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Consider a multiagent system where each agent may either like or dislike another agent. For two agents, there are three distinct cases: the agents (i) mutually like each other; (ii) mutually dislike each other; or (iii) the first likes the second but the second does not like the first. (There is a symmetric second case for (iii).)

My question, which I have been unable to work out the answer to, is: what is the number of distinct cases for $n$ agents? Distinct here means that a case cannot be transformed into another by relabeling the agents. The number of cases (not necessarily distinct) seems to be $n \times 2^{n-1}$.

Shrisha Rao
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You're essentially asking for the number of directed graphs with $10$ nodes. The idea is that each node is an agent and an edge from node $a$ to $b$ means $a$ likes $b$.

Much like in the case for simple graphs, I don't think there's a neat closed formula for this, although asymptotic estimates are known. That said, there's an entry in the OEIS for this: Sloane's $A000273$. According to it, the answer is

$$341260431952972580352$$

Fimpellizzeri
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  • I think the answer is larger than this (depending on you definition of a directed graph). In the quesiton formulation, an edge may be bidirectional. – Arthur Jul 18 '17 at 07:50
  • Directed graphs may have bidirectional edges. Oriented graphs may not. – Fimpellizzeri Jul 18 '17 at 07:54
  • @Arthur Here is OEIS's entry for oriented graphs: link. – Fimpellizzeri Jul 18 '17 at 07:58
  • I did not know that, but it seems you are right. Are loops allowed in Sloane's directed graphs? – Arthur Jul 18 '17 at 07:59
  • I meant loops of length 1. Those shouldn't be counted for this problem, and it seems both conventions are used for digraphs. – Arthur Jul 18 '17 at 08:00
  • I don't see why not. Oriented graphs may also contain loops, provided they are not of length two (because that would simply be a disguised bidirectional edge). – Fimpellizzeri Jul 18 '17 at 08:01
  • Checking small values of $n$, like $n=2$ or $3$, seems to indicate that loops of length one are not allowed on digraphs (nor on oriented graphs for that matter). – Fimpellizzeri Jul 18 '17 at 08:02
  • @Fimpellizieri: I don't think it's quite that. If you want to formulate my problem in terms of graphs, it is to count all distinct graphs of $n$ nodes where there is always an edge between any two nodes, but edges may be unidirectional in either direction, or bidirectional, or have no direction at all. – Shrisha Rao Jul 20 '17 at 07:18
  • I don't think I get what you propose. How exactly does my translation to graphs fail to model the problem, and how does your translation correct that? – Fimpellizzeri Jul 20 '17 at 14:54
  • @Fimpellizieri, first of all the problem is with $n$ nodes, not 10 in particular (i.e., to find a combinatorial expression, not a particular value in a specific case), and secondly you are looking only at directed edges, whilst edges may be undirected as well. – Shrisha Rao Sep 19 '17 at 09:03
  • What does an undirected edge mean? – Fimpellizzeri Sep 19 '17 at 15:55