Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems-cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science.Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse "real-life" situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications.Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence.The three major book publication platforms of the Springer Complexity program are the monograph series "Understanding Complex Systems" focusing on the various applications of complexity, the "Springer Series in Synergetics", which is devoted to the quantitative theoretical and methodological foundations, and the "Springer Briefs in Complexity" which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.Future scientific and technological developments in many fields will necessarily depend upon coming to grips with complex systems. Such systems are complex in both their composition -typically many different kinds of components interacting simultaneously and nonlinearly with each other and their environments on multiple levels -and in the rich diversity of behavior of which they are capable.The Springer Series in Understanding Complex Systems series (UCS) promotes new strategies and paradigms for understanding and realizing applications of complex systems research in a wide variety of fields and endeavors. UCS is explicitly transdisciplinary. It has three main goals: First, to elaborate the concepts, methods and tools of complex systems at all levels of description and in all scientific fields, especially newly emerging areas within the life, social, behavioral, economic, neuro-and cognitive sciences (and derivatives thereof); second, to encourage novel applications of these ideas in various fields of engineering and computation such as robotics, nano-technology...

We present a systematic collection of spectral surgery principles for the Laplacian on a compact metric graph with any of the usual vertex conditions (natural, Dirichlet or δ-type), which show how various types of changes of a local or localised nature to a graph impact on the spectrum of the Laplacian. Many of these principles are entirely new; these include "transplantation" of volume within a graph based on the behaviour of its eigenfunctions, as well as "unfolding" of local cycles and pendants. In other cases we establish sharp generalisations, extensions and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions and introducing new pendant subgraphs.To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a compact metric graph to estimate the lowest nontrivial eigenvalue of the Laplacian with natural vertex conditions. This quantitative isoperimetric-type inequality interpolates between two known estimates -one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound) -and includes them as special cases. Contents2010 Mathematics Subject Classification. 34B45 (05C50 35P15 81Q35).

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature

Abstract. We study the diffusion of epidemics on networks that are partitioned into local communities. The gross structure of hierarchical networks of this kind can be described by a quotient graph. The rationale of this approach is that individuals infect those belonging to the same community with higher probability than individuals in other communities. In community models the nodal infection probability is thus expected to depend mainly on the interaction of a few, large interconnected clusters. In this work, we describe the epidemic process as a continuous-time individual-based susceptible-infected-susceptible (SIS) model using a first-order mean-field approximation.A key feature of our model is that the spectral radius of this smaller quotient graph (which only captures the macroscopic structure of the community network) is all we need to know in order to decide whether the overall healthy-state defines a globally asymptotically stable or an unstable equilibrium. Indeed, the spectral radius is related to the epidemic threshold of the system.Moreover we prove that, above the threshold, another steady-state exists that can be computed using a lower-dimensional dynamical system associated with the evolution of the process on the quotient graph. Our investigations are based on the graph-theoretical notion of equitable partition and of its recent and rather flexible generalization, that of almost equitable partition.Key words. susceptible-infected-susceptible model, hierarchical networks, graph spectra, equitable and almost equitable partitions AMS subject classifications.1. Introduction. Metapopulation models of epidemics consider the entire population partitioned into communities (also called households, clusters or subgraphs). Such models assume that each community shares a common environment or is defined by a specific relationship (see, e.g., [1,2,3]).Several authors also account for the effect of migration between communities [4,5]. Conversely, the model we are interested in suits better the diffusion of computer viruses or stable social communities, which do not change during the infection period; hence we do not consider migration.In this work, we study the diffusion of epidemics over an undirected graph G = (V, E) with edge set E and node set V . The order of G, denoted N , is the cardinality of V , whereas the size of G is the cardinality of E, denoted L. Connectivity of the graph G is conveniently encoded in the N × N adjacency matrix A. We are interested in the case of networks that can be naturally partitioned into n communities: they are represented by a node set partition π = {V 1 , ..., V n }, i.e., a sequence of mutually disjoint nonempty subsets of V , called cells, whose union is V .The epidemic model adopted in the rest of the paper is a continuous-time Markovian individual-based susceptible-infected-susceptible (SIS) model. In the SIS model a node can be repeatedly infected, recover and yet be infected again. The viral state of a node i, at time t, is thus described by a Bernoulli random var...

Key words Operator matrices, semigroups of operators, wave equations with acoustic boundary conditions MSC (2000) 47D05, 47H20, 35L20We define an abstract setting to treat wave equations equipped with time-dependent acoustic boundary conditions on bounded domains of R n . We prove a well-posedness result and develop a spectral theory which also allows to prove a conjecture proposed in [13]. Concrete problems are also discussed.