THE EFFECT OF NETWORK TYPES ON HERD BEHAVIOR
Authors: Bichia Quji 
Published: August 2020

This paper aims to compare spread of an opinion, norm, innovation or a belief in different types of networks. For this purpose, different network metrics are discussed and results of network model are summarized based on simulations. Norms may spread from a single source or multiple sources and these issues require separate analysis.
Networks play an important role in decisions that people make. They determine what information someone will receive and how will he act within this limited information. As it turns out, small number of people can influence decisions of majority. These can be consumption decision, decisions about adopting new technologies, innovations, medical practice, social norms and so on. Mathematical models of net- works help us understand how these processes propagate. There are different types of networks that can emerge within a society or some group and there are characteristics that can describe roles of group members in spreading some idea or innovation.

Networks can be of many kinds but human networks tend to have common characteristics. Therefore, current work focuses on 4 types of networks - small world, single-hub (one central figure), multi-hub (many central figures) and two-component. Small world random networks are observed in different situations and they can be used to describe some human interaction networks. Many networks are described by power law distributions, where new members of a net- work have a preferential attachment and link to other highly connected members. Single-hub and multi-hub networks describe such situations. Two-component network is used to describe polarized groups that have opposing views and are competing with each other. This could be political parties or competing firms.

The present paper analyzes patterns of information flow across different types of networks and compares the conditions for the emergence of group behavior. Contribution of this work is the simulation results that show how different networks exhibit varying outcomes and propagate opinions differently.
Simulations on small world, single-hub, multi-hub and two-component networks with 150 members show that net- work types matter in terms of how fast can group behavior spread within a network. The process of spreading group behavior is as follows: Every individual receives some signal si about a binary decision. Individuals make the first decision based on their signals because they have no other information. In the next step, every individual looks at the decisions of those in his or her neighborhood and updates his or her belief by the Bayes rule. On the next step they observe others’ actions again and decide whether to change own action or not and so on. After some stages, a stable point is reached where no one is willing to change his decision anymore. The study compares the times needed to reach stability in different types of networks.

Simulations have shown that the speed of propagation of a belief varies according to who is the source of this process. However, the difference is not big within a small world network. As it turns out, full distribution occurs in at least 4 and a maximum of 20 periods, and the average time of full distribution varies from 6.5 to 8.6, depending on whether the most connected member is the source or the least connected one. The result is quite different if there is one central figure. The presence of one central figure prevents information from spreading across the network, as there is preferential attach- ment and some members can only acquire one connection. If there are several central figures, the full spread occurs relatively faster. In a two-component network, full adoption oc- curs quite rapidly. Although the connection between components is almost non-existent, a small number of existing links play a critical role in rapidly disseminating a behavior.
Group behavior spreads more rapidly in a random net- work than in a network characteristic of a special society on average. But multi-hub network has the potential for the fast- est spread (although information disseminates faster in a ran- dom network on average). Group behavior is slow to spread in a single-hub network, as some individuals are very weakly connected to other areas of the network. An opinion spread in the neighbourhood of the central figure will soon reach all members of around him or her but it will take a long time to reach far ends of the network. The two-component network in this regard maintains a balance between the speed of dis- tribution and the area of distribution. There is least variation between adoption times in a two-component network (not considering the small-world random network). The high variation in single-hub and multi-hub networks indicates that it is advisable to consider more specific situations for accurate results.
Comparison of adoption times within multi-hub net- works of different size shows that adoption happens at the same speed most of the time regardless of the network size. When two opposing opinions are being spread and one of the opinions is dominated by the other, it takes similar time periods for all sizes of multi-hub networks.