\documentclass[english]{article} \usepackage{mathptmx} \usepackage{amsmath} % improved typesetting of formula \usepackage{array} % tabel commands \usepackage{helvet} \renewcommand{\ttdefault}{lmtt} \renewcommand{\familydefault}{\rmdefault} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \usepackage{geometry} \geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in, headheight=0cm,headsep=0cm,footskip=0.5in} \setlength{\parskip}{\medskipamount} \setlength{\parindent}{0pt} \usepackage{longtable} \usepackage{listings} %\VignetteIndexEntry{NetIndices: network indices and food web descriptors in R} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. \newcommand{\noun}[1]{\textsc{#1}} %% Bold symbol macro for standard LaTeX users \providecommand{\boldsymbol}[1]{\mbox{\boldmath $#1$}} %% Because html converters don't know tabularnewline \providecommand{\tabularnewline}{\\} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage{xspace} \newcommand{\R}{\textbf{\textsf{R}}\xspace} \newcommand{\nw}{\textbf{\textsf{NetIndices}}\xspace} %% \usepackage{babel} \makeatother \usepackage{hyperref} \setlength{\extrarowheight}{0.5cm} \begin{document} \title{Package \nw, network indices and food web descriptors in \R } \author{Karline Soetaert\\ Netherlands Institute of Sea Research\\ the Netherlands \and Julius Kipyegon Kones \\ University of Nairobi \\ Kenya } \maketitle \R package \nw is designed to estimate the most common network indices. It has been created to accompany the following article \cite{Kones09}: \emph{Kones, J.K., Soetaert, K., van Oevelen, D. and J.Owino (2009). Are network indices robust indicators of food web functioning? a Monte Carlo approach. Ecological Modelling, 220: 370-382.} http://dx.doi.org/10.1016/j.ecolmodel.2008.10.012 Please use this reference to cite package \nw in publications. In this vignette we first deal with conventions adopted, after which network functions are briefly discussed. The formulations for all network indices are represented in several tables. This table is more complete than the one in the article. \section {Notations and flow matrix conventions} The descriptions of symbols used in network indices computations are in Table 1. \footnote{As our work generally involves food webs, our notation/terminology will be skewed to this field; hence we will use the term "web" where others might use "network", and "flow" instead of "link"} As in Latham (2006) we adopt for these tables the convention as described in \cite{Hirata84}. We assume that a system has n biotic and abiotic compartments. The flow value $T_{ij}$ is defined as a destination-source flow (i.e. j ->i) . Quantitative flows between compartments of a web are classified into four types \cite{Field89}: \begin{itemize} \item exogenous inputs (imports), \item inter-compartmental exchanges, \item exports of useable medium, and \item dissipation of unusable medium . \end{itemize} The source compartment of imports to the internal network is labeled with number 0 (zero), the destination of usable exports (secondary production) is labeled n+1 and the destination of unusable exports (respiration/dissipation) is labeled n+1 (sensu Hirata and Ulanowicz 1984). The flow matrix, with source compartments in columns and destination compartments in rows, has dimensions $0<=j<=n$ and $1<=i<=n+2$. A matrix containing all flows within a web has dimensions of $1<=i<=n$ and $1<=j<=n$ . \section{Arguments to network index functions} In all functions of \nw, the network can be inputted in two ways: \begin{itemize} \item $Flow$, a matrix defined as source i -> destination j \item $T_{ij}$, the transpose of $Flow$, i.e. a matrix defined as destination i <- source j \end{itemize} Internally the calculation uses $T_{ij}$ If present, the row- and -or column names of $Flow$ or $T_{ij}$ are used to label the compartments. This is recommended. All functions distinguish between internal components and external components. Externals are either specified by their name (more general, only applicable if the compartments have been labelled) or by a number (error-prone): \begin{itemize} \item Import, externals that are a source to the network. If specified by numbers they should refer to *columns* of $T_{ij}$ (or rows of Flow) \item Export, externals that are a sink to the network. If specified by numbers they should refer to *rows* of $T_{ij}$ (or columns of Flow) \end{itemize} \section{Network indices} The R-functions for computing network indices are in Tables 2-8. They fall in several categories: \begin {itemize} \item function \emph{GenInd}. General network indices. In this category we consider a number of general systems' properties. \cite{Latham06} \item function \emph{UncInd}. Network Uncertainty indices, based on communication theory. \cite{Rutledge76} \item function \emph{AscInd}. System's growth and development. They are the ascendancy, development capacity and overhead. e.g. \cite{Ulanowicz80}. They can similarly be defined at four decomposed stages of a system: import (state 0), internal (between the compartments), export and dissipation \cite{Ulanowicz90}. \item function \emph{PathInd} Path analysis. Identifies the direct and indirect pathways in a network. (e.g. \cite{Finn76}) \item function \emph{EnvInd} Environ network indices. (\cite{Fath99}) \item function \emph{TrophInd}. Trophic level and Omnivory index (\cite{Christensen92}. The trophic level of a consumer equals 1 + the weighted average of the trophic levels of its food. Primary producers and the compartments labeled as "detritus" are assumed to have trophic level of 1. The omnivory index measures the variation in trophic levels of the food sources of a consumer. \item function \emph{Dependency} The dependency matrix estimates the direct + indirect dependence of a consumer on a resource. \end {itemize} Note: Most of the index calculations were based on the paper and the software written by Latham (\cite{Latham06}), who did a very commendable (if not heroic) job in gathering all the mathematical formulations of these indices. However, there were a couple of inconsistencies in the paper of Latham: \begin{enumerate} \item The Connectance index (\cite{Gardner70}): The L reported in \cite{Latham06} should be $L_{int}$, because Connectance is only calculated on internal links. \item The value of TSTbar in figure (2) of the article was shown incorrectly (as $T/n$, when it should have been $TST/n$. It was however correctly described in the paper. \item The Synergism index both in the text and the equations were wrong. See Table (7) for how it is correctly estimated. \end{enumerate} \section{Changes in later versions} \subsection{version 1.3} \begin{itemize} \item An alternative way to estimate Finn's cycling index has been added. It is called $FCI_b$. It is more easily interpretable compared to the original index as it scales between 0 and 1. \end{itemize} \begin{table}[t] \caption{Nomenclature for equations}\label{tb:nomen} \centering \begin{tabular}{p{.2\textwidth}p{.75\textwidth}}\\ Term & Description\\ \hline $n$ & Number of internal compartments in the network, excluding 0 (zero),n+1 and n+2\\ $j=0$ & External source\\ $i=n+1$ & Usable export from the network\\ $i=n+2$ & Unusable export from the network (respiration, dissipation)\\ $T_{ij}$ & Flow from compartment j to i, where j represents the columns of the flow matrix and i the rows \\ $T_{ij}^*$ & Flow matrix, excluding flows to and from external \\ $T_{i.}$ & Total inflows to compartment i \\ $T_{.j} $ & Total outflows from compartment j \\ $T_{i}$ & Total inflows to compartment i, excluding inflow from external sources \\ $T_{j} $ & Total outflows from compartment j, excluding outflow to external sinks \\ ${\mathop {\left( {x_i } \right)}\limits^. }_ - $ & A negative state derivative, considered as a gain to the system pool of mobile energy \\ ${\mathop {\left( {x_i } \right)}\limits^. }_ + $ & A positive state derivative, considered as a loss from the system pool of mobile energy \\ $z_{i0} $& Flow into compartment i from outside the network \\ $y_{n +,j} $ & Flow out of the network for compartment j to compartments n+1 and n+2 \\ $c_{ij} $ & The number of species with which both i and j interact divided by the number of species with which either i or j interact \\ $I,\;\delta _{ij} $ & Identity matrix and its elements \\ \hline \end{tabular} \end{table} \begin{table}[t] \caption{General Network indices}\label{tb:general} \centering \begin{tabular}{llll}\\ Index name & Code & Formula & Source(s)\\ \hline Total system throughflow & TST & $ \sum \limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left[ {T_{ij} + z_{i0} - \mathop {(x_i )_ - }\limits^. } \right]} } $\\ && $ = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left[ {T_{ij} + y_{n+,j} + \mathop {(x_j )_ + }\limits^. } \right]} }$ & \cite{Latham06}\\ Total system throughput & $T..$ & $ \sum\limits_{i = 1}^{n + 2} {\sum\limits_{j = 0}^n {T_{ij} } }$ & \cite{Hirata84}\\ Number of links & $L_{tot}$ & $ \sum\limits_{i = 1}^{n + 2} {\sum\limits_{j = 0}^n {(T_{ij} > 0)} }$&\\ Number of internal links & $L_{int}$ & $ \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^n {(T_{ij} > 0)} }$&\\ Link density &LD & $\displaystyle \frac{L_tot}{n}$&\cite{Latham06}\\ Connectance &C & $\displaystyle \frac{L_{int}}{n \cdot (n-1)}$ &\cite{Latham06, {Gardner70}}\\ Average link weight & $\overline T_{ij}$ & $\displaystyle \frac {T_{..}}{L_{tot}}$&\cite{Latham06}\\ Average compartment throughflow &$ \overline {TST}$ & $ \displaystyle \frac{TST}{n}$ &\cite{Latham06}\\ Compartimentalization & $\overline C$ & $ \frac{1}{n \cdot (n-1)} \cdot \sum \limits_{i=1}^{n}{\sum\limits_{j=1,j\neq i}^{n}}{c_{ij}}$ &\cite{Pimm80}\\ \hline \end{tabular} \end{table} \begin{table}[t] \caption{Network uncertainty indices}\label{tb:uncert} \centering \begin{tabular}{l@{$\qquad$}l@{$\qquad$}l@{$\qquad$}l}\\ Index name & Code & Formula & Source(s)\\ \hline Average mutual information & AMI & $ \sum\limits_{i = 1}^{n+2} {\sum\limits_{j = 0}^n {\frac {T_{ij}}{T_{..}}log_2{\frac{T_{ij}T{..}}{T_{i.}T_{.j}}} }} $& \cite{Ulanowicz04,Gallager68,Shannon48,Latham02,Rutledge76}\\ Statistical uncertainty & $H_R$ & $ -\sum\limits_{j = 0}^{n} {\frac{T_{.j}}{T_{..}}log_2 \frac{T_{.j}} {T_{..}}} $ & \cite{Latham06, Ulanowicz90}\\ Conditional uncertainty & $D_R$ & $\displaystyle H_R - AMI$ & \cite{Latham06, Ulanowicz90}\\ Realized uncertainty & $RU_R$ & $\displaystyle \frac{AMI}{H_R}$& \cite{Latham06, Ulanowicz90}\\ Network uncertainty & $H_{max}$ & $\displaystyle \sum\limits _{i=1}^n {log_2 (n+2)}$& \cite{Latham06, Ulanowicz90}\\ Network efficiency& $H_{sys}$ & $\displaystyle -\sum\limits _{i=1}^{n+2}{\sum \limits _{j=1}^n {\frac{T_{ij}}{T_{..}} log_2 \frac{T{ij}}{T_{.j}}}}$& \cite{Latham06, Ulanowicz90}\\ Constraint information& $H_{c}$ & $\displaystyle H_{max}-H_{sys}$& \cite{Latham06, Ulanowicz90}\\ Constraint efficiency & CE & $\displaystyle \frac{H_c}{H_{max}}$ &\cite{Latham06, Ulanowicz90}\\ \hline \end{tabular} \end{table} \begin{table}[t] \caption{System growth and development indices}\label{tb:ascend} \centering \begin{tabular}{l@{$\qquad$}l@{$\qquad$}l@{$\qquad$}l}\\ Index name & Code & Formula & Source(s)\\ \hline Ascendency & A & $ \displaystyle \sum\limits_{i = 1}^{n+2} {\sum\limits_{j = 0}^n {T_{ij} log_2{\frac{T_{ij}T{..}}{T_{i.}T_{.j}}} }} $& \cite{Ulanowicz00,Ulanowicz90}\\ Development capacity & DC & $\displaystyle -\sum\limits_{i = 1}^{n+2} {\sum\limits_{j = 0}^{n} {T_{ij}log_2 \frac{T_{ij}} {T_{..}}}} $ & \cite{Ulanowicz00, Ulanowicz90}\\ Overhead & $\displaystyle \phi$ & $DC - A$ & \cite{Ulanowicz00,Ulanowicz90}\\ Extent of development & AC & $\displaystyle \frac{A}{DC}$& \cite{Ulanowicz00,Ulanowicz90}\\ \hline \end{tabular} \end{table} \begin{table}[t] \caption{Effective measures indices}\label{tb:effect} \centering \begin{tabular}{l@{$\qquad$}l@{$\qquad$}l@{$\qquad$}l}\\ Index name & Code & Formula & Source(s)\\ \hline Effective connectivity & $CZ$ & $\displaystyle \prod \limits_{i,j = 1}^{n} {(\frac {T_{ij}^2}{T_{i.}T_{.j}})}^{-0.5 \cdot T_{ij}/T_{..}} $& \cite{Zorach03}\\ Effective flows & $FZ$ & $\displaystyle \prod \limits_{i,j = 1}^{n} {(\frac {T_{ij}}{T_{..})}^{-T_{ij}/T_{..}}} $& \cite{Zorach03}\\ Effective nodes & $NZ$ & $\displaystyle \prod \limits_{i,j = 1}^{n} {(\frac {T_{..}^2}{T_{i.}T_{.j}})}^{0.5 \cdot T_{ij}/T_{..}} $& \cite{Zorach03}\\ Effective roles & $RZ$ & $\displaystyle \prod \limits_{i,j = 1}^{n} {(\frac {T_{ij}T_{..}}{T_{i.}T_{.j}})}^{T_{ij}/T_{..}} $& \cite{Zorach03} \\ \hline \end{tabular} \end{table} \begin{table}[t] \caption{Pathway analysis}\label{tb:pathway} \centering \begin{tabular}{l@{$\qquad$}l@{$\qquad$}l@{$\qquad$}l}\\ Index name & Code & Formula & Source(s)\\ \hline Total System cycled throughflow & $TST_c$ & $\displaystyle \sum\limits_{j = 1}^{n} {(1-\frac{1}{q_{jj}})\cdot T_j} $ &\cite{Finn76, Finn78,Finn80,Patten84,Patten76}\\ && $\displaystyle Q = \left[ I-G^* \right] ^{-1}$ & \\ && $\displaystyle G^*=\left[ T_{ij}^*/max(T_j,T_i)\right] $&\\ Total System non-cycled throughflow & $TST_S$ & $\displaystyle TST-TST_c $ & \cite{Finn76, Finn78,Finn80,Patten84,Patten76}\\ Finn's cycling index & FCI & $\displaystyle \frac{{TST_c }}{{TST}}$ & \cite{Finn76, Finn78,Finn80,Patten84,Patten76}\\ revised Finn's cycling index & FCIb & $\displaystyle \frac{{TST_c }}{{T..}}$ & \cite{Ulanowicz86,Allesina04}\\ Average pathlength & $\displaystyle \overline {PL}$ & $\frac{TST}{\sum {z_{i0}}-\sum{ {\mathop {\left( {x_i } \right)}\limits^. }_ - }}$ &\\ & & $\displaystyle = \frac{TST}{\sum {y_{n+,j}}+\sum{ {\mathop {\left( {x_i } \right)}\limits^. }_ + }}$ & \cite{Ulanowicz00,Ulanowicz90}\\ \hline \end{tabular} \end{table} \begin{table}[t] \caption{Environ analysis}\label{tb:environ} \centering \begin{tabular}{l@{$\qquad$}l@{$\qquad$}l@{$\qquad$}l}\\ Index name & Code & Formula & Source(s)\\ \hline Transitive closure matrix & $G$ & $\displaystyle \left[ T_{ij}^*/T_j\right]$ & \cite{Patten82,Fath99}\\ Integral nondimensional matrix & $N$ & $\displaystyle I+G+G^2+...=\left[ I-G \right]^{-1}$ & \cite{Patten82,Patten84}\\ Non-dimensional direct flow-based utility matrix & $D$ & $(d_{ij})= \frac{T_{ij}^*-T_{ji}*}{T_i}$ & \cite{Patten84,Higashi89}\\ Utility nondimensional matrix & $U$ & $\displaystyle I+D+D^2+...=\left[ I-D \right]^{-1}$ & \cite{Patten84,Higashi89}\\ Coefficient of variation of N&CV(N)&$\displaystyle \sqrt {{\frac{{\sum\nolimits_{i,j = 1}^n {\left( {\overline N - N_{ij} } \right)^2 } }} {{(n^2 - 1) \cdot {\overline N }^2}}}} $&\cite{Patten81,Fath99b}\\ Coefficient of variation of G&CV(G)&$\displaystyle \sqrt {{\frac{{\sum\nolimits_{i,j = 1}^n {\left( {\overline G - G_{ij} } \right)^2 } }} {{(n^2 - 1) \cdot {\overline G }^2}}}} $&\cite{Patten81,Fath99b}\\ Homogenization & $H_p$ & $\displaystyle \frac{{CV(G)}}{{CV(N)}}$ & \cite{Patten81,Fath99b}\\ Integral Utility Matrix & $\gamma$ & $T_i \cdot U$ & \cite{Patten92,Fath98,Fath04}\\ Synergism Index & $\displaystyle \frac{b}{c}$ & $\frac{\sum{+utility~in~\gamma}}{\sum{-utility~in~\gamma}}$ & \cite{Patten92,Fath98,Fath04}\\ Dominance indirect effects & $\displaystyle \frac{i}{d}$ & $\frac{\sum \limits _{i,j=1}^n{(N_{ij}-I_{ij}-G_{ij})}} {\sum \limits _{i,j=1}^n {G_{ij}}}$ & \cite{Patten92,Fath98,Fath04}\\ \hline \end{tabular} \end{table} \begin{table}[t] \caption{Trophic analysis}\label{tb:trophic} \centering \begin{tabular}{l@{$\qquad$}l@{$\qquad$}l@{$\qquad$}l}\\ Index name & Code & Formula & Source(s)\\ \hline Diet matrix & $P$ & $\left[ \displaystyle \frac{T_{ij}^*}{T_i} \right]$ &\\ Diet dependency matrix & $D$ & $\displaystyle I+P+P^2+...=\left[ I-P \right]^{-1}$ & \\ Trophic level of compartment i & $TL_i$ & $ 1+ \sum\limits_{j = 1}^{n} {(\frac{T_{ij}^*}{T_i}\cdot TL_j)} $ &\cite{Christensen92,Lindeman42}\\ Omnivory index for compartment i & $OI_i$ & $\sum \limits_{j=1}^n{(TL_j-(TL_i-1))^2 \cdot \frac{T_{ij}^*}{T_i}}$ & \cite{Christensen92}\\ \hline \end{tabular} \end{table} \clearpage \bibliographystyle{plain} \bibliography{NetIndices} \end{document}