temp

t$%$\Im$
at time $t$ and an ``homology'' matrix $\mathcal{H}(t)$ between the set of fields at two consecutive time: $t-1$ and $t$, from which fields at time $t-1$ $C

\frac{1}{n}$ and $n\leq3$ over the whole period (\emph{e.g.} the term \emph{protein interaction network} has to appear at least in 5 references to be included in our set of candidate keywords). Stop words were discarded. This list of terms was then checked by science historians to further discard uninformative terms, which finally lead to a set $\mathcal{L}$ of 834 terms (given in SI).

These terms were then indexed from 1950 to 2008 in the 2,69M retrieved abstracts to build the co-occurrence matrix $\mathcal{M}$ giving all co-occurrences for terms in $\mathcal{L}$ from 1950 to 2008. $\mathcal{M}_t(i,j)$ gives the number of articles published during the year $t$ which mentioned both terms $i$ and $j$ in their abstract.

\subsection{Static (multi-level) map reconstruction}
Scientometrics has defined a great number of measures based on co-occurrence data that capture the degree of similarity or proximity between two terms, through the analysis of simple co-occurrences statistics \cite{He1999Knowledge}. Among other, we can mention the two index early introduced in scientometrics the inclusion index $\frac{n_{ij}}{min(n_i,n_j)}$ \cite{Callon1986Qualitative} and the proximity index $\frac{n_{ij}

t(i,j)=((\frac{n_{ij}

t})

t}{n_j

{1/\alpha})

T(i,j)=\Proxm

{T,2}(C_a,C_b)=\frac{1}{\mid C_a \mid} \sum_{i \in C_a}(\frac{1}{\mid C_b \mid}\sum_{j\in C_b}\Proxm

{T,2}$ even if they do not share any terms as the terms they contain are themselves close.

This defines a weighted directed graph on the set of clusters that can be drawn with network visualization tools. The labeling of each node of the map finds here a natural solution since we can characterize within a cluster, each terms on a specificity / genericity dimension (\textit{cf.} \cite{coint08multi} for details). According to what is looking for, one can label the clusters with the most generic terms, the specific ones, an so on.

\subsection{Inter-temporal matching function}

%\noteperso{movie is made of successing photographs, previous work, measure pseudo-inclusion, carte des sciences multi-niveau, how to }
The next step in phylogeny reconstruction is to achieve inter-temporal matching between communities. Given a field, we wish to find the field or union of fields from which it inherits. %% parler plutot d'héritage conceptuel. Arranger avec précedemment. Je veux bien essayer de reformuler cette partie.
We assume that the time scale of scientific fields evolution is slow enough to allow simple similarity measures between two close periods to track the meso-dynamics of a given field.
%We assume that the field evolution is slow enough to allow to track it. that a basic similarity measure should enable to match communities together.

We thus seek to find the field \emph{or} combination of fields that are most similar and therefore the most likely matchable.
%We first define a similarity measure $\delta$ between fields.
One of the most straightforward measure is a Jaccard similarity measure\footnote{This function is the inverse of the ``transformation index'' introduced for similar purposes by Callon in \cite{Callon:1991p2209}} on fields terms, thereafter denoted $\delta$. Given two fields $C_1$ and $C_2$ that can be defined as set of terms %$A =(t_i)_{i\subset I}$ and $B% = (t_j)_{j\subset J}$
then $\delta(C_1,C_2)=\frac{| C_1 \cap C_2 |}{| C_1 \cup C_2 |} $. $\delta$ can be interpreted in terms of the probability that a term belonging to $C_1 \cup C_2 $ also belong to $C_1 \cap C_2$. This is simply a measure of the overlap between $C_1$ and $C_2$.


Given a conceptual field $C_l

{t+1}\}_{b \in B}$ at time $t+1$, its ``fathers'' $\mathcal{F}_l

t\}_{a \in A}$ as:$$ \mathcal{F}

{t}_k ,C

{t+1}_l = \displaystyle \argmax_{K \subset A}\frac{|(\cup_{k \in K} \mathcal{C}

{t+1}_l |}{| (\cup_{k \in K} C

{t+1}_l |}$$


%otherwise one may prefer to adopt other kind of measure like a cost/benefit tradeoff function balancing on one side the possible benefits provided by one field (or union of fields at $t$) by the ratio of overlap obtained with the field at $t+1$ versus the ``cost'' of such a matching provided by the total size (in number of items) of the candidate field: that is $\delta(A,B) = \frac{|A \bigcap B|}{|A|| B|}$.(OR other formulas...) % A élaguer ...
%\COR{il faudrait verifier quand meme que ce schema ne ressemnle pas trop au schema de palla, je crains que si....}
Figure \ref{intertemp_match} illustrates the matching procedure. We plotted two successive sub-networks with the same set of nodes between two time steps. The two successive period present distinct cluster sets : $A$ and $B$ at time $t$ and $C$ and $D$ at time $t+1$. Note that one node belongs to two different clusters at time $t+1$. The aim is to determine from which fields or union of fields $C$ and $D$ may be descending. It is straightforward to check that field $A$ is the closest to cluster $C$ (that is $\mathcal{F}

{t+1}_D = A\cup B$.
\begin{figure}
\center
%\hspace{-2cm}
\includegraphicswidth = 0.5 \linewidthNaV\includegraphicswidth = 0.4 \linewidthNaV{image/phyloex.eps}
\caption{Inter-temporal fields matching.}\label{intertemp_match}
\end{figure}

%\subsection{discussion about the distance}

%\noteperso{intertemporal matching: trouver les pères qui expliquent le mieux (au sens d'une distance à préciser) une descendance.}

%d=Jaccard
As it would seem uncorrect to match two fields that have very few terms in common even though no better matching is possible, we need to define a threshold above which the matching is satisfying. We shall call this threshold $\delta_0$. One can tune a threshold requiring a minimum amount of similarity. As we shall see, activity patterns in the phylogeny (areas of activity burst, areas with emergent fields, branches death, etc.) are robust to variations of $\delta_0$ provided that $\delta_0$ does not take get too close from 0 or 1.

%\noteperso{on va pouvoir repérer des patterns dynamiques sur l'évolution du bidule. Rechercher des ruptures pleins de morts, et de naissances}
%\noteperso{parler de la résolution temporelle et de sa conséquence sur la définition du seuil.}

\subsection{Validation}
As stated before, the aim of phylogeny reconstruction is to discover patterns and regularities in Science evolution. Given this objective, we defined two benchmarks for this reconstruction: theoretical validation and empirical validation.

Theoretical validation is related to the robustness of the detected patterns regarding the parameters of the model ($\delta_0$ in our case). Detected patterns should be robust to parameter change if we want them to be significant.

Empirical validation is related with the adequacy of the reconstruction of scientific fields compared the productions of scientific communities. To reflect the activity of a scientific community, it is important that scientific fields are composed by terms that are indeed mentioned altogether in the literature. We will thus checked for each cluster identified in 2.2, that there are some significant number of papers mentioning all the terms of the clusters in their full text. Moreover, a cluster composed by very common terms (\textit{e.g.} {disease ,molecule,cell,division}) will not be as much informative as a cluster composed of more specific terms (\textit{e.g.} {cancer ,dna damage, apoptosis, checkpoint}). This nuance can be caught by the notion of self-information \cite{shannon1948mathematical} conveyed by the observation of an event composed of independent items $a_1$ ... $a_n$ which have a probability $p_1$ ... $p_n$ to be observed individually. Self-Information is then defined by $I(a_1,...,a_n)=\sum_{i=1...n}-log(p_i)$. These two constraints can be synthesized into the \textit{empirical quality} of a cluster $C$, defined as the products of its self-information with the normalized number $\frac{n_C}{N}$ of papers mentioning all the terms of $C$ in their full text: $$Q_e(C)= \frac{n_C}{N}.\sum_{i \in C}-log(\frac{n_i}{N})$$, where $N$ is the total number of papers in the reference corpus.
%In the following case study, we will take as a reference corpus all the journal sources indexed by scirus.com, which covers more than 20 millon references in most scientific domains.
In the example presented here, clusters with null empirical quality were removed from the phylogeny and patterns in the distribution of empirical quality have been further studied. The empirical quality could also be used as a parameter to filter phylogenies so as to display most informative scientific fields.

\subsection{Qualifying clusters}
Relevance is not a binary judgment but rather lays on a continuum, potentially multidimensional, reflecting what is looked for: well-recognized domains of investigation, emergent domains, highlights on interdisciplinary domains, etc. Empirical quality is one of the index that enables to qualify identified scientific fields. We studied furthermore two other indexes that help to give meaning to science evolution.
\begin{itemize}
\item \textbf{Density.} One of the first index introduced to assess scientific fields evolution is the density of a field \cite{callon91coword}.``It characterizes the strength of the links that tie the words making up the cluster together. The stronger these links are, the more the research problems corresponding to the cluster constitute a coherent and integrated whole. It could be said that density provides a good representation of the duster's capacity to maintain itself and to develop over the course of time in the field under consideration." It is computed by: $$D(C)=\frac{1}{Card(C)}\sum_{(w,w')\in C

\alpha(w)=\frac{1}{card(C)}\sum_{w'\in C}P_{min(\alpha,\frac{1}{\alpha})}(w,w')$$
\textsl{The specificity index} indicates to what extent $w$ is specific to $C$ and is defined by:
$$I_s

\alpha(C)=\min_{w \in C}\frac{1}{2}(I_s

\alpha(w))$$ This index indicates the degree of structuration of $C$. As we will see, this index the pseudo-inclusion provides some perspective in the interpretation of science dynamics.
\end{itemize}

Along with empirical quality, these two indexes will be useful to filter science maps and focus on some particular parts of the phylogeny. Note that whereas pseudo-inclusion and density can be computed without supplementary information, empirical quality needs additional queries to a corpus database. One issue will thus be to see in what extent it is possible to use the first to indexes as proxy to estimate the empirical quality.

\subsection{Software}
We used the Words Evolution software\footnote{http://sciencemapping.com/WE} to process and visualize the phylogenies. This software is interfaced with network visualization tools like (Gephi\footnote{http://Gephi.org} or GraphViz\footnote{http://www.graphviz.org}) and clustering software (CFinder\footnote{ http://www.cfinder.org}).

\section{Results}
\subsection{Mapping network studies in biology}
We performed phylogeny reconstruction on the MedLine database focusing on research in biological and biomedical fields related with network studies. After the constitution of a database concerning a set $\mathcal{L}$ of 834 terms according to the methodology explained in 2.1, we generated maps processed on four years sliding time windows from 2007 to 1987.

As example, a detail of the map obtained on the period 2004-2007 is given in figure \ref{map2003-2007}. We extracted all directed cliques based on the terms list $\mathcal{L}$ composed of at least 4 terms. We labeled the clusters with their most generic terms and we further simplified the map by merging clusters with the same label. Nodes size is related to density of clusters, this value is averaged over the set of merged clusters if the node is made of merged clusters. The value of the link between two sets of merged clusters is the maximum value of the inter-cluster similarity between all pairs of clusters.
NaVWe filtered the map by thresholding the cliques on empirical quality and \COR{comment? par rapport a quel parametre?} so as to display a maximum of 100 labels representing the most relevant fields and put a threshold on links to keep the \COR{ more significant ones PQ strongest ones}. The figure \ref{sciencemap} displays a portion of the map of science over the period 2004-2007 \COR{redite avec ci-dessus, mais bon...}. This map is the context of network research in biology \COR{pareil} . We \COR{ Clusters were labelled with the most generic terms PQ use the labeling with most generic terms } \COR{ RIEN PQ to be understandable for a general audience}. %To get a more focused figure, we can plot only the scientific fields mentioning the term network (fig. \ref{sciencemapnetgen}).

\begin{figure}
\centering
\includegraphicswidth=5in{image/BionetMap.pdf}\
\caption{Portion of a map of the scientific fields related to network studies in biology. Clusters are labeled by their two most generic terms. Size of text and bubbles map the density of the cluster. The inset give a detail of a cluster. \tiny Visualized with Gephi.org}\label{sciencemap}
\label{map2003-2007}
\end{figure}

NaV\begin{figure}
NaV \includegraphicswidth=5in{image/MapNetworkGene.pdf}\
% \caption{Macroscopic map of the ``biological fields mentioning network studies. One can zoom into subfields as illustrated in the inset to get further information. Clusters are labelled by their two most generic terms.}\label{sciencemapnetgen}
NaV
\subsection{Phylogenetic Patterns}

We reconstructed the phylogeny of the domains related to networks studies in biology over the period 1987-2007. Releasing all constraints on the phylogeny except that we required the fields to have at least four elements and a non null empirical quality, the phylogenetic network is made of 7759 nodes.

Within this network, we observed a significant positive correlation between the pseudo-inclusion index and the empirical quality. The Pearson coefficient $r$ lays within the 95\% confidence interval $0.14;0.19$, the probability to obtain a correlation as large as the observed value by random chance being $p=4.10

{-6}$) as well as with the number of papers per cluster ($0.16<r<0.21$, $p=0$).

We categorized the fields according their position in the phylogenetic network: aborted (no father, no child), new born (no father, some children), adult (with father(s) and son(s)) and dying fields (with some father(s) but no child). Note that a cluster may belong to a different category according to the value of $\delta_0$. Scientific fields distribution regarding these categories is particularly interesting. Plotting the variations of the fields' empirical quality, density and pseudo-inclusion indexes against this categorization (fig. \ref{AgeQual}), we found very clear patterns for $\delta_0 \in 0.3 0.6$: aborted, newborn and dying fields tend to have weaker indexes than adult fields, with aborted fields having slightly lower values for their indexes than newborns.

The dependency of the mean of the density, pseudo-inclusion and empirical quality indexes over the position of the fields in the phylogeny suggests trends in the ``life cycle'' of scientific fields: these indexes grow while a new field emerges, and then loose their strength when it begins to be neglected by the community. However, density and pseudo-inclusion index are completely different ways of characterizing scientific fields. On the one hand, fields with high pseudo-inclusion will usually have terms with a large spectrum of specificity and genericity, which means that they are likely to contain very specific terms with few occurrences. These terms have a high probability to be new concepts of in fields or new objects of study. Their presence in the phylogeny will then often be correlated with high rate of branching processes. On the other hand, fields with a high density index rather correspond to well structured scientific domains with a priori lower rate of conceptual renewal.

Further studies based on different databases will confirm or not the relevance of these are general patterns in the study of science evolution. However, these regularities open perspectives for the detection of emergent or dying fields on the basis of some indexes computed on co-occurrence data.

\begin{figure}\center
\includegraphicswidth = 0.31 \linewidth{image/AgeQual.pdf}
\includegraphicswidth = 0.31 \linewidth{image/AgeDensity.pdf}
\includegraphicswidth = 0.31 \linewidth{image/AgeInclus.pdf}
\caption{\COR{c'est quoi en ordonnees?}}\label{AgeQual}
\end{figure}


Beside, the fact that aborted fields tend to be of lower quality suggests a methodology to adjust optimally $\delta_0$ in order to have the most informative phylogeny (in the sense of the empirical quality). Indeed, the ratio between the mean quality of fields belonging to the phylogeny and the mean quality of aborted fields is always higher than 1, and reaches its maximum around the value $\delta_c=0.33$. For this value, connected fields in the phylogeny \textit{i.e. } fields that have at least one father or one son are, on average, almost twice as informative as aborted fields.

% \begin{figure}\center
%\includegraphicswidth = 0.5 \linewidthNaV{image/RatioQ.pdf}
NaV\end{figure}

Inheritance patterns can be studied by classifying fields according to their number of sons in the phylogenetic network. While most fields have less than 2 sons, with 44\% having only one successor, almost 14\% have at least 3 children (cf. fig.~\ref{Outstats}-a). Again, the distribution of the different indexes in function of the number of children is very instructive. Figure~\ref{outvs} shows that, on average, the maximum of density (cf. fig.~\ref{Outstats}-b) and pseudo-inclusion is obtain for fields that have only one son. Again, this observation holds for a large range of $\delta_0$.
The synthesis of all these results suggests that relatively young branches of science are generally bushy with fields having lots of children. This corresponds to an intense exploration of new directions of research. Older fields will generally have a much more linear evolution with a lower rate of conceptual change.
This pattern can clearly be observed on figure~\ref{fullphylo} that represents, for $\delta_0=\delta_c$, the subpart of the phylogenetic network composed of fields with highest empirical quality and at least four terms. Recent branches have also been removed to meet editorial constraints. We can also notice that there is much more hybridation between scientific fields in the domain of formal methods and tools than in the branches corresponding to topics in biology. This transversal domain is also over-represented due to the fact that the targeted thematic is itself a transversal methodology.

\begin{figure}\center
\includegraphicswidth = 0.8 \linewidth
{image/Outstats.pdf}
\caption{\COR{dire quelle valeur de $\delta_0$ a ete choisie pour la figure a, ca change j'imagine?}
\COR{je separerais ces deux figures pour mettre b avec ces copines de la figure 6}}\label{Outstats}
\end{figure}


\begin{figure}\center
\includegraphicswidth = 0.45 \linewidth
{image/OutPaper.pdf}
\includegraphicswidth = 0.45 \linewidth
{image/OutIncl.pdf}
\caption{\COR{j'ai ramene ces deux figures ici} \COR{c'est bien l'empircial quality la figure OutPaper?}}\label{outvs}
\end{figure}

Details of the phylogeny are also very informative. Figure \ref{Cancer} represents the fields of more than five terms for which at least one term contains the words ``cancer" or ``tumor". On this partial phylogeny of this thematic, we can clearly see three distinct sets of branches with very different characteristics. Two sets are quite bushy and deals with \emph{cancer }and \emph{DNA} issues on one side, \emph{cancer, tumor and proliferation} issues on the other side. They appear to have increased their interactions these last years around the concepts of \emph{apoptosis}, \emph{suppressor} and \emph{cell cycle}. The third set has very linear branches and is related to the relations between \emph{tumor} and the \emph{immune system}. These three sets are also quite distinct in terms of the range of their density and pseudo-inclusion indexes. Whereas the bushy branches tend to have a higher pseudo-inclusion index than the linear ones, revealing a higher rate of conceptual renewal, they also have a lower density index, indicating that they should be more recent. The study of the evolution of the pseudo-inclusion index along these branches reveals that this index is increasing along most of the branches although its growth rate is decreasing with time. When \COR{relaxing PQ releasing} the constraints on the empirical quality threshold and on the number of terms in clusters, these characteristics regarding the three sets of branches are preserved, although the branches appear to be older than they appear in this partial phylogeny, the upper-part of the phylogeny having been pruned in the thresholding \COR{process}.

\begin{figure}
\center
\includegraphicswidth = .8 \linewidth
{image/PhyloFull.jpg}
\caption{Extract of the full phylogeny of domains related to networks studies in biology and medical research ($\sim$1400 clusters). We kept fields \COR{made} of more than four terms, set a threshold on the empirical quality and removed shortest branches for editorial purposes. The color map\COR{s} the pseudo-inclusion index of the fields.}\label{fullphylo}
\end{figure}

\begin{figure}
\center
\includegraphicswidth =1 \linewidth
{image/PhyloCancer2.pdf}
\caption{Part of sub-phylogenetic network related \COR{to PQ with} cancer studies. The size of a square is proportional to the pseudo-inclusion index of the field and the color, from blue to red maps the growth rate of this index. Note that the pseudo-inclusion index along the branches is increasing along most of the branches although its growth rate is decreasing with time. Fields are labeled with their most generic term, except for the beginning of a branch or for the most recent period, where all terms are displayed. The labels of inter-period arrows indicate which terms have been lost or gained between two periods.}\label{Cancer}
\end{figure}


\subsection{Discussion}
Seminal work of Callon et. al. \cite{callon91coword} \COR{was PQ constitutes} the first \COR{attempt PQ tentative} to quantify the evolution \COR{of} scientific fields through co-word analysis, \COR{monitoring inter alia the evolution of the density of clusters PQ looking among other at the evolution of the density of clusters}. Our work proposes the first automated methods \COR{for reconstructing PQ to reconstruct} the entire phylogeny of a domain of science and is clearly in line with their approach. We extended their work in several ways, trying to go beyond most classical limitations of scientometrics that have been expressed \COR{hitherto PQ since}:
\begin{enumerate}
\item \textbf{Coverage: } Our methods can cover the largest bibliographic database available. Nowadays, online publishers cover between 30 and 40 million articles, which represent a significant part of worldwide scientific literature. We gave an example on a case study based on MedLine (14M papers) covering most medical and biological research.
\item \textbf{Ambiguity: } Contrary to \cite{callon91coword} and most subsequent works, we used overlapping clustering algorithms in order to ensure that we can handle ambiguity in terms use and \COR{je comprends pas bien les false negative - avoid noise/mistakes in attributing terms to its clusters?} avoid false negatives in terms classification in clusters,
\item \textbf{Asymmetry and multi-level mapping: } Following previous work \cite{chava:scien}, we based our clustering algorithm on an asymmetric proximity measure in order to fully reflect the organization of science into domains and sub-domains. This asymmetry enables to highlight the internal structure of clusters allowing automatic labeling \cite{coint08multi}. This entails a multi-level mapping that proposes multiple view points on the phylogeny \COR{according to PQ in function of} the required degree of specificity. We also introduced \COR{a measure of PQ an indicator of} fields structuration, the \textit{pseudo-inclusion index}, that is based on this new \COR{proximity} measure and we showed that \COR{ the pseudo-inclusion index appears to be very informative when assessing PQ this index is an important indicator of} the evolution of a fields of research.
\item \textbf{Validation: } Complementary to \cite{Heal86AnExp} who \COR{suggested to validate science maps with both ``endogeneous and ``exogeneous criteria PQ proposed to the integration of both "internal" and "external" validation of science maps}, we proposed an \textit{empirical validation} (confrontation with real data) that could be articulated with the two previous ones in future work. We defined the \textit{empirical quality} of a cluster as the degree of its \PQ{ quantity of information?? PQ informativeness} relatively to the associated literature and showed that the pseudo-inclusion index was positively correlated with the empirical quality. The density, on the other hand, was only weakly correlated.
\item \textbf{Dynamics: } Our study took advantage of the availability of diachronic data to reconstruct the phylogeny of scientific fields, and takes into account multiple filiations, contrary to what could have been done in other related fields like social group evolution \cite{Palla:2007p229}. This structure revealed strong \COR{and robust} patterns \COR{which appear to highlight strong regularities PQ that seem to express strong regularities} in science evolution.
\end{enumerate}

This approach opens new promising perspectives both from theoretical and applicative point\COR{s} of view. While we tried to show that these approaches of science dynamics reconstruction are close to be able to corroborate of falsify theories in epistemology and science studies, we can also expect they will considerably renew the way we interact with science \COR{especially when navigating large-scale electronic databases PQ and in particular within electronic archives}. Moreover, the methodology presented here is not specific to scientific corpora and \COR{may PQ can} be applied to a wide range of co-occurrence data \COR{like, online communities} from Web 2.0, patents database, \COR{folksonomies PQ tags}, \COR{ or even experimental data like} micro-array data\COR{,etc. PQ and so on.}


\subsection*{Acknowledgements}
The authors warmly thank Jean-Paul Gaudilli\`{e}re et Christophe Bonneuil for their help in selecting the list of terms. These researches have been supported by the FP7 PATRES project and the Paris Île-de-France Complex Systems Institute and the École Polytechnique and project.


\MISSING{Tes remerciements}

\bibliographystyle{plain}
%\bibliographystyle{authordate2}
\bibliography{phylogeny}


\end{document}

\section{Remarques}
We can define two quantities characterizing a
word $w$ within this cluster : the genericity index $i_g$ and
the specificity index $i_s$ defined as follows:
\begin{description}
\item\textsl{The genericity index} defines to what extent the cluster $C$ is a good neighborhood for the word $w$ with respect to the paradigmatic proximity $\Proxm$. When $\alpha<1$ It is the mean of $w$ ``out-paradigmatic proximity" and is defined by : $$I_g(w)=\frac{1}{card(C)}\sum_{w'\in
C}P_{min(\alpha,\frac{1}{\alpha})}(w,w')$$
\item\textsl{The specificity index} provides the extent to which the word $w$ is specific to the
cluster
$C$ with respect to the paradigmatic
proximity $\Proxm$ considered (\textsl{i.e.} is $w$ relevant for the terms in $C$ ?). It is the mean of $w$ ``in-paradigmatic proximity" -
from term $w'\in C$ to $w$ - and is defined as : $$I_s(w)=\frac{1}{card(C)}\sum_{w'\in
C}P_{min(\alpha,\frac{1}{\alpha})}(w',w)$$. We will note that since $\Proxm(w',w)=P_{\frac{1}{\alpha}}(w,w')$, this index can be rewritten : $$I_s(w)=\frac{1}{card(C)}\sum_{w'\in
C}P_{max(\alpha,\frac{1}{\alpha})}(w,w')$$
\end{description}


\bigskip
\bigskip
$I_\subset(C_a)=\frac{1}{2}\frac{1}{\mid C_a \mid

t}{n_i

{\alpha}(\frac{n_{ij}

t})