Simulate a continuous-time threshold model on static networks.

## Project description

Simulate a continuous-time threshold model on static networks using Gillespie’s stochastic simulation algorithm (SSA). The networks can be directed and/or weighted.

In contrast to the original discrete-time model, nodes whose aggregated inputs exceed their respective thresholds will not flip after the “next time step” because there are no time steps. Instead, a node whose threshold has been exceeded will enter an alert state from which it will enter the activated state with rate $gamma = 1$.

## Install

pip install thresholdmodel

## Example

Simulate on an ER random graph.

import numpy as np
import networkx as nx
import matplotlib.pyplot as plt

from thresholdmodel import ThreshModel

N = 1000
k = 10

thresholds = 0.1
initially_activated = np.arange(100)

G = nx.fast_gnp_random_graph(N, k/(N-1.0))

Thresh = ThreshModel(G,initially_activated,thresholds)
t, cascade_size = Thresh.simulate()

plt.show() ## API

### Simulate

Given a networkx-Graph object G (can be a networkx.DiGraph, too), and values for initially_activated and thresholds, simulate like this

Thresh = ThreshModel(G,initially_activated,thresholds)
t, a = Thresh.simulate()

t is a numpy.ndarray containing the times at which node activations happened. a is a numpy.ndarray containing the relative cascade size at the corresponding time in t. Note that the whole process is modeled as a Poisson process such that the time t will be given in units of the node activation rate gamma = 1.0. If you want to simulate for another node activation rate, simply rescale time as t /= gamma.

When the simulation is started with the save_activated_nodes=True flag, a list of activated nodes per time leap is saved in ThreshModel.activated_nodes.

t, a = Thresh.simulate(save_activated_nodes=True)
print(Thresh.activated_nodes)

You can repeat a simulation with the same initial conditions by simply calling Thresh.simulate() again, all the necessary things will be reset automatically.

### Set initially activated nodes

Set nodes 3, 5, and 8 to be activated initially.

initially_activated = [3, 5, 8] # this could also be a numpy array

Choose 20% of all nodes randomly to be activated initially. When the simulation is restarted, the same nodes will be chosen as initial conditions.

initially_activated = 0.2

Choose 35 randomly selected nodes to be activated initially. When the simulation is restarted, the same nodes will be chosen as initial conditions.

initially_activated = 35

### Set thresholds

Activation thresholds can be set for all nodes

thresholds = np.random.rand(G.number_of_nodes())

Note that thresholds need to lie in the domain [0,1].

You can also set a universal threshold

thresholds = 0.1

Here, 10% of a node’s neighbors need to be activated in order for the node to become active, too.

### Directed networks

A node will become active if the sufficient number of nodes pointing towards the node are active. This means that a node’s in-degree will be the important measure to determine wether this particular node will become active.

### Weighted networks

If you want to simulate on a weighted network, provide the weight keyword

Thresh = ThreshModel(G,initially_activated,thresholds,weight='weight')

Similar to the networkx-documentation: weight (string, optional (default=None)) - The attribute name to obtain the edge weights. E.g.: G.edges[0,1]['weight'].

A focal node will become active when the cumulative edge weights of all activated nodes pointing towards the focal node will reach > threshold*in_degree.

## Docstring

This is the model’s docstring.

>>> help(ThreshModel)
Help on class ThreshModel in module thresholdmodel.model:

class ThreshModel(builtins.object)
|  ThreshModel(G, initially_activated, thresholds, weight=None)
|
|  A simple simulation class that runs
|  a threshold-model activation process
|  on a static network (potentially weighted and directed)
|  in continuous time using Gillespie's
|  stochastic simulation algorithm.
|
|  The temporal dimension is fixed by assuming
|  that every node whose activation threshold
|  has been exceeded by neighboring inputs
|  is activated with constant and uniform
|  rate :math:\gamma = 1.
|
|  Parameters
|  ==========
|  G : networkx.Graph, networkx.DiGraph
|      The network on which to simulate.
|      Nodes must be integers in the range
|      of [0, N-1].
|  initially_activated: float, int, or list of ints
|      Can be either of three things:
|
|      1. float of value 0 < initially_activated < 1.
|         In this case, initially_activated is
|         interpreted to represent a fraction of nodes
|         that will be randomly selected from the
|         set of nodes and set to be activated.
|      2. int of value 1 <= initially_activated < N-1.
|         In this case, initially_activated nodes
|         will be randomly sampled from the node set
|         and set to be activated.
|      3. list of ints. In this case, initially_activated
|         is interpreted to contain indices of nodes
|         that will be activated initially.
|  thresholds: float or iterable of floats
|      Can be either of two things:
|
|      1. float of value 0 < thresholds <= 1.
|         In this case, every node will have the same
|         activation threshold.
|      2. iterable of values 0 < thresholds <=1.
|         In this case, the function expectes a list,
|         tuple, or array with length equal to the
|         number of nodes. Each entry m of this list
|         will be interpreted to be node m's activation
|         threshold.
|  weight: str, default = None
|      A string that represents the weight keyword of a link.
|      If None, the network is assumed to be unweighted.
|
|  Example
|  =======
|
|  >>> G = nx.fast_gnp_random_graph(1000,20/(1000-1))
|  >>> model = TreshModel(G, 100, 0.1)
|  >>> t, cascade_size = model.simulate()
|
|  Attributes
|  ==========
|  G : nx.Graph or nx.DiGraph
|      The network on which to simulate.
|      Nodes must be integers in the range
|      of [0, N-1].
|  N : int
|      The number of nodes in the network
|  weight: str
|      A string that represents the weight keyword of a link.
|      If None, the network is assumed to be unweighted.
|  in_deg : numpy.ndarray
|      Contains the in-degree of every node.
|  thresholds: numpy.ndarray
|      Each entry m of this array
|      represents node m's activation
|      threshold.
|  initially_activated: numpy.ndarray
|      Each entry of this array contains
|      a node that will be activated initially.
|  time: numpy.ndarray
|      Contains every time point at which a node was
|      activates (after simulation() was called).
|      The temporal dimension is given by assuming
|      that every node whose activation threshold
|      has been exceeded by activation inputs
|      is activated with constant and uniform
|      rate :math:\gamma = 1.
|      The relative size of the activation cascade
|      at the corrsponding time value in time
|      (relative to the size of the node set).
|      Only available after simulation() was called.
|  activated_nodes: list
|      A list of lists.
|      Each entry contains a list of integers representing
|      the nodes that have been activated
|      at the corrsponding time value in time.
|      Each list entry will contain only a single node
|      for every other time than the initial time.
|      Only available after simulation() was called.
|
|  Methods defined here:
|
|  __init__(self, G, initially_activated, thresholds, weight=None)
|      Initialize self.  See help(type(self)) for accurate signature.
|
|  reset(self)
|      Reset the simulation.
|
|  set_initially_activated(self, initially_activated)
|      Set the process's initial activation state.
|
|      Parameters
|      ==========
|      initially_activated: float, int, or list of ints
|          Can be either of three things:
|
|          1. float of value 0 < initially_activated < 1.
|             In this case, initially_activated is
|             interpreted to represent a fraction of nodes
|             that will be randomly selected from the
|             set of nodes and set to be activated.
|          2. int of value 1 <= initially_activated < N-1.
|             In this case, initially_activated nodes
|             will be randomly sampled from the node set
|             and set to be activated.
|          3. list of ints. In this case, initially_activated
|             is interpreted to contain indices of nodes
|             that will be activated initially.
|
|  set_thresholds(self, thresholds)
|      Set node activation thresholds.
|
|      Parameters
|      ==========
|      thresholds: float or iterable of floats
|          Can be either of two things:
|
|          1. float of value 0 < thresholds <= 1.
|             In this case, every node will have the same
|             activation threshold.
|          2. iterable of values 0 < thresholds <=1.
|             In this case, the function expectes a list,
|             tuple, or array with length equal to the
|             number of nodes. Each entry m of this list
|             will be interpreted to be node m's activation
|             threshold.
|
|  simulate(self, save_activated_nodes=False)
|      Simulate until all nodes that can be activated
|      have been activated.
|
|      Parameters
|      ==========
|      save_activated_nodes: bool, default = False
|          If True, write a list of activated nodes
|          to the class attribute activated_nodes
|          every time an activation event happens.
|          Such a list will contain only a single node
|          for every other time than the initial time.
|
|      Returns
|      =======
|      time : numpy.ndarray
|          Time points at which nodes were activated.
|      cascade_size : numpy.ndarray
|          The relative size of the activation cascade
|          at the corrsponding time value in time
|          (relative to the size of the node set).

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