NETWORK INFERENCE USING INFORMATION THEORY TOOLS 1
NETWORK INFERENCE USING INFORMATION THEORY TOOLS 2
Network Inference from Time-Series Data Using Information Theory Tools
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Abstract
Network Inference from Time-Series Data Using Information Theory Tools
Analyzing complex systems is a difficult process for many people in the world today. Very few tools have been created to aid in such a process in an effective way. Additionally, network inference and complex system analysis require mathematical and computer skill that are not readily available to everyone. A successful analysis can only be carried out by an individual who is acquainted with the proper mechanisms and has the necessary understanding of the organization’s dynamics (Sameshima & Baccala, 2014). Complex systems are characterized by many interacting components that arise and evolve over time. As such, a proper analysis of the system must entail a progressive approach that takes into account the changes that occur over time. moreover, an ideal complex system analysis tool should be balanced in such a way that it takes into account essential microscopic elements that are of importance to the expected outcome while ignoring other components whose presence or absence should not interfere with the results (Deniz, 2018). Consequently, regardless of the similarities in different complex systems, a modeling tool must be customized to meet the needs of the specific network for proper inference.
Many systems of the world can be referred to as complex. Social networks, political organizations, human cultures, the internet, brains, the stock markets, and the global climate are all examples of complex systems. In each of the mentioned organizations, important information is achieved through the interaction of various components within the system (Dehmer, Streib & Mehler, 2011). While each part is important, none can operate alone to produce the results that an entire system would create. Moreover, the various components that interact to create useful information are not static which makes it hard for the complex systems to be analyzed. Network inference of a time series data in a complex system implies that an individual will need to understand the relationships, if any, that exist between variables and how such can be altered to create the desired change.
Characteristics of a complex system can be coupled up into two concepts, namely emergence, and self-organization. Some system properties appear at different intervals in a process called emergence. Mathematical models allow one to understand the factors and relationships behind these macroscopic properties at a given point in time (Bossomaier, 2016). Analyzing the new occurrences at varied scales gives an individual an idea behind the operations of a system which allows for better planning for the future. Moreover, the properties self-organize over time creating a series of events that form the basis of an organization or process. Mathematical modeling practices help to simplify the complexity of the system thus making it a fundamental practice in everyday life. Since complex systems are characterized by nonlinear dynamics, achieving a possible solution by looking at the inputs alone is not possible. Information theory tools are the only approaches that can help in unraveling the mysteries behind nonlinear combinations and creating unreachable realities (Arabnia & Tran, 2011).
Networking inference is an increasingly growing field with researchers proposing new models each day. To make the right choice, one has to look at the limitations and the advantages of each proposal (Goh, Hasim & Antonopoulos, 2018). While some information theory tools are successful, they are limited in terms of how far or how deep they can unravel the complexities of nonlinear systems. The common structures that are found in diverse networks pose a great challenge when creating a reliable inference method. In information theory, the measure of the dependence between two separate variables creates mutual information (MI). To get MI, one has to quantify the amount of data acquired from one variable by observing the other random variable (Barman & Kwon, 2018). If the correlation coefficient of the two variables is zero, then the two properties are not essentially related to one another and their interactions do not affect the performance of the system. Analyzing and understanding the relationships between the microscopic elements of a complex system is the best easiest and simplest way of understanding the intricacies of a complex system.
In a natural complex system situation, it is hard for one to detect physical methods because of the large size. However, using each of the components as nodes of a network and the physical interactions between the nodes as links helps in understanding the exaggerated behavior of complex systems. To detect the physical methods of a large organization, it is vital to infer network structures that create the physical correlation between time-series acquired from the dynamics of the various nodes. Cross-Correlation or MI dynamics are ideal mechanisms to use while trying to quantify the relationship between variables within a complex system (Budden & Crampin, 2016). As such, the current paper will be based on a mutual information rate (MIR) methodology to infer the structure of a complex system from time-series data.
According to Ta, Yoon, Holm & Ham (2010), a mutual information rate (MIR) shows the relationship between two variables by measuring the time rate at which information is exchanged between two correlated and discriminate variables. The MIR is an appropriate tool for measuring the relationship between variables in a complex system because it allows for long measurements and calculations with arbitrary resolution. The tool makes it possible for an individual to analyze the unique properties of a system to understand the relationship between causes and effects. Through the MIR, the researchers in the current study intend to quantify the amount of information passed between two non-random nodes within a given period. Moreover, the tool will aid the team in understanding the relationship between synchronization and the exchange of information in a system (Timme & Casadiego, 2014). The purpose of the examination is to establish if there are any logical inference between microscopic elements of a complex system and the dependence among the variables.
The network inference in the current study will be founded on the rule-based modeling approach that pays particular attention at microscopic scales within an establishment. Since complex systems are diverse and extremely complicated, the time-series data used in the scrutiny process can be easily simulated in a computer to help the analyst appreciate the emergence and self-organization of properties in the system over time (Shandilya & Timme, 2011). Rule-based modeling allows one to explain the observed behavior in a simple language that is understandable to people without mathematical and computer skills. Further, the modeling process employed by the current paper is important in the sense that it helps the involved parties to make considerable predictions of the future and map a clear path that a system is bound to follow over time.
Main Body
Discussion of the Mathematical Theory
Systems produce information that can be transferred between different components. For such an exchange to happen, two independent variables either directly or indirectly linked must be involved (Zou, Romano, Thiel, Marwan & Kurths, 2011). In the current paper, the mode of transfer studied is time-series data where the amount of information exchanged within a given unit of time is examined to determine the link between the non-random elements. Further, the relationship between information synchronization and the speed of transfer will also be looked at in the paper. A positive outcome (the existence of a link between two units) is an indication of a bidirectional connection between the variables as a result of their interaction. Through such an understanding, it is possible for one to correctly infer a network of a complex system and map the future of the organization with clarity.
Mutual Information: The MI between variables indicates the amount of uncertainty that one has about one variable by observing the other unit (Butte & Kohane, 2000). The MI is given byIxy (N) = Hx+Hy-Hxy. The equation shows the strength of dependence existing between the two observed variables. For instance, when Ixy=0, it means that the strength of dependence between the elements observed is null, an indication that the two variables are independent. As such, the higher the value, the stronger the connection between variables and the higher the chances of their interaction producing a considerable effect on the overall performance of the complex system.
The calculation of Ixy(N) from a time series data is a difficult task. One has to calculate the probabilities computed on an appropriate probable space where a partition can be found (Bianco-Martinez, Rubido, Antonopoulos & Baptista, 2016). Moreover, the MI value measure is only suitable for carrying out a comparison between variables of macroscopic elements of the same system and not different structures. For a time series data to produce verifiable and usable results, the correlation decay times must be constant which is not possible when looking at information in different systems. As such, an MI is only viable if the factors analyzed are of a singular system to avoid the different characteristic time-scales produced via the varied correlated decay times in each organization.
Understanding entropy and conditional entropy is the first step towards having knowledge of how MI works in analyzing time-series data. Qualitatively, entropy is a measure of uncertainty – the higher the entropy, the more uncertain one is about a random variable. This statement was made quantitative by Shannon. He postulated that a measure of uncertainty of a random variable X should be a continuous function of its probability distribution PX(x) and should satisfy the following conditions
· It should be maximal when PX(x) is uniform, and in this case, it should increase with the number of possible values X can take
· It should remain the same if we reorder the probabilities assigned to different values of X
· The uncertainty about two independent random variables should be the sum of the uncertainties about each of them.
The only measure of uncertainty that satisfies all these conditions is the entropy, defined as: H(X) =−∑xPX(x) log P(x) =−EPXlogPX (2). Although not particularly obvious from this equation, H(X) has a very concrete interpretation. Suppose x is chosen randomly from the distribution PX(x), and someone who knows the distribution PX(x) is asked to guess which x was chosen by asking only yes/no questions. If the guesser uses the optimal question-asking strategy, which is to divide the probability in half on each guess by asking questions like “is x greater than x0 ?”, then the average number of yes/no questions it takes to guess x lies between H(X) and H(X)+1. This gives quantitative meaning to “uncertainty”: it is the number of yes/no questions it takes to guess random variables, given knowledge of the underlying distribution and taking the optimal question-asking strategy.
The conditional entropy is the average uncertainty about X after observing a second random variable Y and is given by
H(X|Y)=∑yPY(y)[−∑xPX|Y(x|y)log(PX|Y(x|y))]=EPY[−EPX|YlogPX|Y](3)
Where PX|Y (x|y)(≡PXY(x, y)/PY(y)) is the conditional probability of x given y.
With the definitions of H(X) and H (X|Y), the equation can be written as:
I(X; Y) =H(X) −H (X|Y). (4)
Mutual information is, therefore, the reduction in uncertainty about variable X, or the expected reduction in the number of yes/no questions needed to guess X after observing Y (Dehmer et al., 2011). Note that the yes/no question interpretation even applies to continuous variables: although it takes an infinite number of questions to guess a continuous variable, the difference in the number of yes/no questions it takes to guess X before versus after observing Y may be finite and is the mutual information. While problems can arise when going from discrete to continuous variables since subtracting infinities is always dangerous, they rarely do in practice.
Different approaches to the computation of MI exist. The variations in each method arise as a result of the mechanism used to compute the probabilities involved in the computation. In the histogram method, also called the bin, a suitable partition is found in the 2D space on equal and adaptive size cells. In the density kernels, the kernel estimate of the probability density function is applied. The last MI approach quantifies data by estimating probabilities from the distance between the closest variables (Zou et al., 2011). In the current analysis, the first approach where computation of probabilities is carried out in partitions of equally sized cells in the probabilistic space generated by two variables is used. The process has a tendency of overestimating the values because of two basic reasons, namely the finite resolution of a non-Markovian partition and the finite length of the recorded time series. The systematic errors can be avoided by creating a novel normalization when dealing with MI computations.
For the numerical computation of IXY(N), the paper defines a probabilistic space X, where X is formed by the time-series data observed from a pair of nodes, X and Y, of a complex system. Moreover, a partition X into a grid of N_N fixed-sized cells is created. The length-side of each cell, €, is then set to € = 1/N (Budden & Crampin, 2016).Consequently, the probability of having an event I for variable X, PX(i) is the fraction of points found in row I of the partition X. Similarly, PY(j) is the fraction of points that are found in column j of X, and PXY(i, j) is the joint probability computed from the fraction of points that are found in cell(i, j) of the same partition, where i, j = 1;…; N. The paper emphasize here that IXY(N) depends on the partition considered for its calculation as PX, PY, and PXY attain different values for different cell-sizes €.
Mutual information brings a reduction of uncertainties concerning one variable by observing another element whose performance is believed to affect that of the former unit. High mutual information signifies a great reduction of uncertainty while low mutual information is an indication of a small reduction of ambiguity.
Mutual Information Rate: calculating the MIR of a time series must take into consideration the partition dependence discussed in the definition of MI. MIR is defined as the theoretical mutual information exchange within a given time between variables, say X and Y. while the calculation of MIR using the MI principle can arise into errors in relation to the earlier mentioned partitions, other mechanisms of computing the quantity of information passed between variables at a specific ensure that the measure is invariant with respect to the resolution of the partition (Ta et al., 2010). To estimate the information passed between two finite nodes in the current paper, the observed time-series data at a given point I time is computed followed by a proper normalization for the identification of the connectivity structure of small networks of interacting dynamical systems.
MIR is a powerful concept in the analysis of complicated systems. The quantity (MIR) is calculated from mutual information which is defined by random systems within the organization. In the current paper, the researcher offers a simple way of calculating MIR in diverse networks and looking at the upper and lower bounds within a system without having to take into consideration probabilities.
In the current paper, various topologies for the network and different dynamics for the components of the dimensional systems are considered. The network inference, therefore, is done from times-series data that is observed and recorded for each component to determine the topological structure of the components interaction. The purpose of the paper is to determine if the function of one variable is affected by another non-random element by looking at the amount of information passed between the two nodes in a given unit of time. Moreover, the paper will seek to determine if synchronization of data affects the speed of information exchange between variable. Positive or negative values from this analysis will help in figuring out the type of dependence between microscopic elements of the system if any while providing an avenue for the researcher to map the future of the magnificent system.
Background
The paper introduces a new information-based approach for the analysis of networks within complex systems. The MIR computes data transferred per unit of time between two different nodes whose interaction is believed to cause a series alteration in the performance of the magnificent system (Barman & Kwon, 2018). The normalization of MIR used in the paper is measured based on the developed network for inference. The tool is a reliable measure of interdependency between variables in the presence of other additives such as noise, short-time series, as well as other coupling strength complications. The MIR is designed in a way that it can only detect and react when the most important variables in the system are triggered, especially the correlation decay time.
One of the aspects that make the MIR an essential tool is the fact that it embodies the characteristics of a great modeling and measurement tool. Research has shown that proper analysis mechanisms must be sensitive enough to the necessary variables while ignoring other occurrences within the system (Timme & Casadiego, 2014). As stated earlier, complex systems are characterized by the emergence of new elements at as time progresses. Therefore, it is hard to take into consideration all the new variables at each stage of development when trying to map up the future of the system. A model that is able to discard minor changes is an essential tool in the measurement of new elements at different scales.
To achieve this discriminatory role in network inference, researchers use various modeling mechanisms such as rule-based modeling (Butte & Kohane, 2000). The practice of modeling is an effective one in mathematical and computer science studies because it allows researchers to unravel the unreachable realities in life. Naturally, complex systems are magnificent and quite complicated for anyone to analyze. The amalgamation of elements and the constant interrelation between nodes within the system makes it hard for one to determine if the elements have any relationship and the nature of interactions among the nodes. Modeling helps one to create sustainable and reliable tools that are able to take into account some aspects of the system while ignoring the interactions of others.
Rule-Based Modeling
Modeling a complex system requires one to consider the multiple networks, nonlinearity, emergence and the self-organization characteristics of enormous organizations. In rule-based modeling, particular attention is paid at the microscopic scales because looking at the interaction of variables is the best way of understanding the complexity of the system (Goh et al., 2018). The model helps individuals to explain observed behaviors, in our case, the time series data. Moreover, rule-based modeling helps researchers and analysts to make predictions and map the possible progress of the system with certainty.
Various steps are used when creating a rule-based model for a complex system. First, one has to observe the system for a while. Analysis of systems depends greatly on the experience that a person has with a similar organization. The human body is created in such a way that it tends to link similar instances together (Barman & Kwon, 2018). As such, when a person sees an abnormal or new occurrence, he or she will most likely describe the happening based on his or her past interaction with a similar situation. As such, watching and experiencing complex systems helps a researcher to have an idea of how variables interact within magnificent organizations making it easy for him or her to have a background upon which to build his or her theories in the future.
Observing a particular system when trying to create a specific model for an organization gives one an idea of the possible relationship between nodes. As such, an individual is able to decide on the best measurement tool to use based on the variables that are suspected to have interdependency. One must become aware of the complex systems to model them hence the need for observation as the first step towards an effective analysis. Moreover, observation brings a clear understanding of the cause and effect within a system.
In complex organizations, it is impossible to clearly capture the causes and effects of happenings within the system because microscopic elements do not have any meaning when they are not interacting with one another (Bianco-Martinez et al., 2016). Simply put, the results of a process cannot be attributed to one particular variable in a complex system since information is found between the various parts of the organization and not within the units themselves. Observation, therefore, helps to get a glimpse of what relationships are likely to produce measurable results.
The second step in creating an ideal model for a complex system is reflecting on the possible rules that might cause the characteristics that were seen in the observation. Similar to the first step, reflect on the rules depends on a person’s experience with a similar situation in the past. The rules determine the best tool to use for network inference (Zou et al., 2011). The third step is deriving predictions from the rules and comparing them with reality. For instance, if a researcher thinks that two variables exist in a mutually beneficial process, he or she must compare that understanding with the realities of complex systems. Again, this step requires one to have a better understanding of magnificent organizations for the proper comparison of the observed rules with reality.
The fourth and the last step towards building an ideal model is repeating the rules until one is satisfied with the results. The predictions made must make sense; otherwise, the examination processes become a failure. As such, a researcher has to repeat the first three steps over and over until a reasonable conclusion is achieved (Arabnia & Tran, 2011). On aspect of complicated systems that cut across the board is the fact that they barely change. The complex nature of the systems makes it hard for leaders and innovators to manipulate operations. As such, an analyst cannot produce ambiguous results when inferring networks within a complex system. The repetition of the steps ensures that the results arrived at are in line with the expectations and the understanding of the world in regards to the organizations.
Rule-based modeling uses the dynamic equation, theories, and first principles to determine the performance of a system at a specific time and describe how it will change over time (Bossomaier, 2016). Other models do not go as far as analyzing the evolutionary possibilities of a system which creates the major differentiation between rule-based models and other approaches. Mostly, quantitative methods are used to determine the future paths of an organization. For instance, the MIR used in the current paper fits as a rule-based model because it quantifies the relationship between two variables to determine both the present and future relationships between non-random nodes.
When creating a model for a complex system, one has to consider other important issues that are not related to the characteristics of the organization (Deniz, 2018). For instance, it is vital for an analyst to determine the kind of questions he or she wants to address. Secondly, one should ask himself or herself at what scale should the behaviors of the observed data be described to answer the key questions. Due to the complexity of the systems, many relationships can be derived from a couple of nodes; therefore, a researcher must be keen not to include too many behaviors whose analysis may not be related to the expected results. One has to look at the microscopic elements of the system and define the dynamical rules for their behavior with an understanding of the questions that need to be answered.
Another important aspect to consider is the structure of the system. While the majority of the complex organizations are similar to some extent, it is vital to understand that a few variations are often created to make each system unique (Sameshima & Baccala, 2014). A researcher must have a clear understanding of these variations if he or she is to come up with an ideal model. Looking at the structure entails analyzing the microscopic components and grouping them in terms of the assumed interaction with one another. After that, a researcher must consider the possible state of the system. That is to say, one has to describe each variable and the dynamical state that each component can take during the system’s operations.
Lastly, researchers must consider the state of the system over time. Complex organizations are characterized by emergence and self-organization, processes that occur over time. In emergence, system properties occur at different scales depending on the operation of the components. The new elements arising at each stage of development must be taken into consideration when coming up with a proper model (Dehmer et al., 2011). One has to critically analyze how these emergent microscopic factors will affect the non-random variables chosen for the study. Additionally, elements in complex systems self-organize over time. A researcher should consider such clustering when deciding the right model for the network inference.
The five steps stated above are not an easy task to accomplish. Coming up with the right choices for each question is not a trivial job and it requires a researcher to repeat the processes for a long time until the behaviors can mimic the key aspects of the system (Ta et al., 2010). To loop the questions, a researcher has to answer a set of other related questions to show the interaction of the chosen components. For instance, one has to consider the scale to use in order to achieve the desired results, what components to include in the analysis, the possible connection between the chosen nodes, the unit of measurement that can produce and easily mimic the expected interactions, as well as the changes over time that the observed variables might produce and under what circumstances. Answering these questions helps an analyst to make a mental prediction about the kind of microscopic behaviors that would arise if the examination is carried out.
Characteristics of a Good Model
A model is ideal for the analysis of a complex system if it is simple. Modeling is about simplicity especially when a mega-organization is involved. Researchers create a model to have a shorter and simple way of describing reality. As such, one should always choose the mechanism that is easier to use when looking at two models of equal predictive power. Simplicity in this sense means that a measure must be able to give a correct interpretation of observed data by eliminating parameters, variables, and assumptions without interfering with the expected behavior (Goh et al., 2018). The MIR tool used in the current study qualifies in the simplistic aspect because it is easy to create and manipulate.
The second most important characteristic of a model is validity. From a practical point of view, a model should produce results that are closely related to the observed reality. For instance, if the assumed relationship between nodes is that the increase of causes a similar reaction to the other microscopic element, the model’s predictions should agree with such an observation if it a reliable tool. The reliability of the MIR is undeniable (Zou et al., 2011). The mechanism has been used widely in mathematical and computer science practices and it has always shown a close relationship between its computations and the observed data. In complex systems, face validity is very important; as such, a tool that does not offer that comfort is useless since due to the constant interaction between variables in mega organizations, it is impossible to use a quantitative comparison between the model prediction and the observational data.
However, regardless of the need to have a valid model, it is important for one to avoiding over-fitting the predictions and the observed data. Adjusting the forecasts of the tool so much to closely agree with the observed behavior makes it hard for the results of the analysis to be generalized (Goh et al., 2018). As mentioned earlier, an understanding of a single complex system can help one make an informed judgment about other similar organizations in the future. As such, network inference results are often generalized when dealing with complex systems but this is not possible in the case of forced correlation between predictions and observable outcomes. One has to strike a balance between simplicity and validity because the two characteristics are equally important. Increasing the complexity of the model to achieve a better fit takes away the simplicity nature of the tool thus rendering it useless.
The last characteristic of a good model is robustness. A model must be selective in terms of which factors interfere with its computation. Sensitivity to minor variations of the model assumptions can have unintended consequences and render the tool useless (Deniz, 2018). Errors are always present when creating a useful tool in the inference of complex system networks. As such, an effective tool must be sensitive enough to capture the major variables while ignoring the interference from non-essential factors in the analysis. For instance, in the current paper, noise is an example of an existing variable whose interference should not be considered while quantifying the relationship between information passed between two nodes and time.
The MIR tool chosen for the study is robust in that it is able to factor in the amount of data shared within a specific unit of time while ignoring issues of noise (Timme & Casadiego, 2014). When a model is sensitive to all minor variations, then the conclusions it provides are unreliable. However, in a robust measurement tool, the final results hold under minor variations of the model assumptions and parameters. A researcher can make sure that the model he or she uses for the analysis of a complex system is robust by manipulating various parameters to balance the level of sensitivity and ensure that only the essential factors are considered in the measurement process.
Dynamical System Theory
All rule-based models operate under the assumptions of dynamical system theory, including the tool used for the current study, MIR. The theory focuses on how organizations change over time instead of looking at their static nature. By definition, a dynamical system is one whose state is uniquely characterized by a set of microscopic elements whose interactions are described by predefined rules (Budden & Crampin, 2016). Understanding these rules helps one to clearly map the present situation and the possible future progression of the system. Most complex systems in the world today are dynamical by nature thus requiring the use of a rule-based model for inference of their networks.
The dynamic nature of the complex systems can be described over discrete time steps or a continuous timeline. In the current paper, the latter mechanism is used to determine the amount of information shared between two non-random variables within a given unit of time. The general mathematical formulas used for such a computation are:
Discrete-time dynamical system
Continuous –time dynamical system
In either case, or x is the state variable of the structure at time t, which may take a scalar or vector value. F is a function that determines the rules by which the system changes its state over time (Bossomaier, 2016). The formulas given above are first-order versions of dynamical systems (i.e., the equations don’t involve xt−2, xt−3, . . ., or d2x/dt2, d3x/dt3, . . .). But these first-order forms are general enough to cover all sorts of dynamics that are possible in dynamical systems, as we will discuss later.
In the current situation, the paper explores the effectiveness of MIR versus MI in terms of how successful they are in inferring exactly the network of our small complex systems. In general, the researcher finds that the MIR outperforms the MI when different time-scales are present in the system (Zou et al., 2011). The results also show that both measures are sufficiently robust and reliable to infer the networks analyzed whenever a single time-scale is present. In other words, small variations in the dynamical parameters, time-series length, noise intensity, or topology structure maintain a successful inference for both methods. It remains to be seen the types of errors that are found in these measures when perfect inference is missing or impossible to be done.
The Use of Python Modeling Tools in Network Inference
Technological advancements have made time-resolved data available for many models but this can only be useful if the right tools are used to analyze the data. Python 2.7 helps the analyst to create simulation models that are effective in capturing the actual situation of the network being inferred thus making the examination process of a complex system easy (IJzendoorn, Glass, Quackenbush & Kuijjer, 2016). The python tool used in the current study is effective because it runs faster than other computer science and mathematical versions and it includes additional features that allow the research to manipulate the used tool (MIR) to produce the intended results. In fact, python 2.7 helps in increasing the reliability of a model by providing an easy way for the involved parties to manipulate variables to create a closer relationship between predictions and observable data with ease.
Using a python tool increases the simplicity and the robustness of a mathematical tool and it has effectively done so in the current paper (IJzendoorn et al., 2016). The approach simplifies some of the complexities found in various models making them usable to people with little or no mathematical of computer science skills. In terms of robustness, python creates avenues for the researcher to organize the measurement tool to react only to important variables while remaining neutral in the presence of non-essential factors such as noise in the current paper. As such, the use of the mathematical modeling tool has made MIR more successful in determining the relationship between information passed between two non-random nodes at a given time and analyzing the effects of synchronization on the performance of the said variables.
Models for Our Complex Systems
The paper uses various topologies for the networks to analyze the various microscopic components of the complex system in question. The network inference, therefore, is carried out from a time-series that are recorded for each component. This is to say that the nodes that are considered to have a reliable relationship are observed and the time series data recorded for further analysis. Since various components are involved, the examinations are divided based on discrete and on continues time-series components.
Discrete-Time Units
The variables that are of the discrete class of complex systems are described and analyzed using the following equation in the paper:
where in is the n-th iterate of map i, where i = 1;…;M and M is the number of maps (nodes) of the system, a_ is the coupling strength, is the binary adjacency matrix (with entries 1 or 0, depending on whether there is a connection between nodes i and j or not, respectively) that defines the structural connectivity in the network, r is the dynamical parameter of each map, is the node-degree, and is the considered map. For the logistic map, where the correspondents are not explicitly mentioned the paper uses r=4 to fully develop chaos for the circle map. In some cases, r=0.35 and k ≈ 6.9115. The paper uses these measures to study the robustness of the methodology for different coupling strengths, observational noise, and data length. Further, small sized networks with discrete dynamics ad different decay of correlation times for the nodes are used to test the methodology used in the current paper. The measurements are carried out to ensure the quality of the inference process by guaranteeing the effectiveness of the tools used for examination.
In discrete dynamics networks, the calculation and the relationships of the nodes are given by logistic maps. The researchers construct a network of two clusters and three nodes each to determine the amount of information shared among the variables within a specific unit of time. The clusters, however, are connected by a small coupling strength link for easy analysis. The clusters are constructed by time-series with different correlation decay times, creating a good example to understand how a clustered network with different time-scales can affect the inference capabilities of MI- or MIR-based methodologies. Specifically, the cluster formed by the first three nodes is constructed using r=4 and the dynamics formed by nodes 4, 5 and 6 is created using a third order composition of the logistic map with r being 3.9.
Network with Continuous-Time Units
The paper uses a continuous dynamic for the nodes of the network described by the HR neuron model. The model is given by:
Where p is the membrane potential, q is associated with the fast currents (N or), and n with the slow current, for example, C. The rest of the parameters are defined as, where is a uniformly distributed random number in (0; 0:5) for all.
Methods
Correlation decay time T (N). T (N) is a necessary aspect in the inference of the topology of a network. However, calculating the correlation decay in a real-life situation is always hard because it depends on quantities such as Lyapunov exponents and expansion rates which require a high computational cost. In the current paper, the values are achieved by estimating the number of iterations that take a point in cells of to expand and completely cover. The approach helps the researchers to quickly and simply determine the time it takes for the correlation o decay to zero. The paper introduces a novel way of calculating T (N) from the network diameter which is mapped from one cell to another.
To construct measurable networks, the researchers assume that each equally sized cell occupied by a single point represents one node within the network. Since the correlation being analyzed in the current paper is the kind that requires the transfer of data from one point to another, the paper creates connections between nodes by following the dynamics of points moving from one cell to another. Specifically, a connection between two variables says m and n exit if points in the third variable cause movement from cell m to n. if a link between the measured elements exists then the weight is equal to 1. Alternatively, if the variables are independent, the weight is 0. Therefore a network is defined as a binary matrix with specific microscopic elements. In the current framework, a uniformly random time-series with n correlation results in a complete network, an all-to-all network.
T (N) is defined as the diameter of G in the current study because T (N) is the minimum time taken for the points being observed to spread fully within a network. As such, the diameter of the system is the maximum length for all the shortest paths which are calculated by looking at the minimum distance required to cross the entire network. The approach used in the current study transforms the calculation of T (N) into the computation of the diameter of G by applying Johnson’s algorithm principles.
Calculation of MIR. To calculate the MIR from the time series data collected over the specified time, the research truncates the summation of the results into a finite size depending on the resolution of the data. Moreover, the paper considers small trajectory pieces of the times-series with a length that depends on the total length of the time series. When calculating probabilities, the paper uses Markova partition to get equal right and left side variables. The length L represents also the largest order T that a partition that generates statistically significant probabilities can be constructed from these many trajectory pieces. Now, taking two partitions, K1 and K2, with different correlation decay times, T1 and T2, respectively, and a different number of cells, N1 _ N1 and N2 _ N2, respectively, with N2 > N1, we have T2 1. Moreover, K1 generates K2 in the sense that, where F is the evolution operator and means the pre-iteration of partition K1.
In order to use the partition close to a Markov equation, the divisions must be of a specific size. This condition can be achieved by constructing partitions with a significantly large number of equally sized cells of length €=1/N. the partitions used in the current paper will however not be Markov or generating and that will probably cause systematic errors in the estimation of MIR. A normalization equation is used to correct these errors in the paper. is a partition-independent quantity if the partitions are Markov, which is not the case in the current study. As such, to get the correct figures, the paper uses an equation which requires calculation of probabilities in Ω fulfilling the inequality. The equation is used in the research where [ is the mean number of points inside all occupied cells of the partition of Ω. The equations used in the current study provide similar results that one would get when MIR is calculated using. The used equation guarantees that the results are not biased.
Network Inference Using MIR. In the current analysis, the use of a non-Markovian partition allows the researchers to simplify the calculations. However, the approach makes the MIR values to oscillate around an expected value. Additionally, the MIR for the non-Markovian partitions has a non-trial dependence with the number of cells in the partition but also represents a systematic error. As such, since the for non-Markovian partition of N N equally sized cells is expected to be an independent partition, the paper proposes a way to obtain a measure computed from . The equation provides partition independence which is suitable for the network inference.
The paper uses the equation M (M-1)/2 to calculate MIR for the different nodes in the network. The practice helps in the inference of the system’s structure. Further, the MIR values are also discarded because the researchers are only interested in the exchange of information between nodes and not other variables. Moreover, the symmetric properties of MIR make it possible for the used mechanism to provide the intended results (Zou et al., 2011). exchange between any two nodes in the network is computed by taking the expected value over different partition sizes. To remove the systematic error, the paper uses a weighted average where the finer partitions contribute more to the value than the coarser ones. A smaller N value is likely to create a partition that is further away from a Markovian one than a partition of a larger value. Further, weighing the different partitions differently in the current paper helps the researchers to eliminate systematic errors.
The novel normalization proposed in the current study has the following principles. First, we use an equally sized grid of size N, we subtract from, calculated for all pairs of nodes, its minimum value and denote the new quantity as min . Theoretically, a pair that is disconnected should have a MIR value close to zero; however, in practice, the situation is different because of the systematic errors coming from the use of a non-Markovian partition, as well as from the information flow passing through all the nodes in the network (Goh et al., 2018). For example, the effects of a perturbation in one single node will arrive at any other node in a finite amount of time. This subtraction is proposed to reduce these two undesired overestimations of MIR. After this step, we remain with MIR as a function of N. Normalizing then by max _ min, where again the maximum and minimum are taken over all different pairs, we construct a relative magnitude, namely,
The paper further applies different grids sizes to obtain the MIR value where the maximum number of cells has been established. The formula produces results that would be achieved using the Markov tool but without the troubles associated with the mechanism. Moreover, the approach helps the researchers not just to analyze the amount of information passed between two non-random variables but also examines the effects of synchronization on the performance of the networks within the complex system. The paper also makes a second normalization at this point to eliminate the errors of the system and reduce the interference of external factors with the microscopic factors. The normalization is achieved using the following formulae:
The above equation is applied to each pair of XY to obtain the average. The higher the value, the higher the amount of information exchanged between the two nodes per unit of time. Moreover, the same formula helps to determine if synchronization of information at one variable interferes with the exchange of data between the other two units. The mechanism allows the researchers to identify the pairs of nodes that transfer a considerable range of information than others. Moreover, to perform network inference from the MIR, the researchers fix a threshold of (0, 1) and create a binary adjacent matrix where the value of the MIR is higher than the threshold. Creating the threshold helps the researchers to infer various networks within the organization at the same time separately and in a comparative way. Based on the results, it is evident that there are intervals of thresholds within the set limits that fulfill a band that represents a 100% successful network inference.
In general, the usefulness of our network inference methodology is measured by the supreme difference between the real topology and the one inferred for different threshold values. We find that whenever there is a band of threshold values, there is successful inference without errors. In practical situations, where the underlying network is unknown and the absolute difference is impossible to compute, the ordered values of the MIR or other similarity measures 2, 3 shows a plateau which corresponds to the band of thresholds aforementioned. In particular, if the plateau is small, the paper uses a method to increase the size of the plateau by silencing the indirect connections, hence, allowing for a more robust renewal of the underlying network.
Results for Network Inference
Discrete systems. In the current study, the performance of the various equations for network inference where the dynamics of each node is described by a circle or a logistic map is carried out using three different models. The network structure that makes up the small-network of interacting discrete-time systems is given by comparing the larger and smaller values exhibited by each node. Here, we analyze the effectiveness of the inference as the coupling strength, a, between connected nodes is varied. The researchers have shown that, for the logistic and circle maps, assuming the same topology, the dynamics is quasi-periodic for a > 0:15 and chaotic for 0 a 0:15. We, therefore, choose the coupling strength in the subsequent tests to be equal to 0.03 and 0.12, both values corresponding to chaotic dynamics.
From the analysis, it is evident that the wider the band, the bigger the probability to perform a complete reconstruction, therefore the reconstruction is more robust. When we deal with experimental data, and the correct topology is unknown, the optimal threshold can be determined by the range of consecutive thresholds for which the inferred topology is invariant. The reconstruction percentage decreases by inferring non-existent links or by avoiding them. However, to reduce the effects of systematic errors on the inference process, each time such an occurrence happens, we decrease the percentage by an amount less than the real links in the original network.
In determining the effects of noise on the times-series lengths, the paper starts by analyzing the effectiveness of MIR for different time-series strengths by the use of the dynamics of the logistic map for each node. When the value is closer to 0.15, a relatively shorter length generated by an adjacent matrix is enough to infer correctly the original network. On the other hand, when the value is closer to 0.03, a larger time-series is needed to a considerable reconstruction. The results of the current research indicate that the successful reconstruction for short-length time series depends on the intensity of the coupling strength. However, it is surprising to see that exact inference can always be achieved for this dynamical regime if a sufficiently large time-series is available. The best reconstruction using MIR is for coupling strengths in a dynamic regime where chaotic behavior is prevalent.
Neural Networks. In continuous dynamics analysis given by the HR system, the researchers use two electrical coupling mechanisms both of which consider the time-series strengths of the involved variables. Based on the findings, it is clear that MIR is able to infer the correct network structure for small networks of continuous time interacting components.
Comparing Mutual Information and Mutual Information Rate. Finally, the researchers compare MI and MIRXY to assess the effectiveness of our proposed methodology for network inference. The same normalization process is used for MIR to MI to have an appropriate comparison. In particular, we infer the network structure of the system. The different dynamics of the two groups produce different correlation decay times, T (N), for nodes X and Y, in particular, when the pair of nodes comes from different clusters. The different correlation decay times produce a non-trivial dynamical behavior that challenges the MI performance for network inference.
In this paper, we have introduced new information based mechanism to infer the network configuration of complex systems. The MIR is an information measure that computes the information transmitted per unit of time between pairs of components in a complex system. The results show that MIR is a vigorous measure to perform network supposition in the presence of additive noise, short time-series, and also for systems with different pairing strengths. Since MIR and MI depend on the parallel decay time T, they are suitable for inferring the correct topology of networks with different time-scales. In particular, we have explored the efficacy of MIR versus MI in terms of how triumphant they are in inferring exactly the network of our small complex systems. In general, we find that the MIR outperforms MI when different time-scales are present in the system. Our results also show that both procedures are satisfactorily robust and reliable to infer the networks analyzed every time a single time-scale is present.
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