In the abstract, a network is a collection of nodes, together with a collection of links between them. The links are all of the same type reflecting a single social relation. For example, we might look at the communication network among all employees in an organization. Each link between any pair of persons A and B means the same thing: A and B communicate. Or you can look at the friendship network. Or the conflict network (in which if A and B are tied, it means they have had a conflict with each other). Any social relation between pairs of people form a network -- from who goes to lunch with whom to who is having an affair with whom.
A path is an alternating sequence of nodes and links, starting and ending with a node. For example, A and B might not be directly connected, but there might be a path that links them: A---B---C---D.
The length of a path is defined as the number of links in it. The length of the path from A to D above is 3, because there 3 links between them. The path from A to B has length 1.
The shortest path between any two points is called a geodesic. The distance between any pair of nodes is defined as the length of a geodesic from one to the other. In other words, the distance is the number of links in the shortest path between the nodes.
If there exists a path of any length that connects a pair of points, they are said to be reachable from each other. A maximal subset of nodes that are mutually reachable is called a component.
A link between nodes that would separate the networks into different components is called a bridge. In other words, a bridge is a link which ties together parts fo the network that otherwise would not be connected at all. A local bridge is a link between nodes A and B which, if removed, would mean that the shortest path linking A and B would be of at least length 3.
The density of a network is defined as the number of ties present divided by the number of ties possible. It is the proportion of possible ties that actually exist.
Cliques are maximal subsets of nodes that are completely connected: all members connected with all others.
The speed with which information travels through a communication network from one node to another is a function of the number of links in the paths linking them. Denser networks have shorter paths, so they transmit information more quickly.
The shape of the network is also important: diffuse networks with little structure diffuse information more quickly than others. Networks broken up into subgroups diffuse information quickly within groups, but have trouble getting information moving between groups.
According to Granovetter, weak ties are particularly important in diffusion. Studies show that most jobs are obtained through network connections, and that among those getting a job via connections, the vast majority get them through weak ties. The reason is that strong ties create dense little knots of people who share and reshare the same old information. Novel information comes in from connections with people outside one's clique. Connections with people outside one's clique (local bridges) are rarely strong ties. Hence, weak ties are especially important for network diffusion.
At the individual level, the number of ties that a person has affects how quickly (and whether) information reaches them. The more ties, the more chances of hearing about something. (Note that this is not always a good thing: if what is flowing through a network is a disease, the more ties a person has the more chance of catching the disease.)
Closeness provides a good index for how quickly, on average, a person in a network will hear information: it is an estimate of time-until-arrival of information. Closeness is proportional to the average distance from the person to all others via the fastest possible routes. The smaller the number, the more central the person.
Closeness is also important for predicting the speed with which different network structures can solve problems. See the Bavelas & Leavitt experiments in the text.
Another measure of a person's centrality is betweenness. A person's betweenness is basically the number of geodesic paths that go through that person, divided by the number of geodesic paths in total. It can be seen as a measure of brokerage.
Influence refers to the extent that people directly and indirectly affect others. Suppose we collect data on who seeks work-related advice from whom. If B seeks advice from A, we can speculate that A is in a position to influence B. If B in turn influences C, then A has a direct influence on B and an indirect influence on C. This can get quite complicated: As A influences B and B influences C and C influences D and D influences B who also influences E who influences A, the influence system can become nonlinear. Furthermore, we can assume that influence declines with distance, so that A's influence on B may be quite strong, and B's influence on C may be strong, but A's influence on C is not so strong. To calculate A's total influence on the network, we have to calculate all these possible paths and strengths.