what does it mean by strength of vertex in textual graph?
in reality I don't know by strength of vertex? is it equal with the edge weight or not?
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I've read that betweenness centrality is defined as the number of times a vertex lies on the shortest path of the other pairs of nodes.
However, in case weights have a positive meaning (i.e. the more the weight of an edge the merrier), then how does one define betweenness centrality?
In this case, is there another way to calculate betweenness centrality? Or is it simply interpreted in a different way?
Computing the betweenness centrality of a vertex v relies on the following fraction, for any u and w: s(u,w,v) / s(u,w) where s(u,w,v) is the number of shortest paths between u and w that involve v, and s(u,w) is the total number of shortest paths between u and w.
With positive edge weights, I would suggest that you count each shortest path with its own weight: replace s(u,w,v) by the sum of weights of shortest paths between u and w that involve v; and s(u,w) by the sum of weights of all shortest paths between u and w.
Then, you have to define the weight of paths, and this depends on what you have in mind. You may for instance consider the sum of edge weights, their product, their minimum or maximal value, etc.
Warning: this definition still relies on shortest unweighted paths; if longer paths with higher weights exist, they will be ignored, which means that graph structure prevails. This may not be satisfactory.
Note: this approach is somewhat equivalent, if edges have integer weight and a path weight is its edge weight product, to use the classical definition on a multi-graph (an unweighted graph where several edges may exist between two same vertices).
Is there a graph algorithm for solving the following problem:
Given a weighted undirected graph G (all weights are positive), a start node N and a total weight W*. Generate a random cycle through the graph, starting and ending at node N, of which the total weight (the summed weight of all the edges) approximates the given weight W*.
One could see this as generating the cycle that best approximates W*, but generating a cycle that approximates W* within some margin of error is also fine.
If you want a simple cycle, you want an approximation algorithm for the travelling salesman problem. I believe there are known hardness results, indicating that this is NP-hard for general graphs, but there is a wide range of heuristics; you can check the literature.
Hi so bascially lets say I have a network(A) and I want to find the betweeness centrality of it.
I used: centr_betw(graph, directed = FALSE, normalized = TRUE)
This returned every node with the value:
[1] 1.827102e+04 3.554450e+04 5.000000e-01 9.524383e+04
[5] 0.000000e+00 0.000000e+00 1.078184e+05 4.768125e+04
I really want to know what these numbers mean.
It also shows the between centralization of the whole network and a max value. Lets say the network(A) as a whole has a betweenness centrality of 0.04. What can you say about this network(A) when it is compared to a random network with a betweeness centrality of 0.001?
MUCH THANKS GUYS
Quite a bit of information can be found simply if you type ?centr_betw. In particular, centr_betw returns a list of three components: res, centralization, theoretical_max.
Each element of res is the betweenness centrality of a corresponding vertex i computed in this manner. Specifically, given a shortest path between some vertices j and k (not equal to i), i is considered to be more central if this shortest path includes i. Going over all possible pairs of j and k we can find this betweenness centrality of i.
Further, centralization and theoretical_max concern the Freeman centralization. centralization is C_x, which measures how central network's most central vertex is in relation to how central all the other vertices are. theoretical_max is the denominator of C_x providing the maximal possible value of the numerator across all networks with the same number of vertices.
So, if network A has Freeman centralization 0.04 and network B has 0.001, then we may say that the most central vertex of A is significantly more central than the most central vertex of B. If B is random (i.e., Erdos-Renyi), then that makes sense, because in a big enough network all vertices should play pretty similar role.
Is there a relation between edges and nodes? How could it be expressed in asymptotic notation?
In dynamical networks, one may calculate the Hamming distance to compare the similarity between two graphs, can anyone explain how?
Assuming that the Hamming distance of two graphs have equal edge density, what is the difference between Hamming distance and expected Hamming distance between two independent Erdos-Renyi random graphs? How does the later arise?
The Hamming distance measures the minimum number of substitutions required to change (transform) one mathematical 'object' (i.e. strings or binary) into another.
So in network theory it can be defined as a the number of different connections between two networks (it can be formulated also for not equally-sized networks and for weighted or directed graphs). In a simple case in which you have two Erdos-Renyi networks (the adjacency matrix has 1 if the node pair is connected and 0 if not) the distance is mathematically defined as follows:
The values that are subtracted are the two adjacency matrix. If you take two Erdos-Renyi networks with wiring probability of 0.5 and compute the hamming distance between them you should get a value around 0.5. I generated different Erdos-Renyi graph and their Hamming distances produced a Gaussian curve around 0.5 (as we can expect; see below).
If it is needed I can give you the code I used.