Smooth Graphs for Visual Exploration of
Higher-Order State Transitions
Jorik Blaas, Charl P. Botha, Member, IEEE, Edward Grundy, Mark W. Jones,
Robert S. Laramee, Member, IEEE, and Frits H. Post
Fig. 1. A smooth graph representation of a labeled biological time-series. Each ring represents a state, and the edges between states
visualize the state transitions. This graph uses smooth curves to explicitly visualize third order transitions, so that each curved edge
represents a unique sequence of four successive states. The orange node is part of a selection set, and all transitions matching the
current selection are highlighted in orange.
Abstract—In this paper, we present a new visual way of exploring state sequences in large observational time-series. A key advan-
tage of our method is that it can directly visualize higher-order state transitions. A standard first order state transition is a sequence
of two states that are linked by a transition. A higher-order state transition is a sequence of three or more states where the sequence
of participating states are linked together by consecutive first order state transitions.
Our method extends the current state-graph exploration methods by employing a two dimensional graph, in which higher-order state
transitions are visualized as curved lines. All transitions are bundled into thick splines, so that the thickness of an edge represents
the frequency of instances.
The bundling between two states takes into account the state transitions before and after the transition. This is done in such a way
that it forms a continuous representation in which any subsequence of the timeseries is represented by a continuous smooth line.
The edge bundles in these graphs can be explored interactively through our incremental selection algorithm.
We demonstrate our method with an application in exploring labeled time-series data from a biological survey, where a clustering has
assigned a single label to the data at each time-point. In these sequences, a large number of cyclic patterns occur, which in turn are
linked to specific activities. We demonstrate how our method helps to find these cycles, and how the interactive selection process
helps to find and investigate activities.
Index Terms—State transitions, Graph drawing, Time series, Biological data.
1 INTRODUCTION
One of the common ways to visualize state transition sequences is
by using graphs. Each node represents a state, and an oriented edge
between two nodes represents a transition between those two states.
For the exploration of time-series label data, such a graph can be con-
structed by examining all succeeding pairs of states and generating a
• Jorik Blaas, Charl P. Botha and Frits H. Post are with the Data
Visualization Group, Delft University of Technology, NL.
• Edward Grundy, Mark W. Jones and Robert S. Laramee are with the
Visual Computing Group, Swansea University, UK.
Manuscript received 31 March 2009; accepted 27 July 2009; posted online
11 October 2009; mailed on 5 October 2009.
For information on obtaining reprints of this article, please send
email to: tvcg@computer.org .
set of edges between the nodes representing them.
While these graphs give a good overview of the transitions between
states, one important aspect is lost in the visualization: the context
in which these transitions occur is not visible. Figure 2 shows how
the first-order transition graph may ambiguously represent multiple
underlying sequences. The leftmost graph could either correspond
to the sequence ABCABCABC. . ., CDECDECDE. . ., i.e. multiple repe-
titions of each of the triangles as shown in the rightmost graph, or the
sequence ABCDECABCDECA. . ., i.e. all states in one long sequence
passing multiple times through C as shown in the middle graph. One
of our goals is to visually disambiguate these two situations. In the
following sections, we elaborate how this can be done by taking into
account higher-order state transitions when drawing the edges. The
middle and rightmost graph representations in figure 2 show how our
method uses curved edges to emphasize the order in which the tran-
sitions occur, which gives each of the two sequences a unique visual
representation.