Box 1. Graph-theoretic concepts

A graph is a pair G=(V,E), where V is a set of vertices (or nodes) and E is a set of edges connecting pairs of vertices. In PPI networks, the vertices represent proteins and the edges represent interactions.

The distance between two vertices in a graph is the number of edges on a shortest path between them.

The diameter of a graph is the maximum distance between any two of its vertices.

The neighborhood of a vertex is the set of vertices connected to it.

The n-neighborhood of a vertex is the set of vertices whose distance from it is at most n.

A clique in a graph is a fully connected subgraph, that is, a subgraph in which every two vertices are connected by an edge.

The degree of a vertex is the number of its neighbors.

A cut in a graph is a partition of the vertices into two non-overlapping sets. A multiwaycut is a partition of the vertices into several disjoint sets. The value of the cut is the number of edges going between different sets.

Network flow: Imagine a graph as a network of interconnected pipes. Suppose water gets into one or more vertices (sources) from the outside, and can exit the network at certain other vertices (sinks). Then, it will spread in the pipes and reach other nodes, until it exits at sinks. The capacities of the edges (i.e., how much the pipe can carry per unit time) and the input at the sources determine the amount of flow along every edge (i.e., how much each pipe actually carries) and the amount exiting at each sink. In the context of a PPI network, by considering proteins that have a certain function as sources and simulating flow in the network, the amount of flow at edges and sinks can be used to annotate additional proteins.

The density of a graph is the fraction of edges it actually has out of all possible vertex pairs. Hence, the density of G=(V,E) is 2|E|/(|V|(|V|-1)), and a clique graph has the maximum possible density, that is, 1.

The clustering coefficient of a vertex is the density of its neighborhood (Watts and Strogatz, 1998).

A graph is called a k-core if the minimal degree in it is k.

Core clustering coefficient: For a parameter k, the core clustering coefficient of a vertex is the density of the largest k-core of its immediate neighborhood (Bader and Hogue, 2003).

An adjacency matrix of a graph G=(V,E) is a matrix A_|V||V|={a_ij} where a_ij=1 if and only if v_i and v_j are neighbors. As PPI networks usually do not contain loops, in our context a_ii=0.

A graph is called bipartite if its vertices can be partitioned into two disjoint sets such that no edge connects two vertices of the same set.