GraphBuilder<N1> castBuilder = cast();
    return new ImmutableGraph.Builder<>(castBuilder);
  }

  /**
   * Specifies whether the graph will allow self-loops (edges that connect a node to itself).
   * Attempting to add a self-loop to a graph that does not allow them will throw an {@link
   * UnsupportedOperationException}.
   *
   * <p>The default value is {@code false}.
   */

    // whatever ordering the network's edges do, so ImmutableSortedMap is unnecessary even if the
    // input edges are sorted.
    ImmutableMap.Builder<E, N> edgeToReferenceNode = ImmutableMap.builder();
    for (E edge : network.edges()) {
      edgeToReferenceNode.put(edge, network.incidentNodes(edge).nodeU());
    }
    return edgeToReferenceNode.buildOrThrow();
  }

    ValueGraphBuilder<N1, V1> castBuilder = cast();
    return new ImmutableValueGraph.Builder<>(castBuilder);
  }

  /**
   * Specifies whether the graph will allow self-loops (edges that connect a node to itself).
   * Attempting to add a self-loop to a graph that does not allow them will throw an {@link
   * UnsupportedOperationException}.
   *
   * <p>The default value is {@code false}.
   */

            EndpointPair.ordered(1, 3),
            EndpointPair.ordered(1, 2))
        .inOrder();
  }

  /**
   * Populates the graph with nodes and edges in a star shape with node `1` in the middle.
   *
   * <p>Note that the edges are added in a shuffled order to properly test the effect of the
   * insertion order.
   */
  private void populateStarShapedGraph() {
    putEdge(2, 1);
    putEdge(1, 4);

            EndpointPair.unordered(1, 1),
            EndpointPair.unordered(1, 3))
        .inOrder();
  }

  /**
   * Populates the graph with nodes and edges in a star shape with node `1` in the middle.
   *
   * <p>Note that the edges are added in a shuffled order to properly test the effect of the
   * insertion order.
   */
  private void populateTShapedGraph() {
    putEdge(2, 1);
    putEdge(1, 4);

  }

  @Override
  public EndpointPair<N> incidentNodes(E edge) {
    N nodeU = checkedReferenceNode(edge);
    // requireNonNull is safe because checkedReferenceNode made sure the edge is in the network.
    N nodeV = requireNonNull(nodeConnections.get(nodeU)).adjacentNode(edge);
    return EndpointPair.of(this, nodeU, nodeV);
  }

  @Override

  /**
   * Returns the value associated with the edge connecting the origin node to {@code node}, or null
   * if there is no such edge.
   */
  @Nullable V value(N node);

  /** Remove {@code node} from the set of predecessors. */
  void removePredecessor(N node);

  /**
   * Remove {@code node} from the set of successors. Returns the value previously associated with
   * the edge connecting the two nodes.
   */

    N previousNode = incidentEdgeMap.remove(edge);
    // We're relying on callers to call this method only with an edge that's in the graph.
    return requireNonNull(previousNode);
  }

  @Override
  public void addInEdge(E edge, N node, boolean isSelfLoop) {
    if (!isSelfLoop) {
      addOutEdge(edge, node);
    }
  }

  @Override
  public void addOutEdge(E edge, N node) {

    assertThat(directedGraph.nodes()).isEmpty();
    assertThat(directedGraph.edges()).isEmpty();
    assertThat(directedGraph.addEdge(N1, N2, E12)).isTrue();
    assertThat(directedGraph.edgesConnecting(N1, N2)).isEqualTo(ImmutableSet.of(E12));
    assertThat(directedGraph.edgesConnecting(N2, N1)).isEmpty();

    // By default, parallel edges are not allowed.
    IllegalArgumentException e =

 *       locks---to each of the acquired locks, an edge from the soon-to-be-acquired lock is either
 *       verified or created.
 *   <li>If a new edge needs to be created, the outgoing edges of the acquired locks are traversed
 *       to check for a cycle that reaches the lock to be acquired. If no cycle is detected, a new
 *       "safe" edge is created.
 *   <li>If a cycle is detected, an "unsafe" (cyclic) edge is created to represent a potential