## The graph

Of course, before you can do a whole lot of anything, you've got to have a graph to work with. To keep things dead simple (and breaking from what I normally do) I'm going to draw an arbitrary graph (and by draw I mean draw). Here it is:

This can be represented as a tab-separated-values set of adjacency pairs in a file called 'graph.tsv':

$: cat graph.tsv

A C

B C

C D

C E

E F

C H

G H

C G

D E

A G

B G

I don't think it gets any simpler.

## The Measure

Now, the idea here is to compute the similarity between nodes, eg similarity(A,B). There's already a well defined measure for this called the jaccard similarity. The key take away is to notice that:

## The Pig

This can be broken into a set of Pig operations quite easily actually:

### load data

Remember, I saved the data into a file called 'graph.tsv'. So:

edges = LOAD 'graph.tsv' AS (v1:chararray, v2:chararray);

edges_dup = LOAD 'graph.tsv' AS (v1:chararray, v2:chararray);

It is necessary, but still a hack, to load the data twice at the moment. This is because self-intersections (which I'll be doing in a moment) don't work with Pig 0.8 at the moment.

### augment with sizes

Now I need the 'size' of each of the sets. In this case the 'sets' are the list of nodes each node links to. So, all I really have to do is calculate the number of outgoing links:

grouped_edges = GROUP edges BY v1;

aug_edges = FOREACH grouped_edges GENERATE FLATTEN(edges) AS (v1, v2), COUNT(edges) AS v1_out;

grouped_dups = GROUP edges_dup BY v1;

aug_dups = FOREACH grouped_dups GENERATE FLATTEN(edges_dup) AS (v1, v2), COUNT(edges_dup) AS v1_out;

Again, if self-intersections worked, the operation of counting the outgoing links would only have to happen once. Now I've got a handle on the |A| and |B| parts of the equation above.

### intersection

Next I'm going to compute the intersection using a Pig join:

edges_joined = JOIN aug_edges BY v2, aug_dups BY v2;

intersection = FOREACH edges_joined {

--

-- results in:

-- (X, Y, |X| + |Y|)

--

added_size = aug_edges::v1_out + aug_dups::v1_out;

GENERATE

aug_edges::v1 AS v1,

aug_dups::v1 AS v2,

added_size AS added_size

;

};

Notice I'm adding the individual set sizes. This is to come up the |A| + |B| portion of the denominator in the jaccard index.

### intersection sizes

In order to compute the size of the intersection I've got to use a Pig GROUP and collect all the elements that matched on the JOIN:

intersect_grp = GROUP intersection BY (v1, v2);

intersect_sizes = FOREACH intersect_grp {

--

-- results in:

-- (X, Y, |X /\ Y|, |X| + |Y|)

--

intersection_size = (double)COUNT(intersection);

GENERATE

FLATTEN(group) AS (v1, v2),

intersection_size AS intersection_size,

MAX(intersection.added_size) AS added_size -- hack, we only need this one time

;

};

There's a hack in there. The reason is that 'intersection.added_size' is a Pig data bag containing some number of identical tuples (equal to the intersection size). Each of these tuples is the 'added_size' from the previous step. Using MAX is just a convenient way of pulling out only one of them.

### similarity

And finally, I have all the pieces in place to compute the similarities:

similarities = FOREACH intersect_sizes {

--

-- results in:

-- (X, Y, |X /\ Y|/|X U Y|)

--

similarity = (double)intersection_size/((double)added_size-(double)intersection_size);

GENERATE

v1 AS v1,

v2 AS v2,

similarity AS similarity

;

};

That'll do it. Here's the full script for completeness:

edges = LOAD '$GRAPH' AS (v1:chararray, v2:chararray);

edges_dup = LOAD '$GRAPH' AS (v1:chararray, v2:chararray);

--

-- Augment the edges with the sizes of their outgoing adjacency lists. Note that

-- if a self join was possible we would only have to do this once.

--

grouped_edges = GROUP edges BY v1;

aug_edges = FOREACH grouped_edges GENERATE FLATTEN(edges) AS (v1, v2), COUNT(edges) AS v1_out;

grouped_dups = GROUP edges_dup BY v1;

aug_dups = FOREACH grouped_dups GENERATE FLATTEN(edges_dup) AS (v1, v2), COUNT(edges_dup) AS v1_out;

--

-- Compute the sizes of the intersections of outgoing adjacency lists

--

edges_joined = JOIN aug_edges BY v2, aug_dups BY v2;

intersection = FOREACH edges_joined {

--

-- results in:

-- (X, Y, |X| + |Y|)

--

added_size = aug_edges::v1_out + aug_dups::v1_out;

GENERATE

aug_edges::v1 AS v1,

aug_dups::v1 AS v2,

added_size AS added_size

;

};

intersect_grp = GROUP intersection BY (v1, v2);

intersect_sizes = FOREACH intersect_grp {

--

-- results in:

-- (X, Y, |X /\ Y|, |X| + |Y|)

--

intersection_size = (double)COUNT(intersection);

GENERATE

FLATTEN(group) AS (v1, v2),

intersection_size AS intersection_size,

MAX(intersection.added_size) AS added_size -- hack, we only need this one time

;

};

similarities = FOREACH intersect_sizes {

--

-- results in:

-- (X, Y, |X /\ Y|/|X U Y|)

--

similarity = (double)intersection_size/((double)added_size-(double)intersection_size);

GENERATE

v1 AS v1,

v2 AS v2,

similarity AS similarity

;

};

DUMP similarities;

## Results

Run it:

$: pig -x local jaccard.pig

...snip...

(A,A,1.0)

(A,B,1.0)

(A,C,0.2)

(B,A,1.0)

(B,B,1.0)

(B,C,0.2)

(C,A,0.2)

(C,B,0.2)

(C,C,1.0)

(C,D,0.25)

(C,G,0.25)

(D,C,0.25)

(D,D,1.0)

(E,E,1.0)

(G,C,0.25)

(G,G,1.0)

And it's done! Hurray. Now, go make a recommender system or something.

To get around the hack couldn't you use 'added_size' in the group?

ReplyDelete> GROUP intersection BY (v1, v2, added_size);

Since it has a constant value for all occurrences of a given (v1, v2) pair this should give you the same grouping as on (v1,v2) but now you don't have to pull the size out of the bag.

Many thanks for this. I'm doing something similar for a tutorial I'm writing for the Hadoop UK Users' Group, and it would have taken much longer without your example to follow.

ReplyDeleteOne suggestion -- maybe it would be a bit faster to load the edges file once, do the degree counting once, and then do a

edges_dup = foreach edges generate *;

Re. Kurt's comment, I once wrote a "First" UDF for exactly these cases -- where you know all the elements in the bag are the same, so finding their MAX (or copying into the grouping key) is just waited cycles...

Hah, not just me:

ReplyDeletehttps://github.com/linkedin/datafu/blob/master/src/java/datafu/pig/bags/FirstTupleFromBag.java

Great and Useful Article.

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