Community structure refers to the local structures that form within social networks as some agents interact more frequently and intensely forming specific substructures within the overall network. The basic constituents of a social graph are nodes and edges, nodes are people or groups of people, edges also called ties, represent the relationships between these social actors, which can come in many different kind, such as friendship, kinship, colleague etc. These edges may be weighted meaning that we can ascribe some quantitative value to them, such as the amount of time one person spends talking to another, we can also ascribe positive and negative values to this weight to depict positive and negative relations, such as trust or lack of trust, loans, and debts etc. These edges can also be directed giving us an idea to which direction the resource being exchanged is flowing, with this net flow being depicted by an arrow, here we can have undirected relations that go only in one direction, such as the influence that a celebrity might have over others without this influence being reciprocated, or it may be a bidirectional relation like a typical friendship with each influencing each other.
Centrality & Influence
A primary question we are often interested in when looking at the individual agents within a network is not to do with their properties in isolation, but instead asking how influential are they within that network based upon their connections, this measurement of how influential or powerful an agent is within a given network is called centrality. Almost all sociologists would agree that power and influence are fundamental properties of social structures. Network thinking has contributed a number of important insights about social power and influence. Perhaps most importantly, the network approach emphasizes that power is inherently relational. An individual does not have power in the abstract, they have power because they can dominate others — an ego’s power is an alter’s dependence and this metric of centrality is a primary tool for helping us in modelling how the social structure of relations give agents influence and power. Social network analysis has made important contributions in providing precise definitions and concrete measures to this idea of power and influence based upon an agent’s position within a social structure of relations.
Because a network can be considered a description of the paths along which something flows, the significance of any agent to that network can be understood in terms of how much of the network’s resources flow through that node and how critical is it to that flow as both of these factors will give them the capacity to influence whatever resource is flowing and it is from this that they get their influence within the network. Whereas influence and power are well defined within a hierarchical social structure, networks are not so orderly, influence is often context dependent and of course we should remember that being central within a network is not always a good thing it works both ways, centrality measures are really just telling us how embed an agent is within that social network.
Network analysis often describes the way that an actor is embedded in a relational network as both imposing constraints on the actor, and offering the actor opportunities. Actors that face fewer constraints, and have more opportunities than others are said to be in more favorable structural position. Having a favored position means that an actor may extract better bargains in exchanges, have greater influence, and that the actor will be a focus for deference and attention from those in less favored positions. But, what do we mean by “having a favored position” and having “more opportunities” or “fewer constraints?” There are no single correct and final answers to these difficult questions. Trying to capture how influential an agent is within a network is not trivial, it is quite complex in reality and thus researchers use a number of different metrics, including, degree centrality, closeness centrality, betweenness centrality, and prestige centrality.
Actors who have more ties to other actors may be in an advantaged position. Because they have many ties, they may have alternative ways to satisfy needs, and hence are less dependent on other individuals. Because they have many ties, they may have access to, and be able to call on more of the resources of the network as a whole. Because they have many ties, they are often third-parties and deal makers in exchanges among others, and are able to benefit from this brokerage. And thus the primary measure to the significance of any social actor within a network is his or her degree of connectivity, which is simply how many connections they have and the weight of those connections if relevant. This tells us the likelihood of a node contacting or being able to effect in some way whatever is being exchanged within their immediate network, it tells us something about their embeddedness within that network, thus a higher degree of connectivity may be a positive or negative thing depending on what is spreading within the network. A node with a high degree of connectivity is termed a hub. But this simple degree of connectivity measurement is a very blunt way of interpreting a node’s significance that can often be misleading, we will need a number of other metrics to support it.
Closeness centrality is another metric for interpreting a node’s significance, one that looks at how far it is to any other node in the network, as distance is assumed to be a restriction on transmission, thus whichever agent is closest to all others can have the greatest capacity to affect them. Betweenness centrality is a third metric quantifying how often a node acts as a bridge along the shortest path between any other nodes in the network, this gives the agent influence in that it is playing a role to reduce the distance between any two nodes, thus significantly helping to hold the network together by reducing transaction costs, institutions that work as market makers within the financial system are a good example of this, they are working as critical bridges between agents and organizations, holding the network together and thus they can demand significant transaction fees.
This is also called occupying a structural whole, meaning that the agent who is working as a link between two clusters, is filling some gap within the network that is critical in maintaining its overall integration, this actor is bridging two communities and may play a critical role in transferring information or some other valued resource, for example, they may be transferring information between two scientific domains or playing a critical role as mediator during periods of conflict between two clustered communities. Lastly, prestige centrality which is really looking at how connected the nodes that you are connected to are. These prestige metrics such as eigenvector centrality, assign relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes. so your centrality and influence is greater if the people you are connected to are well connected. The assumption is that each node’s centrality is the sum of the centrality values of the nodes that it is connected to.
The local connections that agents make are called interpersonal ties. Making connections typically costs something in terms of resources, laying cables to transport information costs money, making new friends or developing a diplomatic relation with another country takes time and some effort. Added to this we can recognize that making connections between different components typically requires more resources than making those same connections between similar component, whether we are talking about connections between computers with different operating systems, trade between countries with different import procedures or communications between different cultures. The fact that it requires less resource to make connections between components with similar attributes is a key factor in the makeup of many networks and particularly so with social networks.
It in many ways defines the difference between strong and weak ties that describe the intensity of interpersonal ties between actors. A strong tie is between two agents that interact frequently and typically share similar attributes, thus they are connections that are typically easier for us to enact. Inversely a weak tie connects people to different social circles, they can be more challenging in that they require the agent to overcome some difference between groups but they also expose the person to novel phenomena and information. The “strength” of an interpersonal tie is a product of many different factors, it may be a combination of the amount of time, the emotional intensity, the intimacy, or some other reciprocal service that is exchanged in that relation, the greater the exchange the stronger the tie.
Most of the time, most people interact through strong ties with a fairly small subset of others, many of whom know one another and this creates a distinct substructure within the network, what we call a cluster and this clustering pattern is an almost universal feature to social networks. The clustering coefficient of a graph is a measure of the degree to which nodes in a graph tend to cluster together, social clustering can be understood by simply asking how many of the people that someone is connected to are also connected to each other. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterized by a relatively high density of ties and clustering. These closely knit clustered communities can maintain their diversity in the face of homogeneity within the larger network. The extent to which these subpopulations are open or closed may be a telling dimension of social structure. With too many strong ties we can get strong clustering and a network that tends to be fragmented into local communities, these clumpy networks will have longer distances relative to other networks with the same density and these clusters slow the even flow across the network.
Weak ties in contrast to strong ties, connect people to different social circles, as such, they are bridging ties that expose people to new information and novel phenomena. Specifically, more novel information flows to individuals through weak rather than strong ties. Because our close friends tend to move in the same circles that we do, the information they receive overlaps considerably with what we already know. Acquaintances, by contrast, know people that we do not, and thus receive more novel information.
When we combine both strong links within clusters and these weak bridging links we get an effective network for spreading information even though it may have high clustering, this type of social graph that has both high clustering and some random bridging links, giving it a low average path length is called a small world network. These characteristics result in networks with the unique property of regional specialization and efficient long range information transfer. Social networks are intuitive examples of this small world phenomenon, in which cliques or clusters of friends are strongly interconnected, but also people often have some random acquaintances within other far of groups, by using these weak ties we find that even within very large social networks consisting of many millions or even billions of people any person may be only five or six links away from anyone else within the system, giving us the famous six degrees of separation theory. The “small world” phenomenon seem to have evolved independently in many large networks. Thus we can see how these micro-level interactions of agents choosing to make strong or weak ties can give rise to overall macro-level properties to the network such as its average path length which we can see is important to its overall cohesion.
Simple graphs allow for just one type of connection between nodes but we can also have multiplex graphs that allow us to model a number of different relations between nodes, so in a multiplex graph we would draw two different edges between people to describe how they are say, work colleagues as well as friends. Of course this adds a significant amount of complexity to our model but this gives us a much more realistic representation as social actors are often embedded within a multiplicity of different networks, social, political, cultural, economic and so on. With a multiplex network we can try and capture how these different connections interact and affect each other. This is a much more realistic picture that lies behind many social phenomena and a lot more faithful to one of the basic premise of complexity theory, that is that many phenomena are in fact the product of a multiplicity of nonlinear interacting forces. As a quick example we might think about the recent uprisings in Egypt. When we first look at this phenomenon we would consider it political in nature and start analysing the political network, but research has shown a robust correlation between spikes in the price of basic foods and the occurrence of these riots, thus these events are an emergent phenomena of different interacting networks, social, political and economic all putting stress on the social system. Thus in this situation it would be of use to use a multiplex network to try and model the overall dynamic. Phenomena like this are very complex they are embedded within many different overlapping networks, simply modeling one of these networks can only ever give us a partial insight, this is the nature of complex systems of all kind, they are multi-dimensional.