Calculates the graph level secrecy index of a graph and its vertices
g_secrecy(g, type = 0, p = 0.25, digits = 3)
v_secrecy(g, type = 1, p = 0.25, digits = 3)data.frame
The secrecy measure is useful for covert networks where its
members want to remain undetected to the law enforcement agencies (LEA's),
even when some of their peers are detected.
The secrecy measure is defined as the fraction of the network that
remains unexposed if a single member of the network is detected.
Hence, the score runs between 0 (= everybody gets exposed as soon as 1
person is exposed) to 1 (= nobody gets exposed when 1 person is exposed).
In real cases, the boundary values of 0 and 1 will only occur in pathological
networks.
This relies on two things: 1. the probability of an individual becoming exposed when the LEA conducts a surveillance; 2. the fraction of the network that is exposed when a member of the network becomes detected in that surveillance.
There are several ways of computing the network's secrecy.
S1 assumes that every actor has the same probability of being
detected under a surveillance (with p = 1/number_of_vertices) and
defines the secrecy that individual i 'contributes' to the
network as the fraction of individuals that remain unexposed when
upon monitoring individual i all his links with his neighbors are
detected.
S2 assumes that whenever an individual in
the network is being monitored, communication between him and
one of his neighbors is detected independently with probability p.
The case where p = 1 therefore corresponds to measure S1.
If individual i has di neighbors the
number of neighbors that will be detected is binomially distributed.
As with S1, it is assumed that every actor has the same probability
of being detected under a surveillance (with p = 1/number_of_vertices).
The calculation of S2 requires the user to specify a reasonable
value for p, which may not be obvious.
S3 no longer assumes that i = 1/n for all i in V (as in S1
and S2).
It can be argued that i = 1/n is a fair assumption when a covert operation
is in its initial phase.
However, if an operation passed its initial stage the probability of
exposure will vary among network members.
This happens because certain individuals, due to a more central position
in the network, are more likely to be discovered.
In S3 this is captured by the equilibrium distribution of a random
walk on the graph.
This random walk chooses its next vertex at random from the neighbors
of the current vertex including itself.
NOTE: measure S3 can break down in certain graphs and is no longer
bound to \[0,1\]. Discard of S3 in those situations.
This function either returns one of the three secrecy measures (specified
by type) or a data.frame containing all three.
When type == 0 or type == 2, a reasonable value for p
should be specified.
The function g_secrecy returns the overall secrecy score of the
graph: this is the fraction of vertices in the graph expected to remain
unexposed under a single surveillance when a single vertex is exposed by the
LEA.
Here, the default scenario is type = 0.
The function v_secrecy returns the secrecy score per vertex,
this is the fraction of vertices in the graph that are to remain
unexposed under a surveillance when that vertex is exposed multiplied by
the probability of that vertex being exposed to begin with.
In other words: if 1 vertex in the graph is exposed, what is the
fraction of vertices that that is likely to remain
unexposed, due to vertex i? This is
indeed determined by the number of neighbors of
i and the risk of i being the one
to be exposed.
In scenarios 1 and 2, all vertices have the same probability of being exposed,
so this secrecy score is proportional to the fraction that they leave unexposed
when captured.
For scenario 3, more central vertices are more likely to be exposed themselves,
so more central vertices will have lower secrecy than other vertices (with
the same number of neighbors): these central vertices are more easily exposed
and therefore threaten the secrecy of the graph more.
Here, the default scenario is type = 1.
Note that the graph-level secrecy score is equal (barring rounding differences) to the sum of the vertex-level secrecy scores.
The formulas come from Lindelauf, R., Borm, P., & Hamers, H. (2009). The influence of secrecy on the communication structure of covert networks. Social Networks, 31(2), 126-137.
if (FALSE) { # \dontrun{
data(Madrid_bombing, package = "snafun")
g <- Madrid_bombing[[25]]
g_secrecy(g) # all three measures, p = .25
g_secrecy(g, p = .1) # all three measures, p = .1
g_secrecy(g, type = 1) # only S1
v_secrecy(g, type = 0)
# almost the same, difference only due to rounding of each score
v_secrecy(g, type = 0) |> colSums()
} # }