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If you're thinking of weighted, directed graphs, a binary relation doesn't quite fully represent it. In the weighted directed case, a graph is a function from

If you're thinking of weighted, directed graphs, a binary relation doesn't quite fully represent it. In the weighted directed case, a graph is a function from Nodes × Nodes → Weights, (where Weights ⊆ ℤ or Weights ⊆ ℝ or something like that)... Which makes weighted graphs perfect for representation by matrices! Studying adjacency matrix of weighted, directed graphs gave me a profound realization matrices are a table of relationships between their "dimensions". I never looked at matrices the same way again. I realized that the identity column was literally the entries that represented a relationship between a dimension/node and itself. I went down many rabbit holes... particularly the question of graph isomorphism. That problem is a rabbit hole. Another rabbit hole is the Hamiltonian cycle problem. P.S. check out hypergraphs, which are not necessarily binary relations and generalize graphs. Incidentally, hypergraphs have an adjacency tensor. I've started to understand that tensors are like 'tables' of relationships between dimensions + dimensions representing some or all of their possible combinations. By the way, if you can figure out a way to do fast multiplication of matrices over degree-truncated polynomial rings, I'll show you a fast way to count the number of hamiltonian cycles in a graph. These problems are intricately linked.

theoh 10y ago
Sometimes the matrix, like for example the adjacency matrix for Wikipedia internal links, will be very sparse. Another reason to use relations rather than an inefficient, mostly empty implicit data structure. That Wikipedia adjacency matrix would be interesting to visualize, though. There are tools out there for visualizing sparse matrices as graphs: http://yifanhu.net/GALLERY/GRAPHS/ http://yifanhu.net/GALLERY/GRAPHS/
- maweki 10y ago
  "using relations" or a particular data structure wasn't at all the point of this particular discussion. The point was that, for a simple directed graph, the edges represent a binary relationship. While for a weighted directed graph, it represents a relationship-matrix. For the graph being non-directed this would mean, that the relationship(-matrix) is symmetric. Meaning that any algebra, algorithm, proof, etc. on graphs of that type can be applied for the other interpretation as well. Implementation is a different question. "Most" binary relations we happen upon (less-than as the obvious) have an infinite domain and infinite cardinality (when represented as a set of tuples), making it very hard to "visualize" as a graph.
  - macawfish 10y ago
    Your last sentence is pretty much a summary of my attempt to understand the amplituhedron, and as far as I've gottan
- rspeer 10y ago
  I've never seen a visualization of a graph at the scale of Wikipedia that conveys anything useful in the visualization. You just get a big hairball.
  - theoh 10y ago
    Yeah, actually that page has a Wikipedia graph on it which looks exactly like a spiky sun. I'd like to see more work like the paper from a few years ago which ran PageRank and HITS on Wikipedia to discover the most central pages. The PageRank result indicated church hierarchy and nation states were important, while HITS had things like "television" "animal" as the most authoritative/central pages. There is so much structure there to be investigated.
  - ygra 10y ago
    The best you can usually do is clustering, but at that point you can just as well render clusters as single nodes and significantly lessen the strain on the visualization or the layout. I'm always astonished when customers ask about visualizing graphs with tens of thousands to millions of nodes. Thy usually have never thought about the UX implications of that (let alone the time needed to calculate a layout which results in an image so large that each node is smaller than a pixel anyway).
    - MurrayHill1980 10y ago
      This is a very good point. In the early days of Graphviz we put a lot of effort into making very readable diagrams of fairly small graphs, mainly in drawing directed graphs (dot). We aimed at graphs with under a hundred objects. We agree with the viewpoint that it is better to apply some sort of analysis to reduce a larger graph to a smaller one that can be read and understood. People still want to draw much larger networks. In fact it is valid to want an overview of all the data, it's just not clear how to do this with many thousands of random connections. Graphviz sfdp (based on Yifan Hu's earlier work) and recent work on Maxent (which has a decent theoretical justification) and from Ulrik Brandes group are good examples. One reason graphviz neato does not make such great layouts of large graphs out of the box is that it is straight statistical multidimensional scaling, and we just drop shapes on top of the points, so they can overlap badly. We felt since people are relying on it for data visualization we did not want to post process the MDS layouts by default in a way that could be misleading. But you can turn on overlap removal (neato or sfdp -Goverlap=false). We really should document all this in one place that is easier to find and understand. About the external shape loader issue - this code is somewhat centralized in graphviz/lib/gvc/gvusershape.c and in gvrender.c which calls it, and I thought it could be disabled at compile time (because we did address the security concerns at one point) and there's a lot of other machinery to control compile time features. Maybe John Ellson can comment here. Kudos to John for recognizing the problems with the shape loader as soon as I proposed it but apparently that didn't stop us at the time. Stephen North
fibo 10y ago
Here It Is a package that implements some hypergraphs features. http://g14n.info/iper/ http://g14n.info/iper/

theoh 10y ago

Sometimes the matrix, like for example the adjacency matrix for Wikipedia internal links, will be very sparse. Another reason to use relations rather than an inefficient, mostly empty implicit data structure. That Wikipedia adjacency matrix would be interesting to visualize, though. There are tools out there for visualizing sparse matrices as graphs: http://yifanhu.net/GALLERY/GRAPHS/ http://yifanhu.net/GALLERY/GRAPHS/

maweki 10y ago
"using relations" or a particular data structure wasn't at all the point of this particular discussion. The point was that, for a simple directed graph, the edges represent a binary relationship. While for a weighted directed graph, it represents a relationship-matrix. For the graph being non-directed this would mean, that the relationship(-matrix) is symmetric. Meaning that any algebra, algorithm, proof, etc. on graphs of that type can be applied for the other interpretation as well. Implementation is a different question. "Most" binary relations we happen upon (less-than as the obvious) have an infinite domain and infinite cardinality (when represented as a set of tuples), making it very hard to "visualize" as a graph.
- macawfish 10y ago
  Your last sentence is pretty much a summary of my attempt to understand the amplituhedron, and as far as I've gottan
rspeer 10y ago
I've never seen a visualization of a graph at the scale of Wikipedia that conveys anything useful in the visualization. You just get a big hairball.
- theoh 10y ago
  Yeah, actually that page has a Wikipedia graph on it which looks exactly like a spiky sun. I'd like to see more work like the paper from a few years ago which ran PageRank and HITS on Wikipedia to discover the most central pages. The PageRank result indicated church hierarchy and nation states were important, while HITS had things like "television" "animal" as the most authoritative/central pages. There is so much structure there to be investigated.
- ygra 10y ago
  The best you can usually do is clustering, but at that point you can just as well render clusters as single nodes and significantly lessen the strain on the visualization or the layout. I'm always astonished when customers ask about visualizing graphs with tens of thousands to millions of nodes. Thy usually have never thought about the UX implications of that (let alone the time needed to calculate a layout which results in an image so large that each node is smaller than a pixel anyway).
  - MurrayHill1980 10y ago
    This is a very good point. In the early days of Graphviz we put a lot of effort into making very readable diagrams of fairly small graphs, mainly in drawing directed graphs (dot). We aimed at graphs with under a hundred objects. We agree with the viewpoint that it is better to apply some sort of analysis to reduce a larger graph to a smaller one that can be read and understood. People still want to draw much larger networks. In fact it is valid to want an overview of all the data, it's just not clear how to do this with many thousands of random connections. Graphviz sfdp (based on Yifan Hu's earlier work) and recent work on Maxent (which has a decent theoretical justification) and from Ulrik Brandes group are good examples. One reason graphviz neato does not make such great layouts of large graphs out of the box is that it is straight statistical multidimensional scaling, and we just drop shapes on top of the points, so they can overlap badly. We felt since people are relying on it for data visualization we did not want to post process the MDS layouts by default in a way that could be misleading. But you can turn on overlap removal (neato or sfdp -Goverlap=false). We really should document all this in one place that is easier to find and understand. About the external shape loader issue - this code is somewhat centralized in graphviz/lib/gvc/gvusershape.c and in gvrender.c which calls it, and I thought it could be disabled at compile time (because we did address the security concerns at one point) and there's a lot of other machinery to control compile time features. Maybe John Ellson can comment here. Kudos to John for recognizing the problems with the shape loader as soon as I proposed it but apparently that didn't stop us at the time. Stephen North

fibo 10y ago

Here It Is a package that implements some hypergraphs features. http://g14n.info/iper/ http://g14n.info/iper/