# eugenio-valdano/threshold

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# Computing the Epidemic Threshold on Temporal Networks

Provides Python tools 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.

## Content

• test_system.py checks if your system has all the needed libraries.
• threshold.py main module.
• threshold_util.py additional methods for network handling.

## Required external modules

• numpy
• scipy
• networkx
• pandas (for threshold_util.py)

Run test_system.py to check if you have everything you need.

# Overview

The package consists of two objects: the class tnet for uploading and managing the temporal network, and the class threshold, for the actual computation of the threshold.

## import

The directory containing threshold.py must be in your Python search path. You can temporarily add it using

from sys import path
path.append('<dir to threshold.py>')

Then actually import the module as, for instance,

import threshold as thr # main module
import threshold_util as thu # additional utils

## tnet: manage your temporal network

Class tnet is able to load a temporal network given in different formats:

• 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.
• (Python) list of networkx Graph or DiGraph objects. If the network is weighted, weights must be assigned to edges as weight keywords.

The network can then be loaded in class tnet as follows:

R = thr.tnet(my_network)

### Arguments for tnet, with their default values

• my_network: where to look for the network, according to supported formats (see above);
• period = None: set period like this, if only a part of the network is to be used, up to period T (less than the one inferred from time stamps);
• dtype = 'float128': the bit length of the used float. 'float128' is the default because it is often needed. Every string that is not 'float64' is interpreted as 'float128'.
##### other optional keywords
• directed: it may be used when loading from text file. If directed=True, then the edge list is assumed to be directed. If not specified, treated as directed=False. When loading from a list of networkx graphs, it inherits from them the fact of being (un)directed.
• attributes=None: with this keyword you can provide a dictionary for assigning node attributes. Imagine your nodes are people, you could set attributes={'id1':'male','id2':'female'}. The dictionary does not have to be exhaustive. Nodes without attribute are allowed.
• separator: it may be used when loading from text file, to specify the separator. If not specified, treated as separator='\t'.

### Attributes

name description
N number of nodes.
T period. You can manually reduce it. It will drop the time steps in excess from the end.
weighted True/False
lG list of networkx graphs
lA list of adjacency matrices in scipy.sparse.csr_matrix format
attributes node attributes
nodelist list of nodes

## threshold: compute the threshold

Intstantiate a threshold object like this:

myth = th.threshold(X)

Where X can be either a tnet object or a list of adjacency matrices in scipy.sparse.csr_matrix. Additional optional arguments are

##### related to power method:
• eval_max=20000: maximum number of eigenvalue evaluations.
• tol=1e-6 : tolerance for power method convergence.
• store=10 : number of eigenvector(value) values to use to check convergence.
• convergence_on_eigenvector=True. If True uses the algorithm that checks convergence on the L1 norm of the principal eigenvector (probably more accurate). If False, checks the convergence of the eigenvalue estimate itself.
##### related to the temporal network:
• weighted=None. You have to specify it when you provide a list of adjacency matrices instead of a tnet object. You can specify it also with a tnet object if you want to override the .weighted attribute of the tnet object. If the network itself is weighted, you still can set weighted=False here. It simply means it multiplies transmissibility directly to the adjacency matrices. To know more about weights, read this article. weighted=False is more time-efficient than weighted=True.
• attributes=None. It is ignored when X is a tnet object, as it will inherit the attributes from X. When X is a list of matrices, you can use this to provide a list of length N containing the attribute of each node. If you do not wish to set an attribute for node i, put None in the list at place i.

You can access and edit eval_max, tol, store and weighted as class attributes.

The class has also the attribute convergente_on which is either eigenvector or eigenvalue. You can access it and edit it.

For instance:

myth.tol = 1e-5
myth.convergence_on = 'eigenvalue'

The class has the attribute lA which is the list of adjacency matrices. You can access it and set it safely.

Finally, the attribute avg_k returns the average (weighted) degree of the network, i.e., \frac{\sum_{t=1}^T\sum_{i,j}A_{t,ij}}{NT}

### compute method

This carries out the actual computation of the threshold.

x = th.compute(mu, vmin=1e-3, vmax=1, maxiter=50, root_finder='brentq', **kwargs)
• mu is the only compulsory argument. It can be either a single value (recovery probability) or a dictionary having a recovery probability for every attribute: {'attr 1': 0.1, 'attr 2': 0.3, 'default':0.6}. It must always have a 'default' value, which will be assigned to nodes with no attribute.
• vmin and vmax are the boundaries of the intervals in which to look for the threshold.
• maxiter is the maximum number of iterations of the root finding algorithm.
• root_finder can be either 'brentq' or 'bisect', referring to the functions in scipy.optimize. For further details see, for instance, scipy documentation.
• Other keyword arguments are directly sent to the root finding scipy function (e.g. xtol and rtol).

## 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.