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Complex systems play an essential role in our daily lives. These systems consist of many connected components that interact with each other. Consider, for example, society with billions of collaborating individuals, the stock market with numerous buyers and sellers that trade equities, or communication infrastructures with billions of phones, computers and satellites.

The key to understanding complex systems is to understand the interaction patterns between their components - their networks. To create the network, we need data from the system and a model that organizes the given data in a network representation. Today's increasing availability of data and improved computational capacity for analyzing networks have created great opportunities for the network approach to further prosper. However, increasingly rich data also gives rise to new challenges that question the effectiveness of the conventional approach to modeling data as a network. In this thesis, we explore those challenges and provide methods for simplifying and highlighting important interaction patterns in network models that make use of richer data.

Using data from real-world complex systems, we first show that conventional network modeling can provide valuable insights about the function of the underlying system. To explore the impact of using richer data in the network representation, we then expand the analysis for higher-order models of networks and show why we need to go beyond conventional models when there is data that allows us to do so. In addition, we also present a new framework for higher-order network modeling and analysis. We find that network models that capture richer data can provide more accurate representations of many real-world complex systems.

We live in an interconnected world full of complex systems that cannot be understood simply by analyzing their components. From how genes regulate biological functions to the distribution of life on Earth, we need methods that can analyze systems as a whole.

Networks are abstractions of complex systems, helping capture properties that emerge from patterns of interactions rather than from the individual parts. To understand the patterns of interactions in large networks, we need to simplify them by discovering their modular structure that often characterizes complex systems. A hierarchical modular structure functions as a map that lets us navigate relational data efficiently and helps us see the general patterns. But how reliable is the map if it is based on incomplete data?

This thesis applies and builds upon the map equation, which is an information-theoretic method for detecting modular regularities in the flow patterns on networks. To robustly map incomplete data, we have developed three general approaches: (1) Adaptive resolution in both sampling of and dynamics on networks better fits the data. (2) Regularization avoids overfitting to random patterns. (3) Richer data can be included into the network for a more complete map. Methods that can include evolutionary relationships and handle incomplete data provide more powerful tools for mapping biodiversity in space and time.

Vi lever i en sammankopplad värld full av komplexa system som inte låter sig förstås enbart genom att analysera dess komponenter. Från hur gener reglerar biologiska funktioner till livets utbredning på jorden behöver vi metoder som kan analysera system som en helhet.

Nätverk är abstraktioner av komplexa system som hjälper till att fånga egenskaper som uppstår genom interaktionsmönster snarare än hos de enskilda delarna. För att förstå dessa mönster i stora nätverk måste vi förenkla dem genom att upptäcka dess modulära stuktur som präglar komplexa system. En hierarkisk modulär struktur fungerar som en karta som låter oss navigera effektivt i relationsdata och hjälper oss att se de allmänna mönstren. Men hur tillförlitlig är kartan om den baseras på inkompletta data?

Den här avhandlingen applicerar och bygger vidare på kartekvationen som är en informationsteoretisk metod för att upptäcka modulära regelbundenheter i flödesmönstren på nätverk.För att robust kartlägga inkompletta data har vi utvecklat tre övergripande tillvägagångssätt: (1) Adaptiv upplösning i båda sampling av och dynamik på nätverk ger bättre anpassning till data. (2) Regularisering undviker överanpassning till slumpmässiga mönster. (3) Rikare data kan inkluderas i nätverket för en mer komplett karta. Metoder som kan inkludera evolutionära relationer och hantera inkompletta data ger kraftfullare verktyg för att kartlägga den biologiska mångfalden i rum och tid.