc. Therefore agent based modeling consists of a set of agents, their relationship, the interaction rules, network structure, and a framework to simulate the agents’ behaviors [10].

A challenge in agent based simulation is managing the complexity of the model. Since, in many cases the number of agents is large; mechanisms that can potentially reduce thecomputational complexity are appreciated. Therefore, in this paper our focus is on identification of the neutral nodes, i.e., nodes that show similar behavior in their relationship with other nodes in the network. We argue that through identifying and eliminating these nodes the complexity of the network and the algorithms to analyze them will be decreased.

Agent based modeling is used here to model the relationship between entities in social networks. Our focus here is certain kinds of applications like e-commerce applications in which there are certain types of individuals that collaborate with each other in specific roles. These roles can be considered as agent types. Then through an agent modeling we try to investigate the agents that are neutral in relation with othertypes. This can help reduce the complexity of investigating the social networks. On the other hand, this modeling can help to apply the rules defined on the relationship of different roles.Another advantage of this method other than its usage in social networks with specific roles is its application in distributed systems, where different agents work with each other and there is no central control. By considering the agents and their relationship as social networks, the proposed method can help investigating the collaboration of different agents in such a network regarding the rules defined for their relationships.

In the proposed method, each node of a social network is represented by an instance of an agent type with a given set of roles and actions that can be performed by that node. The social context of the network is modeled by collaboration diagrams that show the defined relationship (i.e., rules of interaction) between different agent types. The collaboration diagrams are like UML diagrams that show the relationship between different components of the system. We then apply clustering on the vectors of relationship of each agent type with

2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

978-0-7695-4799-2/12 $26.00 © 2012 IEEE

DOI 10.1109/ASONAM.2012.222

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2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

978-0-7695-4799-2/12 $26.00 © 2012 IEEE

DOI 10.1109/ASONAM.2012.222

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the others and devise an algorithm that detects the neutral agents. The proposed method detects neutral nodes with respect to inter-type communication and suggests eliminating them from later stages of analysis since they don’t show new behavior in existing networks. Considering the roles and the different types of the agents and investigating the inter-type relationships in the social networks is the innovation of this work.

The structure of this paper is as follows. In Section 2 the proposed method is explained in detail. The related definitions and the algorithm are presented in Section 3. A case study that shows how the proposed method helps reduce the complexity of existing algorithms by detecting the similarity in agents’ behaviors is presented in Section 4. Results, discussions and conclusions are given in Sections 5.

II. PROPOSED METHOD

The method explained in this part can be used for social networks of different software agents or in networks in which individuals play different roles. In each case, by using the agent based modeling, we can investigate the inter-type relationship among agents. In case of software agents we mean agents thatare autonomous software entities that interact with each other in an environment. Each agent has its’ own behavioral rules and can be considered in an environment [11]. The agents are also modeled for their interactions and relationships showing how the agents are connected or what are the mechanisms for their communications [11, 12]. These rules that are defined in the agent based modeling and analysis, show the agents’ behaviors or the communication rules between the agents. These rules are referred as user defined rules in this paper.

The proposed method has the following steps:

Step 0: Agent types

In this step different agent types that correspond to different nodes of the social network are identified. Each agent type will be associated with a set of roles, and actions that they can perform in the network. For example, send a message and share are actions available to User agent types and deletemessage is action available to Moderator agent type.

Step 1: User defined rules

For each social network there are defined rules of interactions (or protocols) that govern the social behavior of the nodes. In many networks the rules are given in declarative form. For example, in some networks, friends of friends are friends and in some they are not. These rules define the social context for the network. In this step, the rules are analyzed and converted to the procedural form that are applicable to the different agent types. Each rule will be represented by a collaboration diagram among agent types that representsimplementation of the rule in the network. Apparently, agents of one type can have complex collaborations, although the collaboration of different types of agents may not have that much complexity. The rules defined here help us to illustrate the collaboration diagrams in Step 2.

Step 2: Collaboration diagrams

In this step the collaboration diagrams are illustrated based on the user defined rules. Based on the extracted information, the collaboration diagrams are created which show the possible interaction path between different types of agents and how they can communicate with each other. Definition related to this diagram is given in Definition 1.

The advantage of using collaboration diagrams is that thecomplex social network which shows the relationship between all the agents is reduced to a set of networks which shows the relationship between different types of agents. This helps to reduce the complexity of analyzing the social networks, since the overview of relationships is easier to investigate.

Step 3: Collaboration matrices

In this step, the relationships between different types of agents are transformed to collaboration matrices explained in Definition 2.

Step 4: Clustering the behaviors

After constructing the matrices, which show the relationships of agents, the clustering algorithm is applied to find the behavior of an agent type in different collaboration diagrams.

Step 5: Detect neutral agents

In this step, the results from previous step are analyzed and the agents which are neutral are extracted. The neutral agents show the same behavior in all of the possible collaborations. It means that they almost behave the same with other agent types. Therefore the relations between this agent type and other agent types may not cause the network to change.

The algorithm and definitions are explained in next section.

III. DEFINITIONS AND ALGORITHM

In this section, the related definitions for the proposed method and the algorithm are explained. Definitions for collaboration diagrams, collaboration matrices and vectors related to each agent are as follow:

Definition 1: A collaboration diagram is a directed graph G= (V, E, W) in which the vertices are denoted by V(G) and show the agent types. In a collaboration diagram |V(G)| equals to the total number of agent types. The edges show the relationship between the agents and are denoted by E(G). In graph G there is an edge eij = (vi ,vj) from vi to vj if there is a message sent from vi to vj. W(G) shows the weights of the edges E(G). For each edge eij, w(eij) is determined by the number of messages sent from vertex i to vertex j.

Definition 2: A collaboration matrix M related to a collaboration diagram G is a matrix M = [mij]n×n where nshows the number of vertices in graph G or n = |V(G)|. In matrix M, an entry mij is non-zero if there is a corresponding edge eij = (vi ,vj) in the related graph and the value of mij equals to the weight of that edge, mij = w(eij). Otherwise mij = 0.

Definition 3: A vector vecak related to an agent type Ak in a collaboration diagram Ga shows the whole incoming and outgoing edges of that agent in the related collaboration diagram. This vector can be derived from the related

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collaborated matrix Ma. Therefore, a vector vecak for an agent Ak related to a collaboration matrix Ma is defined as vecak =[rowak , (colak )T] where rowak is the entries of kth row of related matrix and (colak)T is the transpose of the kth column of that matrix. The kth row in matrix M shows the outgoing messages of agent Ak of and the kth column shows the incoming messages from other agents to agent Ak in the related collaboration diagram.

An example of a collaboration diagram composed of 6 agent types is shown in Figure 1.

Figure 1:An example of a collaboration diagram G1 with six agent types

The collaboration matrix for this diagram is shown in matrix M1 as below. By this matrix one can define, for example, agent A2 has sent messages to agents A1, A3, A4, A5,and A6. This is clarified by considering the entries in second row. Also, the incoming edges for agent A2 or the received messages can be defined, too. This is obtained by considering the second column. Therefore, for this agent the messages are received from agents A1, A3, A4, and A6.

For the diagram shown in Figure 1 the vector which shows the relationships of agent A2 in matrix M1 is defined as vec12 =[row12 + (col12 )T]. The vector row12= {2,0,6,2,1,2} and (col12)T= {2,0,2,5,0,7}. Therefore we have vec12 ={2,0,6,2,1,2,2,0,2,5,0,7}. This vector shows the whole relationships of agent A2 in collaboration diagram G1 with other agents (the incoming, outgoing and the number of messages in their communication).

A. Algorithm For Detecting Neutral Agents In this part, the algorithm for the proposed method is

defined. The types of agents are counted and considered as an input to the algorithm. Another input of the algorithm are collaboration diagrams which are extracted based on the user-defined rules or based on the rules that are regulated for the communication between different types of agents. These

diagrams somehow can be extracted from the existing scenarios that show the relationship between the agents too. Definition of collaboration diagrams is explained in Definition 1.

In the first step of the algorithm the collaboration matrices are made from the collaboration diagrams with Definition 2.All the relationships that are shown with a collaboration diagram can be extracted from the collaboration matrices. These matrices, as mentioned before, show all the incoming and outgoing messages from one agent to the other agents and vice-versa.

The next step is the fourth step of the proposed method in which the agent types are clustered into different clusters. Each agent type is clustered based on its relationship in different networks with other agents. For this, all the vectors related to that agent are extracted from the matrices. A vector shows the whole relations of one agent in a collaboration diagram (network). Clustering is applied on the non-zero vectors related to each type of agent. Zero vectors should be omitted because they show that this agent is not participated in a collaboration diagram. In other word, the existing relationships are important for us and therefore we omit the zero vectors (steps 2 to 10 of the algorithm). By this clustering one can understand the changes that exist in the relations of one agent to other agents in different networks. If there is no change in this relationship, then the agent is neutral in communication with other types (steps 11 to 16 of the algorithm). Therefore these neutral agents are not mentioned in the analysis of inter relational behavior of different roles in social networks and can be excluded from the analysis phase.

Input: number of agent types (n), d collaboration diagrams in the form of G= (V, E, W) Output: list of neutral agents 1. Transform collaboration diagrams into d collaboration

matrices Ma = [maij]n×n a 2. For each agent type do the following 3. Make all the related vectors of the d matrices in the

form of vecak = [rowak , (colak )T] where and

4. End for 5. For each of the n agent types do the following 6. If vector vecak equals to zero 7. Eliminate that vector 8. End if 9. Cluster all nonzero vectors related to that agent with

Manhattan Distance 10. End for 11. For each of the n agent types 12. If the vectors remain in the same cluster 13. Print that agent type as a neutral agent 14. Else 15. Print the number of clusters for each agent type and

the cluster numbers for each collaboration diagram 16. End for

IV. CASE STUDY

As a case study, an e-business social network implementing an e-commerce system is considered. This is an open auction systems that individual sellers and buyers can participate and provide goods and/or bid on goods. The auction system is

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modeled with agent based modeling and simulation as explained above. In this system, learning or changing in the behavior can happen [11]. It is assumed that agents are rational,with defined objectives (e.g. maximizing their individual utility) and follow rules of behavior as decided by the auction hosting authority [10]. The agents, such as buyers and sellers,can show new behaviors based on their previous experiences and competition factors which exist for each of them [12]. An auction system consists of multiple agents with different types that communicate with each other autonomously, while most of the agents are of the same type.

Modeling and analysis of such a system is important regarding the fraud detection or other trust and security issues. But the complex social network of the existing relationship between the agents taking part in an online auction make the modeling and analysis of such a network complicated. In this case study it is shown how the proposed method can reduce the complexity related to analyzing such a network and analyze the relationship between different types of the agents before analyzing the whole network.

For explanation simplicity six types of agents are considered here:

Controllers or market place agents (A1): These agentsare responsible for keeping the auction types, the protocol for each auction, different protocols for each auction and other related rules.

Auctioneers (A2): Agents responsible for taking the auction actions. Declaring the protocol of auctions, declaring the sellers, buyers, and items in each auction, taking the time, accepting bids from the buyers, and deciding about the winner of an auction.

Registrars (A3): Agents responsible for the communications between the sellers and buyer with the auctioneer. These agents introduce the sellers and buyers to the auctioneers, register new users, and take the log in functions. Also the search for each item or auction type is passed to these agents.

Sellers (A4): Agents which have some items and decide to sell their items in each type of auction. Sellers should register first to the registrar auction for selling the items. These agents declare the items and their first price to the registrars.

Buyers (A5): These agents look for items to buy and bid for each item they want. Buyers should register first and then log in each time they want to participate in an auction.

Trust managers (A6): These agents assess and represent trust criteria for other agents taking part in online auctions. The trust managers help the other agents in decision making.

The relationship between these agent types is defined with scenarios that indicate the rules of behavior for participant agents. For example, for registering a new buyer into an auction system, a message is sent from the buyer to the registrar agent. The request contains the buyer’s information. A

few messages are communicated between the two agents e.g. security question, and user name and password, and after a few messages the buyer is registered. The same procedure is followed when a new seller wants to register with the system. In either case, the information of auctions is passed from the auctioneer to registrar and shown on the website. This process is named scenario 1 for registering a new user into the auction system.

A few of these scenarios are named in the below list:

Scenario 1: As explained above, registering a new user (buyer or seller agent) into the auction system is done in this scenario.

Scenario 2: In this scenario the registered users (either sellers or buyers) try to sign in based on the information declared on the website about the auctions.

Scenario 3: The registrar agent registers seller and buyer agents into a certain auction. The registrar agent then introduces them to the auctioneer agent.

Scenario 4: The registered users in a certain type of auction take part in this scenario. The actions related to performing an auction e.g. bidding, declaring the winner, and introducing the sold items information are done in this scenario. The trust manager agent also participates and has an important role here. The protocols and criteria found and described to calculate the amount of trust to other agents are executed by the trust manager agents. The messages between different seller and buyer agents to the trust manager agents are defined in this scenario.

Scenario 5: In this scenario the rules on how the buyer agents can search for items are defined.

Scenario 6: In an auction system seller agents can search for certain types of auctions and get the related information and protocols of each auction. The related communication rules are defined in this scenario.

Scenario 7: In this scenario, the seller agents can declare the items they want to sell, prices of each of them, and the type of auction to the registrar. The registrar agent puts this information on the website.

This set of scenarios is open and expandable. In real world, there are more scenarios that can be defined for the communications between different types of agents. An advantage of the method proposed in this paper is that it allows seamless inclusion and integration of new scenarios and new agent types.

For better understanding the communications between agent types and the rules on the order of messages, sequence diagrams, as shown in Figure 2, can be used. In this figure each vertical line represents an agent instance. The communications are presented in the form of messages and shown with arrows from one agent to another. The message labels are written above the arrows. The order of messages communicated between the agents is shown with order of arrows in the vertical line. It means that the first messages are upper than the other ones or the time line starts from top of the vertical lines

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and ends at the bottom. Figure 2 shows the communications of agents related to Scenario 2.

Figure 2: users sign in based on the auctions information on the website

The collaboration diagram of this scenario is shown in Figure 3. This diagram is produced using the Definition 1.

Figure 3: Collaboration diagram related to scenario 2

The collaboration matrix made from the collaboration diagram of Scenario 2 is shown below in matrix M2.

Some of the collaboration diagrams for other scenarios are shown in Figure 4 and Figure 5 that together represent some possible interaction networks for this system. Note that there are more networks of the relationship between these agents but just a few of them are shown here.

Figure 4: Collaboration diagram of scenario 4

Figure 5: Collaboration diagram of scenario 7

Applying the algorithm, we can conclude that the controllerand trust manager agents (agents 1 and 6) remain in the same cluster. It means the communication of these two agents with other agents in the given networks will show the same behavior. For controller agent (A1) the vectors in the seven scenarios are extracted from the collaboration matrices of the diagrams by Definition 3. From this definition, the vectors related to this agent in all networks are in this format: veca2 ={0,2,0,0,0,0,0,2,0,0,0,0} where . The results of clustering the related vectors show that the seven vectors are in the same cluster.

For trust manager agent (A6) again we have veca6 ={0,5,0,4,4,0,0,3,0,2,3,0} for a=3,4 and veca6={0,0,0,0,0,0,0,0,0,0,0,0} for a= 1,2,5,6,7. By following the algorithm, the non-zero vectors should be considered. Therefore again the results show that this agent remains in the same cluster and it shows the same behavior in all the networks it has communication with others.

The results of proposed method in this case study shows two types of agents behave similarly in collaboration with other agent types. For example, the trust manager agent’s communications with other five agents is almost similar in the diagrams that it collaborates in. In other words, it shows a neutral behavior in the networks and in communication with the other agents. Therefore, it is unlikely that this agent changes the model of network in future and show a new behavior. The reduction of two agent types means approximately 33% decrease of the size of different types in the network for further analysis. Note that this conclusion is dynamically evolvable when new agent types and scenarios are added to the system.

The reduction in size of the types of nodes in the social network depends on the number of roles or types and the relationship between different types and can be more or less in various networks. The key point here is considering the roles and the different types of the agents and investigating the inter-type relationships in the social networks. By considering the

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types of agents the investigation of their inter-type relationship becomes less complex.

Also it should be mentioned that the results of clustering can be helpful in next steps of analysis and modeling of the social networks. For example, in this case study the result of clustering shows same clusters for some agents such as agents 2, 4 and 5 (Auctioneers, Sellers and Buyers) in some of the participating networks. For example, the sellers are in the same cluster in collaboration diagrams of scenarios 1 and 7. Results of such clustering can be helpful in analysis of networks since it shows in which possible networks, the behavior of one type of agent is similar and they can be treated almost the same in the modeling and analysis of social networks. Also it is found that the agents that have more number of clusters are more possible to show an emergent behavior in the network, since they have a new behavior in each possible network.

V. DISCUSSION AND CONCLUSION

There are numerous communications in the social networks and a large number of them are the communications between instances of one agent type, while others are the communications between different types of agents. Althoughintra-type communications are possible in a generic social network, they are governed by rules of interaction in the network. For example, in the auction system, communication between individual buyers or sellers may be prohibited.

The method proposed in this paper detects neutral nodes in a social network with respect to inter-type communication and suggests eliminating them from later stages modeling and analysis since they don’t show new behavior in networks. Therefore, they are unlikely to change the topology of network or cause the whole network to change over time. In the analysis of social networks these agents are not a point of attention. Therefore, by finding the neutral agents, we can have more focus on other agents and also reduce the complexity or time and cost of analyzing the networks. This reduction depends on the relationships and number of agent types in each network. The key point here is investigating the inter-type communications separately and find the neutral agent types in relation with other agent types.

As for future works, we are working to improve the algorithm by clustering the agents with the messages in their

collaborations. This kind of clustering can be helpful in modeling and analysis of social networks by their semantics.

Acknowledgements This research was partially supported by a grant from

Natural Sciences and Engineering Research Council of Canada (NSERC).

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