Layered Graph Force-driven Vertex Positioning

Radek Ma

ık

Faculty of Electrical Engineering, Czech Technical University, Technicka 2, Prague, Czech Republic

Keywords:

Force-driven Layout, Layered Graph, Multitree, Spanning Tree, Phylogenetic Network, Genealogical

Network.

Abstract:

We propose a new method of node positioning for huge layered graphs speciﬁed by layers and ﬁxed ordering

of nodes within layers. We assume that the assignments of nodes to the layers and the order of nodes within

the layers are provided by other suitable methods capable of processing multitree like networks. The node

positioning method is based on the force-driven approach with barrier-like repulsive forces that avoids the

quadratic complexity of traditional methods. We demonstrate achievements on several datasets containing

up to millions of people or species. The proposed layout method of layered graphs that are close to acyclic

multitrees creates aesthetically acceptable layouts in linear time.

1 INTRODUCTION

Although it has been more than 55 years since Tutte

introduced barycentric embedding, research of graph

visualization techniques remains a highly active ﬁeld

attracting much attention (Tutte, 1963). Working with

tree-like structures such as genealogical or phyloge-

netic networks is no exception in this sense. We will

use a genealogical graph as an exemplar of general

tree-like networks to simplify our further discussion.

Applications to other tree-like network domains are

provided in the section dedicated to experiments.

In many situations the resulting network layout is

produced by state of the art tools as required (a brief

overview of such tools is provided in Section 2). Nev-

ertheless, the tools can often only process networks

up to several thousand nodes. The tools and methods

that are able to cope with millions of nodes and edges

are not able to cover additional constraints, such as

the layering of node children into the same layer and

keeping the order of nodes within a given layer as ex-

pected in genealogical network visualizations.

In this paper, we propose a new node positioning

method that keeps both assignments of nodes to layers

and the order of nodes within each layer. The method

is very fast and therefore can operate with networks

having more than a million nodes and edges. The

method is based on the force-driven approach with

barrier-like repulsive forces that avoids the quadratic

complexity of traditional methods.

Thus, in this paper we focus on the third most

critical aspect discussed by Sugiyama in (Sugiyama

et al., 1981), also related to the third step of the main

algorithm proposed in (Gansner et al., 1993), node

positioning:

1. determination of generations (layers, node ranks),

2. enforcing node orders within the layers,

3. setting the actual layout coordinates of nodes

(node positioning),

4. design of edges.

In other words, we assume that node layers and or-

ders in the layers are ﬁxed in an optimal way (they

should not be changed) and it is necessary to accom-

plish only node positioning. The rest of the steps,

such as layering and ordering, is performed by other

methods able to operate within huge networks, e.g.

such as those proposed for multitree-close networks

in (Marik, 2017).

The remainder of the paper is organized in the fol-

lowing way. In the next section, a brief overview

of related tools and methods is given. Then in Sec-

tion 3, we provide a summary of important mathe-

matical concepts used in this paper. We continue with

a description of the proposed method’s technical de-

tails and its steps in Section 4. Finally, we discuss

achieved results tested on datasets with up to millions

of nodes in Section 5.

2 RELATED METHODS

In the following paragraphs, we provide a brief

overview of the methods related to the method pro-

posed in this article. Sections 2.1 and 2.2 review

rík, R.

Layered Graph Force-driven Vertex Positioning.

DOI: 10.5220/0006624703010308

In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 3: IVAPP, pages

301-308

ISBN: 978-989-758-289-9

301

methods related to layering and ordering that serve

as preprocessing steps to node positioning. Having

this context established, node positioning methods are

then described in Section 2.3.

2.1 Multitree Close Networks Layering

and Node Ordering

Tree based drawing methods for phyloge-

netic/genealogical graphs have been one of the

standard techniques for centuries. Ancestor trees,

descendant trees and Hourglass charts are part

of the traditional tools employed by the major-

ity of freeware, shareware, or commercial tools,

for example Gramps (Gramps, 2016), MyHer-

itage (MyHeritage, 2016), TreePlus (Lee et al.,

2006), GraphViz (Gansner and North, 2000), and

Phylo.io (Robinson et al., 2016). There are also

other more space-efﬁcient representations such as

fan charts or H-charts (Tuttle et al., 2010; Kieffer

et al., 2016). As any pure tree representation enables

any ordering of node predecessors/successors, it is

possible to specify the type of ordering, such as

children ordered by their birth dates. Furthermore,

tree representations can be laid out in such a way that

family members are grouped together. The tools are

not able to process huge networks, which are not pure

trees, with millions of nodes while still satisfying

constraints on node layering and ordering.

However, the situation with family member

grouping changes signiﬁcantly if the assumptions of

one main person and direct ancestors/descendants are

dropped. In a number of cases, it is highly ben-

eﬁcial if the entire network of families, or at least

a signiﬁcant part, can be displayed in one layout.

Then we often deal with structures close to multitrees

and face difﬁcult issues linked with edge crossing

and preferences on node clustering (Warﬁeld, 1977;

Sugiyama et al., 1981) with occasional cycles usu-

ally caused by mistakes in datasets. Therefore, the

standard techniques for planar graph layouts (Lem-

pel et al., 1967; Hopcroft and Tarjan, 1974; Shih and

Hsu, 1999) including planarization techniques (Re-

sende and Ribeiro, 2001; Chimani et al., 2008) are

not suitable in all cases because of a higher than lin-

ear complexity.

As we assume multitree-like networks, we would

like to stress the signiﬁcance of layers, so we con-

sider a layout design targeting layered drawing (Healy

and Nikolov, 2013). A layering and ordering method

targeting huge tree-close networks with up to several

millions of nodes that ensures some node layer and

order constraints was introduced in (Marik, 2017). A

two-level approach when a given quasi-tree (a graph

having O(|V |) biconnected components) is decom-

posed into a tree of biconnected components where

biconnected components are drawn with a force-

directed approach with subtrees placed onto concen-

tric rings was proposed in (Archambault et al., 2006).

Both approaches (Archambault et al., 2006; Marik,

2017) use a spanning tree to drive the high level lay-

out, but they differ in optimization criteria. The ma-

jority of algorithms for node layering are derived from

the topological order computation, O(|V | + |E|) time

complexity (Cormen et al., 2009).

2.2 Constrained Graph Layout

There is also a class of methods that tackle graph

layout while satisfying different types of constraints

targeting, for example, node separation, drawing cy-

cles, and non-overlapping clusters (Dwyer and Koren,

2005; Dwyer, 2009). Although the methods are ap-

plicable to more general networks than those close to

multitree treated in this paper, they have a higher com-

plexity (often quadratic) than linear. The experiments

presented networks of up to 10,000 nodes only. The

given node layers and the order of nodes within layers

assumed in this paper can be expressed using about

O(|V |

) node separation constraints that leads to an

unfeasible problem for networks with the number of

nodes of magnitude |V | ≈ 10

2.3 Node Positioning

In this paper, we assume that layers of nodes and

a node order within the layers are given and ﬁxed.

Therefore, we focus only on a new node positioning

force-driven method. At ﬁrst, we refer to the tradi-

tional reference by Tutte (Tutte, 1963). His paper

focuses mainly on 3-connected planar graphs. Tutte

considered barycentric graph embedding. The paper

is also referenced as the ﬁrst “force-directed” algo-

rithm because suitable vertex positions can be found

by solving a system of linear equations, so that it

exhibits features of force-directed methods (Battista

et al., 1999). Sugiyama in (Sugiyama et al., 1981)

proposed vertex positions calculated using quadratic

programming with an objective function combining

distances among vertices from neighboring layers and

distances from lower and upper layer barycenters that

is minimized. In 1984, Eades (Eades, 1984) intro-

duced the spring-embedder model in which graph ver-

tices are denoted by a set of rings, each pair of rings

is connected by a spring and the spring is associated

with two types of forces: attraction forces and repul-

sive forces. Fruchterman and Reingold (Fruchterman

and Reingold, 1991) added the even vertex distribu-

IVAPP 2018 - International Conference on Information Visualization Theory and Applications

302

tion criterion to the previous two, aiming for the same

length of edges and a layout symmetry and also exper-

imented with a number of force forms. They selected

a model in which nodes at a distance d are attracted

to each other by the attractive force f

following the

quadratic form and they are pushed apart by the re-

pulsive force f

(d) = d

/k, f

(d) = −k/d

, (1)

where k is the optimal distance between vertices de-

ﬁned as

k = C

area

|V |

(2)

Both algorithms (Eades, 1984) and (Fruchterman and

Reingold, 1991) compute attractive forces between

adjacent vertices and repulsive forces between all

pairs of vertices. Thus, these classical algorithms ex-

hibit a quadratic complexity and do not keep a given

node ordering. We will propose a suitable modiﬁca-

tion of the force-directed method to position ordered

nodes in layers in almost linear time.

3 TREE-LIKE NETWORKS

To begin, we need to demarcate the structures we

treat in this paper. We aim for domain data that

cannot be modeled as pure tree structures. We as-

sume directed tree-like networks that can form hier-

archies, occasionally exhibit sharing of subtrees, and

ones in which strongly connected components very

rarely occur. We do not attempt to assess to which

level such a structure deviates from a tree although

we are aware of some quantiﬁcation methods (Doro-

govtsev and Mendes, 2003; Gu and Sun, 2010). We

follow the usual graph theory terminology (Diestel,

2005; Bondy and Murty, 2008).

A layering L = (L

,. ..,L

) of a graph, G =

(V,E) is an ordered partition of V into non-empty lay-

ers L

such that adjacent vertices are in different lay-

ers, i.e. if {u,v} ∈ E, where u ∈ L

and v ∈ L

, then

i 6= j (Brandes and K

opf, 2002). An ordering of a

layered graph is a partial order ≺ of V such that either

u ≺ v or v ≺ u if and only if L(u) = L(v).

4 NODE POSITIONING

We assume that the partitions of the vertices into lay-

ers and node orders in each graph layer are precom-

puted and ﬁxed, e.g. by a method in (Marik, 2017)

following the ﬁrst two phases proposed by Sugiyama.

The order of nodes in layers is the main force driv-

ing the resulting graph layout. However, there is still

Figure 1: A private family tree (Mykiska’s network). A

visualization of a sample private family tree with 2192 indi-

viduals and 765 marriages created using the uniform spac-

ing method. The layout is created very quickly (below 0.5

second with a Python script on DELL XPS 13 using an In-

tel i7 2GHz processor). The visualization demonstrates the

unpleasant zig-zag layout that might be difﬁcult to read for

large networks. The layers are drawn vertically from left to

right.

a large number of possibilities in which to assign 2D

coordinates to the nodes. In the following paragraphs

we discuss and propose two techniques for node po-

sition calculations that also satisfy the constraints on

node orders. As the order of nodes in layers can be

determined in O(|V |) time for multitree-like networks

we focus only on techniques that scale in a similar

way. All the following methods assume that graph

layers are placed uniformly in a horizontal direction.

Additionally, the nodes are linked with straight-lined

edges.

We start with a simple positioning technique that

produces results that might not be easily readable. It is

based on uniform spacing. Then, we provide a tech-

nique inspired by force-directed methods. We show

that there is a combination of special attractive and

repulsive forces that enables the readjustment of node

positions quickly, even for millions of nodes.

4.1 Uniform Spacing

A very simple technique is based on uniform spacing.

Nodes in each graph layer are placed with a uniform

distance in a vertical direction. A layout generated us-

ing the ordering method proposed in the paper (Marik,

2017) and using uniform node spreading in both di-

rections is shown in Fig 1. Family clans (subtrees

of the undirected spanning tree of the processed net-

work) with more than 150 nodes in presented visual-

izations are emphasized by different colors. An ideal

layout would result in edges creating colored horizon-

Layered Graph Force-driven Vertex Positioning

303

tal strips with only a minimum number of crossings.

Rapid zig-zag changes (waves) in color strips in verti-

cal directions make graph reading more difﬁcult. On

the other hand, the computation can be performed in

linear time O(|V |) and it provides a kind of top-level

overview for very large graphs with millions of nodes.

4.2 Force-directed Node Positioning

Although node ordering based on uniform spacing

produces useable results for very large graph data, its

zig-zag form makes reading difﬁcult. Another method

we have investigated is based on force-directed node

positioning.

We assume again that nodes are layered and or-

dered within layers. Therefore, we need a method that

is capable of reducing the wavy format while retain-

ing the order of nodes in layers.

Force-directed node positioning uses two types of

forces: attractive and repulsive ones. Our goal is to

make waves straight so we deﬁne attractive forces be-

tween nodes linked by edges and belonging to differ-

ent layers. Thus, attractive forces push linked nodes

to each other which in turn decreases the range of

the zig-zag wavy form of strips (subtrees). We use

attractive forces following the quadratic form intro-

duced by Fruchterman and Reingold (Fruchterman

and Reingold, 1991), see Equation (1). We also need

to use repulsive forces to keep vertices apart. We

propose a local barrier-like force instead of f

(d) ac-

cording to Equation (1) leading to the quadratic com-

plexity. The resulting force for each node must hold

the node ordering and the two node separation con-

straints. Forces are applied iteratively while nodes

move signiﬁcantly.

The diagram in Figure 2(d) depicts forces

schematically. Both types of forces are projected in

the direction along a given layer (a vertical in our

case). The optimum distance k is deﬁned as a dis-

tance of nodes in a layer with the largest number of

nodes L

max

if the nodes are placed uniformly, i.e.

k = δ = C/|L

max

|, C is the predeﬁned height of the

layout. If the actual deﬁnition of the forces is taken as

proposed in (Fruchterman and Reingold, 1991) then

the actual distance of nodes might converge to a very

small fraction of δ because there are only one or two

sources of the repulsion. In addition, a number of

neighboring nodes can generate attraction in one di-

rection and the nodes are prescribed to stay in the

same order. The variant a) in Figure 2 captures an

instance in which the vertices v

and v

i+1

in the same

layer are pushed so close to each other that the to-

tal sum of all projected attractive forces f

shown as

a given value and pulling the vertex v

is in equilib-

rium with the repulsive force f

generated by the ver-

tex v

i+1

. In real layouts the resulting distance of both

vertices might be a tiny fraction of the minimum sepa-

ration constraint δ. A modiﬁcation of repulsive forces

compensating such cases can be deﬁned to generate

an inﬁnite repulsive force at distance δ by shifting

the repulsive force ﬁeld beyond the minimum sepa-

ration distance as is shown in variant b). However,

in both cases, displacements of nodes in each itera-

tion become very small after a few iterations and then

the algorithm converges very slowly. The repulsive

force from the neighbors of a given node is necessary

to propagate forces through the network in order to

reach their equilibrium.

However, there is another way to propagate re-

pulsive forces. We can imagine a repulsive force

with a proﬁle in which a repulsive force between

two neighbor nodes v

(i)

∈ L

is 0 if their distance

= |x(v

) − x(v

)| ≥ δ and ∞ if d

< δ. In other

words, such a proﬁle models a barrier at distance δ,

see Figure 2 variant c). If d

< δ, we say that the

barrier between v

and v

is active. While this vari-

ant of repulsive force reﬂects the shifted modiﬁcation

discussed above, case b), it cannot be easily used for

force propagation. Nevertheless, having neighboring

nodes in the layer order, we can approximate the re-

pulsive force of a given node by the attractive forces

of its neighboring nodes if their barriers are active.

In fact, we only consider the two direct neighboring

nodes in the rest of the paper.

We will show that if repulsive forces are modeled

as barriers then the problem leads to a solution com-

bining barycenters of vertex clusters that can be per-

formed iteratively again in linear time. In fact, repul-

sive forces are modeled as decision procedures realiz-

ing barriers. In each iteration, each node is moved to

its equilibrium position or the nearest barrier if such a

position is not reachable.

As vertices are strictly tied to a given layer, we

can only focus on projections of forces towards the

direction of the layer. Thus, we can denote the union

of the lower and upper neighbors of the given vertex

v as N

= N

−

∪ N

. Furthermore, we drop the upper

index of the k layer in the following steps for vertices

= v

(k)

∈ L

with the exception of v

∈ N

. The coor-

dinate of vertex v

will be denoted as x

= x(v

). Let f

be a total attractive force pulling the vertex v

and hav-

ing the quadratic form f

) = 1/δ

∑

∈N

− x

)

The optimum position µ

= arg min

of the vertex

can be found as an minimum of the function f

us-

ing

d f

= 0. Thus, the resulting equilibrium position

= 1/|N

∑

∈N

(3)

IVAPP 2018 - International Conference on Information Visualization Theory and Applications

304

1.0

i+1

(a)

i+1

(b)

distance

i+1

Figure 2: Repulsive force variants. Three different variants of repulsive force models. Nodes v

and v

i+1

of the same layer

mark the resulting position of v

i+1

relatively to the position of v

. f

is the sum of all repulsive forces. a) The variant

proposed by Fruchterman and Reingold breaking the minimum separation δ constraint. b) Its shifted modiﬁcation satisfying

the minimum separation constraint. c) The barrier variant proposed and used in this article. d) Forces on a vertex. Attractive

(black) forces from the lower and upper vertex neighbors pull the vertex v

towards their barycentric position. Repulsive (red)

forces from the vertex predecessor and successor in the layer L

push the vertex v

away to maintain the minimum separation

constraint δ. The vertices might be equipped by a solid separation barrier schematically drawn as red rectangles. Only force

projections to the layer L

need to be treated because of the layer constraint.

is the barycentric position of v

. The vertex v

should

be placed into the equilibrium position µ

if no barrier

to the predecessor v

i−1

or the successor v

i+1

is active.

If a barrier is active then we need to combine two or

three force sources as we have a node and at most two

of its neighbor barriers active. Let us assume that the

active barrier is between the vertex v

and its succes-

sor v

i+1

. The position of v

is x

and the position of

i+1

is x

i+1

= x

+ δ. The total attractive force f

i+1

pulling the successor v

i+1

pushes also the vertex v

the repulsive force. Thus, if only this one barrier is

active, then the total force pulling the vertex v

) = f

) + f

i+1

) = f

) + f

i+1

+ δ)

(4)

In this case equilibrium can be reached at the position

| + |N

i+1

| + |N

i+1

(µ

i+1

− δ)

(5)

In other words, the equilibrium position µ

is a con-

vex combination of no barrier equilibrium positions

with the successor displaced by −δ. Similarly, we

can obtain the equilibrium position of v

, only if the

barrier with the predecessor is active

| + |N

i−1

−1

| + |N

i−1

(µ

i−1

+ δ)

(6)

If both the barriers with the predecessor and the suc-

cessor are active, we start with the total force pulling

the vertex v

pIs

) = f

i−1

) + f

i+1

) = (7)

= f

i−1

− δ) + f

) + f

i+1

+ δ) (8)

for which the equilibrium position of the vertex v

pIs

−1

|(µ

i−1

+ δ) + |N

|µ

+ |N

|(µ

i+1

− δ)

i−1

| + |N

i+1

(9)

Furthermore, we need to determine what case of ac-

tive barriers occurs to calculate the new optimum po-

sition x

of the vertex v

. This can be accomplished

using a simple decision procedure:

1. if µ

i−1

≤ µ

− δ and µ

+ δ ≤ µ

i+1

, then x

:= µ

;

2. if µ

i−1

> µ

− δ and µ

+ δ ≤ µ

i+1

, then x

:= µ

;

3. if µ

i−1

≤ µ

− δ and µ

+ δ > µ

i+1

, then x

:= µ

;

4. otherwise x

:= µ

pIs

We update the vertices positions layer by layer fol-

lowing the order of vertices in the given layer. We

perform two such sweeps in the forward direction and

then two sweeps in the backward direction (both lay-

ers and orders) to speed up node position changes

propagation. The initial positions of vertices are de-

termined using the uniform spacing strategy. As we

move only one node, although its new position is the

result of all its neighboring nodes, we also need to

satisfy the order of vertices. We again use another de-

cision procedure to determine a new feasible position

for the vertex v

1. if x

≤ x

i−1

+ δ then x

:= x

i−1

+ δ,

2. if x

≥ x

i+1

− δ then x

:= x

i+1

− δ,

3. otherwise x

:= x

The new feasible positions might result in position os-

cillations. Thus, the new position x

of the vertex

Layered Graph Force-driven Vertex Positioning

305

Table 1: Sample datasets and their basic network statistics: a node number |V | of the complete network , a people number |V

of the complete network, a marriage number |V

| of the complete network, a node number |V

max

| of the maximum component,

an edge number |E

max

| of the maximum component, a number of strongly connected components |SCC| of a size larger than

1, a number of layers |L|, a number of source nodes |V

src

Dataset |V | |V

| |V

max

| |E

max

| |SCC| |L| |V

src

Mykiska’s network 2952 2192 765 2913 2917 0 27 609

USA presidents 3186 2145 1042 1589 1602 0 77 480

WeMightBeKin 52783 38486 14297 52672 54210 0 82 12716

ITIS 945352 472676 65799 615342 615341 0 36 1

Stobie’s network 996055 706794 289268 995522 1038192 2 187 218593

MGP 200832 188351 0 188351 213865 0 52 7678

is determined conservatively as a convex combi-

nation of the current and feasible positions as x

0.6x

+ 0.4x

. The constants 0.6 and 0.4 were deter-

mined experimentally, but the approach is not sensi-

tive to their modiﬁcations. The constant values mean

that the correction of the vertex position is reﬂected

partially by displacement of the current vertex and

partially by its neighbors. We stress again that the

strict and solid barriers with the attractive forces as-

sumed to have the quadratic form enable the combin-

ing of barycenters of vertices only. The related sums

of coordinate values and their counts are sufﬁcient to

calculate new positions.

5 IMPLEMENTATION,

EXPERIMENTS, AND

DISCUSSION

The algorithm was implemented as an experimental

non-optimized Python script with a number of addi-

tional procedures evaluating the layout process and

used on an ultrabook DELL XPS 13 with 16GB of

RAM using an Intel i7 2GHz processor. The layering

and ordering is very fast, taking from mere seconds

up to a dozen minutes for networks with a million ver-

tices. The positioning procedure uses 10×4 iterations

lasting a similar amount of time.

Although tree-like networks might be observed

in a number of domains, such as telecommunica-

tions, forensic and cause-effect networks, in this pa-

per we performed experiments mainly using phylo-

genetic and genealogical datasets exhibiting multi-

tree properties. We selected 20 datasets to evaluate

the proposed methods (Pruitt, 2017; Leskovec and

Krevl, 2017; GoogleFFT, 2017). Example datasets

and their statistics are shown in Table 1. Except for

one, all datasets we used in our experiments had no

cycles. Stobie’s dataset (Stobie, 2017) contains 2

small strongly connected components. Some datasets

represent genealogical networks, the ITIS (Integrated

Taxonomic Information System) dataset is a snapshot

of the Catalog of Life in the GEDCOM format (ITIS,

2017). The ITIS dataset exhibits a special feature: the

total number of ’person’ and ’marriage’ records cov-

ers only 57% of the entire graph nodes.

Another dataset consists of 3057 people from a

database created by Egyptologists (Dul

ıkov

a, 2016).

Experiments with families from the database did not

exhibit any layout deﬁciencies as the families are

quite simple and no larger than 50 family mem-

bers. Similar results were obtained for the rest of the

datasets which are GEDCOM ﬁles downloaded from

the Internet containing between 400 and 1,000,000 in-

dividual records, such as Stobie’s network, the Math-

ematics Genealogy Project (MGP) (MGP, 2017), and

the ITIS dataset (ITIS, 2017).

For example, the diagram in Figure 3(a) de-

picts the result of an almost hierarchical graph struc-

ture capturing a genealogical network of US presi-

dents (Pruitt, 2017). The performance of the method

can be demonstrated on the phylogenetic network lay-

out in Figure 3(b) with 615342 nodes as an overview

of taxonomic information on plants, animals, fungi,

and microbes as developed by the Integrated Taxo-

nomic Information System (ITIS) (ITIS, 2017). One

can easily observe the overall structure of such a mul-

titree.

Example layouts shown in this section dedicated

to experiments demonstrate that the wavy form of

the resulting layout is only partially removed. As

has been observed with a number of different force-

directed layout method variants, the convergence of

a vertex’s position after dozens of iterations is very

slow. The slow convergence is attributable to several

reasons. First, if the position of lower neighbors is

opposite to the position of upper neighbors (layers),

then the current vertex is not moved because it stays

in the optimum position. Thus, the vertices of wavy

subgraphs are dragged into new positions only by un-

balanced forces for vertices located near the wave ex-

trema. Furthermore, there are stable conﬁgurations of

vertices even when their local layout is bent. A so-

IVAPP 2018 - International Conference on Information Visualization Theory and Applications

306

(a) (b)

Figure 3: a) Genealogy of US presidents. A resulting layout of a network with an almost hierarchical structure. While the

rapid zig-zag subtree layouts disappeared, the large range of wavy formats for subtrees is still present. Thus, the node positions

are still not optimal as the locally force-driven algorithm is caught in a local extreme. b) An ITIS snapshot. A snapshot of

a phylogenetic network with almost one million nodes. One can easily identify that the entire network consists of one large

submultitree on the right side and a forest of smaller ones on the left side of the diagram.

lution might be found in a special treatment of entire

subtrees that we are currently investigating.

6 CONCLUSIONS

In this work we proposed a new method for layered

graph node positioning, i.e. layered tree-like network

layouts with constraints on node order with regard to

their partition into layers and their order in layers. We

restricted and demonstrated its application to tree-like

networks as we are aware of methods that are capa-

ble of layering such large graphs with almost linear

complexity. The produced graph layouts are more ac-

ceptable for the user if they deal with large networks

combining many trees into a single acyclic graph. We

are not aware of any other tool that is able to layout

multitree-like networks with millions of nodes and

edges while maintaining the given layering and order-

ing of nodes. The layout can be computed using an

unoptimized Python script during a processing time

in which other tools would utilize for drawing net-

works of two order magnitude smaller. Thus, the ex-

periments clearly demonstrate a signiﬁcant improve-

ment in graph comprehension. They also indicate that

results provided by existing state of the art tools are

quite far from computing an optimum layout with re-

gard to the number of edge crossings, constraints on

node layers and orders, at least for special types of

graphs such as genealogical ones.

The proposed method enables graphical highlights

that assist with any type of initial analysis and data

processing treating datasets with a large number of

entities. Technical details on how a suitable software

tool providing additional utilities like zooming, pan-

ning, searching, ﬁltering can be easily implemented

are beyond the scope of this article. It is also obvious

that the wavy character of the layout can still be im-

proved. We are investigating possibilities that would

assign node positions based on forces taking into ac-

count entire subtrees of the undirected spanning trees.

ACKNOWLEDGEMENTS