Download Nodes centralities interference tutorial Example 1 Interference

Giovanni Scardoni, Carlo Laudanna
University of Verona, CBMC Center for BioMedical Computing
(http//www.cbmc.it)
contact: [email protected]
First version: September 2011.
Nodes centralities interference tutorial
All definitions consider connected networks (i.e. networks where each node
is reachable from all the others), which remain connected even after node
removal. The definitions are introduced for Betweenness centrality and are
then extended to other centrality measures.
Given a network (a graph) G = (N, E) where N is the set of nodes and
E is the set of edges, betweenness of node n is defined as
Btw(G, n) =
X
X σst (n)
s6=n∈N t6=n∈N
σst
(1)
where σst is the number of shortest paths between s and t and σst (n) is the
number of shortest paths between s and t passing through the node n. The
relative value of betweenness is obtained by normalizing it as
Btw(G, n)
j∈N Btw(G, j)
relBtw(G, n) = P
(2)
in order to have the fraction of betweenness of each node with respect to
the rest of the network.
Example 1
Consider the network in figure 1a. Node a is connected to the rest of the
network through nodes b and k. If we remove node k from the network, node
b become the only node connecting a to all the other nodes of the network
(figure 1b), consequently its betweenness value will increase (see Table. 1).
This is a case of betweenness interf erence of node k with respect to node b
since there is “interference” of node k with respect to the betweenness value
of node b.
Interference definition
Such interference, and the interference of node k with respect to all the
other nodes, is detected by removing node k from the network and can be
measured as follow: G|i is the network obtained from G removing node i
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Figure 1: a. Node k and b are in the shortest paths from node a to the other
nodes. b. Node k have been removed. Node b is now essential for connecting
node a to the rest of the network: it is the only node in the shortest paths
connecting a to the other nodes: node b betweenness increases.
node
name
a
b
c
d
e
k
relative betwenness
(with node k)
4.00
19.00
4.00
54.00
0.00
19.00
relative betwenness
(node k removed)
0.00
50.00
0.00
50.00
0.00
interference
value
+4.00
-31.00
+4.00
+4.00
0.00
Table 1: Interference values of the network in figure 1, expressed as percentage. As expected nodes k, b and d have high betweenness value. k has
negative interference with respect to node b: if node k is removed from the
network, node b gains more than 30% of its betweenness value. This is reflected by the negative sign of interference. Node k has negative interference
with respect to node b meaning that the presence of node k is negative for
node b to play a central role in the network.
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and all its edges from the network. The betwenness interf erence of a node
i with respect to another node n in the network G is:
IntBtw (i, n, G) = relBtw(G, n) − relBtw(G|i , n)
(3)
The measure shows which fraction of the total betweenness value a node
loses or gains with respect to the rest of the network when the node i is removed. Notably, expressing interference values as percentage may facilitate
understanding the meaning of the calculated data. The complete analysis
of the network in the example is shown in Table 1.
Potential applications
• Biological networks where one or more nodes (genes, proteins, metabolites) are possibly removed from the network because of gene deletion,
pharmacological treatment or protein degradation. Interference can
be used to:
– Pharmacological treatment: one can potentially predict side
effects of the drug by looking at the topological properties of
nodes in a drug-treated network, meaning with that a network in
which a drug-targeted node (protein) was removed. To inhibit a
protein (for instance a kinases) corresponds to removing the node
from the network
– Gene deletion: gene deletion implies losing encoded proteins,
thus resulting in the corresponding removal of one or more nodes
from a protein network
• Social and financial networks where the structure of the network is
naturally modified over time
• Power grid failures
• Traffic jam or work in progress in a road network
• Temporary closure of an airport in an airline network
Positive and negative interference
As in the example of figure 1, the interf erence value of a node i with respect
to a node n can be positive or negative.
Positive Interference
If a node (A), upon removal from the network of a specific node (B) or
of a group of nodes, decreases its value for a certain centrality index, its
interference value is positive. This means that this node (A), topologically
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speaking, takes advantage (is positively influenced) by the presence in the
network of the node (B) or of that group of nodes. Thus, removal of node (B)
or of that group of nodes from the network, negatively affects the topological
role of the node (A). This is called positive interference.
Negative Interference
If a node (A), upon removal from the network of a specific node (B) or
of a group of nodes, increases its value for a certain centrality index, its
interference value is positive. This means that this node (A), topologically
speaking, is disadvantaged (is negatively influenced) by the presence in the
network of the node (B) or of that group the nodes. Thus, removal of
node (B) or of that group of nodes from the network, positively affects the
topological role of node (A). This is called negative interference.
Example 2
The following example explains the difference between the notions of negative and positive interference. Let consider the network in figure 2.
Figure 2: a Node3 and node4 are the nodes connecting the top of the network
with the bottom. b Node6 has been removed: node4 becomes a peripheral
node, its betweenness decreases. The presence of node6 is important for
node4 to play a central role (positive interference). c Node3 has been removed: node4 is the only connection between the top and the bottom, its
betweenness increases. Node4 has a more central role if node3 is not part of
the network. The presence of node3 limits the central role of node4 (negative
interference).
Positive interference
Node4 has high value of betweenness (12), since it is important to connect
the top of the network with the bottom. But this importance strictly de4
Figure 3: Node6 interference values
Figure 4: Node3 interference values
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pends on node6. Indeed, by removing node6, node4 results a peripheral
node, as shown figure 2b. This is a case of “positive interference”, since the
high betweenness of node4 depends on the presence of node6: if node6 is
part of the network node4 has higher betweenness value. The presence of
node6 positively “interferes” with the role of node4. The interference values
of node6 with respect to the overal network are reported in figure 3. Node6
has the highest value of positive interference with respect to node4.
Negative interference
Consider node3. It is evident from the graphical representation (figure 2a)
that node3 plays a role similar to node4: they both connect the top of
the network with the bottom, and they can be considered “competitors”
in playing such a role. If we remove node3 from the network (figure 2c),
node4 remains the only node connecting the top with the bottom and its
betweenness value increases. This is a case of negative interference of node3
with respect to node4, since the presence of node3 is negative for node4 to
play a central role in the network: node4 is more central if node3 is not part
of the network thus node3 negatively interferes with node4. The interference
values of node3 with respect to the overall network are reported in figure 4.
Node3 has the highest value of negative interference with respect to node4.
Definition generalized to other centralities
Interference centralities definitions are generalized to other centralities as
Stress, Radiality, Closeness, Eccentricity as follows. Given a centrality C,
we define the relative interference (or simply interference) of the node i with
respect to node n in the network G as:
C(G|i , n)
C(G, n)
−P
j∈N C(G, j)
j∈N C(G|i , j)
IntC (i, n, G) = P
Contacts
If you have any question about the interference notion, you encounter a bug
with the interference plugin, or have feature suggestions, we encourage you
to use the Cytoscape-discuss:
https://groups.google.com/group/cytoscape-discuss
or the Cytoscape-helpdesk:
https://groups.google.com/group/cytoscape-helpdesk
mailing-list or write an e-mail to:
[email protected]
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Acknowledgement
Funding: Fondazione Cariverona, CBMC, AIRC, MIUR 2009.
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