Survival Mixed Membership Blockmodel

Figures & data

Table 1 Time complexity of MCMC steps.

Variables	U	$(\mathit{Q},\mathit{R})$	$\mathbf{\Pi }$	$\mathbf{\Lambda }$	$\mathit{\beta }$	$\mathit{\xi }$
Time complexity	$O(L(p+M))$	$O(L{K}^2(p+M))$	$O(E+NK)$	O(LM)	$O(Lp(p+M))$	O(NK)

NOTE: Here, E denotes the number of active edges in the network, and $L=\sum _{(i,j)\in \mathcal{E}}{n}_{ij}$ represents the total number of observed survival times.

Fig. 1 Boxplot of coverage probabilities among 100 synthetic replicated datasets (a) when K = 3, $n_{ij}\sim \text{Pois}(25)+5$ and the connectivity probabilities p_c varies over $(0.2,0.3,0.4,0.5)$; (b) when K = 3, $p_c=0.3$, $n_{ij}\sim \text{Pois}(\mu )+5$ and μ varies over $(10,15,20,25,30,35)$; (c) when $p_c=0.3,n_{ij}\sim \text{Pois}(25)+5$ and K varies over $(2,3,4,5)$.

Fig. 2 Running time of 100,000 iterations of the MCMC algorithm on one core of an Intel Xeon Gold 6226R Processor. The number of observations per edge $n_{ij}\sim \text{Pois}(\mu )+5$. (a) We fix the connectivity probability $p_c=0.3$ and μ = 25 while varying the number of roles K from 2 to 5. (b) We fix K = 3 and μ = 25 but vary p_c from 0.2 to 0.5. (c) We fix K = 3 and $p_c=0.3$ but vary μ from 10 to 35.

Fig. 3 Patterns learned from the Enron E-mail corpus. (a), (b) The scatterplots of the employee-specific probability ${\pi }_{i1}$ of belonging to the first role and the log-scale E-mail numbers (a) when the SMMB is applied to time-to-event data, and (b) when the MMSB is applied to relational data. Each node represents an employee, with the color indicating his or her position. The size of each node represents the number of E-mails related to the employee. (c) The estimated baseline survival curve $\hspace{0.17em}\exp \hspace{0.17em}\{-\hspace{0.17em}\exp \hspace{0.17em}({\widehat{\beta }}_1^{lk})[1-S_0(t|{\widehat{\mathit{\lambda }}}^{lk})]\}$ of role l replying E-mails to role k when $\mathit{x}={(1,0,0,\dots ,0)}^T$.

Table 2 The estimated role probabilities ${\mathit{\pi }}_i$ by the SMMB from time-to-event data and by the MMSB from binary data of the four CEOs, respectively.

User ID	User Name	SMMB		MMSB
		${\widehat{\pi }}_{i1}$	${\widehat{\pi }}_{i2}$	${\widehat{\pi }}_{i1}$	${\widehat{\pi }}_{i2}$
23	David Delainey	0.049	0.951	0.478	0.522
62	John Lavorato	0.011	0.989	0.835	0.165
63	Kenneth Lay	0.208	0.792	0.463	0.537
117	Jeffrey Skilling	0.014	0.986	0.145	0.855

Table 3 The posterior mean, posterior standard deviation (SD), and 95% credible interval (CI) of coefficient ${\beta }_r^{lk}$’s for the SMMB learned from the Enron E-mail corpus without considering confidential information.

Covariates	Parameters	Posterior mean	Posterior SD	95% CI
Weekend effect	${\beta }_2^{11}$	–2.6923${}^{*}$	1.1340	[–5.4631, –0.7872]
	${\beta }_2^{12}$	–0.9623	1.6866	[–4.6645, 1.6364]
	${\beta }_2^{21}$	0.1191	0.1813	[–0.2510, 0.4611]
	${\beta }_2^{22}$	0.2810	0.2051	[–0.1370, 0.6667]
Forwarded effect	${\beta }_3^{11}$	–1.6664${}^{*}$	0.2363	[–2.1566, –1.2218]
	${\beta }_3^{12}$	–0.9986${}^{*}$	0.5689	[–2.2854, –0.0571]
	${\beta }_3^{21}$	–0.6419${}^{*}$	0.1321	[–0.9028, –0.3852]
	${\beta }_3^{22}$	–0.4578${}^{*}$	0.0960	[–0.6499, –0.2718]
Receiver number	${\beta }_4^{11}$	–1.2633${}^{*}$	0.1600	[–1.5760, –0.9518]
effect	${\beta }_4^{12}$	–1.4220${}^{*}$	0.4894	[–2.4875, –0.5576]
	${\beta }_4^{21}$	–0.7037${}^{*}$	0.0917	[–0.8965, –0.5373]
	${\beta }_4^{22}$	–0.8923${}^{*}$	0.0822	[–1.0588, –0.7376]
Word count effect	${\beta }_5^{11}$	–0.1086${}^{*}$	0.0477	[–0.2033, –0.0140]
	${\beta }_5^{12}$	0.1578	0.1532	[–0.1421, 0.4717]
	${\beta }_5^{21}$	0.0256	0.0399	[–0.0517, 0.1062]
	${\beta }_5^{22}$	–0.0113	0.0334	[–0.0781, 0.0504]

NOTE: The superscript * denotes that the corresponding 95% credible interval does not contain zero.