<｜begin▁of▁sentence｜>### Instruction:
Imagine you're architecting a web infrastructure where various paths represent connections between servers. The tuples in your 'paths' dataset function like the virtual cables connecting these servers. As a forward-thinking web developer, you're interested in evaluating the robustness of your network under different levels of failure tolerance.

You've decided to model your server network using NetworkX. Now, you need to analyze the stability of your system by determining the k-edge-connected components, specifically for scenarios where k equals 1, 2, 3, and 4. This would mean seeing how your network would behave under one, two, three, or four connection failures.

To achieve this, you plan to utilize the EdgeComponentAuxGraph feature from the NetworkX library. This tool will help you simulate and understand which clusters of servers would remain interconnected despite a certain number of connection losses. How would you proceed with this analysis using NetworkX, given the connection paths of your server network embedded in your 'paths' variable? Remember, you're looking to retain the exact meaning behind your question but want it to relate directly to the task at hand as a web developer.
### Response:
import itertools as it
from networkx.utils import pairwise
from networkx.algorithms.connectivity import EdgeComponentAuxGraph

# Define a graph with multiple levels of k-edge-connected components
paths = [
    (1, 2, 3, 4, 1, 3, 4, 2),  # a 3-edge-connected component (a 4 clique)
    (5, 6, 7, 5),              # a 2-edge-connected component (a 3 clique)
    (1, 5),                    # combine the first two ccs into a 1-edge-connected component
    (0,),                      # add an additional disconnected 1-edge-connected component
]
G = nx.Graph()
G.add_nodes_from(it.chain(*paths))
G.add_edges_from(it.chain(*[pairwise(path) for path in paths]))

# Construct the auxiliary graph
aux_graph = EdgeComponentAuxGraph.construct(G)

# Query k-edge-connected components
components_k1 = sorted(map(sorted, aux_graph.k_edge_components(k=1)))
components_k2 = sorted(map(sorted, aux_graph.k_edge_components(k=2)))
components_k3 = sorted(map(sorted, aux_graph.k_edge_components(k=3)))
components_k4 = sorted(map(sorted, aux_graph.k_edge_components(k=4)))

# Output the components
print('1-edge-connected components:', components_k1)
print('2-edge-connected components:', components_k2)
print('3-edge-connected components:', components_k3)
print('4-edge-connected components:', components_k4)
<|EOT|>
