PRÁCTICA R REDES SCRIPT

---

title: "Handout 4"

output: html_document

---

**Students' names**

* Ramírez Mercau, Estanislao.

* Said, Larabes.

* ...

```{r global_options, include=FALSE}

#adapt it at your taste and convenience

knitr::opts_chunk$set(fig.width=8, fig.height=8, fig.align="center", echo=TRUE, warning=FALSE, message=FALSE)

```

```{r}

library(igraph)

library(igraphdata)

data(UKfaculty)

```

*1)* The graph *UKfaculty* from **igraphdata** describes the personal friendship network of 81 UK university professors. It is a directed and weighted network (where an arc X->Y means that X reports to be friend of Y with relative friendship weight w>0).

*A)* Extract from it an undirected and unweighted network where an edge X-Y means that both individuals X, Y have declared to be friends of each other. Plot it.

```{r}

UKfaculty_U <- as.undirected(UKfaculty, mode = "mutual")

plot(UKfaculty_U)

```

*A.1)* Compute its order (it should be 81), its size and its density. Would you consider it sparse?

```{r}

ord = gorder(UKfaculty_U)

siz = gsize(UKfaculty_U)

den = edge_density(UKfaculty_U)

den

round(1/81,4)

```

El orden del grafo $UKfaculty_U$ es `r ord`, el tamaño es `r siz`, y la densidad `r round(den,4)`. En relación a estos parámetros podemos afirmar que eel grafo es poco denso, si utilizamos el criterio de compararlo con 1/n no podemos afirmar que sea disperso, aunque la densidad es del orden de 1/n.

*A.2)* Is it connected? If not, what percentage of the whole set of nodes covers the largest connected component?

Would you consider it a giant connected component?

```{r}

com <- components(UKfaculty_U)$csize

lcom <- length(components(UKfaculty_U)$csize)

per <- round(max(components(UKfaculty_U)$csize)/sum(components(UKfaculty_U)$csize)*100,2)

```

El grafo $UKfaculty_U$ no es conexo, tienen `r lcom` componentes conexas, de tamaño `r com`. El porcentaje de nodos de la componente de mayor grado es de `r per` %. Podemos concluir que se trata de una componente gigante.

*A.3)* Compute its mean degree. What is the (theoretical) relationship between the average degree and the density of a graph? Confirm it on this network

```{r}

mdeg <- round(mean(degree(UKfaculty_U)),4)

mdeg

mdeg2 <- round((ord-1)*den,4)

mdeg2

```

El grado medio del grafo es `r mdeg`. La relación entre el grado medio y la densidad es: $<DEG>$UKfaculty_U = ($n$-1)*$den$(UKfaculty_U), donde $<DEG>$ es grado medio, $n$ es el número total de nodos (orden) y $den$ es la densidad del grafo. Para el grafo se verifica el cálculo que da como resutado `r mdeg2`.

*A.4)* Compute the range and the median of the degrees. What are the meanings of these indices? Compare the median and the mean, and guess from their relationship the shape of the degree distribution (is it symmetric, does it have a right tail, or a left tail?).

```{r}

ran <- range(degree(UKfaculty_U))

ran

medg <- median(degree(UKfaculty_U))

medg

```

Para la distribución de los grados de nuestro grafo vemos que la media es `r mdeg` y que es mayor que la mediana, que es `r medg`, esto implica que la distribución es asimétrica con cola a la derecha (sesgada a la derecha). El rango es `r ran`, lo que indica que el/los nodo/s de menor grado, tiene/n grado `r min(ran)` y el/los de mayor, grado `r max(ran)`.

*A.5)* Plot its degree distribution in linear, log and log-log scale (do not take into account the nodes of degree 0, if any). Simply by looking at these graphics, answer the following two questions: Does this distribution seem to follow a power-law? And a Poisson (i.e., exponential) law?

```{r}

UKfaculty_U0 <- delete.vertices(UKfaculty_U,V(UKfaculty_U)[degree(UKfaculty_U) == 0])

plot(degree_distribution(UKfaculty_U0, cumulative=T, mode="all"),type="l")

plot(degree_distribution(UKfaculty_U0, cumulative=T, mode="all"),type="l", log = "x")

plot(degree_distribution(UKfaculty_U0, cumulative=T, mode="all"),type="l", log = "xy")

```

Mirando la distribución de grados acumulativa puede concluirse por la forma de la gráfica de que existe una ley de potencia respecto al grado en la distribución.

*A.6)* Compute

...