Foundations of Data Science

are likely to be balanced given that a ij are all between 0 and 1. In this case, there will
be O(1) groups only and the log factors disappear.
Another measure of density is based on similarities. Recall that the similarity between
objects represented by vectors (rows of A) is defined by their dot products. Thus, similarities
are entries of the matrix AA T . Define the average cohesion f(S) of a set S of rows
of A to be the sum of all pairwise dot products of rows in S divided by |S|. The average
cohesion of A is the maximum over all subsets of rows of the average cohesion of the subset.
Since the singular values of AA T are squares of singular values of A, we expect f(A)
to be related to σ 1 (A) 2 and d(A) 2 . Indeed it is. We state the following without proof.
Lemma 8.12 d(A) 2 ≤ f(A) ≤ d(A) log n. Also, σ 1 (A) 2 ≥ f(A) ≥ cσ 1(A) 2
log n .
f(A) can be found exactly using flow techniques as we will see later.
8.11 Community Finding and Graph Partitioning
Assume that data are nodes in a possibly weighted graph where edges represent some
notion of affinity between their endpoints. In particular, let G = (V, E) be a weighted
graph. Given two sets of nodes S and T , define
E(S, T ) = ∑ i∈S
j∈T
We then define the density of a set S to be
e ij .
d(S, S) =
E(S, S)
.
|S|
If G is an undirected graph, then d(S, S) can be viewed as the average degree in the
vertex-induced subgraph over S. The set S of maximum density is therefore the subgraph
of maximum average degree. Finding such a set can be viewed as finding a tight-knit
community inside some network. In the next section, we describe an algorithm for finding
such a set using network flow techniques.
8.11.1 Flow Methods
Here we consider dense induced subgraphs of a graph. An induced subgraph of a
graph consisting of a subset of the vertices of the graph along with all edges of the graph
that connect pairs of vertices in the subset of vertices. We show that finding an induced
subgraph with maximum average degree can be done by network flow techniques. This
is simply maximizing the density d(S, S) over all subsets S of the graph. First consider
the problem of finding a subset of vertices such that the induced subgraph has average
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