updated references and fixed typos

README.md

@@ -10,7 +10,7 @@
 # *PyGenStability*
 
 This ``python`` package is designed for multiscale community detection with Markov Stability (MS) analysis [1, 2] and allows researchers to identify robust network partitions at different resolutions. It implements several variants of the MS cost functions that are based on graph diffusion processes to explore the network (see illustration below). Whilst primarily built for MS, the internal architecture of *PyGenStability* has been designed to solve for a wide range of clustering cost functions since it is based on optimising the so-called generalized Markov Stability function [3]. To maximize the generalized Markov Stability cost function, *PyGenStability* provides a convenient ``python`` interface for ``C++`` implementations of Louvain [4] and Leiden [5] algorithms.
-We further provide specific analysis tools to process and analyse the results from multiscale community detection, and to facilitate the autmatic selection of robust partitions [6]. *PyGenStability* is accompanied by a software paper that further details the implementation, result analysis, benchmarks and applications [7].
+We further provide specific analysis tools to process and analyse the results from multiscale community detection, and to facilitate the automatic selection of robust partitions [6]. *PyGenStability* is accompanied by a software paper that further details the implementation, result analysis, benchmarks and applications [7].
 
 ![illustration](docs/artwork/diffusion_schematic.png)
 
@@ -35,7 +35,7 @@ pip install .
 ```
 using a fresh `virtualenv` in python3 may be recommanded to avoid conflict of python packages. 
 
-To use plotly for interacting plos in browser, install this package with 
+To use plotly for interacting plots in browser, install this package with 
 ```
 pip install .[plotly]
 ```
@@ -52,15 +52,15 @@ pip install .[all]
 
 ## Using the code
 
-The code is simple to run with the default settings. We can input our graph (of type scipy.csgraph), run a scan in scales with a chosen Markov Stability construcotr and plot the results in a summary figure presenting different partition quality measures across scales (values of MS cost function, number of communities, etc.) with indication of optimal scales.
+The code is simple to run with the default settings. We can input our graph (of type scipy.csgraph), run a scan in scales with a chosen Markov Stability constructor and plot the results in a summary figure presenting different partition quality measures across scales (values of MS cost function, number of communities, etc.) with indication of optimal scales.
 
 ```
 import pygenstability as pgs
 results = pgs.run(graph)
 pgs.plot_scan(results)
 ```
 
-Altough it is enforced in the code, it is advised to set environement variables
+Although it is enforced in the code, it is advised to set environment variables
 ```
 export OPENBLAS_NUM_THREADS=1
 export OMP_NUM_THREADS=1
@@ -69,7 +69,7 @@ export NUMEXPR_MAX_THREADS=1
 to ensure numpy does not use multi-threadings, which may clash with the parallelisation and slow down the computation. 
 
 There are a variety of further choices that user can make that will impact the partitioning, including:
-- Constructor: Generalized Markov Stability requires the user to input a quality matrix and associated null models. We provide an object oriented module to write user-defined constructors for these objects, with several already implemented (see `pygenstability/constructors.py` for some classic examples).
+- Constructor: Generalized Markov Stability requires the user to input a quality matrix and associated null models. We provide an object-oriented module to write user-defined constructors for these objects, with several already implemented (see `pygenstability/constructors.py` for some classic examples).
 - Generalized Markov Stability maximizers: To maximize the NP-hard optimal generalized Markov Stability we interface with two algorithms: (i) Louvain and (ii) Leiden.
 
 While Louvain is defined as the default due to its familiarity within the research community, Leiden is known to produce better partitions and can be used by specifying the run function.
@@ -78,11 +78,11 @@ While Louvain is defined as the default due to its familiarity within the resear
 results = pgs.run(graph, method="leiden")
 ```
 
-There are also additional postprocessing and analysis functions, including:
+There are also additional post-processing and analysis functions, including:
 - Plotting via matplotlib and plotly (interactive).
 - Automated optimal scale selection.
 
-Optimal scale selection [6] is performed by default with the run function but can be repeated with different parameters if needed, see `pygenstability/optimal_scales.py`. To reduce noise, e.g., one can increase the parameter values for `block_size` and `window_size`. The optimial network partitions can then be plotted given a NetworkX nx_graph.
+Optimal scale selection [6] is performed by default with the run function but can be repeated with different parameters if needed, see `pygenstability/optimal_scales.py`. To reduce noise, e.g., one can increase the parameter values for `block_size` and `window_size`. The optimal network partitions can then be plotted given a NetworkX nx_graph.
 
 ```
 results = pgs.identify_optimal_scales(results, block_size=10, window_size=5)
@@ -94,12 +94,12 @@ pgs.plot_optimal_partitions(nx_graph, results)
 We provide an object-oriented module for constructing quality matrices and null models in `pygenstability/constructors.py`. Various constructors are implemented for different types of graphs:
 
 - `linearized` based on linearized MS for large undirected weighted graphs [2]
-- `continuous_combinatorial` based on combinatorial Lablacian for undirected weighted graphs [2]
+- `continuous_combinatorial` based on combinatorial Laplacian for undirected weighted graphs [2]
 - `continuous_normalized` based on random-walk normalized Laplacians for undirected weighted graphs [2]
 - `signed_modularity` based on signed modularity for large signed graphs [8]
 - `signed_combinatorial` based on signed combinatorial Laplacian for signed graphs [3]
 - `directed` based on random-walk Laplacian with teleportation for directed weighted graphs [2]
-- `linearized_directed` based on random-walk Laplacian with teleportation for large  directed weighted graphs
+- `linearized_directed` based on random-walk Laplacian with teleportation for large directed weighted graphs
 
 For the computationally efficient analysis of **large** graphs we recommend using the `linearized`, `linearized_directed` or `signed_modularity` constructors instead of `continuous_combinatorial`, `continuous_normalized`, `directed` or `signed_combinatorial` that rely on the computation of matrix exponentials.
 
@@ -108,13 +108,13 @@ For those of you that wish to implement their own constructor, you will need to
 - take a scipy.csgraph `graph` and a float `time` as argument
 - return a `quality_matrix` (sparse scipy matrix) and a `null_model` (multiples of two, in a numpy array)
 
-## Contributers
+## Contributors
 
 - Alexis Arnaudon, GitHub: `arnaudon <https://github.com/arnaudon>`
 - Robert Peach, GitHub: `peach-lucien <https://github.com/peach-lucien>`
 - Dominik Schindler, GitHub: `d-schindler <https://github.com/d-schindler>`
 
-We are always on the look out for individuals that are interested in contributing to this open-source project. Even if you are just using *PyGenStability* and made some minor updates, we would be interested in your input. 
+We always look out for individuals that are interested in contributing to this open-source project. Even if you are just using *PyGenStability* and made some minor updates, we would be interested in your input. 
 
 ## Cite
 
@@ -171,7 +171,7 @@ Other examples can be found as jupyter-notebooks in the `examples/` directory, i
 * Example 6: Signed networks
 
 Finally, we provide applications to real-world networks in the `examples/real_examples/` directory, including:
-* Powergrid network
+* Power grid network
 * Protein structures
 
 
@@ -183,7 +183,7 @@ If you are interested in trying our other packages, see the below list:
 * [MSC](https://github.com/barahona-research-group/MultiscaleCentrality) : MultiScale Centrality: A scale dependent metric of node centrality.
 * [DynGDim](https://github.com/barahona-research-group/DynGDim) : Dynamic Graph Dimension: Computing the relative, local and global dimension of complex networks.
 * [RMST](https://github.com/barahona-research-group/RMST) : Relaxed Minimum Spanning Tree: Computing the relaxed minimum spanning tree to sparsify networks whilst retaining dynamic structure.
-* [StEP](https://github.com/barahona-research-group/StEP) :  Spatial-temporal Epidemiological Proximity: Characterising contact in disease outbreaks via a network model of spatial-temporal proximity.
+* [StEP](https://github.com/barahona-research-group/StEP) : Spatial-temporal Epidemiological Proximity: Characterising contact in disease outbreaks via a network model of spatial-temporal proximity.
 
 ## References
 
@@ -197,11 +197,11 @@ If you are interested in trying our other packages, see the below list:
 
 [5] V. A. Traag, L. Waltman, and N. J. van Eck, ‘From Louvain to Leiden: guaranteeing well-connected communities’, *Sci Rep*, vol. 9, no. 1, p. 5233, Mar. 2019, doi: 10.1038/s41598-019-41695-z.
 
-[6] D. J. Schindler, J. Clarke, and M. Barahona, ‘Multiscale Mobility Patterns and the Restriction of Human Movement’, *arXiv:2201.06323 [physics.soc-ph]*, Jan. 2023, Available: https://arxiv.org/abs/2201.06323
+[6] D. J. Schindler, J. Clarke, and M. Barahona, ‘Multiscale Mobility Patterns and the Restriction of Human Movement’, *Royal Society Open Science*, vol. 10, no. 10, p. 230405, Oct. 2023, doi: 10.1098/rsos.230405.
 
-[7] Preprint incoming ...
+[7] A. Arnaudon, D. J. Schindler, R. L. Peach, A. Gosztolai, M. Hodges, M. T. Schaub, and M. Barahona, ‘PyGenStability: Multiscale community detection with generalized Markov Stability‘, *arXiv pre-print*, Mar. 2023, doi: 10.48550/arXiv.2303.05385.
 
-[8] Gómez, S., Jensen, P., & Arenas, A. (2009). ‘Analysis of community structure in            networks of correlated data‘. *Physical Review E*, 80(1), 016114.
+[8] S. Gómez, P. Jensen, and A. Arenas, ‘Analysis of community structure in networks of correlated data‘. *Physical Review E*, vol. 80, no. 1, p. 016114, Jul. 2009, doi: 10.1103/PhysRevE.80.016114.
 
 ## Licence