System minimal paths and cuts

  • Thread starter brad sue
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In summary, the conversation is about finding minimal paths and cuts for a given figure. The speaker has identified some cuts and paths but is unsure if they have missed any. They also mention needing an expression for the structure function using minimal paths and cuts. The responder explains that there should be ten paths because the graph is strongly connected and there is a path between each pair of vertices. The labels in the figure represent components of the system with logical connections.
  • #1
brad sue
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Hi,
please I need help with finding the minimal paths and cuts of the attached figure.

Here for the cuts I found: {1,6}, {7,3}, {6,4,2}, {2,5,7}
for the paths I get: {6,7},{1,2,3},{1,2,5,7},{1,4,7}
Do i miss some paths or cut please?

Also i cannot find the an expression for the structure function, using minimal paths and cuts.

Thank you
 

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  • #2
I am pretty sure there should be ten paths. What do the labels mean? Are they just labels for the edges or distances?
 
  • #3
hi,
the labels are components of the system with their logical connections. how do you know , i need to have 10 paths?
 
  • #4
The graph is strongly connected, there is a path between each two vertices. There are five vertices, so there are 4 paths from an arbitrary vertex to another (not including paths to itself), then from another arbitrary vertex there are 3 paths to other vertices not including the first counted path, etc. 4+3+2+1= 10.
 
  • #5
for your question. It seems like you have correctly identified the minimal cuts and paths for the attached figure. However, without knowing the specific structure and properties of the figure, it is difficult to determine if any paths or cuts are missing. It is possible that there could be alternative paths or cuts that could be considered minimal. I would recommend double checking your work and possibly consulting with other experts in the field for their input.

As for finding an expression for the structure function using minimal paths and cuts, this can be a complex and challenging task. The structure function is a mathematical representation of the relationship between nodes in a network or system, and it can vary depending on the specific system being studied. It may be helpful to break down the figure into smaller sub-networks and analyze the paths and cuts within each sub-network to build a larger structure function. Again, consulting with other experts in the field may also be beneficial in finding an appropriate expression for the structure function.
 

1. What are system minimal paths and cuts?

System minimal paths and cuts refer to the shortest paths and smallest sets of edges or nodes in a network or graph that connect specific points or divide the network into two parts. These paths and cuts are important for analyzing the structure and behavior of complex systems.

2. How are system minimal paths and cuts calculated?

System minimal paths and cuts can be calculated using algorithms such as Dijkstra's algorithm, which finds the shortest paths between two nodes, or the Ford-Fulkerson algorithm, which finds the maximum flow in a network. These algorithms use mathematical principles to efficiently determine the minimal paths and cuts in a system.

3. What is the significance of system minimal paths and cuts?

System minimal paths and cuts are significant because they help us understand the critical components and connections within a complex system. By identifying these minimal paths and cuts, we can determine the most efficient ways to navigate or manipulate the system, and also identify potential vulnerabilities or points of failure.

4. How are system minimal paths and cuts used in real-world applications?

System minimal paths and cuts have various applications, such as in transportation systems to find the shortest routes between two points, in computer networks to optimize data flow, and in social networks to identify key influencers. They are also used in biology to understand metabolic pathways and in economics to analyze supply chains.

5. Are there any limitations to using system minimal paths and cuts?

While system minimal paths and cuts are useful for analyzing complex systems, they do have limitations. These methods may not account for all factors or variables in a system and may not accurately predict behavior in dynamic systems. Additionally, the computation of these paths and cuts can be time-consuming for large networks.

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