Cannot change member variable via std::set iterator

I have two friend classes:

class Node {
private:
    unsigned id_;
    bool marked_;
    std::set<Node> neighbors_;

    friend class Graph;
    ......

public:
bool operator == (const Node & other) const { return id_ == other.id(); }
bool operator < (const Node & other) const { return id_ < other.id(); }
......
};

class Graph {
private:
    std::set<Node> vertices_{};
    void reach_set_helper(unsigned id, std::vector<unsigned> &reach_set);
    ......
};

I am trying to create a function that can first find a specific node in the graph vertices_, say node v. And then I want to change the marked_ property of the neighbors of v. To find v, I have to use std::find() method. However, the return iterator of this method does not allow me to change member variables of neighbors. This is what I have tried:

Node & s = v;
std::set<Node>::iterator pos = vertices_.find(s);
const std::set<Node> & neighbors = (pos->neighbors_);
for (std::set<Node>::iterator it = neighbors.begin(); it != neighbors.end(); it++) {
    if (!it->is_marked()) {
        reach_set.push_back(it->id());
        it->set_marked(true);
    }
    this->reach_set_helper(it->id(), reach_set);
}

Notice here that I have to use const std::set<Node> & neighbors because I want to change neighbors in place. But I can't change neighbors of v through const iterator it, so this method does not work. I have a way to make changes to vertices_ by erase and copy back through iterator, that's not a problem. But here it is different, I am doing operations on elements of vertices, which is another set. Any suggestions?