My friend Jamie convinced me that now would be a good time to implement the info about interfaces that he mentioned a while ago:
Until this code is actually running, I have no way to know what the most efficient way to code the data behind my networks will be. Even a priori, I could choose between storing the basic data of ‘how many edges are there between each pair of nodes’ as an adjacency matrix or as a (decorated) adjacency list, and that’s even before deciding how those matrices or lists will be stored in the C# framework. I have asked for advice on StackExchange, but in the mean time I have created an interface (code pictured above) and will be producing one implementation of it to work with for the time being.
I hope that eventually the implementation chosen will vary with the network (as edge density changes beyond certain thresholds, say), and having an interface will allow me to do this with minimal changes to the code and without losing earlier solutions to the problems I encounter.
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Write a C Program for Creation of Adjacency Matrix
Creation of Adjacency Matrix Write a C Program for Creation of Adjacency Matrix. Here’s simple Program for adjacency matrix representation of graph in data structure in C Programming Language. Adjacency Matrix: Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i…
Calculating adjacency matrix from randomly generated graphs
Calculating adjacency matrix from randomly generated graphs
I have developed small program, which randomly generates several connections between the graphs (the value of the count could be randomly too, but for the test aim I have defined const value, it could be redefined in random value in any time).
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Once upon a time, I said I finished my combinatorics series. Once upon a time. I lied. I'll give you a hint: those were the same time.
I just remembered a cool homework problem we did (graph theory related) so I'm just going to do a little exposé and call it a day.
Take an undirected simple graph, G, with n vertices. (All of this stuff would technically work with a graph with infinite vertices, but let's keep it simple.) The adjacency matrix of G is the n x n matrix where (G)ij = 1 if there is edge between vertex i and vertex j; it is 0 otherwise.
First, notice that the matrix is symmetric. In an undirected graph, if there is an edge between i and j, there is of course an edge between j and i. Also, of course, the matrix itself depends on how you label the vertices of G but this detail isn't too important to the problem.
Okay. The problem asks us to prove that (Gk)ij is the number of walks of length k between i and j. In general, if C = A × B, then (C)ij = Σ(A)ip(B)pj. Let's use this definition and a proof by induction.
Clearly, when k = 1, then the result is pretty trivial. It's almost the definition of an edge.
Now, let's assume (Gk)ij is the number of walks of length k between i and j. What do we get for (Gk+1)ij? Well, using the definition, we get (Gk+1)ij = Σ(Gk)ip(G)pj. A single term in this sum, (Gk)ip(G)pj, represents the number of paths of length k from i to p, times 0 if there is no edge from p to j, or times 1 if there is an edge from p to j. If there is no edge from p to j, then there is no walk of length k + 1 between i and j which goes through p as the second last vertex. If there is an edge from p to j, then we can do a walk of length k from i to p, then go from p to j in one step, to get a walk of length k + 1. We repeat this process for all vertices in the graph (since p ranges from 1 to n), so the whole sum results in the number of walks from i the j of length k + 1.
I started off the problem by assuming G is an undirected graph. You will realise that the identical setup and proof works for a directed graph. In this case, the adjacency matrix is not necessarily symmetric.
That was brief! A nice change of pace following my really long and dense posts recently.
