For the last few decades researchers from different fields gather their findings andknowledge trying to give a shape to the new science of complex systems. To address thisproblem， new tools and methods have to be established. We propose a new， or more precisely analternative， framework for the characterization of complex system. In this book we will give anintroduction of applicability of simplicial complexes in the science of complex systems. Afterintroducing of main definitions and properties of simplicial complexes necessary forrepresentation and analysis of complex systems， we will illustrate the usefulness and versatilityof tools and concepts related to the simplicial complexes. In the broad sense the exposition willcover several typical manifestations of complex systems， that is， emerging psychological spaceof opinions and their exchange， topological description of complex networks， and topologicalinvariability in dynamical systems. This book sets a stage and serves as a starting point toinspire readers. This is always the authors' intention and wish.
1 Introduction 1．1 Complex system 1．2 Complex networks 1．3 Simplicial Complexes Bibliography 2 The World of Simplicial Complexes 2．1 Geometric simplicial complex 2．2 Abstract simplicial complex 2．3 Simplicial complex via relation 2．4 Characterization of simplicial complexes 2．5 Summary Bibliography 3 How Do We Build Simplicial Complexes 3．1 From complex networks （graphs） 3．2 From data in metric space 3．3 From time series 3．4 Summary Bibliography 4 Several Applications of Simplicial Complexes 4．1 Opinion exchange modeling 4．2 Topological description of the air traffic control network 4．3 Topological properties of dynamical systems 4．4 Instead of summary： Potential contributions to aeronautical science Bibliography 5 Take-home Messages 5．1 Brief book summary 5．2 Suggestions for further reading
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