samples/texts/2058102/page_5.md

Fix $c \in [K]$. The difference of the corresponding terms the hypotheses means can be written as

$$E_1[{\epsilon }_c]-E_0[{\epsilon }_c]=D_1^{(c)}+D_2^{(c)}=\left(\genfrac{}{}{0px}{}{m}{1}\right)\left(\genfrac{}{}{0px}{}{n-m}{1}\right){\nu }_{(c)}^{(1)}({\mu }_{(c)}^{(1)}-{\nu }_{(c)}^{(1)})+\left(\genfrac{}{}{0px}{}{m}{2}\right)([{\mu }_{(c)}^{(1)}{]}^2-[{\nu }_{(c)}^{(1)}{]}^2)E\_1[\backslash varepsilon\_c]\; -\; E\_0[\backslash varepsilon\_c]\; =\; D\_1^\{(c)\}\; +\; D\_2^\{(c)\}\; =\; \backslash binom\{m\}\{1\}\; \backslash binom\{n-m\}\{1\}\; \backslash nu\_\{(c)\}^\{(1)\}\; (\backslash mu\_\{(c)\}^\{(1)\}\; -\; \backslash nu\_\{(c)\}^\{(1)\})\; +\; \backslash binom\{m\}\{2\}\; \backslash left(\; [\backslash mu\_\{(c)\}^\{(1)\}]^2\; -\; [\backslash nu\_\{(c)\}^\{(1)\}]^2\; \backslash right)$$

(here $D_i^{(c)}$ corresponds to the edge-count that includes edges with exactly $i$ anomalous vertices). The reader can verify that $\frac{Var_1(T^*)}{Var_0(T^*)} \to 1$. Moreover, given that the limiting distribution (under the null) is normal, we write

$$c_{\alpha }=z_{\alpha }\sqrt{Va{r}_0(T^{\ast })}+E_0[T_2^W]c\_\{\backslash alpha\}\; =\; z\_\{\backslash alpha\}\; \backslash sqrt\{Var\_0(T^*)\}\; +\; E\_0[T\_2^W]$$

and thus

$${\beta }_2^W=\mathbb{P}\{Z>z_{\alpha }-\underset{n\to \mathrm{\infty }}{\lim }\left(\frac{E_1[T_2^W]-E_0[T_2^W]}{\sqrt{Va{r}_0(T^{\ast })}}\right)\}\mathrm{.}\backslash beta\_2^W\; =\; \backslash mathbb\{P\}\; \backslash left\backslash \{\; Z\; >\; z\_\backslash alpha\; -\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash left(\; \backslash frac\{E\_1[T\_2^W]\; -\; E\_0[T\_2^W]\}\{\backslash sqrt\{Var\_0(T^*)\}\}\; \backslash right)\; \backslash right\backslash \}.$$

Recall that $Var_0(T^*) = \Theta(n^3)$; thus, if

$$\sum _c{w}_c{D}_1^{(c)}\mathrm{\ne }0\backslash sum\_c\; w\_c\; D\_1^\{(c)\}\; \backslash neq\; 0$$

(i.e. there is signal in the null-to-anomaly connectivity) then the limiting power $\beta_2^W > \alpha$ when $\frac{m(n-m)}{\sqrt{n^3}} \to 0$ or, equivalently, when $m = \Omega(\sqrt{n})$ (similarly, if $m = \omega(\sqrt{n})$ then $\beta_2^W \to 1$). Furthermore, if

$$\sum _c{w}_c{D}_1^{(c)}=0\backslash sum\_c\; w\_c\; D\_1^\{(c)\}\; =\; 0$$

(i.e. there is no signal in the null-to-anomaly connectivity) the limiting power $\beta_2^W > \alpha$ when $\sum_c w_c D_2^{(c)} \neq 0$ and $\frac{m^2}{\sqrt{n^3}} \to 0$ (which is equivalent to $m = \Omega(\sqrt[4]{n^3})$). Moreover, if $m = \omega(\sqrt[4]{n^3})$ under these conditions then $\beta_2^W \to 1$.

It follows that the optimal choice of weights ($w_1, \dots, w_K$) is the one which maximizes the expression

$$\underset{n\to \mathrm{\infty }}{\lim }\left(\frac{E_1[T_2^W]-E_0[T_2^W]}{\sqrt{Va{r}_0(T^{\ast })}}\right)\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash left(\; \backslash frac\{E\_1[T\_2^W]\; -\; E\_0[T\_2^W]\}\{\backslash sqrt\{Var\_0(T^*)\}\}\; \backslash right)$$

in either of the two above-mentioned cases.

3.2 The Attributed Number of Triangles Fusion

We begin by writing

$$\tau =\sum _{c\in [K]}{\tau }_c+\sum _{b\mathrm{\ne }c}{\tau }_{b,c}+\sum _{d\mathrm{\ne }b,c}{\tau }_{b,c,d}\mathrm{.}\backslash tau\; =\; \backslash sum\_\{c\; \backslash in\; [K]\}\; \backslash tau\_c\; +\; \backslash sum\_\{b\; \backslash neq\; c\}\; \backslash tau\_\{b,c\}\; +\; \backslash sum\_\{d\; \backslash neq\; b,c\}\; \backslash tau\_\{b,c,d\}.$$

We denote the number-of-triangles fusion invariant to be

$$T_3^W=\sum _{c\in [K]}{w}_c{\tau }_c+\sum _{b\mathrm{\ne }c}{w}_{b,c}{\tau }_{b,c}+\sum _{d\mathrm{\ne }b,c}{w}_{b,c,d}{\tau }_{b,c,d}\mathrm{.}T\_3^W\; =\; \backslash sum\_\{c\; \backslash in\; [K]\}\; w\_c\; \backslash tau\_c\; +\; \backslash sum\_\{b\; \backslash neq\; c\}\; w\_\{b,c\}\; \backslash tau\_\{b,c\}\; +\; \backslash sum\_\{d\; \backslash neq\; b,c\}\; w\_\{b,c,d\}\; \backslash tau\_\{b,c,d\}.$$

Similar to what was done in the previous section, we obtain

$$E_0[T^{\ast }]=\left(\genfrac{}{}{0px}{}{n}{3}\right)[\sum _{c\in [K]}{w}_c[{\nu }_{(c)}^{(2)}{]}^3+\sum _{b\mathrm{\ne }c}{w}_{b,c}3{\nu }_{(b)}^{(2)}[{\nu }_{(b,c)}^{(1,1)}{]}^2+\sum _{d\mathrm{\ne }b,c}{w}_{b,c,d}3{\nu }_{(b,c)}^{(1,1)}{\nu }_{(b,d)}^{(1,1)}{\nu }_{(c,d)}^{(1,1)}]E\_0[T^*]\; =\; \backslash binom\{n\}\{3\}\; \backslash left[\; \backslash sum\_\{c\; \backslash in\; [K]\}\; w\_c\; [\backslash nu\_\{(c)\}^\{(2)\}]^3\; +\; \backslash sum\_\{b\; \backslash neq\; c\}\; w\_\{b,c\}\; 3\backslash nu\_\{(b)\}^\{(2)\}\; [\backslash nu\_\{(b,c)\}^\{(1,1)\}]^2\; +\; \backslash sum\_\{d\; \backslash neq\; b,c\}\; w\_\{b,c,d\}\; 3\backslash nu\_\{(b,c)\}^\{(1,1)\}\; \backslash nu\_\{(b,d)\}^\{(1,1)\}\; \backslash nu\_\{(c,d)\}^\{(1,1)\}\; \backslash right]$$