IONTW Reference Guide
Version 1.0

M. Drew LaMar¹ \ Winfried Just² \ Hannah Callender³

1  Infection Model Types

2  Disease Transmission Parameters

• infection-rate: Rate of making effective contacts with an infectious host (β)
• end-infection-rate: Rate of loss of infectiousness (α = [1/(〈τ^I〉)])
• end-latency-rate: Rate of onset of infectiousness (γ)
• infection-prob: Probability of at least one effective contact with an infectious host over time ∆t (Model parameter b). The formula relating the continuous infection rate β to the discrete probability b is given by
  

  b = 1 − e−β∆t

  (1)
• end-infection-prob: Probability of loss of infectiousness over time ∆t (Model parameter a). The formula relating the continuous end infection rate α to the discrete probability a is given by
  

  a = 1 − e−α∆t

  (2)
• end-latency-prob: Probability of onset of infectiousness over time ∆t. This is derived from the continuous end latency rate γ using the formula
  

  1 − e−γ∆t

  (3)

3  Numerical Simulation Parameters

• model-time: Denotes whether the model is a Continuous time model or a Discrete time model
• time-step: Time step used in discrete time models (∆t)
• Discrete Approx: Calculates discrete probabilites from continuous rates using equations -

4  Network Parameters

4.1  Network Creation

• network-type: Types of supported networks. See Section 5 for a descriptive list.
• num-nodes: Number of nodes (N)
• lambda: One of two possible parameters to vary properties of a specific network type. See Section 5 for its use in each type of network.
• d: One of two possible parameters to vary properties of a specific network type. See Section 5 for its use in each type of network.

4.2  Network Modification and Visualization

• spawn-kill: Specifies whether the user wants to spawn (create) or kill (remove) a node or link in the network. Used in conjunction with the Node and Link buttons.
• Node: When pressed, a node is created or destroyed (depending on selection of spawn-kill) when a user clicks in the World window. If spawn-kill is set to kill, the nearest node to where the user clicks is destroyed.
• Link: When pressed, the user can create or destroy links (depending on selection of spawn-kill) as follows: on the first click of the mouse in the World window, the closest node to this click is selected and set to white; on the second click of the mouse, a link is created or destroyed between the first selected node and the node that is closest in space to the second click.
• Clear: Clears the network, setting num-nodes to 0.
• Randomize: While pressed, performs random edge swaps between nodes without changing the degree sequence. Useful for exploring results of different network realizations of fixed degree sequences.
• Metrics: Displays network and simulation metrics in the Command Center. Current metrics include:
  • Simulation parameters
    • <tau>: Mean time of infectiousness 〈τ^I〉
    • R0: Basic reproductive ratio R₀
    • Maximum number of simultaneous infections
  • Network parameters
    • Mean degree: If the degree of node i is denoted by k_i, then this is given by
      

      1 N N ∑ i=1  ki.
    • Edge density: This is the number of links L divided by the number of all possible links, given by
      

      2L N(N−1) .
    • Clustering coefficient: If N > 1, computes the mean node clustering coefficient over all nodes. For a node i, let N₁(i) denote the set of nodes that link to i. Let tr(i) denote the number of links {j₁,j₂} such that j₁, j₂ ∈ N₁(i). The node clustering coefficient is given by
      

      C(i) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 2tr(i) ki(ki−1) if  ki > 1, 2L N(N−1) if  ki ≤ 1.

      The network clustering coefficient is then given by
      

      C = 1 N N ∑ i=1  C(i).
    • Normalized clustering coefficient: If N > 1, computes the mean normalized node clustering coefficient. The normalized node clustering coefficient is given by
      

      Cnorm(i) = ⎧ ⎪ ⎨ ⎪ ⎩ C(i) N(N−1) 2L if  L > 0, 1 if  L = 0,

      and thus the normalized network clustering coefficient is given by
      

      Cnorm = 1 N N ∑ i=1  Cnorm(i).
    • Number of connected components
    • Largest component (as proportion of network)
    • Average path length in largest component: If {i₁,…,i_n} denotes the nodes in the largest component, then the average path length in the largest component is given by
      

      2 n(n−1) ∑ j > m  d(ij,im),

      where d(i_j,i_m) is the number of edges in the shortest path between i_j and i_m.
    • Diameter of largest component: The diameter of the largest component is the maximum shortest path in the largest component, i.e.
      

      max j > m  d(ij,im),

      where, as above, d(i_j,i_m) is the number of edges in the shortest path between two nodes i_j and i_m in the largest component.
• Labels: Toggles node labels on and off.
• Scale: Scales up the network to fill the world window. Useful in conjunction with Spring, which tends to contract everything down.
• Spring: Useful to "clean up" network. Simulates links as springs. While this is pressed, the network will approach an equilibrium configuration. We recommend using Spring with the fastest setting of the speed control slider. Particularly useful for trees (although fun to use with all network types).
• plot-metric: In conjunction with the Network Metrics plot window, selects the type of information to display. Current choices include:
  • Degree Distribution: Distribution of node degrees
  • Clustering Coeffs: Histogram of node clustering coefficients
  • Normalized Coeffs: Histogram of normalized node clustering coefficients
  • Shortest Paths: Histogram of shortest paths in largest component
  • Probability Distribution: Distribution corresponding to the Custom Distribution network type.
• Update: Updates the Network Metrics plot window to plot the metric chosen in plot-metric. Note that when loading a distribution from file, plot-metric is automatically set to Probability Distribution and the distribution is displayed. Also, any change in the currently displayed network that changes the number of nodes or links updates automatically the Degree Distribution plot.

5  Network Types

• Complete graph
  Parameters: num-nodes (N)
  Description: Graph K_N consisting of N nodes with edges between every pair of distinct nodes.
• Empty graph
  Parameters: num-nodes (N)
  Description: Graph ―K_N consisting of N isolated nodes with no edges.
• Erdos-Renyi
  Parameters: num-nodes (N), lambda (λ)
  Description: Implements the version of Erdös-Rényi random graphs where
  

  p = λ N−1

  is the probability that each possible edge is included in the graph, and λ is the expected mean degree.
• Nearest neighbor 1
  Parameters: num-nodes (N), d
  Description: Creates a graph with N nodes in a ring, connecting the d nearest neighbors to each side of every node. This creates a particular instance of a 2d-regular graph.
• Nearest neighbor 2
  Parameters: num-nodes (N), d
  Description: Creates an n ×m grid of nodes, where N = nm with m the largest integer factor of N. The integer d connects the nodes in a nearest-neighbor fashion as follows:
  • d = 0: Empty graph
  • d = 1: Each node connects to the nodes that are to the north, south, east and west.
  • d ≥ 2: If we denote the index of a node by (i,j), then this node will connect to all nodes with indices (i±k,j±l), where 0 ≤ k, l ≤ d-1.
• Small world 1
  Parameters: num-nodes, lambda, d
  Description: Creates the union of an Erdos-Renyi graph with parameters num-nodes and lambda and a Nearest-neighbor 1 graph with parameters num-nodes and d.
• Small world 2
  Parameters: num-nodes, lambda, d
  Description: Creates the union of an Erdos-Renyi graph with parameters num-nodes and lambda and a Nearest-neighbor 2 graph with parameters num-nodes and d.
• Preferential attachment
  Parameters: num-nodes (N), lambda (m₀), d (m)
  Description: Create a graph using a modified form of the preferential attachment algorithm. We start with an initial graph G⁽⁰⁾ as the complete graph on m₀ nodes. We then perform the following iteration over i=0,…,N−m₀:
  1. Let ―k⁽ⁱ⁾ be the vector of degrees of all nodes in the graph G⁽ⁱ⁾
  2. Add a new node and connect this node to m nodes in G⁽ⁱ⁾, choosing these m nodes randomly in proportion to their degrees in ―k⁽ⁱ⁾.
  3. Increment i by 1, and repeat 1-2 until number of nodes is N.
  Note: This algorithm is animated in the World window for illustrative purposes. If you would like to speed up the process, you can either unselect view updates or adjust with the speed slider.
• Generic scale-free
  Parameters: num-nodes (N), lambda (λ)
  Description: Creates a random scale-free graph with degrees drawn from the distribution
  

  f(k) = c kλ ,

  where k = 1,…,N−1 and λ ≥ 1. The value c is chosen so that f(k) sums to 1.
• Spatially clustered
  Parameters: num-nodes (N), lambda (λ)
  Description: Creates a spatially clustered graph by first assigning N nodes x and y coordinates uniformly at random in the World window. While the edge density of the graph is less than λ, nodes are chosen at random and connected to their nearest neighbor.
• Random regular
  Parameters: num-nodes (N), lambda (λ)
  Description: Creates a random graph with N nodes such that every node has degree λ.
• Regular tree
  Parameters: lambda, d
  Description: Creates a rooted tree with depth lambda and d branchings per interior node.
• Custom distribution
  Parameters: None
  Description: When a degree distribution has been pre-loaded (see Load below) and this network type is selected, each press of the New button will draw a new degree sequence from this distribution. You can randomize the graph while keeping the degree sequence fixed by pressing the Randomize button. You can move between the specific network's degree distribution and the custom distribution by changing plot-metric and pressing Update.

6  Setup & Go

• New: Creates a new network based on the network-type menu and corresponding network parameters lambda and d, if appropriate. When using BehaviorSpace, the command new-network performs the same actions.
• Last: Resets the node states to the previous initial conditions. When using BehaviorSpace, the command last-init performs the same actions.
• Go: Run the simulation. Press again to stop a simulation. When using BehaviorSpace, the command go performs the same actions.
• Defaults: Sets all simulation, disease and network parameters to defaults.
• Load: Loads information from file. The types of information that can be loaded include:
  • Degree sequence (see degree.txt for an example)
  • Degree distribution (see distribution.txt for an example)
  • Network (this is "Network Parameters" as referred to in this document, as well as nodes, links and node states - see sample-network-detailed.txt for an example with a detailed format, and sample-network.txt for an example with a simple format).
  • Parameters (this is "Simulation and Disease Parameters" as referred to in this document - see parameters.txt for an example).
  • All (Networks and Parameters - see all.txt for an example).
  Note: File text is sensitive to both case and whitespace.
• Save: Saves information to file. The types that can be saved include:
  • Network (this is "Network Parameters" as referred to in this document, as well as nodes, links and node states - see sample-network-detailed.txt for an example).
  • Parameters (this is "Simulation and Disease Parameters" as referred to in this document - see parameters.txt for an example).
  • All (Networks and Parameters) - see all.txt for an example).

6.1  Initialize States

• set-state-to: Can be Infectious or Removed. Specifies the state you would like to initialize nodes to upon pressing the Set button.
• set-state-by: Specifies the way in which you would like to change the node states. Choice include:
  • Number of nodes: Needs to be an integer.
  • Fraction of nodes: Between 0 and 1.
  • Vector from input: Specify node labels in list format (e.g. [0 1 10 16]).
  • Vector from file: File should have one line with a list of node labels (e.g. [0 1 10 16]).
• num/frac: Stands for "number or fraction," depending on the choice of set-state. This is how you specify the number or fraction of nodes to infect or immunize.
• min-deg: Specifies the minimum degree for the population of nodes to apply the set-state algorithm to.
• Set: Applies the state initialization specified to all susceptible nodes with all of the above choices. By performing state initializations in serial, you can, for example, immunize 50% of the susceptible nodes and infect 10% of the remaining susceptible nodes.
• Reset: Sets the state of all nodes to susceptible.
• Select: When pressed, a user can cycle through the states for a node by clicking on the node.
• auto-set: When set to On, every network created with the New button will automatically press the Set button.

7  Helpful Commands

Footnotes: