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(1) Is there any tool by which we can know the current number of nodes (participants as payer/payee) and number of channels in Lightning Network with an accurate estimation?

(2) As a more advanced query, is it also possible to know the topology of this graph? (vertices as participants and edges as channels)

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As for (1) yes you can use tools like the recksplorer as suggested by JBaczuk but you can also query the gossip store of your lightning node directly. Each node has to know the entire topology of the network for source based routing. C-Lightning provides a command line interface to retrieve this data.

However nodes can have private channels or be even private themselves. So with the Recksplorer which utilizes the Gossipstore and with the Gossip store you will only get a fraction of the Lightning Network. People Using Bitcoin Lightning Wallet or Eclaire mobile Wallet will currently not be public.

As for (2) I have created an autopilot for c-lightning with the python client library which analyzed the graph with the networkx module of python. you can check out the code at this pull request: https://github.com/ElementsProject/lightning/pull/1888 This should give you a starting point for doing more statistics about the topology.

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You can try recksplorer at https://rompert.com/recksplorer/ to get a node count, or drill down on individual nodes:

Recksplorer

Source is at: https://github.com/chemicstry/recksplorer

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Since https://rompert.com/recksplorer/ apparently is no longer available (screen shot), I add the following link including a mostly visual collection of the Lightning Network:

https://gist.github.com/bretton/798ec38165ffabc719d91e0f4f67552d

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