V - the graph vertex typeE - the graph edge typepublic class WattsStrogatzGraphGenerator<V,E> extends Object implements GraphGenerator<V,E,V>
The generator is described in the paper: D. J. Watts and S. H. Strogatz. Collective dynamics of small-world networks. Nature 393(6684):440--442, 1998.
The following paragraph from the paper describes the construction.
"The generator starts with a ring of $n$ vertices, each connected to its $k$ nearest neighbors ($k$ must be even). Then it chooses a vertex and the edge that connects it to its nearest neighbor in a clockwise sense. With probability $p$, it reconnects this edge to a vertex chosen uniformly at random over the entire ring with duplicate edges forbidden; otherwise it leaves the edge in place. The process is repeated by moving clock-wise around the ring, considering each vertex in turn until one lap is completed. Next, it considers the edges that connect vertices to their second-nearest neighbors clockwise. As before, it randomly rewires each of these edges with probability $p$, and continues this process, circulating around the ring and proceeding outward to more distant neighbors after each lap, until each edge in the original lattice has been considered once. As there are $\frac{nk}{2}$ edges in the entire graph, the rewiring process stops after $\frac{k}{2}$ laps. For $p = 0$, the original ring is unchanged; as $p$ increases, the graph becomes increasingly disordered until for $p = 1$, all edges are rewired randomly. For intermediate values of $p$, the graph is a small-world network: highly clustered like a regular graph, yet with small characteristic path length, like a random graph."
The authors require $n \gg k \gg \ln(n) \gg 1$ and specifically $k \gg \ln(n)$ guarantees that a random graph will be connected.
Through the constructor parameter the model can be slightly changed into adding shortcut edges instead of re-wiring. This variation was proposed in the paper: M. E. J. Newman and D. J. Watts, Renormalization group analysis of the small-world network model, Physics Letters A, 263, 341, 1999.
| Constructor and Description |
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WattsStrogatzGraphGenerator(int n,
int k,
double p)
Constructor
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WattsStrogatzGraphGenerator(int n,
int k,
double p,
boolean addInsteadOfRewire,
Random rng)
Constructor
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WattsStrogatzGraphGenerator(int n,
int k,
double p,
long seed)
Constructor
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| Modifier and Type | Method and Description |
|---|---|
void |
generateGraph(Graph<V,E> target,
Map<String,V> resultMap)
Generates a small-world graph based on the Watts-Strogatz model.
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clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, waitgenerateGraphpublic WattsStrogatzGraphGenerator(int n,
int k,
double p)
n - the number of nodesk - connect each node to its k nearest neighbors in a ringp - the probability of re-wiring each edgeIllegalArgumentException - in case of invalid parameterspublic WattsStrogatzGraphGenerator(int n,
int k,
double p,
long seed)
n - the number of nodesk - connect each node to its k nearest neighbors in a ringp - the probability of re-wiring each edgeseed - seed for the random number generatorIllegalArgumentException - in case of invalid parameterspublic WattsStrogatzGraphGenerator(int n,
int k,
double p,
boolean addInsteadOfRewire,
Random rng)
n - the number of nodesk - connect each node to its k nearest neighbors in a ringp - the probability of re-wiring each edgeaddInsteadOfRewire - whether to add shortcut edges instead of re-wiringrng - the random number generator to useIllegalArgumentException - in case of invalid parameterspublic void generateGraph(Graph<V,E> target, Map<String,V> resultMap)
generateGraph in interface GraphGenerator<V,E,V>target - the target graphresultMap - not used by this generator, can be nullCopyright © 2019. All rights reserved.