# Transportation Geography and Network Science/Social networks and travel behavior

**Social Network Models**

**Introduction**

Social networks are an integral part of a functioning society and economy. A social network can be thought of as a group of individuals as well as the connections between those individuals. They spread a wide variety of information and services. For example, an individual searching for a job will search through formal postings, but will also likely let his network of friends and associates know that he is searching for a job and in turn they may let him know of positions they are aware of. In addition, an employer searching for a candidate for an open position may ask other employees if they know of anyone who could be a good fit for the position. Alternately invitations to social gatherings and professional events often spread via word of mouth from an original formal notice to a select group of individuals. Due to the extent and impacts of social networks, several attempts have been made to generate models of social networks and their structure. The following is a brief overview of the types of problems that are typically considered in the context of social networks, as well as the theory utilized in the formation of social networks.

**Common Queries**

The majority of studies involving social networks can be categorized into one of four potential goals or queries. The first involves the ability to detect communities or groups within larger social networks. The second is the determination of the most influential individuals in a social network for marketing purposes. The third involves the use of incentives in social networks to improve the likelihood of receiving an answer to a posed question. The fourth regards the formation and development of social networks.

*Community Detection*

Communities are subgroups of larger social networks in which individuals are generally densely connected. An example would be the drama club in a high school. Members of the drama club would likely spend a considerable amount of time with one another, and as such it could be anticipated that any individual in the drama club would directly be connected with most if not all of the other members of the drama club. A second community in the high school might be the football team. Again, members of the football team would likely be directly connected to most or all of the other member of the team. However, it could be anticipated, that their would be very few direct connections between the drama club and the football team. This illustrates the other distinction of communities: there tend to be relatively few connections between communities in comparison to within communities. According to Narahari and Narayanam, the detection of communities or subgroups in a social network can aid in the understanding of the larger network^{[1]}.

*Discovering Influential Individuals*

Imagine you have a new product, and you would like to infiltrate as much of the market as you can. In order to do so, you must first make people aware of your product. One method could be to utilize the so called 'word of mouth' of existing social networks. Basically, you give information or potentially free samples to a few individuals. They then recommend the product to their connections, who may or may not accept their influence. If so, the recommendations continue through the network. For each set of initial individuals, a subset of the network will utilize the product. However, if you wish to maximize your profit, you need to find the most influential set of initial individuals to give your free samples to. This set of individuals will result in the largest subset of the social network utilizing your product. Kempe Kleinberg and Tardos suggest the use of a greedy heuristic and a general threshold and cascade model of diffusion in order to estimate the k most influential individuals in a social network^{[2]}.

*Query Incentive Networks*

Kleinberg and Raghavan initially discussed this problem in the context of internet protocol. The idea concerns the decentralized request for either information goods or services. By decentralized, it is meant that the originator of the request does not pose their question to a central index such as Google or a library catalog. Instead the request is forwarded to a group of peers, with the expectation that they will forward the request through their social networks, and report back an answer when it is found. However, it is theorized that there is an effort associated with the forwarding of a request and reporting back an answer. In order to get past this, the request must be sent with an associated reward for a returned answer or service. In addition, each individual includeS in the answer chain, must receive a portion of the reward for their efforts, or they are unlikely to make the effort. Klienberg and Raghavan, suggest that this process occurs in three steps: the propagation of the request, indication to the requester when answers are found, and the selection of an answerer and the report of the answer back to the requester. It is argued that most individuals associate minimal effort with the first two stages, and thus will likely only require compensation if their effort is required for the third step. They suggest that at each level of propagation the reward is reduced by the propagators desired cut in the event an answer is found. As such, at some point the request will not continue to be propagated, due to a zero remaining value of the reward offer for 'child' nodes. So, if a reward or incentive is not large enough, an answer may not be reached. It is therefore desirable to determine the reward level required to produce a certain probability of achieving an answer. Raghavan and Kleinberg utilized game theory in their modeling of this process. Basically each individual is expected to strategically attempt to maximize their payoff, and the network is modeled as an infinite tree with a branching parameter, in which any given node has a probability of having the answer. This formulation allows for a unique Nash equilibrium^{[3]}.

*Social Network Formation*

Social networks are generally formed via the interactions of a large number of individuals with their own needs and desires. Networks resulting from such interactions are generally considered to be equilibrium networks; however they are not necessarily efficient networks. Efficient networks are generally thought of as centrally enforced networks. These networks are also frequently referred to as star networks and are characterized as having a central node i that every other node is connected to. In other words, every link in the social network has node i at one end.^{[4]} In contrast, equilibrium networks are highly variable and dependent on assumptions about individual behavior as well as mechanisms for the formation and severance of links and the manner in which links are valued that the individual agents are autonomous, self-interested and strategic interacted. For example some authors assume that the number of links between individuals impacts the value of the link and any potential information that could be shared between the two nodes. Watts explores the results of different assumptions about the values and costs of indirect and direct connections on the types of networks that a network formation process has as its equilibrium. She concludes that efficient networks, star networks, are unlikely to be the result of formation process in which the value of an indirect link is assumed to be greater than the net value of a direct link in cases where there are four or more individuals in the network, with decreasing probability as the number of individuals in the network increases^{[5]}. Richards on the other hand explores a model parameterization of social networks based on levels of leadership, bonding, and diversity which he compares to small world model analysis, random graphs, and scale free graphs^{[6]}.

**Theoretical Frameworks**

According to Narahari and Narayanam, social network analysis is based in techniques developed mainly in four other fields: graph theory, spectral theory, optimization theory, and game theory^{[7]}.

*Graph Theory*

In general, a graph consists of a number of nodes (or vertices) which are connected via links (or edges). In the context of social network models, individuals are considered to be nodes and they are connected to other individuals via links. Three graph constructions are used to model social networks: Erdos-Renyi random graphs, small world constructions, and scale-free graphs^{[8]}. Erdos-Renyi random graphs are randomly generated with each link between any given pair of nodes having equal probability, and have properties similar to local road networks. Small world graphs on the other hand tend to have high levels of clustering with any two nodes linked via a small chain of nodes^{[9]}. Scale free graphs on the other hand tend to exhibit power-law distributions, wherein a few individuals have many connections and most individuals have few connections. Brown et. al. argue that real world social networks as a general rule exhibit both power-law distributions, dense clustering, and short chains of connections between any two individuals. For example, consider the well-known study by Stanley Milgram, in which he finds that on average the shortest path between any two individuals had a chain of six links^{[10]}.

*Spectral Theory*

Spectral theory is a mathematical theory concerning the simplifiction of large operators, such as integrals for example, into summations of simpler operators. These processes are particularly helpful in the analysis of matrices and their properties^{[11]}. In regards to social networks, there is a specific subset of spectral theory which uses matrix representation of graphs, such as those used in the definition of social networks. Spectral graph theory, allows for the analysis of graphs and the properties of graphs based on the matrix representations of those graphs^{[12]}.

*Optimization Theory*

Optimization theory basically works to either maximize or minimize objective functions via both linear and nonlinear programming techniques. Frequently, as the case with social network modeling and analysis, realistic optimization problems do not have closed form solutions and are to large to solve in a straightforward or brute force computational manner. As such, a large set of heuristics have been developed in order to find near optimal solutions. One such heuristic the greedy heuristic, is utilized by Kempe, Klienberg, and Tardos in their paper discussing the identification of the set of the k most influential individuals^{[13]}.

*Game Theory*

Narahari and Narayanam argue that game theory is fundamental to accurate understanding and modeling of social networks, because the three theories mentioned thus far cannot account for the strategic behavior and decisions of individuals in the social network^{[14]}. Game theory, is a mathematical theory focused on the behavior of a set of players who are assumed to be rational and independent of each other. As such, it can be very useful in modeling the strategic decisions making of the individuals in social networks, whether they are deciding to form a new link, sever an old link, forward a query, or accept the suggestions made by their peers.

**Recommended Additional Readings**

Alison Watts. A Dynamic Model of Network Formation. In Games and Economic Behavior, 2001.

D. Kempe, J. Kleinberg, and E. Tardos. Maximizing the spread of influence through a social network. In ACM SIGKDD, 2003.

Matthew O. Jackson and Asher Wolinsky. A Strategic Model of Social and Economic Networks. In Journal of Ecomonic Theory, 1996.

Prabhakar Raghavan and Jon Kleinberg. Query Incentive Networks. In IEEE FOCS, 2005.

William Richards. Social Network Models. In IEEE ICPSRT and IEEE ICSC, 2011.

**References**

- ↑ Y. Narahari and Ramasuri Narayanam. Tutorial: Game Theoretic Models for Social Network Analysis. In ACM, 2011.
- ↑ D. Kempe, J. Kleinberg, and E. Tardos. Maximizing the spread of influence through a social network. In ACM SIGKDD, 2003.
- ↑ Prabhakar Raghavan and Jon Kleinberg. Query Incentive Networks. In IEEE FOCS, 2005.
- ↑ Matthew O. Jackson and Asher Wolinsky. A Strategic Model of Social and Economic Networks. In Journal of Economic Theory, 1996.
- ↑ Alison Watts. A Dynamic Model of Network Formation. In Games and Economic Behavior, 2001.
- ↑ William Richards. Social Network Models. In IEEE ICPSRT and IEEE ICSC, 2011.
- ↑ Y. Narahari and Ramasuri Narayanam. Tutorial: Game Theoretic Models for Social Network Analysis. In ACM, 2011.
- ↑ William Richards. Social Network Models. In IEEE ICPSRT and IEEE ICSC, 2011.
- ↑ Chloe Brown, Anastasious Noulas, Cecilia Mascolo, and Vincent Blondel. A place-focused model for social networks in cities. 2013.
- ↑ Chloe Brown, Anastasious Noulas, Cecilia Mascolo, and Vincent Blondel. A place-focused model for social networks in cities. 2013.
- ↑ "Spectral Theory." Wikipedia. Wikimedia Foundation, n.d. Web. 20 Apr. 2015.
- ↑ Edo Liberty. A Very Short Intro to Spectral Graph Theory. Lecture notes. <http://www.cs.yale.edu/homes/el327/datamining2012aFiles/12_intro_to_spectral_graph_theory.pdf>
- ↑ D. Kempe, J. Kleinberg, and E. Tardos. Maximizing the spread of influence through a social network. In ACM SIGKDD, 2003.
- ↑ Y. Narahari and Ramasuri Narayanam. Tutorial: Game Theoretic Models for Social Network Analysis. In ACM, 2011.