thresholdmodel

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.

importnumpyasnpimportnetworkxasnximportmatplotlib.pyplotaspltfromthresholdmodelimportThreshModelN=1000k=10thresholds=0.1initially_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.plot(t,cascade_size)
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`.
| 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).
| 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).