<｜begin▁of▁sentence｜>### Instruction:
In an experimental setting where we simulate a complex network using a stochastic model, we've generated a graph with 100 vertices, invoking the principles of random graph theory. With an edge formation probability set at 0.05, our graph model emulates the random interconnections one might find in real-world networks.

Within this network, we've identified three nodes – 10, 20, and 30 – that represent critical elements in our study, perhaps akin to key species in an ecosystem or pivotal chemical agents in a reaction network.

To advance our understanding of the graph's structure and the potential influence of these critical nodes, we are considering employing an algorithmic technique from the domain of spectral graph theory. Specifically, we're interested in utilizing the SpectralVertexNomination class from the graspologic API to categorize the remaining 97 nodes. Our goal is to arrange these nodes in a hierarchy, evaluating and quantifying their connectivity or relevance concerning nodes 10, 20, and 30.

Can we reformulate our inquiry to ask how, within this context of a synthesized network, we may harness the capabilities of the SpectralVertexNomination methodology to systematically rank all subsidiary nodes based on their spectral relationship to our nodes of particular interest? The integrity of our query should persist, focusing solely on the rephrasing of the method by which we might conduct such an analysis.
### Response:
import networkx as nx
from graspologic.nominate import SpectralVertexNomination
from graspologic.embed import AdjacencySpectralEmbed

# Generate a random graph
G = nx.fast_gnp_random_graph(n=100, p=0.05)

# Assume we somehow know that nodes 10, 20, 30 are of a particular type we're interested in
seed_vertices = [10, 20, 30]

# Spectral Vertex Nomination using Adjacency Spectral Embedding
vertex_nomination = SpectralVertexNomination(input_graph=True, embedder=AdjacencySpectralEmbed(), n_neighbors=None, metric='euclidean')

nomination_list, distance_matrix = vertex_nomination.fit_predict(X=nx.to_numpy_array(G), y=seed_vertices)

print('Nomination List:\n', nomination_list)
print('Distance Matrix:\n', distance_matrix)
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