Dynamical processes occurring on the edges in complex networks are relevant to a variety of real-world situations. Despite recent advances, a framework for edge controllability is still required for complex networks of arbitrary structure and interaction strength. Generalizing a previously introduced class of processes for edge dynamics, the switchboard dynamics, and exploit- ing the exact controllability theory, we develop a universal framework in which the controllability of any node is exclusively determined by its local weighted structure. This framework enables us to identify a unique set of critical nodes for control, to derive analytic formulas and articulate efficient algorithms to determine the exact upper and lower controllability bounds, and to evaluate strongly structural controllability of any given network. Applying our framework to a large number of model and real-world networks, we find that the interaction strength plays a more significant role in edge controllability than the network structure does, due to a vast range between the bounds determined mainly by the interaction strength. Moreover, transcriptional regulatory networks and electronic circuits are much more strongly structurally controllable (SSC) than other types of real-world networks, directed networks are more SSC than undirected networks, and sparse networks are typically more SSC than dense networks.

We formulate an in silico model of pathogen avoidance mechanism and investigate its impact on defensive behavioural measures (e.g., spontaneous social exclusions and distancing, crowd avoidance and voluntary vaccination adaptation). In particular, we use SIR(B)S (e.g., susceptible-infected-recovered with additional behavioural component) model to investigate the impact of homo-psychologicus aspects of epidemics. We focus on reactionary behavioural changes, which apply to both social distancing and voluntary vaccination participations. Our analyses reveal complex relationships between spontaneous and uncoordinated behavioural changes, the emergence of its contagion properties, and mitigation of infectious diseases. We find that the presence of effective behavioural changes can impede the persistence of disease. Furthermore, it was found that under perfect effective behavioural change, there are three regions in the response factor (e.g., imitation and/or reactionary) and behavioural scale factor (e.g., global/local) factors ρ–α behavioural space. Mainly, (1) disease is always endemic even in the presence of behavioural change, (2) behavioural-prevalence plasticity is observed and disease can sometimes be eradication, and (3) elimination of endemic disease under permanence of permanent behavioural change is achieved. These results suggest that preventive behavioural changes (e.g., non-pharmaceutical prophylactic measures, social distancing and exclusion, crowd avoidance) are influenced by individual differences in perception of risks and are a salient feature of epidemics. Additionally, these findings indicates that care needs to be taken when considering the effect of adaptive behavioural change in predicting the course of epidemics, and as well as the interpretation and development of the public health measures that account for spontaneous behavioural changes.

Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed.

Recent works revealed that the energy required to control a complex network depends on the number of driving signals and the energy distribution follows an algebraic scaling law. If one implements control using a small number of drivers, e.g. as determined by the structural controllability theory, there is a high probability that the energy will diverge. We develop a physical theory to explain the scaling behaviour through identification of the fundamental structural elements, the longest control chains (LCCs), that dominate the control energy. Based on the LCCs, we articulate a strategy to drastically reduce the control energy (e.g. in a large number of real-world networks). Owing to their structural nature, the LCCs may shed light on energy issues associated with control of nonlinear dynamical networks.

Given a complex geospatial network with nodes distributed in a two-dimensional region of physical space, can the locations of the nodes be determined and their connection patterns be uncovered based solely on data? We consider the realistic situation where time series/signals can be collected from a single location. A key challenge is that the signals collected are necessarily time delayed, due to the varying physical distances from the nodes to the data collection centre. To meet this challenge, we develop a compressive-sensing-based approach enabling reconstruction of the full topology of the underlying geospatial network and more importantly, accurate estimate of the time delays. A standard triangularization algorithm can then be employed to find the physical locations of the nodes in the network. We further demonstrate successful detection of a hidden node (or a hidden source or threat), from which no signal can be obtained, through accurate detection of all its neighbouring nodes. As a geospatial network has the feature that a node tends to connect with geophysically nearby nodes, the localized region that contains the hidden node can be identified.

Locating sources of diffusion and spreading from minimum data is a significant problem in network science with great applied values to the society. However, a general theoretical framework dealing with optimal source localization is lacking. Combining the controllability theory for complex networks and compressive sensing, we develop a framework with high efficiency and robustness for optimal source localization in arbitrary weighted networks with arbitrary distribution of sources. We offer a minimum output analysis to quantify the source locatability through a minimal number of messenger nodes that produce sufficient measurement for fully locating the sources. When the minimum messenger nodes are discerned, the problem of optimal source localization becomes one of sparse signal reconstruction, which can be solved using compressive sensing. Application of our framework to model and empirical networks demonstrates that sources in homogeneous and denser networks are more readily to be located. A surprising finding is that, for a connected undirected network with random link weights and weak noise, a single messenger node is sufficient for locating any number of sources. The framework deepens our understanding of the network source localization problem and offers efficient tools with broad applications.

Through the mathematical study of two models we quantify some of the theories of co-development and co-existence of focused groups in the social sciences. This work attempts to develop the mathematical framework behind the social sciences of community formation. By using well developed theories and concepts from ecology and epidemiology we hope to extend the theoretical framework of organizing and self-organizing social groups and communities, including terrorist groups. The main goal of our work is to gain insight into the role of recruitment and retention in the formation and survival of social organizations. Understanding the underlining mechanisms of the spread of ideologies under competition is a fundamental component of this work. Here contacts between core and non-core individuals extend beyond its physical meaning to include indirect interaction and spread of ideas through phone conversations, emails, media sources and other similar mean.

This work focuses on the dynamics of formation of interest groups, either ideological, economical or ecological and thus we explore the questions such as, how do interest groups initiate and co-develop by interacting within a common environment and how do they sustain themselves? Our results show that building and maintaining the core group is essential for the existence and survival of an extreme ideology. Our research also indicates that in the absence of competitive ability (i.e., ability to take from the other core group or share prospective members) the social organization or group that is more committed to its group ideology and manages to strike the right balance between investment in recruitment and retention will prevail. Thus under no cross interaction between two social groups a single trade-off (of these efforts) can support only a single organization. The more efforts that an organization implements to recruit and retain its members the more effective it will be in transmitting the ideology to other vulnerable individuals and thus converting them to believers.

We describe a multi-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of the corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions and the corresponding photon statistics are discussed. Some applications to quantum optics, cavity quantum electrodynamics and superfocusing in channelling scattering are mentioned. Explicit solutions of the Heisenberg equations for radiation field operators with squeezing are found.

Background
In 2015, the Zika arbovirus (ZIKV) began circulating in the Americas, rapidly expanding its global geographic range in explosive outbreaks. Unusual among mosquito-borne diseases, ZIKV has been shown to also be sexually transmitted, although sustained autochthonous transmission due to sexual transmission alone has not been observed, indicating the reproduction number (R0) for sexual transmission alone is less than 1. Critical to the assessment of outbreak risk, estimation of the potential attack rates, and assessment of control measures, are estimates of the basic reproduction number, R0.
Methods
We estimated the R0 of the 2015 ZIKV outbreak in Barranquilla, Colombia, through an analysis of the exponential rise in clinically identified ZIKV cases (n = 359 to the end of November, 2015).
Findings
The rate of exponential rise in cases was ρ = 0.076 days[superscript −1], with 95% CI [0.066,0.087] days[superscript −1]. We used a vector-borne disease model with additional direct transmission to estimate the R0; assuming the R0 of sexual transmission alone is less than 1, we estimated the total R0 = 3.8 [2.4,5.6], and that the fraction of cases due to sexual transmission was 0.23 [0.01,0.47] with 95% confidence.
Interpretation
This is among the first estimates of R0 for a ZIKV outbreak in the Americas, and also among the first quantifications of the relative impact of sexual transmission.