# expected value of sum of weights in a random directed graph

Assuming we have a random directed weighted graph with $$n$$ nodes. Furthermore let us assume the nodes are divided into two categories:

1. A node $$i$$ is of category C if there are only outgoing edges or only ingoing edges from/to $$i$$
2. Otherwise a node $$i$$ is of category D

Let the number of nodes in category C be $$c$$,then the number of nodes in category D is simply $$n-c$$.
Let $$p_1$$ be the probability that there is an edge between two nodes of category $$C$$ (and we pick each direction with $$p_1/2$$) and let $$p_2$$ be the probability of an edge between two nodes $$i$$ and $$j$$ for all other cases (again picking each direction with probability $$p_2/2$$). We denote the weight going from a node $$i$$ to a node $$j$$ by $$e_{ij}$$, where the $$e_{ij}$$'s follow three different distributions (with means $$\mu_1, \mu_2, \mu_3$$ respectively) depending on following cases:

• $$(i,j)\in D\times D$$
• $$(i,j)\in D\times C$$ or $$(i,j)\in C\times D$$
• $$(i,j)\in C\times C$$

I am interested in calculating the expected value of $$\sum_{j=1}^{n-1} e_{ij}-e_{ji}$$. For the cases where $$i$$, $$j$$ are both in the same category, the values will simply be zero by symmetry. But how can I calculate the other cases?

My idea was to first calculate the expected value of $$\sum_{j=1}^{n-1} e_{ij}$$ for the two cases of $$i$$:

i.e. if $$i \in C$$, then $$\sum_{j=1}^{n-1} e_{ij}=\sum_{j\in C} e_{ij} +\sum_{j\in D} e_{ij}=\frac{ \mu_3(c-1)p_1}{2} +\frac{ \mu_2(n-c)p_2}{2}$$ And if if $$i \in D$$, then $$\sum_{j=1}^{n-1} e_{ij}=\sum_{j\in C} e_{ij} +\sum_{j\in D} e_{ij}=\frac{ \mu_2 c p_2}{2} +\frac{ \mu_1(n-c-1)p_2}{2}$$

Then since if a node $$i$$ is in C it only has arrows in one direction, the expected 'net position' would be $$\frac{ \mu_3(c-1)p_1}{2} +\frac{ \mu_2(n-c)p_2}{2}$$

And for the other case, if $$i$$ is in D the expected 'net position' would be simply $$0$$ by symmetry. Is this correct until now or am I missing something?

Thanks for any help!

• if $$i \in C$$: $$\quad \pm (\mu_3 (c-1)p_1/2 + \mu_2 (n-c) p_2/2)$$
• if $$i \in D$$: $$\quad 0$$