FIGURE 2. Changes in flow capacity robustness of the BA scale-free network. (A) Change in flow capacity robustness under different damage rates and increase rates for BA scale-free network under deliberate attack; (B) change in flow capacity robustness under different damage rates and increase rates for BA scale-free network under random attack; (C) change in the difference between the flow capacity robustness under random attack minus the flow capacity robustness under deliberate attack.
FIGURE 9. Changes of flow recovery robustness of the BA scale-free network. (A) Change in flow recovery robustness under different damage rates and increase rates for BA scale-free network under deliberate attack; (B) change in flow recovery robustness under different damage rates and increase rates for BA scale-free network under random attack; (C) change in the difference between the flow recovery robustness under random attack minus the flow capacity robustness under deliberate attack.
FIGURE 15. Difference between the flow recovery robustness and the flow capacity robustness of four typical networks. (A, B), respectively, show the difference between the flow recovery robustness and the flow capacity robustness in BA scale-free network under deliberate attack and random attack; (C, D), respectively, show the difference between the flow recovery robustness and the flow capacity robustness in ER random network under deliberate attack and random attack; (E, F), respectively, show the difference between the flow recovery robustness and the flow capacity robustness in the NNC regular network under deliberate attack and random attack; (G, H), respectively, show the difference between the flow recovery robustness and the flow capacity robustness in the WS small-world network under deliberate attack and random attack.
Employment dynamics are the product of complex interactions taking place inside and between firms. In labor markets, human resources are continuously reallocated across firms, industries, and regions. In labor economics it is conventional to aggregate job hirings and job separations (both voluntary and involuntary) across companies to get pools of job changers and the unemployed . The sizes of these pools are then conceived of as being determined by rate processes over these pools . In reality, hiring and separation occur at individual companies and important information about the varieties of firm behavior is lost in the process of aggregating labor data into pools, with otherwise comparable firms experiencing quite different labor turnover. For instance, understanding how micro-dynamics affect aggregate variables (such as employment growth) from a disaggregate perspective is an ongoing challenge. We demonstrate that the science of complex networks can be helpful in tackling this problem.
Here we blend these motivations for studying networks, using newly available micro-data and the ability to work with large-scale, complex networks computationally, to study labor dynamics. Here we characterize a LFN for an entire economy. We also provide a model that generates many of the properties of the empirical LFN from economic behavior. The data do not tell us about the motivations of individual workers for changing jobs. However, we are able to develop a model that is consistent with the data in which workers act in their own self-interest strategically in seeking better employment opportunities.
The construction of a LFN is rather simple. For a selected period, we count the total flows of labor between every two firms in both directions. Although this is a directed network, we found that the most interesting insights come from studying its structural properties as an undirected graph. Therefore, our analysis uses algorithms for undirected networks (with exception of in-degree and out-degree centralities).
A different but related type of firms network is the supplier-customer ones. They have been studied comprehensively for Japan by , , where it was found that a network of 800,000 Japanese firms connected through economic transactions has the scale-free and hierarchical properties that we have found in our LFNs. Although these networks are of a different nature, they share common features with LFNs, suggesting the important role of firms' dynamics in economic systems.
In fluid dynamics, pipe network analysis is the analysis of the fluid flow through a hydraulics network, containing several or many interconnected branches. The aim is to determine the flow rates and pressure drops in the individual sections of the network. This is a common problem in hydraulic design.
To direct water to many users, municipal water supplies often route it through a water supply network. A major part of this network will consist of interconnected pipes. This network creates a special class of problems in hydraulic design, with solution methods typically referred to as pipe network analysis. Water utilities generally make use of specialized software to automatically solve these problems. However, many such problems can also be addressed with simpler methods, like a spreadsheet equipped with a solver, or a modern graphing calculator.
For these reasons, a probabilistic method for pipe network analysis has recently been developed, based on the maximum entropy method of Jaynes. In this method, a continuous relative entropy function is defined over the unknown parameters. This entropy is then maximized subject to the constraints on the system, including Kirchhoff's laws, pipe friction properties and any specified mean flow rates or head losses, to give a probabilistic statement (probability density function) which describes the system. This can be used to calculate mean values (expectations) of the flow rates, head losses or any other variables of interest in the pipe network. This analysis has been extended using a reduced-parameter entropic formulation, which ensures consistency of the analysis regardless of the graphical representation of the network. A comparison of Bayesian and maximum entropy probabilistic formulations for the analysis of pipe flow networks has also been presented, showing that under certain assumptions (Gaussian priors), the two approaches lead to equivalent predictions of mean flow rates.
In this chapter we will examine some of the most obvious and least complex ideas of formal network analysismethods. Despite the simplicity of the ideas and definitions, there are good theoretical reasons (and some empiricalevidence) to believe that these basic properties of social networks have very important consequences. For bothindividuals and for structures, one main question is connections. Typically, some actors have lots of of connections,others have fewer. Some networks are well-connected or "cohesive,"others are not. The extent to which individuals are connected to others, and the extent to whichthe network as a whole is integrated are two sides of the same coin.
Differences among individuals in how connected they are can be extremely consequential for understanding theirattributes and behavior. More connections often mean that individuals are exposed to more, and more diverse, information.Highly connected individuals may be more influential, and may be more influenced by others. Differences among wholepopulations in how connected they are can be quite consequential as well. Disease and rumors spread more quicklywhere there are high rates of connection. But, so to does useful information. More connected populations may bebetter able to mobilize their resources, and may be better able to bring multiple and diverse perspectives to bearto solve problems. In between the individual and the whole population, there is another level of analysis -- thatof "composition." Some populations may be composed of individuals who are all pretty much alike in theextent to which they are connected. Other populations may display sharp differences, with a small elite of centraland highly connected persons, and larger masses of persons with fewer connections. Differences in connections cantell us a good bit about the stratification order of social groups. Agreat deal of recent work by Duncan Watts, Doug White and many others outside ofthe social sciences is focusing on the consequences of variation in the degreeof connection of actors.
Figure 7.2 Knoke information exchange adjacency matrixUsing Data>Display, we can look at thenetwork in matrix form. There are ten rows and columns, the data are binary, and the matrix is asymmetric. As we mentioned in the chapteron using matrices to represent networks, the row is treated as the source of information and the column as thereceiver. By doing some very simple operations on this matrix it is possible to develop systematic and useful indexnumbers, or measures, of some of the network properties that our eye discerns in the graph.
Individuals, as well as whole networks, differ in these basic demographic features. Individual actors may havemany or few ties. Individuals may be "sources" of ties, "sinks" (actors thatreceive ties,but don't send them), or both. These kinds of very basic differences among actors immediate connections may becritical in explaining how they view the world, and how the world views them. The number and kinds of ties thatactors have are a basis for similarity or dissimilarity to other actors -- and hence to possibledifferentiation andstratification. The number and kinds of ties that actors have are keys to determining how much their embeddednessin the network constrains their behavior, and the range of opportunities, influence, and power that they have.
Another way of thinking about each actor as a source of information is to look at the row-wise variance or standarddeviation. We note that actors with very few out-ties, or very many out-ties have less variability than those withmedium levels of ties. This tells us something: those actors with ties to almost everyone else, or with ties toalmost no-one else are more "predictable" in their behavior toward any given other actor than those withintermediate numbers of ties. In a sense, actors with many ties (at the center of a network) and actors at theperiphery of a network (few ties) have patterns of behavior that are more constrained andpredictable. Actorswith only some ties can vary more in their behavior, depending on to whom they are connected. 2b1af7f3a8