Paths/Walks/Cycles

A network is a set of connected actors, which are displayed with points and lines on a graph. These points and lines may form different configurations of relations between each other, which are analyzed through the relation-based concepts of walks, paths, and cycles. All these concepts correspond with particular options of connecting points with lines and thus display direct and indirect linkages among the actors.

Two points connected with a line have direct connection, although there may be other points indirectly connected with them with several lines. This sequence of lines is called a walk. It characterizes indirect connection between two points, A and B, where lines and other points in between may repeat on the way from A to B or vice versa. The length of a walk is a sum of all lines it contains. For instance, the walk A–B–C–D–B includes lines A–B, B–C, C–D, and D–B; thus, if each line is 1, then the length of A–B–C–D–B is four. Although points A and B are also connected directly with a line at a length one, they are also linked with the walk A–B–C–D–B, which is another option of indirect connection.

A path is a walk that connects two network actors with distinct points and lines. In other words, it satisfies the rule that a flow avoids passing one junction more than once. This concept sometimes might be more efficient than a walk and thus appears to be one of the basics in network analysis; a range of theoretical measures and indicators are calculated using the concept of path as described below. An overall number of existing paths show the extent to which actors are connected into a single network. Length of the path is calculated according to a similar principle as that of the walk. Thus, the length of the path A–B–C–D, from the point A to D, will be three, counting as a total number of lines A–B, B–C, and C–D at a length of one each. In order to define whether two persons are reachable based on the path between them, the direction of lines can be either ignored (as in an undirected network) or it can be obeyed. In the second case, a path should contain reciprocated ties and paths (those going in both directions, from A to D and from D to A). Each path is a walk but not vice versa.

The concept of a path allows defining to what extent the directed network is connected, which refers to the concept of social cohesion. In a weakly connected network, all linked points can reach each other using paths but neglecting the directionality of ties. A network is considered to be strongly connected if its actors can reach their counterparts following paths of particular direction.

These procedures for defining connected and dense parts in the network with the high concentration of ties leads to the concept of a network component—a [Page 696]particular cohesive group of actors, even if the network is not connected into a single set. Hence, the subnetwork is a component when it contains a maximal number of connected points and when any additional point as well as exclusion of any point within it would destroy its connectedness. This principle is used to define weak components in the network. Moreover, according to the principle mentioned above, the subnetwork is considered to be a strong component when its points are strongly connected. In an undirected network, the one type of connectedness can be explored, which is similar to the weak type in directed networks. Consequently, only weak components can be defined in undirected and disconnected networks, and they would be isolated from each other. If all points of the undirected network are connected into a single set, they form one component.

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