A package to study complex networks based on the temporal evolution of their Dynamic Communicability and Flow.

Graph theory constitutes a widely used and established field providing powerful tools for the characterization of complex networks. However, the diversity of complex networks studied nowadays overcomes the capabilities of classical graph metrics (originally developed for binary adjacency matrices) to provide with information to understand networks and their function. Also, in several domains, networks are often inferred from real-data-forming dynamic systems and thus, their analysis requires a different angle. The tools given in this package allow to overcome those limitations for a variety of complex networks, specially those that are weighted and whose structure is associated with dynamical phenomena.

Dynamic Flow characterises the transient network response over time, as the network dynamics relax towards their resting-state after a pulse perturbation (either independent or correlated Gaussian noise) has been applied to selected nodes. On the other hand, Dynamic Communicability corresponds to the special case where uncorrelated Gaussian noise has initially been applied to all nodes. The behaviour of the interactions during this transition allows to uncover properties of networks and their function. From a computational point of view dynamic communicability and flow are characterised by a series of matrices, encoding the temporal evolution of the pair-wise interactions between nodes.

• M. Gilson, N. Kouvaris, G. Deco and G. Zamora-Lopez "Framework based on communicability and flow to analyze complex networks" Phys. Rev. E 97, 052301 (2018).
• M. Gilson, N. Kouvaris, et al. "Analysis of brain network dynamics estimated from fMRI data: A new framework based on communicability and flow" bioRxiv (2018). DOI: 10.1101/421883

Installation of NetDynFlow is simple. An existing python distribution and the pip package manager need to be installed. If Python was installed via the Canopy or the Anaconda distributions, then pip is surely installed. To check, open a terminal and type:

$ pip --help

NetDynFlow is still not registered in PyPI (the Python Packages Index) and installation follows directly from GitHub. However, pip will automatically take care of the dependencies (see the requirements.txt file). There are two alternative manners to install: the easy and the lazy.

- The easy installation: Visit the GitHub repository https://github.com/gorkazl/NetDynFlow/ and click on the "Clone or download" button at the right hand side (the green button). Select "Download ZIP". Unzip the file, open a terminal and move to the folder, e.g.,

$ cd ~/Downloads/NetDynFlow-master/

Once on the folder that contains the setup.py file, type the following

$ pip install .

Do not forget the "." at the end which means "look for the setup.py file in the current directory." This will check for the dependencies and install NetDynFlow. To confirm the installation open an interactive session and try to import the library by typing import netdynflow.

NOTE: After installation the current folder "~/Downloads/NetDynFlow-master/" can be safely deleted, or moved somewhere else if you want to conserve the examples and the tests.

- The lazy installation: If git is also installed in your computer, then open a terminal and type:

$ pip install git+https://github.com/gorkazl/NetDynFlow.git@master

This will install the package, that is, the content in the folder netdynflow/. Other files (Examples/, README.md, LICENSE.txt, etc.) need to be downloaded manually, if wanted.

NOTE: If you are using Python 2 and Python 3 environments, NetDynFlow needs to be installed in each of the environments.

The package is organised into two modules:

• core.py: Functions to obtain the temporal evolution of dynamic communicability and flow.
• metrics.py: Network descriptors to analyse the temporal evolution of the dynamic communicability and flow.

To see the list of all functions available use the standard help in an interactive session, e.g.,

>>> import netdynflow as ndf
>>> ndf.core?
>>> ndf.metrics?

NOTE: Importing NetDynflow brings all functions in the two modules into its local namespace. Thus, functions in each of the two modules are called as ndf.func() instead of ndf.core.func() or ndf.metrics.func(). Details of each function is also found using the usual help, e.g.,

>>> ndf.DynCom?
>>> ndf.Diversity?

Create a simple weighted network of N = 4 nodes (a numpy array) and compute its dynamic communicability over time:

>>> net = np.array((	(0, 1.2, 0, 0),
						(0, 0, 1.1, 0),
						(0, 0, 0, 0.7),
						(1.0, 0, 0, 0)), float)
>>> tau = 0.8
>>> dyncom = ndf.DynCom(net, tau, tmax=15, timestep=0.01)

The resulting variable dyncom is an array of rank-3 with dimensions ((tmax x tstep) x N x N) containing tmax / tstep = 1500 matrices of size 4 x 4, each describing the state of the network at a given time step.

NOTE: NetDynFlow employs the convention in graph theory that rows of the connectivity matrix encode the outputs of the node. That is, net[i,j] = 1 implies that the node in row i projects over the node in column j.

Now we calculate the total communicability and diversity of the network over time as:

>>> totalcom = ndf.TotalEvolution(dyncom)
>>> divers = ndf.Diversity(dyncom)

totalcom and divers are two numpy arrays of length (tmax / tsteps) = 1500.

Copyright 2019, Gorka Zamora-López, Matthieu Gilson and Nikos E. Kouvaris. E-mail: [email protected]

Licensed under the Apache License, Version 2.0 (the "License"); you may not use this software except in compliance with the License. You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

First release of NetDynFlow (Beta).