*   <li>Nodes being visited multiple times (if multiple paths exist from any start node to any
   *       node reachable from any start node)
   * </ul>
   *
   * <p><b>Performance notes</b>
   *
   * <ul>
   *   <li>Traversals require <i>O(n)</i> time (where <i>n</i> is the number of nodes reachable from
   *       the start node).

  int outDegree(N node);

  /**
   * Returns true if there is an edge that directly connects {@code nodeU} to {@code nodeV}. This is
   * equivalent to {@code nodes().contains(nodeU) && successors(nodeU).contains(nodeV)}.
   *
   * <p>In an undirected graph, this is equal to {@code hasEdgeConnecting(nodeV, nodeU)}.
   *
   * @since 23.0
   */
  @Override
  boolean hasEdgeConnecting(N nodeU, N nodeV);

  /**

      return true;
    }
    if (!(obj instanceof Graph)) {
      return false;
    }
    Graph<?> other = (Graph<?>) obj;

    return isDirected() == other.isDirected()
        && nodes().equals(other.nodes())
        && edges().equals(other.edges());
  }

  @Override
  public final int hashCode() {
    return edges().hashCode();
  }

  /** Returns a string representation of this graph. */

    MutableGraph<N> subgraph =
        (nodes instanceof Collection)
            ? GraphBuilder.from(graph).expectedNodeCount(((Collection) nodes).size()).build()
            : GraphBuilder.from(graph).build();
    for (N node : nodes) {
      subgraph.addNode(node);
    }
    for (N node : subgraph.nodes()) {
      for (N successorNode : graph.successors(node)) {

      @Override
      public Set<N> adjacentNodes(N node) {
        return AbstractValueGraph.this.adjacentNodes(node);
      }

      @Override
      public Set<N> predecessors(N node) {
        return AbstractValueGraph.this.predecessors(node);
      }

      @Override
      public Set<N> successors(N node) {
        return AbstractValueGraph.this.successors(node);
      }

      @Override

   * Returns true if there is an edge that directly connects {@code nodeU} to {@code nodeV}. This is
   * equivalent to {@code nodes().contains(nodeU) && successors(nodeU).contains(nodeV)}.
   *
   * <p>In an undirected graph, this is equal to {@code hasEdgeConnecting(nodeV, nodeU)}.
   *
   * @since 23.0
   */
  boolean hasEdgeConnecting(N nodeU, N nodeV);

  /**

   * given nodes is not present in this network.
   *
   * @since 33.1.0
   */
  protected final <T> Set<T> nodePairInvalidatableSet(Set<T> set, N nodeU, N nodeV) {
    return InvalidatableSet.of(
        set,
        () -> nodes().contains(nodeU) && nodes().contains(nodeV),
        () -> String.format(NODE_PAIR_REMOVED_FROM_GRAPH, nodeU, nodeV));
  }

    GraphConnections<N, V> connectionsU = nodeConnections.get(nodeU);
    return (connectionsU != null) && connectionsU.successors().contains(nodeV);
  }

  private final @Nullable V edgeValueOrDefaultInternal(N nodeU, N nodeV, @Nullable V defaultValue) {
    GraphConnections<N, V> connectionsU = nodeConnections.get(nodeU);
    V value = (connectionsU == null) ? null : connectionsU.value(nodeV);

  int outDegree(N node);

  /**
   * Returns true if there is an edge that directly connects {@code nodeU} to {@code nodeV}. This is
   * equivalent to {@code nodes().contains(nodeU) && successors(nodeU).contains(nodeV)}.
   *
   * <p>In an undirected graph, this is equal to {@code hasEdgeConnecting(nodeV, nodeU)}.
   *
   * @since 23.0
   */
  @Override
  boolean hasEdgeConnecting(N nodeU, N nodeV);

  /**

        writeElastisearchJsonApi(zos, id, "nodes", "stats");
        writeElastisearchJsonApi(zos, id, "nodes", "_all");
        writeElastisearchJsonApi(zos, id, "nodes", "usage");
        writeElastisearchJsonApi(zos, id, "remote", "info");
        writeElastisearchJsonApi(zos, id, "tasks", "");
        writeElastisearchJsonApi(zos, id, "nodes", "hot_threads");
    }

    /**