BFL: a node and edge betweenness based fast layout algorithm for large scale networks
© Hashimoto et al; licensee BioMed Central Ltd. 2009
Received: 01 September 2008
Accepted: 15 January 2009
Published: 15 January 2009
Network visualization would serve as a useful first step for analysis. However, current graph layout algorithms for biological pathways are insensitive to biologically important information, e.g. subcellular localization, biological node and graph attributes, or/and not available for large scale networks, e.g. more than 10000 elements.
To overcome these problems, we propose the use of a biologically important graph metric, betweenness, a measure of network flow. This metric is highly correlated with many biological phenomena such as lethality and clusters. We devise a new fast parallel algorithm calculating betweenness to minimize the preprocessing cost. Using this metric, we also invent a node and edge betweenness based fast layout algorithm (BFL). BFL places the high-betweenness nodes to optimal positions and allows the low-betweenness nodes to reach suboptimal positions. Furthermore, BFL reduces the runtime by combining a sequential insertion algorim with betweenness. For a graph with n nodes, this approach reduces the expected runtime of the algorithm to O(n2) when considering edge crossings, and to O(n log n) when considering only density and edge lengths.
Our BFL algorithm is compared against fast graph layout algorithms and approaches requiring intensive optimizations. For gene networks, we show that our algorithm is faster than all layout algorithms tested while providing readability on par with intensive optimization algorithms. We achieve a 1.4 second runtime for a graph with 4000 nodes and 12000 edges on a standard desktop computer.
Advances in biotechnology have made it possible to collect vast amounts of genetic data. Although extensive research has been done on numerical and statistical methods to infer the relationship among genes, which we call gene networks, methods for analyzing such data visualizing large gene networks has received less attention.
There exists significant former literature on general graph layout algorithms such as orthogonal drawing, planar embedding, force-directed layout . Similarly, metabolic networks with relatively small numbers of nodes (<100) have received significant attention, with notable algorithms being proposed by Karp , and . However, these algorithms are designed with fundamentally different goals than those for gene networks. Well known fast graph-theoretic algorithms such as Sugiyama , radial tree  are capable of drawing large graphs, but give degenerate results for large and dense graphs. On the other hand, force based algorithms such as spring embedder  are able to produce symmetric and aesthetic results, but become intractable in the case of large datasets, and fail to represent biological datasets. Former work in incorporating biologically information [7, 8] applies simple positional constraints, but do not scale well to large networks. It has also been noted that such algorithms fail to produce compact layouts .
Optimizing algorithms rely on minimizing an underlying metric, and have been used to great success. Grid-layout [3, 9, 10] has been used in cellular circuits to incorporate complex constraints, while multidimensional scaling  along with planar subgraph extraction , which maps an artificial metric to Euclidean space, has been used to create fast algorithms incorporating biological attributes.
All of the above approaches have their drawbacks; they either fail to reflect biological relationships in the layout or fail to scale for large problems. This problem arises because utilizing biological facts is a computationally expensive operation [9, 10] which most algorithms are not designed for. Grid layout  for example, requires satisfying biologically meaningful component placements.
This paper introduces a fast, biologically relevant layout algorithm using the concept called betweenness.
Betweenness is most commonly used as a way of analyzing social networks [14, 15]. This metric was first proposed by Freeman et al.  as a way to characterize sparsely connected graphs. Betweenness centrality for certain types of flow is known as an indicator of traffic through a certain node or edge [14, 16, 17]. The index has previously been used in ranking websites and clustering in social networks .
Biologically, betweenness is useful when the digraph relationship correspond to information flow. In this case modules from betweenness represents informationally isolated modules while the high betweenness nodes and edges are hubs and links with high betweenenss.
BFL is of interest in large gene, and protein networks. Protein and gene networks allow for a straightforward attack with BFL, in that a straightforward weighted layout will produce biologically relevant results, as they represent interaction networks. Metabolic networks on the other hand should first be analyzed with modularity analysis . This is because the genes of interest are often not those with the highest information content (which would be common ATP, NADH pathways) but rather those which function uniquely.
Specifically, given σ st defined as the number of shortest paths between nodes s and t, and σ st (v) defined as the number of shortest paths passing through v (and s, t ≠ v), betweenness is defined as the sum of σ st (v)/σ st for all nodes s and t in the graph. In other words, betweenness is the sum of the probabilities of v being in the shortest path between any two nodes. This definition of node betweenness has been extended to edges by Newman et al. as a way to extract community structures . Edge betweenness is similarly defined as above by taking the sum of σ st (e)/σ st , where σ st (e) is defined as the number of shortest paths between s and t passing through edge e.
Recently, betweenness has become of interest in bioinformatics because of its biological importance in gene and protein networks. Specifically, it has been shown that betweenness values correctly identify bridge proteins , protein modules [11, 16, 19], and essential proteins .
Although there are other measures which fulfill the above measures, such as random walk betweenness  and eigenvector centrality , these measures have a higher runtime complexity and produce similar values. In some isolated cases, such as an extremely dense graph, these measures may result in better layout, although we consider that in general the runtime trade-off is unnecessary.
These results imply first that clusters generated with Girvan-Newman's algorithm  using edge betweenness accurately represent clusters in protein function . Second, high betweenness value nodes are biologically important to the function of the gene network. Finally, betweenness based layout correctly identifies bridges, which is valuable to graph layout techniques. We attach a standard biological dataset by Luo et al.  to show these properties.
The remainder is organized as follows. In the methods section, we first define betweenness and then we demonstrate an efficient parallel algorithm for calculating betweenness. We then present a new node and edge betweenness based fast layout algorithm (BFL) and the specific score methods. Lastly, we present the expected runtime of the layout methods. In the results and discussion section, we show the effects of graph size and confirm the effectiveness of our approach on runtime. We then compare the run-times and outputs of various networks with other layout algorithms, and also show that betweenness is crucial to our algorithm.
Definition of betweenness
We will use the same notation originally developed by Brandes  to describe node and edge betweenness calculations. First, let G = (V, E) be a connected directed graph. We define σ ab to be the number of shortest paths between nodes a and b in G. We then define σ ab (n) as the number of shortest paths between a and b which go through n ∈ V. In this paper, for each edge e ∈ E, we denote e p and e c to be the originating and destination nodes respectively.
If the graph is a TSP, following property is satisfied for a, b ∈ V with a as an ascendant of b.
- (ii)Similarly for each edge e ∈ E, we define the sigma operator σ ab (e) to be the number of paths from a to b which pass through edge e. In a TSP, we have(4)
We propose a new forward algorithm where we start at the root node and propagate downwards. This allows us to parallelize the operations in a much more straightforward way compared to the backwards algorithm as in the next section.
Parallelized betweenness calculation
Brandes  previously showed an implementation for calculating edge betweenness values. We show that the forward algorithm operates upon the same principle while allowing for parallelism.
Given a graph G, we start by running Dijkstra's algorithm on each node v and storing all shortest paths from node v to all other nodes. This gives us a TSP T comprises of shortest paths from v.
Our algorithm attempts to break down the betweenness calculation for shortest paths starting at each node v in a recurrent relation.
The first term can be seen as the additional contribution made from the new edge between v and k. The latter term can be seen as the contributions of all nodes downstream of k. A proof of the correctness of the backwards form of this equation is given in .
This equation implies that the operations involved in calculating node betweenness can be used for edge betweenness values.
This algorithm can be parallelized for each v ∈ V since the operations from each TSP T are independent. Therefore, this is a more efficient algorithm than many of the current methods, which depend on calculating node betweenness before edge betweenness.
The method presented in  relied upon this approach, but started at v j rather than v i . This is an obstacle for parallelization since the values of some v j cannot be fixed unlike those for v i .
For large networks, betweenness values become extremely large for central nodes, while terminal nodes with no children have zero centrality. In order to make this metric more suited for layout, we take the log centrality for both edge and betweenness (we add value one to the original betweenness value in order to avoid log(0)).
Edge and node betweenness based fast layout algorithm BFL
Variable legend and overall layout algorithm
Queue of nodes with largest node betweenness first
N odeBC [v]
Betweenness values for node v
EdgeBC [v1] [v2]
Edge betweenness from v1 to v2
Empty set (contains nodes already placed)
Empty multi-value map (Key – node, Value – orphans with v as parent)
Tree contains a quadtree containing nodes in Table 4
controls how far from the parent nodes are initially placed
lower values force tighter convergence constraints
controls how far nodes are moved each iteration
controls when to cut off simulated annealing loops
controls how much density is used in score calculation
controls how much edge lengths are used
controls how important edge crossings are
sets how large each bucket can be in the quadtree
Fast Edge and Node Betweenness Based Layout Algorithm
1: procedure LAYOUT
2: Deque v ← Q
3: Set v. coordinate ← (0, 0)
4: Push v → S
5: while Q is not empty do
6: Deque v ← Q
7: if S contains a v' connected to v then
10: for all Neighbors n of v do
11: Put (n, v) → H
12: end for
13: end if
14: end while
15: end procedure
16: procedure PLACENODE(v, v')
17: Set v. coordinate ← Coordinate generated by a Gaussian centered at v'. coordinate with variance c1N odeBC [v]2
19: Push v → S
20: Get nodes ← values in H with key v ▹ Get orphans which can now be placed
21: for all node in nodes do
23: end for
24: end procedure
As described in introduction section, for the BFL layout algorithm, we mainly care following two points; (i) the important elements (high betweenness nodes and edges) should be emphasized in the resulting layout, (ii) the layout algorithm should run in real-time for large scale gene networks (around 10000 elements).
A naïve implementation of betweenness would scale scores as part of an optimizing algorithm. Such a naïve method was initially investigated, but incorporating betweenness in a global optimization algorithm caused significant slowdowns (conflicting with (ii)). The global optimization incurred a large penalty because the scaling forced high-betweenness nodes with strict tolerances to be optimized in a sea of lower-betweenness nodes.
For the above reason, instead of using a global optimization approach in BFL, we created a local optimization procedure which took advantage of the properties of betweenness and minimized the loss of quality. BFL places one node at a time in order of descending betweenness instead of placing all nodes at once (see lines 5 to 14 in Table 2). In BFL, simulated annealing is used to place the inserted node by minimizing the score function (the detail of this function is defined in the next section).
The execution steps of BFL in Tables 1 and 2 is summarized as follows. Initially, BFL stores all nodes to the priority queue Q, in which each node is prioritized with the node betweenness value (highest value is at the head) and creates an empty set S that is used to store the already placed nodes. In the first step, dequeue the top node in Q, put the node to S, and set the position of the node to (0,0). In the main recursive loop (lines 5 to 14), dequeue the current head node v in Q and check whether or not S contains a neighbor of v. (i) If S does not contain a neighbor, all edges connected to v are inserted into a map H (lines 9–13) and is not added the v to S. (ii) If S contains a neighbor of v (lines 7–9), the v is inserted to S (line 19) and one node v' in S is with the highest edge betweenness is selected and placed the v at the initial coordinate randomly drawn from a Gaussian centered at v with variance c1·NodeBC[v]2 (line 17). The initial coordinate is optimized by using the annealing method described in the next section (line 18). In (ii), all nodes connected to v in H are also placed in the same manner (line 21). We separate execution branches (i) and (ii) in order to resolve orphan nodes. Since insertion order depends only on betweenness, there are some nodes which are disconnected from the currently placed graph S. In this case we put this node on a dependency queue, and place the node as soon as the dependency is fulfilled.
This significantly speeds up BFL layout since only nodes directly connected to the newly inserted node is needed to calculate the score function in each insertion step, i.e. pairwise effects elsewhere on the graph do not need to be calculated at all.
BFL runtime is further shorten by using following two properties. In the beginning phase, few nodes are already placed and score calculation proceeds quickly. Later, on the other hand, the inserted nodes will have few edges (since the value of betweenness is low), leading to much looser score tolerances and fewer children to process (the effectiveness of those properties are confirmed in later section with simulation test).
Simulated annealing with betweenness based score function
1: procedure ENERGY OF(v,currentCoord)
2: Add energy ← c3 density(v,currentCoord)
3: Add energy ← c4 edgeLength(v,currentCoord)
4: Add energy ← c5 edgeCrosses(v,currentCoord)
5: Return energy
6: end procedure
7: procedure DENSITY(v,currentCoord)
8: for all node in Set s do
9: Add density ←
10: if v and node overlap then 11: Return ∞
12: end if
13: end for
14: Return density
15: end procedure
16: procedure EDGE LENGTH(v,currentCoord)
17: for all Edges of v connected to a destination in S do
18: Add length ← EdgeBC [v] [destination] * Distance(v, destination)2
19: end for
20: Return length
21: end procedure
22: procedure EDGE CROSSES(v,currentCoord)
23: for all Edges in v connected to a destination in S do
24: Add crosses ← CountIntersections(Edges) * EdgeBC [v] [destination] ▹ Intersections should be counted with an efficent Ray-Shooting algorithm
25: end for
26: Return crosses
27: end procedure
where is the connected edges to the node v i , is the subgraph before inserting node v i , and k1 + k2 + k3 = 1 (NodeDensity, EdgeLength and EdgeCross are defined later).
Similar metrics have been used in  and there have been aesthetic justifications for their use. Simulated annealing is even more suited in this case because of its robustness and single-point performance. While there are very efficient algorithms such as genetic, particle swarm, or ant colony optimization for parallel optimization procedures, BFL reduces global optimization to a series of local optimization problems, which removes the need for parallel optimization. In this case, nearly all stochastic optimization problems become a variant of simulated annealing. On the other hand, hill climbing and BFGS based numerical optimization procedures are not robust enough for this problem. The optimization landscape is extremely multimodal (as each vertex becomes a local mode) and therefore the chance of local minima are extremely high.
In our score function, values of node and edge betweennesses are effectively used to ensure that high betweenness nodes are given more emphasis than low-betweenness ones with low calculation cost.
Node density function
where is the Euclidean distance of nodes v i and .
The density function will create a multi-scale layout; high betweenness nodes are separately positioned as core nodes and low-betweenness nodes are positioned around them.
Fast density modifications
1: procedure FAST DENSITY(v,currentCoord)
2: Set currentNode ← head of tree T
3: while currentNode is not a leaf do
4: if currentCoord > currentNode.vpartition then
5: if currentCoord > currentNode.hpartition then
6: Set currentNode ← currentNode.topRight
7: else Set currentNode ← currentNode.bottomRight
8: end if
10: if currentCoord > currentNode.hpartition then
11: Set currentNode ← currentNode.topLeft
12: else Set currentNode ← currentNode.bottomLeft
13: end if
14: end if
15: end while
16: for all node in Set currentNode do
17: Add density ←
18: if v and node overlap then
19: Return ∞
20: end if
21: end for
22: Return density
23: end procedure
24: procedure FAST PLACE NODE(v,v')
25: Set v.coordinate ← Coordinate generated by a Gaussian centered at v'.coordinate with variance c1N odeBC [v]2
27: Push v → S
28: InsertNode(v) ▹ Node insertion to tree added
29: Get set of nodes ← values in H with key v ▹ Get orphans which can now be placed
30: for all node in nodes do
32: end for
33: end procedure
34: procedure INSERT NODE(v)
35: Set currentN ode ← head of tree T
36: while currentN ode is not a leaf do
37: if currentCoord > currentN ode.vpartition then
38: if currentCoord > currentN ode.hpartition then
39: Set currentN ode ← currentN ode.topRight
40: else Set currentN ode ← currentN ode.bottomRight
41: end if
43: if currentCoord > currentN ode.hpartition then
44: Set currentN ode ← currentN ode.topLef t
45: else Set currentN ode ← currentN ode.bottomLef t
46: end if
47: end if
48: end while
49: currentN ode.add(v)
50: while currentN ode.size > maxT reeSize do22
51: Set currentN ode.topRight ← currentN ode.partitionT opRight
52: Set currentN ode.topLeft ← currentN ode.partitionT opLef t
53: Set currentN ode.bottomRight ← currentN ode.partitionBottomRight
54: Set currentN ode.bottomLeft ← currentN ode.partitionBottomLef t
55: end while
56: end procedure
Edge length function
In , the average edge length is used to counterbalance the density and prevent a space-inefficient layout. In BFL, each edge length is scaled by its betweenness score, which forces nodes to shorten high betweenness edges over low betweenness ones. Edge lengths therefore as a aesthetic measure of the contribution of an edge to the node betweenness.
where is the Euclidean length of the edge .
Edge crossing function
where is an indicator function which returns 1 if and cross and otherwise returns 0. In order to calculate the number of crossings, we use the efficient ray shooting algorithm proposed by .
For our simulated annealing loop, a polynomial cooling scheme is specified by defining the temperature t as
t = (k max - k) n ,
Annealing and optimization algorithm
1: procedure ANNEAL(v,threshold)
2: Set currentCoord ← v.coordinate
3: Set currentEnergy ← EnergyOf(v, currentCoord)
4: Set bestCoord ← currentCoord
5: Set bestEnergy ← currentEnergy
6: Set k ← 0
7: while currentEnergy – tempEnergy > threshold AND k < kmax do ▹ kmax is some constant to cut off runaway calculations
8: Set tempCoord ← newNeighbor(currentCoord,v)
9: Set tempEnergy ← EnergyOf(v,tempCoord)
10: if bestEnergy > tempEnergy then
11: Set bestEnergy ← tempEnergy
12: Set bestCoord ← tempCoord
13: end if
14: if transition(currentEnergy, tempEnergy, k) then
15: Set currentCoord ← tempCoord
16: Set currentEnergy ← tempEnergy
17: end if
18: end while
19: Set v.coordinate ← bestCoord
20: end procedure
21: procedure NEW NEIGHBOR(currentCoord,v)
22: Return Gaussian centered at currentCoord with deviation c2Ln(N odeBC [v] + 1)3
23: end procedure
24: procedure TRANSITION(currentEnergy,tempEnergy,k)
25: Set temp ← (kmax - k)3
26: Set transition ← e(currentEnergy-tempEnergy)/temp
27: Set rand ← random value from 0 to 1
28: if rand < transition then
29: Return true
31: Return false
32: end if
33: end procedure
Layout algorithm runtime analysis
Since the betweenness calculation can be cached into the network file for repeated uses, we only consider the runtime of the layout algorithm itself. The runtime of the layout is dominated by evaluation of the scoring function which is called |V| times.
where ζ(x,y)is the 2nd derivative of generalized Zeta function with respect to x.
Which shows that, the runtime grows slower than d log(d) with respect to degree d and slower than |V|2 for node size |V|, or in little o notation,
ζ(2,0)(-1/2, |V|) = O(|V|2).
The estimated result O(|V|2) implies that our algorithm has an asymptotic complexity better than many fast optimizing algorithms with respect to node size. Edge crossing calculation can be ignored in many cases leading to an even faster runtime log(Gamma(|V| + 1)) if degree is constant, which is asymptotically equal to |V|log(|V|) and is the current standard for the fastest layout algorithms.
This speedup would not be possible without the sequential layout from the betweenness algorithm.
Results and discussion
Methods and datasets
The algorithm was implemented in Java with files stored in Cell System Markup Language (CSML) format . A Fibonacci heap was used for the priority queue, all other data structures used library implementations available in the JDK.
Runtime tests were done on a 8-core Intel Xeon 4800 X5450 3 GHz machine with 16 GBs RAM, with random graphs generated by methods given by Rodionov et al. . Comparisons to other programs were made on one sparse graph (2000 nodes and 7000 edges), two dense graphs (2000 nodes and 11000/47000 edges) and one estimated gene regulatory network (1897 nodes and 2849 edges) on an Athlon X2 3.3 GHz machine with 4 GBs RAM running on Windows XP. For the last gene regulatory network (calls UO Analysis), the microarray data of the ultradian oscillation (UO) clock in mouse presomitic mesoderm cells by Dequeant et al  is used. We generated graphs in CSML, GML, NET, and TLP files for various programs and used the same graph to compare run-times and outputs. These graph data files used in our simulation are available in the additional file 1.
For all tests, cached costs were not calculated (including loading, preprocessing, and betweenness calculations), as we are concerned with the time to layout.
BFL runtime dependency on node size
BFL runtime dependency on betweenness
BFL runtime dependency on graph density
Runtime Comparisons Between Algorithms
Dataset Number of Node/Edge
Fruchterman Reingold (Pajek)
BFL compared to existing algorithms
Table 6 shows the various run-times of our algorithm against those of four other competing algorithms. Force-directed and optimization algorithms similar to our own were chosen from possible candidates. By this criterion we compared against Kamada-Kawai  and Fruchterman Reingold  energy based algorithms in Pajek , GEM (Generalized Expectation-Maximization) based optimization in Tulip , and the RSDFP layout algorithm in InterViewer . It is worth noting that our algorithm is implemented in Java while the competing algorithms are native applications. Thus, if our application were implemented efficiently in C, we would be able to achieve even faster run-times with even more drastic results. A future goal is to implement K-K, FR or GEM algorithms using the sequential insertion and betweenness weight functions used in the BFL algorithm. We hope to be able to get true force-directed algorithms which can produce better results with no increase in runtime.
Resulting Layout of BFL compared to those of others
We also note that the runtime of this set was significantly lower (800 ms) for the betweenness algorithm compared to the others. The betweenness algorithm performs drastically better than others with sparse and multiscale datasets, while the competing algorithms have similar performance in randomly generated graphs.
Betweenness is critical to BFL layout structure
Our BFL layout algorithm mainly achieved following two points: (i) the important elements (high betweenness nodes and edges) are emphasized in the resulting layout, (ii) the layout algorithm runs in real-time for large scale gene networks (around 10000 elements). For a graph with n nodes, this approach reduces the expected runtime of the algorithm to O(n2) when considering edge crossings, and to O(n log n) when considering only density and edge lengths. We also compared against fast graph layout algorithms and approaches requiring intensive optimizations. For gene networks, our algorithm was faster than all layout algorithms tested while providing readability on par with intensive optimization algorithms. We achieve a 1.4 second runtime for a graph with 4000 nodes and 12000 edges on a standard desktop computer. We will develop an effective tuning method for scaling parameters automatically in response to change in graph degree and optimize the algorithm further. We also intend to show that the layout algorithm provides a rough metric of functional relations, where betweenness separates functionally unrelated units and identifies hub genes.
Computation time was provided by the Super Computer System, Human Genome Center, Institute of Medical Science, University of Tokyo.
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