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Table 2 List of symbols and notation

From: Social learning for resilient data fusion against data falsification attacks

Network properties

 N

\(\triangleq\)

Size of the sensor network

 \(N^*\)

\(\triangleq\)

Number of Byzantine nodes

 \(p_{\text{b}}\)

\(\triangleq\)

Probability of a given node being compromised

Sensor and social signals

 \(S_n\)

\(\triangleq\)

Signal measured by the n-th node

 \(\mathcal {S}\)

\(\triangleq\)

Set of values that \(S_n\) can take

 \(\mu _w\)

\(\triangleq\)

Distribution of \(S_n\) given \(W=w\)

 \(\Lambda _S(s)\)

\(\triangleq\)

Log-likelihood of \(S_n\) with respect to W

 \(F_w^\Lambda (s)\)

\(\triangleq\)

c.d.f. of \(\Lambda _S(s)\) conditioned on \(W=w\)

 \(\varvec{G}_n\)

\(\triangleq\)

Social observations of the n-th node

 \(\mathcal {G}_n\)

\(\triangleq\)

Set of values that \(\varvec{G}_n\) can take

 \(\Lambda _{\varvec{G}_n}(\varvec{g})\)

\(\triangleq\)

Log-likelihood of \(\varvec{G}_n\) with respect to W

 \(\beta _w^n(\varvec{g} | x_n,\varvec{g'})\)

\(\triangleq\)

Transition probabilities from \(\varvec{G}_{n-1}\) to \(\varvec{G}_{n}\) given \(X_n\) and W

Data fusion variables

 W

\(\triangleq\)

Target of the networked inference

 \(u(\pi _n,w)\)

\(\triangleq\)

Node’s utility function for deciding \(\pi _n\) when \(W=w\)

 \(\tau _n\)

\(\triangleq\)

Decision threshold used by the n-th node

 \(\pi _n(s,\varvec{g})\)

\(\triangleq\)

Data fusion strategy of the n-th node given \(S_n\) and \(\varvec{G}_n\)

 \(X_n\)

\(\triangleq\)

Signal broadcasted by the n-th node

 \(C(\pi _n), c_{0|0}, c_{0|1}\)

\(\triangleq\)

Corruption function, which links \(\pi _n\) and \(X_n\)

 \(\mathbb {P}\{\text {MD};p_b\}\)

\(\triangleq\)

Network miss-detection rate

 \(\mathbb {P}\{\text {FA};p_b\}\)

\(\triangleq\)

Network false alarm rate

Simulation parameters

 r

\(\triangleq\)

Ratio of the area of interest within the sensing range of a single node

 m

\(\triangleq\)

Number of quantization levels of a node’s sensor

 k

\(\triangleq\)

Node’s memory size