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 net- works 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 inter- actions within gene regulatory networks and food webs. Tech- nological networks such as the Internet have changed the way the world seeks information and does business. And technology- enabled 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 surpris- ing 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 telecommu- nications service offerings according to expected user behaviour. Yet the scale, complexity and dynamics of today’s technology- based 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 phe- nomena of networked structures have been widely studied for many years, mainly via simulations and theoretical analysis. Un- fortunately, this work often yields unsatisfactory real-world re- sults. Most models addressing the growth of complex networks strive to reproduce certain global characteristics of those net- works: e.g. node degree distribution, 1 clustering coefficient 2 and network diameter. 3 At the same time, models developed specif- ically 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 lo- cal 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 ap- proach toward quantifying network evolution schemes which, though rooted in graph analysis, uses inherent network dynam- ics, 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 cen- sus, involves slicing the history of the network into time win- dows, 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 (= 4 3 ) 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 ac- cumulated across successive time windows. Given any two pos- sible 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 Transi- tion Matrix stores these probabilities for every possible before- and-after configuration pair. Continued on next page