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# Pairwise Disconnectivity Index

#### Definition

The pairwise disconnectivity defined as index of vertex v, Dis(v), as the fraction of those initially
connected pairs of vertices in a network which become
disconnected if vertex v is removed from the network
Here, N

The pairwise disconnectivity index quantifies how crucial an individual element is for sustaining the communication ability between connected pairs of vertices in a network that is displayed as a directed graph.

σ

It immediately detects the fraction of connected ordered pairs of vertices different from v for whose reachability vertex v is necessary.

In a directed network, an ordered pair of vertices (i,j) consists of two distinct nodes i and j. Such a pair is connected if there exists at least one path of any length from vertex i to vertex j. The pair (i,j) differs from the ordered pair (j,i), which is connected if there is a path from vertex j to vertex i. In an undirected network all pairs are unordered, i.e. (i,j) = (j,i).

Let G be a (directed) graph and N the number of connected (ordered) pairs of vertices in G. The PDI of a network entity is defined as:

and always ranges between 0 and 1. The maximum PDI of 1 means that no ordered pair of vertices in G is linked anymore due to the deletion of the entity whereas 0 indicates that all ordered pairs in G are still linked.

Note, that the meaning of N' depends on the chosen network entity. An entity may be

_{0}is the total number of ordered pairs of vertices in a network that are connected by at least one directed path of any length. It is supposed that N_{0}> 0, i.e., there exists at least one edge in the network that links two different vertices. N_{-v}is the number of ordered pairs that are still connected after removing vertex v from the network, via alternative paths through other vertices.The pairwise disconnectivity index quantifies how crucial an individual element is for sustaining the communication ability between connected pairs of vertices in a network that is displayed as a directed graph.

**Mediative Disconnectivity Index**σ

_{st}(v) expresses the number of ordered pairs {s,t} ¦ s ≠ t ≠ v and s, t, v ∈ V that are exclusively linked through vertex v.It immediately detects the fraction of connected ordered pairs of vertices different from v for whose reachability vertex v is necessary.

### From DiVa site:

The pairwise disconnectivity index (PDI) evaluates the topological significance of a network entity (vertex, edge, groups of vertices/edges) based on its influence on the connectedness of a network. This can be quantified by estimating how the deletion of an entity affects the existing number of connected ordered pairs of vertices in a network. The more of these pairs are being disconnected upon the removal of a network entity the more important it is.In a directed network, an ordered pair of vertices (i,j) consists of two distinct nodes i and j. Such a pair is connected if there exists at least one path of any length from vertex i to vertex j. The pair (i,j) differs from the ordered pair (j,i), which is connected if there is a path from vertex j to vertex i. In an undirected network all pairs are unordered, i.e. (i,j) = (j,i).

Let G be a (directed) graph and N the number of connected (ordered) pairs of vertices in G. The PDI of a network entity is defined as:

### PDI(entity) = 1 - N'/N

and always ranges between 0 and 1. The maximum PDI of 1 means that no ordered pair of vertices in G is linked anymore due to the deletion of the entity whereas 0 indicates that all ordered pairs in G are still linked.

Note, that the meaning of N' depends on the chosen network entity. An entity may be

- a single vertex v: N' refers to the number of connected
(ordered) pairs of vertices in G without v and its edges.
- a group of vertices, e.g. {a,b,c}: N' is the number of connected (ordered) pairs of vertices in G without the vertices a,b,c and their edges.
- a single edge e: N' is the number of connected (ordered) pairs of vertices in G without e.
- a group of edges, e.g. {a -> b,c -> a}: N' is the number of connected (ordered) pairs of vertices in G without the edges a -> b and c -> a.
- a topological pattern or a pattern instance: in a directed graph G, a pattern is the joint feature of every n connected vertices and describes the way how n vertices are connected with each other. A pattern always comprehends all existing edges between n vertices. None of the n vertices is isolated from the others, i.e. each of the n vertices must be directly attached to at least another one. If a pattern is over-represented in G it is usually called a motif.

#### Requirements

Require directed network.

#### Software

- DiVa

DiVa - Disconnectivity Valuation tool - DiVa online

DiVa online

#### References

- POTAPOV, A. P., GOEMANN, B. & WINGENDER, E. 2008. The pairwise disconnectivity index as a new metric for the topological analysis of regulatory networks. BMC bioinformatics, 9, 227.