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public static GetBetweennessPrefMatrix ( TemporalNetwork temp_net, string x, int>.Dictionary |
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| temp_net | TemporalNetwork | The temporal network to compute betweeness preference for |
| x | string | The node for which to compute the betweenness preference matrix |
| index_pred | int>.Dictionary | A mapping of nodes to columns in the betweenness preference matrix |
| index_succ | int>.Dictionary | A mapping of nodes to rows in the betweenness preference matrix |
| normalized | bool | |
| return | ].double[ | |
public static IDictionary <int, double> RunRW_BWP(TemporalNetwork temp_net, int max_steps = 100000, bool null_model = false)
{
Random r = new Random();
var cumulatives = new Dictionary <Tuple <string, string>, Dictionary <double, string> >();
var sums = new Dictionary <Tuple <string, string>, double>();
// Dictionary<string, Dictionary<double, string>> cumulatives =
// Dictionary<string, double> sums = new Dictionary<string, double>();
Dictionary <string, Dictionary <string, int> > indices_pred = new Dictionary <string, Dictionary <string, int> >();
Dictionary <string, Dictionary <string, int> > indices_succ = new Dictionary <string, Dictionary <string, int> >();
Dictionary <string, double[, ]> matrices = new Dictionary <string, double[, ]>();
Dictionary <string, int> visitations = new Dictionary <string, int>();
Dictionary <string, double> stationary = new Dictionary <string, double>();
Dictionary <Tuple <string, string>, int> edge_visitations = new Dictionary <Tuple <string, string>, int>();
Dictionary <Tuple <string, string>, double> edge_stationary = new Dictionary <Tuple <string, string>, double>();
Dictionary <int, double> tvd = new Dictionary <int, double>();
// Aggregate network
WeightedNetwork network = temp_net.AggregateNetwork;
// Read analytical stationary distribution (i.e. flow-corrected edge weights) from disk
string[] lines = System.IO.File.ReadAllLines("stationary_dist_RM.dat");
foreach (string x in lines)
{
string[] split = x.Split(' ');
string[] nodes = split[0].Split('.');
var edge = new Tuple <string, string>(nodes[0], nodes[1]);
// Extract stationary dist, set visitations to zero and adjust edge weights ...
edge_stationary[edge] = double.Parse(split[1], System.Globalization.CultureInfo.GetCultureInfo("en-US").NumberFormat);
edge_visitations[edge] = 0;
network[edge] = edge_stationary[edge];
}
// Compute stationary dist of vertices ...
double total = 0d;
foreach (string x in network.Vertices)
{
stationary[x] = 0d;
foreach (string s in network.GetPredecessors(x))
{
stationary[x] += edge_stationary[new Tuple <string, string>(s, x)];
}
total += stationary[x];
}
foreach (string x in network.Vertices)
{
stationary[x] = stationary[x] / total;
}
// Compute betweenness preference matrices
if (!null_model)
{
Console.Write("Computing betweenness preference in temporal network ...");
}
else
{
Console.Write("Calculating null model betweenness preference ...");
}
foreach (string x in network.Vertices)
{
var ind_p = new Dictionary <string, int>();
var ind_s = new Dictionary <string, int>();
if (!null_model)
{
matrices[x] = BetweennessPref.GetBetweennessPrefMatrix(temp_net, x, out ind_p, out ind_s, false);
}
else
{
matrices[x] = BetweennessPref.GetUncorrelatedBetweennessPrefMatrix(temp_net, x, out ind_p, out ind_s);
}
indices_pred[x] = ind_p;
indices_succ[x] = ind_s;
}
Console.WriteLine("done.");
// Initialize visitations, stationary distribution and cumulatives ...
foreach (string x in network.Vertices)
{
visitations[x] = 0;
stationary[x] = 0d;
foreach (string s in indices_pred[x].Keys)
{
Tuple <string, string> key = new Tuple <string, string>(s, x);
stationary[x] += network.GetWeight(s, x);
// Compute the transition probability for a edge (x,t) given that we are in (s,x)
cumulatives[key] = new Dictionary <double, string>();
double sum = 0d;
foreach (string t in indices_succ[x].Keys)
{
double transition_prob = 0d;
string two_path = s + "," + x + "," + t;
transition_prob = matrices[x][indices_pred[x][s], indices_succ[x][t]];
if (transition_prob > 0)
{
sum += transition_prob;
cumulatives[key][sum] = t;
}
}
sums[key] = sum;
}
}
// Draw two initial nodes ...
string pred = network.RandomNode;
string current = network.GetRandomSuccessor(pred);
visitations[pred] = 1;
visitations[current] = 1;
edge_visitations[new Tuple <string, string>(pred, current)] = 1;
// Run the random walk (over edges)
for (int t = 0; t < max_steps; t++)
{
// The edge via which we arrived at the current node
Tuple <string, string> current_edge = new Tuple <string, string>(pred, current);
// If this happens, we are stuck in a sink, i.e. there is no out edge
System.Diagnostics.Debug.Assert(sums[current_edge] > 0, string.Format("Network not strongly connected! RW stuck after passing through edge {0}", current_edge));
// Draw a sample uniformly from [0,1] and multiply it with the cumulative sum for the current edge ...
double sample = rand.NextDouble() * sums[current_edge];
// Determine the next transition ...
string next_node = null;
for (int i = 0; i < cumulatives[current_edge].Count; i++)
{
if (cumulatives[current_edge].Keys.ElementAt(i) > sample)
{
next_node = cumulatives[current_edge].Values.ElementAt(i);
break;
}
}
pred = current;
current = next_node;
visitations[current] = visitations[current] + 1;
edge_visitations[new Tuple <string, string>(pred, current)] = edge_visitations[new Tuple <string, string>(pred, current)] + 1;
tvd[t] = TVD(visitations, stationary);
}
return(tvd);
}
/// <summary>
/// This method creates a random sequence of two paths, where the betweenness preferences as well as edge weights match those of a given temporal network.
/// The model implemented here can be viewd in two different ways:
/// 1.) It can be seen as a reshuffling of two paths realized in a given temporal network, while destroying other correlations like bursty activity patterns
/// 2.) Alternatively, it can be seen as a random sampling based on the betweenness preference matrices of nodes as computed from an empirical network
/// </summary>
/// <param name="temp_net">The temporal network based on which a randomized sequence of two paths will be created</param>
/// <param name="length">The length of the sequence in terms of the number of time-stamped interactions</param>
/// <param name="precision">The numerical precision that will be used when sampling from the distribution. This
/// at the same time affects the memory requirements of the procedure, which is at most precision*two_paths in the input network</param>
/// <returns>A temporal network that preserves betweenness preferences as well as the aggregated network of the input</returns>
public static TemporalNetwork ShuffleTwoPaths(TemporalNetwork temp_net, int length = 0, int precision = 1000)
{
// If no length is specified (i.e. length has the default value of 0), use the length of the original sequence
length = length > 0 ? length : temp_net.Length;
// This will take the betweenness pref. matrices of all nodes ...
Dictionary <string, double[, ]> betweenness_matrices = new Dictionary <string, double[, ]>();
Dictionary <string, Dictionary <string, int> > pred_indices = new Dictionary <string, Dictionary <string, int> >();
Dictionary <string, Dictionary <string, int> > succ_indices = new Dictionary <string, Dictionary <string, int> >();
int lcm = 1;
// Compute unnormalized betweenness preference matrices of all nodes in the aggregate network
foreach (string x in temp_net.AggregateNetwork.Vertices)
{
Dictionary <string, int> preds = new Dictionary <string, int>();
Dictionary <string, int> succs = new Dictionary <string, int>();
betweenness_matrices[x] = BetweennessPref.GetBetweennessPrefMatrix(temp_net, x, out preds, out succs, normalized: false);
pred_indices[x] = preds;
succ_indices[x] = succs;
// Compute the least common multiple of the denominators of all entries ...
// Eventually we will multiply ALL entries of the matrices of ALL nodes with the LCM in order to
// ensure that all two paths are represented with the correct relative frequencies
foreach (string s in temp_net.AggregateNetwork.GetPredecessors(x))
{
foreach (string d in temp_net.AggregateNetwork.GetSuccessors(x))
{
int whole, numerator, denominator;
MathHelper.RoundToMixedFraction(betweenness_matrices[x][pred_indices[x][s], succ_indices[x][d]], precision, out whole, out numerator, out denominator);
lcm = MathHelper.LCM(lcm, denominator);
}
}
}
List <string> sampling_set = new List <string>();
// Create a list of two_paths whose relative frequencies match the betweenness preference matrix...
foreach (string v in betweenness_matrices.Keys)
{
// Add source and target of two paths to list according to their relative frequencies in the matrices
foreach (string s in temp_net.AggregateNetwork.GetPredecessors(v))
{
foreach (string d in temp_net.AggregateNetwork.GetSuccessors(v))
{
for (int k = 0; k < Math.Round(betweenness_matrices[v][pred_indices[v][s], succ_indices[v][d]] * lcm); k++)
{
sampling_set.Add(s + "," + v + "," + d);
}
}
}
}
// Create an empty temporal network
TemporalNetwork microstate = new TemporalNetwork();
int time = 0;
// Draw two-paths at random from the sampling set, this is equivalent to reshuffling existing two paths of the original sequence
// However, it additionally correctly accounts for continued two paths (in which case an edge overlaps) and for multiple edges in
// a single step (such two paths will be counted fractionally)
for (int l = 0; l < length / 2; l++)
{
// Draw a random two path
int r = rand.Next(sampling_set.Count);
string tp = sampling_set[r];
string[] nodes = tp.Split(',');
// Add two temporal edges
microstate.AddTemporalEdge(time++, nodes[0], nodes[1]);
microstate.AddTemporalEdge(time++, nodes[1], nodes[2]);
}
return(microstate);
}