<｜begin▁of▁sentence｜>### Instruction:
Given a kernel function with a mean expected degree 'c' that determines the probability of edge creation between nodes in a graph, can you generate a random graph with 10 nodes, label the nodes and edges, and produce a JSON object describing the graph?
### Response:
import networkx as nx

def integral(u, w, z):
    c = 1  # Mean expected degree
    return c * (z - w)

def root(u, w, r):
    c = 1  # Mean expected degree
    return r / c + w

# Create a random graph using the kernel functions defined above
n = 10  # Number of nodes in the graph
seed = 42  # Seed for the random number generator
graph = nx.random_kernel_graph(n, integral, root, seed=seed)

# Generate the JSON object
nodes_list = [{'id': node, 'label': f'node_{node}'} for node in graph.nodes()]
edges_list = [{'source': edge[0], 'target': edge[1], 'label': f'edge_{edge[0]}_{edge[1]}'} for edge in graph.edges()]
graph_json = {'graph nodes': nodes_list, 'graph edges': edges_list}

print(graph_json)
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