Hyperassociative Map Explanation
Introduction
Almost 8 years ago, on Aug 15, 2009, I invented a new game-changing algorithm called the Hyperassociative Map algorithm. It was released as part of the dANN v2.x library. The HAM algorithm, as it is often called, has since been used by countless developers and in hundreds of projects. HAM is a Graph Drawing algorithm that is similar to force-directed algorithms but in a class all its own. Unlike other force-directed algorithms HAM does not have a sense of momentum or acceleration which makes it debatable if it can even still be called force-directed.
Below is a video demonstration of HAM in action. In this 3D visualization the vertices of the graph are displayed as grey spheres but the edges are not rendered. The graph's topology is relatively simple containing 128 nodes in groups of 16 layered such that each group is fully connected to each adjacent group. This results in 256 edges between each adjacent group. Since the groups on either end only have one other group they are adjacent to that means there is a total of 1,792 edges. Despite this the graph aligns quickly and smoothly on a single 1 Ghz processor as demonstrated in the video. It starts with randomized locations for each vertex and then aligns. After each alignment the graph is reset with new random starting positions to show that the same alignment is achieved every time.
What makes HAM so special is that it retains many of the advantages that have made force-directed algorithms so popular while simultaneously addressing their short comings. Wikipedia describes the following advantages to using force-directed algorithms, all of which hold true for the HAM algorithm.
- Good-quality results - The output obtained usually have very good results based on the following criteria: uniform edge length, uniform vertex distribution and showing symmetry. This last criterion is among the most important ones and is hard to achieve with any other type of algorithm.
- Flexibility - Force-directed algorithms can be easily adapted and extended to fulfill additional aesthetic criteria. This makes them the most versatile class of graph drawing algorithms. Examples of existing extensions include the ones for directed graphs, 3D graph drawing, cluster graph drawing, constrained graph drawing, and dynamic graph drawing.
- Intuitive - Since they are based on physical analogies of common objects, like springs, the behavior of the algorithms is relatively easy to predict and understand. This is not the case with other types of graph-drawing algorithms.
- Simplicity - Typical force-directed algorithms are simple and can be implemented in a few lines of code. Other classes of graph-drawing algorithms, like the ones for orthogonal layouts, are usually much more involved.
- Interactivity - Another advantage of this class of algorithm is the interactive aspect. By drawing the intermediate stages of the graph, the user can follow how the graph evolves, seeing it unfold from a tangled mess into a good-looking configuration. In some interactive graph drawing tools, the user can pull one or more nodes out of their equilibrium state and watch them migrate back into position. This makes them a preferred choice for dynamic and online graph-drawing systems.
- Strong theoretical foundations - While simple ad-hoc force-directed algorithms often appear in the literature and in practice (because they are relatively easy to understand), more reasoned approaches are starting to gain traction. Statisticians have been solving similar problems in multidimensional scaling (MDS) since the 1930s, and physicists also have a long history of working with related n-body problems - so extremely mature approaches exist. As an example, the stress majorization approach to metric MDS can be applied to graph drawing as described above. This has been proven to converge monotonically. Monotonic convergence, the property that the algorithm will at each iteration decrease the stress or cost of the layout, is important because it guarantees that the layout will eventually reach a local minimum and stop. Damping schedules cause the algorithm to stop, but cannot guarantee that a true local minimum is reached.
However the two disadvantages described of force-directed algorithms, namely high running time and poor local minima, have been corrected in the HAM algorithm. As described earlier HAM is not a true force-directed algorithm because it lacks any sense of momentum. This was intentional as it ensures there is no need for a dampening schedule to eliminate oscillations that arise from the momentum of nodes. This has the added advantage that the algorithm does not prematurely come to rest at a local minima. It also means fewer processing cycles wasted on modeling oscillations and vibrations throughout the network.
These properties alone already make HAM a worthwhile algorithm for general study and real-world applications, however it is important to note that HAM was originally designed with a very specific use case in mind. Originally HAM was designed to facilitate the distribution of massive real-time graph processing networks. The sort of scenario where each vertex in a graph had to process some input data and produce some output data and where each vertex is part of a large interdependent graph working on the data in real time. When distributing the tasks across a cluster of computers it is critical that vertices that are highly interconnected reside on the same computer in the cluster and physically close to the computers housing the vertices that will ultimately receive the data, process it and then carry it throughout the rest of the network. For this purpose HAM was created to model graphs such that each node in a compute cluster took ownership of tasks associated with vertices that were spatially close to each other according to the HAM's drawing of the compute graph.
In order for HAM to be successful at it's job it needed to exhibit a few very specific properties. For starters the interactivity property mentioned earlier was a must. HAM needed to be able to work with a graph that is constantly changing its topology with new vertices able to be added, removed, or reconfigured in real time. This is ultimately what led the algorithm to be modeled in a way that made it similar to force-directed algorithms.
The other requirement is that the motion of the vertices had to be smooth without any oscillations as they align. This was critical because if oscillations occurred on a vertex as it was near the border that distinguishes one compute node from another then those oscillations across that border would cause the task in the compute cluster to be transferred between the nodes in the cluster each time. Since this is an expensive operation it is important that as HAM aligned the vertices didn't jitter causing them to cross these borders excessively.
Finally HAM needed to be able to be parallelized and segmented. That means that it needed to scale well for multi-threading but in such a way that each thread didn't need to be aware of the entire graph in order to process it; instead each thread had to be capable of computing the alignment of HAM on an isolated section of the graph. This is obviously critical because of the distributed nature of the compute graph, particularly if we want something capable of unbounded scaling. I basically wanted an algorithm that could be successful on even massively large graphs.
With almost 8 years of testing it has become evident that HAM is top in its class compared to many graph drawing algorithms. Despite this it is still scarcely understood by those studying graph drawing algorithms. For this reason I wanted to write this article to share some of its internal workings so others can adapt and play with the algorithm for their own projects.
The Algorithm
In this section I want to get into the internal workings of the Hyperassociative Map algorithm, HAM. Below is the pseudocode breakdown explaining the algorithm. Notice I use some math notation here for simplicity. Most notably I use vector notation where all variables representing vectors have a small arrow above the variable and the norm, or magnitude, of the vector is represented by double vertical bars on either side, for example \(||\vec{p}||\). If you have trouble with vector notation or just want to see a concrete example the full working java code can be found at the end of this article for reference.
\begin{algorithm}
\caption{Hyperassociative Map}
\begin{algorithmic}
% equilibrium distance
\REQUIRE $\tilde{\chi} > 0$
%repulsion strength
\REQUIRE $\delta > 1$
% learning rate
\REQUIRE $\eta = 0.05$
% alignment threshold (determines when graph is aligned)
\REQUIRE $\beta =$ 0.005
\PROCEDURE{HAM}{Vertex Set \textbf{as} $g$}
\STATE \CALL{Randomize}{$g$}
\WHILE{\CALL{AlignAll}{$g$} $> \beta \cdot \tilde{\chi}$}
\STATE optionally recenter the graph
\ENDWHILE
\ENDPROCEDURE
\PROCEDURE{AlignAll}{Vertex Set \textbf{as} $g$}
\STATE $\zeta = 0$
\FORALL{$v$ \textbf{in} $g$}
\STATE $\vec{{\scriptsize \triangle} p} =$ \CALL{Align}{$v$}
\IF{$||\vec{{\scriptsize \triangle} p}|| > \zeta$}
\STATE $\zeta = ||\vec{{\scriptsize \triangle} p}||$
\ENDIF
\STATE \CALL{Place}{$v$, \CALL{Position}{$v$} $+ \vec{{\scriptsize \triangle} p}$}
\ENDFOR
\RETURN $\zeta$
\ENDPROCEDURE
\PROCEDURE{Align}{Vertex \textbf{as} $v$}
\STATE $\vec{p} =$ \CALL{Position}{$v$}
\STATE $\vec{{\scriptsize \triangle} p} = 0$
\FORALL{$m$ \textbf{in} \CALL{Neighbors}{$v$}}
\STATE $\vec{q} =$ \CALL{Position}{$m$} - $\vec{p}$
\STATE $\vec{{\scriptsize \triangle} p} = \vec{{\scriptsize \triangle} p} + \vec{q} \cdot \frac{(||\vec{q}|| - \tilde{\chi}) \cdot \eta}{||\vec{q}||}$
\ENDFOR
\FORALL{$m$ \textbf{in} \CALL{NotNeighbors}{$v$}}
\STATE $\vec{q} =$ \CALL{Position}{$m$} - $\vec{p}$
\STATE $\vec{c} = \vec{q} \cdot \frac{-\eta}{{||\vec{q}||}^{\delta + 1}}$
\IF{$||\vec{c}|| > \tilde{\chi}$}
\STATE $\vec{c} = \vec{c} \cdot \frac{\tilde{\chi}}{||\vec{c}||}$
\ENDIF
\STATE $\vec{{\scriptsize \triangle} p} = \vec{{\scriptsize \triangle} p} + \vec{c}$
\ENDFOR
\RETURN $\vec{{\scriptsize \triangle} p}$
\ENDPROCEDURE
\PROCEDURE{Randomize}{Vertex Array \textbf{as} $g$}
\STATE randomise position of all vertex in $g$
\ENDPROCEDURE
\PROCEDURE{Place}{Vertex \textbf{as} $v$, Vector \textbf{as} $\vec{p}$}
\STATE sets the position of $v$ to $\vec{p}$
\ENDPROCEDURE
\PROCEDURE{Neighbors}{Vertex \textbf{as} $v$}
\RETURN set of all vertex adjacent to $v$
\ENDPROCEDURE
\PROCEDURE{NotNeighbors}{Vertex \textbf{as} $v$}
\STATE $s =$ set of all vertex not adjacent to $v$
\STATE $w =$ set of all vertex whose position is close to that of $v$
\RETURN $s \cap w$
\ENDPROCEDURE
\PROCEDURE{Position}{Vertex \textbf{as} $v$}
\RETURN vector representing position of $v$
\ENDPROCEDURE
\end{algorithmic}
\end{algorithm}
Obviously the pseudocode packs a lot of information into only a few lines so I'll try to explain some of the more important parts so you have an idea at what you're looking at.
Constants
First, lets consider the constants defined at the beginning. the variable \(\tilde{\chi}\) is called the Equilibrium Distance. It defines the ideal distance between two vertices connected by an edge. If a pair of vertices connected by a single edge are the only vertices present then they will align such that they are approximately as far apart as the value of \(\tilde{\chi}\). For simplicity here we have represented \(\tilde{\chi}\) as a single constant but in practice it is also possible to assign a different value to this constant for each edge, resulting in a graph with different aesthetic qualities. This value must of course always be a positive number greater than \(0\). The default value, and the one used in the demonstration video, is \(1\).
The second constant is called the Repulsion Strength and is represented by the \(\delta\) variable. This constant determines how strong the repulsion between two unconnected vertices are, that is two vertices not connected by an edge. Lower values for \(\delta\) result in a stronger repulsion and larger numbers represent a weaker repulsion. The default value is \(2\) and this is the value used in the demonstration video.
Next is the Learning Rate constant, \(\eta\). This is simply a scaling factor applied when the vertices are aligned to ensure the graph is aligned to each node with equivalent effect rather than over-fitting to the last node processed.
The last constant is the Alignment threshold, \(\beta\), this represents the minimum movement threshold. Once the vertices move less than this value during an alignment cycle it is presumed the graph is sufficiently aligned and the loop ends.
Align Procedure
The algorithm itself is represented such that it is broken up into three major procedures. The procedure named HAM is the entry point for the algorithm, the procedure named Align calculates the incremental alignment for a single vertex, and the procedure named AlignAll calculates alignment once for every vertex in the graph.
Lets first explain what is going on in the Align procedure. Here we have a single value being passed in, the vertex to be aligned, \(v\). On line 19 the current position of the vertex is obtained and represented as a euclidean vector, \(\vec{p}\). Next on line 20 a zero vector is initialized and represented as \(\vec{{\scriptsize \triangle} p}\). The vector \(\vec{{\scriptsize \triangle} p}\) is calculated throughout the procedure and represents the desired change to the current position of the vector \(\vec{p}\). In other words once \(\vec{{\scriptsize \triangle} p}\) is calculated it can be added to the current position of the vertex and will result in the new position for the vertex. Just as if calculating forces the \(\vec{{\scriptsize \triangle} p}\) will be the sum of all the composite influences acting on the vertex; so it represents the overall influence exerted on the vertex at any time.
When calculating \(\vec{{\scriptsize \triangle} p}\) the procedure must iterate through all the other vertices that have an effect on the vertex being aligned. There are two type of vertices each with different effects: neighbors, and non-neighbors. Neighbors are all the vertices connected directly to the current vertex by an edge, non-neighbors are all the other vertices not connected by an edge.
First the influence from the neighbor vertices is calculated on lines 21 - 24. The influence two neighbor vertices have on each other is different depending on how far apart they are. If they are closer than the Equilibrium Distance, \(\tilde{\chi}\), then the effect is repulsive. If they are farther apart than \(\tilde{\chi}\) then the effect is attractive. The calculation for this is represented by line 23. It basically calculates the vector that represents the difference between the position of the vertex being aligned and its neighbor and reduces the magnitude of the vector back to \((||\vec{q}|| - \tilde{\chi}) \cdot \eta\). To look at it another way if the equation was just \(||\vec{q}|| - \tilde{\chi}\) then the new position of the vector would be exactly at the Equilibrium Distance, but instead it is scaled to a fraction of this by \(\eta\) which adjusts how quickly the vertex will approach its equilibrium point.
Next the influence between non-neighbor vertices is calculated on lines 25 - 32. Non-neighbor vertices, that is vertices not connected by an edge, always exhibit a purely repulsive influence. Line 27 calculates this in a similar technique as before. That is the difference between the position of the two vertices is represented by \(\vec{q}\) and then its magnitude is scaled. Of course it's also negative to indicate that the force is repulsive. The equation just seems confusing in its simplified and compacted form. Initially it was derived by calculating the new magnitude of \(\vec{q}\) as the following.
$$ \frac{-1}{{\mid\mid\vec{q}\mid\mid}^{\delta}} \cdot \eta $$
This makes a lot more sense as we know in nature repulsive forces are the inverse square of their distance. So this accurately represents a repulsive influence that diminishes with distance. Once we actually apply that magnitude to the vector and simplify we arrive at our final equation.
$$ \vec{q} \cdot \frac{\frac{-1}{{\mid\mid\vec{q}\mid\mid}^{\delta}} \cdot \eta}{\mid\mid\vec{q}\mid\mid} \Rightarrow \vec{q} \cdot \frac{-\eta}{{||\vec{q}||}^{\delta + 1}}$$
The only caveat to this is seen in lines 28 to 30 where it checks the distance moved as a result of the repulsive influence. If it is greater than the Equilibrium Distance, \(\tilde{\chi}\), then its magnitude is scaled back to be \(\tilde{\chi}\). This is done because at very close distances the exponential nature of the repulsive influence becomes overwhelming and we want to ensure the majority of this influence works at a distance to allow the graph to spread apart but still allow the other influences to be the dominate influences on the graph.
At this point the computed change in position for the vertex is simply returned at line 33 for further processing by the AlignAll procedure.
AlignAll Procedure
The AlignAll Procedure is extremely straight forward. It is passed in the set of all vertices in the graph as \(g\) and iterates over the set while aligning them one at a time. Each vertex will get aligned once per call to the procedure, this means the procedure will usually need to be called multiple times.
On line 8 the Maximum Vertex Movement variable, represented as \(\zeta\), is initialized to \(0\). This variable represents the greatest distance any vertex moved during the alignment; after being calculated it's value is returned on line 15. The Maximum Vertex Movement is important for determining when the HAM algorithm has finished processing.
Other than that this procedure doesn't do anything special, the vertex alignment vector is calculated on line 10 and the new position for the vertex is set on line 14.
HAM Procedure
The HAM procedure is another rather straight forward procedure to explain. It starts by assigning some initial random coordinates to each vertex in the graph. After that it continually loops calling AlignAll until the graph is sufficiently aligned.
On line 3 the AlignAll procedure is called in a loop until the Max Vertex Movement returned is less than \(\beta \cdot \tilde{\chi}\). This is just the Alignment Threshold normalized by the Equilibrium Distance constant. The Alignment Threshold is sufficiently small such that if the movements in the graph are less than this value then they are considered negligible and the alignment can end.
As an optional step after each alignment iteration it may be desired to translate the entire graph so it is centered around the zero vector. There is a small amount of drift as the alignment of the graph is calculated and by doing this it ensures the graph remains in the center of the view when rendered. The drift is usually negligible however so this step is entirely optional. In the full java example below the logic for centering the graph is included.
Appendix: Full Java Code
public class HyperassociativeMap<G extends Graph<N, ?>, N> implements
GraphDrawer<G, N> {
private static final double REPULSIVE_WEAKNESS = 2.0;
private static final double DEFAULT_LEARNING_RATE = 0.05;
private static final double EQUILIBRIUM_DISTANCE = 1.0;
private static final double EQUILIBRIUM_ALIGNMENT_FACTOR = 0.005;
private final G graph;
private final int dimensions;
private Map<N, Vector> coordinates = Collections.synchronizedMap(new
HashMap<N, Vector>());
private static final Random RANDOM = new Random();
private final boolean useWeights;
private double equilibriumDistance;
private double learningRate = DEFAULT_LEARNING_RATE;
private double maxMovement = 0.0;
public HyperassociativeMap(final G graph, final int dimensions, final
double equilibriumDistance, final boolean useWeights) {
if (graph == null)
throw new IllegalArgumentException("Graph can not be null");
if (dimensions <= 0)
throw new IllegalArgumentException("dimensions must be 1 or more");
this.graph = graph;
this.dimensions = dimensions;
this.equilibriumDistance = equilibriumDistance;
this.useWeights = useWeights;
// refresh all nodes
for (final N node : this.graph.getNodes()) {
this.coordinates.put(node, randomCoordinates(this.dimensions));
}
}
@Override
public G getGraph() {
return graph;
}
public double getEquilibriumDistance() {
return equilibriumDistance;
}
public void setEquilibriumDistance(final double equilibriumDistance) {
this.equilibriumDistance = equilibriumDistance;
}
public void resetLearning() {
maxMovement = 0.0;
}
@Override
public void reset() {
resetLearning();
// randomize all nodes
for (final N node : coordinates.keySet()) {
coordinates.put(node, randomCoordinates(dimensions));
}
}
@Override
public boolean isAlignable() {
return true;
}
@Override
public boolean isAligned() {
return isAlignable()
&& (maxMovement < (EQUILIBRIUM_ALIGNMENT_FACTOR *
equilibriumDistance))
&& (maxMovement > 0.0);
}
@Override
public void align() {
// refresh all nodes
if (!coordinates.keySet().equals(graph.getNodes())) {
final Map<N, Vector> newCoordinates = new HashMap<N, Vector>();
for (final N node : graph.getNodes()) {
if (coordinates.containsKey(node)) {
newCoordinates.put(node, coordinates.get(node));
} else {
newCoordinates.put(node, randomCoordinates(dimensions));
}
}
coordinates = Collections.synchronizedMap(newCoordinates);
}
maxMovement = 0.0;
Vector center;
center = processLocally();
// divide each coordinate of the sum of all the points by the number of
// nodes in order to calculate the average point, or center of all the
// points
for (int dimensionIndex = 1; dimensionIndex <= dimensions;
dimensionIndex++) {
center = center.setCoordinate(center.getCoordinate
(dimensionIndex) / graph.getNodes().size(), dimensionIndex);
}
recenterNodes(center);
}
@Override
public int getDimensions() {
return dimensions;
}
@Override
public Map<N, Vector> getCoordinates() {
return Collections.unmodifiableMap(coordinates);
}
private void recenterNodes(final Vector center) {
for (final N node : graph.getNodes()) {
coordinates.put(node, coordinates.get(node).calculateRelativeTo
(center));
}
}
public boolean isUsingWeights() {
return useWeights;
}
Map<N, Double> getNeighbors(final N nodeToQuery) {
final Map<N, Double> neighbors = new HashMap<N, Double>();
for (final TraversableCloud<N> neighborEdge : graph.getAdjacentEdges
(nodeToQuery)) {
final Double currentWeight = (((neighborEdge instanceof Weighted)
&& useWeights) ? ((Weighted) neighborEdge).getWeight() :
equilibriumDistance);
for (final N neighbor : neighborEdge.getNodes()) {
if (!neighbor.equals(nodeToQuery)) {
neighbors.put(neighbor, currentWeight);
}
}
}
return neighbors;
}
private Vector align(final N nodeToAlign) {
// calculate equilibrium with neighbors
final Vector location = coordinates.get(nodeToAlign);
final Map<N, Double> neighbors = getNeighbors(nodeToAlign);
Vector compositeVector = new Vector(location.getDimensions());
// align with neighbours
for (final Entry<N, Double> neighborEntry : neighbors.entrySet()) {
final N neighbor = neighborEntry.getKey();
final double associationEquilibriumDistance = neighborEntry
.getValue();
Vector neighborVector = coordinates.get(neighbor)
.calculateRelativeTo(location);
double newDistance = Math.abs(neighborVector.getDistance()) -
associationEquilibriumDistance;
newDistance *= learningRate;
neighborVector = neighborVector.setDistance(newDistance);
compositeVector = compositeVector.add(neighborVector);
}
// calculate repulsion with all non-neighbors
for (final N node : graph.getNodes()) {
if ((!neighbors.containsKey(node)) && (node != nodeToAlign)
&& (!graph.getAdjacentNodes(node).contains(nodeToAlign))) {
Vector nodeVector = coordinates.get(node).calculateRelativeTo
(location);
double newDistance = -1.0 / Math.pow
(nodeVector.getDistance(), REPULSIVE_WEAKNESS);
if (Math.abs(newDistance) > Math.abs(equilibriumDistance)) {
newDistance = Math.copySign(equilibriumDistance,
newDistance);
}
newDistance *= learningRate;
nodeVector = nodeVector.setDistance(newDistance);
compositeVector = compositeVector.add(nodeVector);
}
}
Vector newLocation = location.add(compositeVector);
final Vector oldLocation = coordinates.get(nodeToAlign);
double moveDistance = Math.abs(newLocation.calculateRelativeTo
(oldLocation).getDistance());
if (moveDistance > maxMovement) {
maxMovement = moveDistance;
}
coordinates.put(nodeToAlign, newLocation);
return newLocation;
}
public static Vector randomCoordinates(final int dimensions) {
final double[] randomCoordinates = new double[dimensions];
for (int randomCoordinatesIndex = 0; randomCoordinatesIndex <
dimensions; randomCoordinatesIndex++) {
randomCoordinates[randomCoordinatesIndex] = (RANDOM.nextDouble()
* 2.0) - 1.0;
}
return new Vector(randomCoordinates);
}
private Vector processLocally() {
Vector pointSum = new Vector(dimensions);
for (final N node : graph.getNodes()) {
final Vector newPoint = align(node);
for (int dimensionIndex = 1; dimensionIndex <= dimensions;
dimensionIndex++) {
pointSum = pointSum.setCoordinate(pointSum.getCoordinate
(dimensionIndex) + newPoint.getCoordinate
(dimensionIndex), dimensionIndex);
}
}
return pointSum;
}
}
