Calculating SimRank using NetworkX?

Question

I was wondering how can we can use the python module networkX to implement SimRank to compare the similarity of 2 nodes? I understand that networkX provides methods for looking at neighbors, and link analysis algorithms such as PageRank and HITS, but is there one for SimRank?

Examples, tutorials are welcomed too!

Answer 1

Update I implemented an networkx_addon library. SimRank is included in the library. Check out: https://github.com/hhchen1105/networkx_addon for details.

Sample Usage:

    >>> import networkx
    >>> import networkx_addon
    >>> G = networkx.Graph()
    >>> G.add_edges_from([('a','b'), ('b','c'), ('a','c'), ('c','d')])
    >>> s = networkx_addon.similarity.simrank(G)

You may obtain the similarity score between two nodes (say, node 'a' and node 'b') by

    >>> print s['a']['b']

---

SimRank is a vertex similarity measure. It computes the similarity between two nodes on a graph based on the topology, i.e., the nodes and the links of the graph. To illustrate SimRank, let's consider the following graph, in which a, b, c connect to each other, and d is connected to d. How a node a is similar to a node d, is based on how a's neighbor nodes, b and c, similar to d's neighbors, c.

    +-------+
    |       |
    a---b---c---d

As seen, this is a recursive definition. Thus, SimRank is recursively computed until the similarity values converges. Note that SimRank introduces a constant r to represents the relative importance between in-direct neighbors and direct neighbors. The formal equation of SimRank can be found here.

The following function takes a networkx graph $G$ and the relative imporance parameter r as input, and returns the simrank similarity value sim between any two nodes in G. The return value sim is a dictionary of dictionary of float. To access the similarity between node a and node b in graph G, one can simply access sim[a][b].

    def simrank(G, r=0.9, max_iter=100):
      # init. vars
      sim_old = defaultdict(list)
      sim = defaultdict(list)
      for n in G.nodes():
        sim[n] = defaultdict(int)
        sim[n][n] = 1
        sim_old[n] = defaultdict(int)
        sim_old[n][n] = 0

      # recursively calculate simrank
      for iter_ctr in range(max_iter):
        if _is_converge(sim, sim_old):
          break
        sim_old = copy.deepcopy(sim)
        for u in G.nodes():
          for v in G.nodes():
            if u == v:
              continue
            s_uv = 0.0
            for n_u in G.neighbors(u):
              for n_v in G.neighbors(v):
                s_uv += sim_old[n_u][n_v]
            sim[u][v] = (r * s_uv / (len(G.neighbors(u)) * len(G.neighbors(v))))
      return sim

    def _is_converge(s1, s2, eps=1e-4):
      for i in s1.keys():
        for j in s1[i].keys():
          if abs(s1[i][j] - s2[i][j]) >= eps:
            return False
      return True

To calculate the similarity values between nodes in the above graph, you can try this.

    >> G = networkx.Graph()
    >> G.add_edges_from([('a','b'), ('b', 'c'), ('c','a'), ('c','d')])
    >> simrank(G)

You'll get

    defaultdict(<type 'list'>, {'a': defaultdict(<type 'int'>, {'a': 0, 'c': 0.62607626807407868, 'b': 0.65379221101693585, 'd': 0.7317028881451203}), 'c': defaultdict(<type 'int'>, {'a': 0.62607626807407868, 'c': 0, 'b': 0.62607626807407868, 'd': 0.53653543888775579}), 'b': defaultdict(<type 'int'>, {'a': 0.65379221101693585, 'c': 0.62607626807407868, 'b': 0, 'd': 0.73170288814512019}), 'd': defaultdict(<type 'int'>, {'a': 0.73170288814512019, 'c': 0.53653543888775579, 'b': 0.73170288814512019, 'd': 0})})

Let's verify the result by calculating similarity between, say, node a and node b, denoted by S(a,b).

S(a,b) = r * (S(b,a)+S(b,c)+S(c,a)+S(c,c))/(2*2) = 0.9 * (0.6538+0.6261+0.6261+1)/4 = 0.6538,

which is the same as our calculated S(a,b) above.

For more details, you may want to checkout the following paper:

G. Jeh and J. Widom. SimRank: a measure of structural-context similarity. In KDD'02 pages 538-543. ACM Press, 2002.

Answer 2

No, simrank is not implemented in networkx.

If you were to add this to networkx, you could shorten the code given by user1036719 by using numpy and itertools:

def simrank(G, r=0.8, max_iter=100, eps=1e-4):

    nodes = G.nodes()
    nodes_i = {k: v for(k, v) in [(nodes[i], i) for i in range(0, len(nodes))]}

    sim_prev = numpy.zeros(len(nodes))
    sim = numpy.identity(len(nodes))

    for i in range(max_iter):
        if numpy.allclose(sim, sim_prev, atol=eps):
            break
        sim_prev = numpy.copy(sim)
        for u, v in itertools.product(nodes, nodes):
            if u is v:
                continue
            u_ns, v_ns = G.predecessors(u), G.predecessors(v)

            # evaluating the similarity of current iteration nodes pair
            if len(u_ns) == 0 or len(v_ns) == 0: 
                # if a node has no predecessors then setting similarity to zero
                sim[nodes_i[u]][nodes_i[v]] = 0
            else:                    
                s_uv = sum([sim_prev[nodes_i[u_n]][nodes_i[v_n]] for u_n, v_n in itertools.product(u_ns, v_ns)])
                sim[nodes_i[u]][nodes_i[v]] = (r * s_uv) / (len(u_ns) * len(v_ns))


    return sim

Then, taking the toy example from the SimRank paper (University graph), reproduces the paper results:

G = networkx.DiGraph()
G.add_edges_from([('1','2'), ('1', '4'), ('2','3'), ('3','1'), ('4', '5'), ('5', '4')])
pprint(simrank(G).round(3))

Which outputs:

array([[ 1.   ,  0.   ,  0.   ,  0.034,  0.132],
       [ 0.   ,  1.   ,  0.   ,  0.331,  0.042],
       [ 0.   ,  0.   ,  1.   ,  0.106,  0.414],
       [ 0.034,  0.331,  0.106,  1.   ,  0.088],
       [ 0.132,  0.042,  0.414,  0.088,  1.   ]])