02-ER.Rmd

# 随机网络

*根据不同的N和P值生成ER随机网络，并且统计相关性质的变化*

我们选取了N为100、500、1000的三种情况，并对三种情况分别生成平均度为1，10和15的ER随机网络。表\@ref(tab:config-ER)给出了ER随机网络配置情况，图\@ref(fig:visualize-ER)则给出了每种配置的ER随机网络的可视化图形。

```{r define-ER-generation}
generate_random_network <- function(n, p) {
  expand_grid(n1 = 1:n, n2 = 1:n) %>%
    filter(n1 < n2) %>%
    mutate(
      connected = if_else(
        runif(nrow(.)) < p,
        TRUE, FALSE
      )
    ) %>%
    filter(connected) %>%
    select(n1, n2) %>%
    as.matrix() %>%
    graph_from_edgelist(directed = FALSE)
}
```

```{r config-ER}
config_er <- expand_grid(
  tibble(
    mean_k = c(1, 10, 15)
  ),
  n = c(100, 500, 1000)
) %>%
  mutate(p = mean_k / n)
config_er %>%
  select(n, p, mean_k) %>%
  knitr::kable(
    digits = 4,
    col.names = c("$N$", "$p$", "$<k>$"),
    caption = "ER随机网络类型配置"
  )
```

```{r generate-ER, cache=TRUE}
set.seed(20191112)
random_graphs <- config_er %>%
  mutate(
    graph = map2(
      n, p, generate_random_network
    )
  )
```

```{r visualize-ER, fig.width=6, fig.height=4, fig.cap="可视化ER随机网络"}
par(mar = c(0, 0, 2, 0) + 0.1, mfrow = c(3, 3))
for (i_row in 1:nrow(random_graphs)) {
  with(
    random_graphs,
    plot(
      graph[[i_row]],
      layout = layout_nicely,
      vertex.color = "black",
      vertex.size = 1,
      vertex.label = NA,
      edge.curved = 0,
      main = str_c("N: ", n[i_row], ", p: ", round(p[i_row], 4))
    )
  )
}
```

图\@ref(fig:degree-distribution-ER)画出了每种ER随机网络的度分布图，以及与其度分度的渐进分布的比较。可以看出来，随着节点数的增加，ER随机网络的度分度越来越接近$\lambda$值（平均值）等于度分布的泊松分布。

```{r degree-distribution-ER, fig.height=4, fig.width=6, fig.cap="不同随机网络的度分布[^1]"}
random_graphs %>%
  mutate(
    dat_plt = map2(
      graph, mean_k,
      ~ tibble(
        freq = degree_distribution(.x),
        k = seq(0, by = 1, length.out = length(freq)),
        freq_expected = dpois(k, .y)
      )
    )
  ) %>%
  unnest(dat_plt) %>%
  mutate(
    n = str_c("N: ", n) %>%
      factor(levels = c("N: 100", "N: 500", "N: 1000")),
    mean_k = str_c("<k>: ", mean_k)
  ) %>%
  ggplot(aes(k, freq)) +
  geom_point(color = "blue", size = 1) +
  geom_line(aes(y = freq_expected), color = "lightblue") +
  facet_grid(n ~ mean_k, scales = "free") +
  labs(y = "P(k)") +
  theme_few()
```

[^1]: 其中浅蓝色线是$\lambda$等于平均度$<k>$的泊松分布（即相应随机网络的渐进分布）。