# eugenio-valdano/threshold

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# Computing the Epidemic Threshold on Temporal Networks

A Python library for computing the epidemic threshold on temporal network, as explained in paper

Analytical Computation of The Epidemic Threshold on Temporal Networks

Valdano E, Ferreri L, Poletto C, Colizza V, Phys Rev X 5, 021005 2015.

When you use this code, please cite the above reference.

# Intro

This version has been restructured using the Python module setuptools. This means that the library will install just like one you would get through pip.

The package will look for the needed dependencies, and try to install them if necessary, through pip. However, there can be issues sometimes. For instance, if you're using Anaconda (and I recommend you do use it, especially if you're starting with Python), there are conflicts, as packages are usually installed through conda. So it is always better to check beforehand if you have the packages needed. They are listed in requirements.txt, along with their versions.

# Install

• cd inside it (where setup.py is),
• execute python setup.py test. You should get an output like this:
running test
running egg_info
writing requirements to Epidemic_Threshold.egg-info/requires.txt
writing Epidemic_Threshold.egg-info/PKG-INFO
writing top-level names to Epidemic_Threshold.egg-info/top_level.txt
writing manifest file 'Epidemic_Threshold.egg-info/SOURCES.txt'
running build_ext
copying build/lib.macosx-10.6-x86_64-2.7/threshold/utilc.so -> threshold
test_comput (tests.1.myTest) ... threshold: 0.045617
ok
test_comput (tests.cython.myTest) ... threshold with CYTHON: 0.045945
ok

• execute python setup.py install

## Cython

Some functions exists both in pure Python and in Cython. Cython translates these functions into C, greatly increasing peformance. When installing, the program tries to understand if you have what Cython needs (the Cython module, a C compiler) and if so, you will have both versions: pure Python and C. If you do not have what Cython needs, the program will install only the Python versions. A keyword in threshold.threshold.threshold.compute will let you choose between Python and Cython. You should choose cython=False if you want something more numerically stable, more versatile (only unweighted computation is implemented in Cython). You should choose cython=True if your concern is performance (large networks, and/or little time available).

import threshold.threshold as thr
import threshold.threshold_util as thu

# Content

Module threshold.threshold contains two main classes:

• tnet (handles temporal networks),
• threshold (computes the threshold).

Module threshold.threshold_util contains some useful functions for converting formats. At this stage, it contains

• DataFrame_to_lA (from pandas.DataFrame to a list of scipy.sparse.csr_matrix),
• DataFrame_to_lG (from pandas.DataFrame to a list of networkx.Graph or networkx.DiGraph).

## tnet

The constructor has only one compulsory argument: thr.tnet(my_network).

my_network can be

• a path to a text file containing the whole edge list. First two columns represent edges' origin and destination, while last column is the time stamp. Time stamps are assumed to be integers from 0. If there are more than 3 columns, then 3rd column is interpreted as edge weight. Further columns between the 3rd and the last (time) are disregarded. Default separator is \t; different separators (e.g. separator=',') can be input via the optional keyword separator in the tnet constructor. By default the edge list is assumed undirected; this can be changed via the optional keyword directed in the tnet constructor.
• a list of networkx.Graph or networkx.DiGraph objects. If the network is weighted, weights must be assigned to edges as weight keywords.

Other optional keywords of the thr.tnet constructor are

• period (default None): if not None, will override the computation of the period resulting from the input, by taking the first period snapshots. For example, if my_network is a list of 20 networkx.Graph snapshots, and period=15, then the last 5 snapshots are discarded. If the specified period is longer than the period of the dataset, you will get an AssertionError.
• dtype (default float128): it is a str argument. Set it to float64 if you really want to use 64-bit floating point numbers. Any other value of dtype, including the default one, will lead to using 128-bit floating points.
• attributes (default None): None, or a dict with node IDs as keys, and arbitrary node attributes as attributes.

thr.tnet has the following members, accessible through @property decorator syntax (some them can be manually set):

• lA: list of scipy.sparse.csr_matrix adjacency matrices,
• lG: list of networkx graphs,
• weighted: boolean,
• N: number of nodes,
• T: number of time steps. It can be set,
• nodelist: list of all nodes
• attributes: returns list of attributes, ordered as nodelist. Can be set by providing a dict,

For instance this works

R = thr.tnet(my_network)
print R.T # say we get 15
R.T = 10 # set it to 10
print R.T # now we get 10. The last 5 snapshots have been discarded.


If you try to set members other than the settable ones, the program will simply tell you you can't do it.

## threshold

The constructor has again one compulsory argument: thr.threshold(my_network), where my_network can be

• A tnet object,
• A list of scipy.sparse.csr_matrix objects.

Other keyword arguments:

• eval_max=20000, tol=1e-6, store=10: parameters of the modified power methods
• additional_time (default 0): It allows to add an arbitrary number of empty snapshots (empty means no edges in them). This is a convenient way to do it, as the order of empty snapshots inside the sequence does not matter.
• weighted (default None). This is important: the meaning of weighted is different in thr.tnet and in thr.threshold. In the former it means if the edges have weights or not. In the latter it refers to the way the infection propagator is computed. When weighted=False in threshold, the t-th term in the infection propagator is $1-\mu + \lambda A(t)$ regardless of the nature of $A(t)$, which is itself binary if tnet.weighted is False, or real-valued otherwise. If threshold.weighted is instead True, then binomial transmission is assumed, so that the t-th term in the infection propagator is $1-\mu + [1-(1-\lambda)^E(t)]$, with $E(t)$ being the entry-wise exponential of $A(t)$. Hence, when tnet.weighted is False, the result is the same regardless of threshold.weighted. However, the algorithm is much faster when threshold.weighted is False. Despite the difference between weighted in the two classes, if my_network is a tnet object and weighted=None, then thr.threshold will inherit the weighted attribute from my_network. weighted=True/False overrides the inheritance. If my_network is a list of matrices, weighted must be explicitly set to either True or False. If all this gives you headache, always set weighted=False in thr.threshold.
• convergence_on_eigenvector (default True) check the convergence of the algorithm on the stability of the eigenvector, rather than the eigenvalue (recommended).
• attributes (default None): see tnet. Inherited from tnet when applicable.
• cython (default False): in addition to pure Python, the power method algorithm is implemented in Cython, in order to make it faster. Cython requires a C compiler, which must be present on your machine.

This class hass many methods/variables (@property style) you can access and (sometimes) set. They are (when not explained, similar to tnet's, or repetitions of the keywords of the constructor)

• N
• T
• avg_k: time average of the average degree of the snapshots
• avg_A: time-averaged adjacency matrix
• avg_sr: spectral radius of the time-averaged adjacency matrix
• weighted
• convergence_on
• eval_max, tol, store
• additional_time
• lA
• l_indptr, l_indices, l_data, l_place (see doc of scipy.sparse.csr_matrix for some of them)

thr.threshold has two methods:

### thr.threshold.compute

This function computes the threshold, using optimization algorithms in scipy.optimize. It needs one compulsory argument: mu. mu can be either a (floating point) number, in which case it is interpreted as the recovery probability, or a dict. This dict must have attributes as keys (the same node attributes of the network), pointing to their corresponding values of recovery probability. This implements heterogeneous recovery rates. Optional keyword arguments are

• vmin ( default 1e-3), vmax (default 1) : range of transmissibility in which to look for the threshold
• root_finded (default 'brentq'). It can be either 'brentq' or 'bisect'
• maxiter (default 50) and arguments inherited from thr.threshold.__init__: see documentation of scipy.optimize

### thr.threshold.sr_point

Computes one point of the spectral radius. Its arguments are

• transmissibilty
• mu (see above)
• other optional keywords (see above)

# Minimal example

# import threshold modules
import threshold.threshold as thr
import threshold.threshold_util as thu

import networkx as nx
import numpy as np

# create a sequence of ER random graphs
N,T = 500,400
lG = []
for t in range(T):
lG.append(nx.gnm_random_graph(N,N))

# load it as a tnet object, and print
R = thr.tnet(lG)
print R

# threshold object, and print
Z = thr.threshold(R)
print Z

# compute the threshold
mu = 0.01
lc = Z.compute(mu,vmin=0.003,vmax=0.005)
print mu, lc


## threshold_util

This module contains two functions: DataFrame_to_lG and DataFrame_to_lA. They turn a pandas.DataFrame object into a list of networkx graphs or scipy.sparse CSR matrix. The former is a suitable input for threshold.tnet, the latter for threshold.threshold.

### DataFrame_to_lG

lG = thu.DataFrame_to_lG(df, directed=False, weight=None, source='source', target='target', time='time')
• df is a pandas.DataFrame.
• directed bool variable about (un)directedness.
• source name of the column of source nodes.
• target name of the column of target nodes.
• time name of the column with timestamps.
• weight can be None (unweighted network) or a string with the name of the column to be interpreted as weights.

It returns a list of networkx Graph or DiGraph objects.

### DataFrame_to_lA

Assumes node id's are integers from 0 to N-1, where N is the number of nodes.

lA = thu.DataFrame_to_lA(df, directed=False, source='source', target='target', time='time', weight='weight', dtype=np.float128, force_beg=None, force_end=None)
• df is a pandas.DataFrame.
• directed bool variable about (un)directedness.
• source name of the column of source nodes.
• target name of the column of target nodes.
• time name of the column with timestamps.
• weight can be None (unweighted network) or a string with the name of the column to be interpreted as weights.
• force_beg if not None, will discard all timesteps smaller than this.
• force_end if not None, will discard all timesteps larger than this.