10.2417/3201209.004369
Triad transition probabilities
characterize complex networks
Katarzyna Musial, Krzysztof Juszczyszyn, and Marcin Budka
A data-driven approach to analysing and characterizing complex networks outperforms simulations based on traditional network models.
Complex networked systems are present in every aspect of our
lives. Life itself is made possible by the intricate biological interactions within gene regulatory networks and food webs. Technological networks such as the Internet have changed the way
the world seeks information and does business. And technologyenabled social networks have drastically altered how people
meet and communicate.
In general, complex networks feature large numbers of highly
interconnected units exhibiting time-dependent behaviour. A
relatively simple interactivity between neighbouring units may
often compound into emergent collective behaviour of surprising sophistication. The need to analyse complex systems and to
predict changes in them is crucial: from assessing the potential
effects of human activity on food webs, to defining telecommunications service offerings according to expected user behaviour.
Yet the scale, complexity and dynamics of today’s technologybased complex networks have proven resistant to traditional
network analysis methods. Predicting structural changes in such
networks remains a challenge.
Self-organization, synchronization and other emergent phenomena of networked structures have been widely studied for
many years, mainly via simulations and theoretical analysis. Unfortunately, this work often yields unsatisfactory real-world results. Most models addressing the growth of complex networks
strive to reproduce certain global characteristics of those networks: e.g. node degree distribution,1 clustering coefficient2 and
network diameter.3 At the same time, models developed specifically for social networks naturally focus on features typical of
those networks.4–8 But both approaches tend to ignore the local
structure of complex systems.
Inspired by the modern-day ‘data explosion,’9 we propose
using data-driven techniques to infer dynamic structure from
local characteristics. We start from the observation that the local topologies of social networks differ significantly from those
of standard network models (see Figure 1). Indeed, when local
topology is expressed in terms of network motifs (e.g. subgraphs
Figure 1. The triad as a tool for analysing network structure. Within
complex social networks, standard network models are inadequate to
describe even static local topologies, let alone their evolution over time.
containing three to seven nodes), self-evolving networks show
visibly biased frequency distributions that contrast sharply with
those of artificially generated networks. We then propose an approach toward quantifying network evolution schemes which,
though rooted in graph analysis, uses inherent network dynamics, as revealed through observations across recorded history.
The smallest non-trivial subgraph of a network is a triad:
a directed network of exactly three nodes. One technique for
quantifying the connections within triads, called a triad census, involves slicing the history of the network into time windows, determining the configurations of multiple triads in each
period and tallying the results. Since the three nodes in a triad
form three pairs, and since each pair is able to connect in either,
neither or both directions, there are 64 (= 43 ) possible distinct
triad configurations. (If no distinction is made among the three
nodes, this number decreases to 16.) We propose an enhanced
procedure called the Triad Transition Method,10–12 in which the
configurations of a large number of triads in the network are accumulated across successive time windows. Given any two possible triad configurations, the triad census is used to estimate the
probability that a triad in the first configuration will transition to
the second in the next time window. The resulting Triad Transition Matrix stores these probabilities for every possible beforeand-after configuration pair.
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10.2417/3201209.004369 Page 2/3
Figure 2. Flowchart and example of a predictive procedure that relies on Triad Transition Matrices.
Figure 2 presents the Triad Transition Method, broken down
into individual steps. Although its statistical analysis of topological connection patterns assumes no a priori knowledge about
the nature of relations among nodes, we find that its ability to
predict link dynamics outperforms existing methods, especially
in the case of sparse networks analysed across short timescales.
Moreover, to the extent that the computed triad transition probabilities capture local connection pattern dynamics, the approach
provides a useful tool for classifying distinct dynamic networks
(social, biological etc.) according to the local evolutionary trajectories of a collection of nodes.
In real-world complex networks, global structure arises from
local interaction patterns in a non-obvious way. The Triad Transition Method represents one of the first attempts to use historical data about local topology to predict global dynamics. As
such, it may be seen as an initial step toward not only understanding and classifying these systems, but also predicting their
evolution. Our future work on predictive methods will focus
on considering the context of the analysed system and on nonstructural network features. This will enable us to tighten the
connection between machine learning, autonomy, predictive analytics and complex networked systems research.
Author Information
Katarzyna Musial
PAIS
King’s College London
London, UK
Katarzyna Musial is a lecturer in computer science in the Department of Informatics. Her research is interdisciplinary, involving
theoretical foundations as well as applications of complex networked systems. Her recent work has focused on the dynamics
and evolution of networked systems.
Krzysztof Juszczyszyn
Wroclaw University of Technology
Wroclaw, Poland
Krzysztof Juszczyszyn is an assistant professor at the Institute
of Computer Science. His research concentrates on dynamic
network models applied to social networks based on communication technologies, local topology analysis and Semantic Web
systems.
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10.2417/3201209.004369 Page 3/3
Marcin Budka
Bournemouth University
Bournemouth, UK
Marcin Budka is a lecturer in computational intelligence at the
School of Design, Engineering and Computing. His research
interests encompass machine learning, data mining, predictive
modelling and computational intelligence. The evolution of
complex systems is one relatively little-explored application of
his expertise.
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c 2012 Awareness