# NetComp: Python Network Comparison

(NetComp source code is available on GitHub.)

As I worked on my research on network data analysis, it became clear that there was a need for a Python library that implemented the analytical tools I was interested in. The ubiquitous NetworkX package contains quite a few metrics, but since it is such a popular package, it does not implement research algorithms until they reach a high level of maturity. I decided to build NetComp in order to bridge this gap. You can get it locally via

pip install netcomp


## Usage

In this demo, we’ll compare two Erdos-Renyi random graphs, and then compare an Erdos-Renyi graph to a planted partition graph. We expect that the distance between the latter pair should be greater than the distance between the former, as they are drawn from distinct models. We’ll use graphs of size 100, and set the parameters for the planted partition graphso that it has the same volume as the Erdos-Renyi graph.

>>> import netcomp as nc
... import networkx as nx

>>> G1 = nx.erdos_renyi_graph(100,0.1)
... G2 = nx.erdos_renyi_graph(100,0.1)
... G3 = nx.planted_partition_graph(2,50,0.19,0.01)

>>> A1,A2,A3 = [nx.adjacency_matrix(G) for G in [G1,G2,G3]]

>>> d0 = nc.lambda_dist(A1,A2,kind='laplacian_norm')
... d1 = nc.lambda_dist(A1,A3,kind='laplacian_norm')

>>> print({:.03f}.format(d1/d0))

3.548


The first principal eigenvalue of the adjacency matrix is a signature for two-community structure. Therefore, we expect performance to improve when we compare using only the first principal eigenvalue. Let’s check:

>>> d0 = nc.lambda_dist(A1,A2,kind='laplacian_norm',k=2)
... d1 = nc.lambda_dist(A1,A3,kind='laplacian_norm',k=2)

>>> print({:.03f}.format(d1/d0))

37.409


Performance increases by an order of magnitude. This sort of comparison can be done using the adjacency or Laplacian spectrum (set using the kind keyword argument), or via non-spectral distances. A complete list of included graph distances is provided below.

## Design Principles

The guiding principles behind the library are:

• Flexibility. Graphs can be input as either dense NumPy or sparse SciPy arrays/matrices.

• Speed. All algorithms must run in linear or near-linear time, and have efficient implementations leveraging sparse data structures when appropriate.

I opted to utilize arrays as the fundamental data structure of the library, rather than creating a custom Graph class or extending that of NetworkX. This has some drawbacks in terms of flexibility; NetworkX supports arbitrary vertex labels and edge properties, whereas using an array implicitly assumes integer vertex labels. The benefit of such an approach, however, is that we avoid the costly conversion to and from arrays, which is the format in which networks are often encountered in application.

## Distances Included in Library

The graph distances included are:

• Resistance Distance. Compares networks by looking at differences in their resistance structure. Also included is the renormalized resistance distance, which extends this metric to disconnected networks and networks of different sizes. Our paper studying the properties of the renormalized resistance distance can be found on the arXiv.

• DeltaCon. Another popular distance, DeltaCon is comparable to the resistance distance. It uses an alternative method of calculating node affinities, and then looks compares the resulting structure for the two networks in question.

• NetSimile. NetSimile examines the statistics of a variety of local network measures, and compares statistical properties (mean, standard deviation, etc.) of these measures. We implement it as originally proposed, but this method is highly extensible, and can include any network measure of interest.

• Spectral Distances. These distances compare graphs via looking at the spectrum (eigenvalues) of the various matrix representations of the graph. The most common matrix representation of the graph is the adjacency matrix, but this method is also implemented on the Laplacian, both in its normalized and unnormalized forms.

• Graph Edit Distance. Trivial to calculate, and implemented in NetworkX, but we often use it as a baseline, so I included it.

Note that the vertex affinity matrices calculated by the resistance distance and DeltaCon are themselves directly accessible. Also useful is a helper function _eigs which calculates the k largest (or smallest) eigenvalues of a symmetric matrix, using the appropriate method depending on the input data type (sparse or dense) and the number of eigenvalues to be calculated.1

## Future Work

The library is still in alpha, and is incomplete in its implementation of the various algorithms. In particular, it lacks the fast implementation of the resistance distance and DeltaCon, both of which are linear-time approximations of the quadratic-time exact algorithms. The resistance distance approximation in particular will rely on proper implementation of an algebraic multigrid solver for linear systems, which will be an involved process to build. I look forward to working on it.

1. This function must be imported directly via from netcomp.linalg import _eigs

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