Community characterization of heterogeneous complex systems

We introduce an analytical statistical method to characterize the communities detected in heterogeneous complex systems. By posing a suitable null hypothesis, our method makes use of the hypergeometric distribution to assess the probability that a given property is over-expressed in the elements of a community with respect to all the elements of the investigated set. We apply our method to two specific complex networks, namely a network of world movies and a network of physics preprints. The characterization of the elements and of the communities is done in terms of languages and countries for the movie network and of journals and subject categories for papers. We find that our method is able to characterize clearly the identified communities. Moreover our method works well both for large and for small communities.Comment: 8 pages, 1 figure and 2 table

Correlation between the number of common answers and the geometric mean of votes of pairs of candidates belonging to different parties.

<p>An asterisk indicates statistically significant correlation at the threshold. Pairs of parties are sorted in decreasing order of correlation.</p

Summary statistics of the number of common answers present in the political profile of elected and non-elected candidates, disaggregated by party.

<p>The second column is the number of respondents in each group. is the average number of common answers within members of each party, is its standard deviation, and the -value is obtained through a t-test testing the null hypothesis that the the values of for elected and non-elected candidates come from a distribution with the same mean. An asterisk indicates a -value less than 0.001.</p

Probability mass functions of the number of common answers.

<p>Comparison between the probability mass function of the number of common answers between respondents, separately for candidates and elected (top panel) and disaggregated by gender (bottom panel). The probability mass functions are obtained by considering the answers of all 1,793 candidates who provided information about the gender, and the subset of 181 elected candidates.</p

Over-expressed (OE) and under-expressed (UE) endorsement of a specific answer (second column) given to the related question (first column) by candidates of different gender.

<p>The letter F indicates female candidates and M male candidates respectively.</p

Summary statistics of the degree of similarity of political profiles of females and males, for both candidates and elected members, disaggregated by party.

<p>The second column is the number of respondents in each group. is the average number of common answers, is its standard deviation, and the -value is obtained through a t-test testing the null hypothesis that the values of for males and females come from a distribution with the same mean. An asterisk indicates the cases when the -value is less than 0.001.</p

Scatter plot of votes VS number of common answers within each of four parties.

<p>Scatter plot of the geometric mean of the votes obtained versus the number of common answers for each pair of candidates belonging to the four parties KESK (top left), KD (top right), SDP (bottom left) and VIHR (bottom right). Yellow triangles denote pairs of non elected candidates, whereas blue circles denote pairs of candidates that have been elected.</p

Summary statistics of the inter-party average number of common answers present in the political profile of elected (E) and non-elected (NE) candidates, disaggregated by pairs of parties.

<p>The second (third) column is the number of elected (non elected) respondents in the first party (P1), while the fifth (sixth) column is the number of elected (non elected) respondents in the second party (P2). is the average number of common answers between the members of the two parties, is its standard deviation, and the -value is obtained through a t-test testing the null hypothesis that the the values of for elected and non-elected candidates come from a distribution with the same mean. An asterisk indicates a -value less than 0.001.</p

Over-expressed (OE) and under-expressed (UE) endorsement of a specific answer (second column) given to the related question (first column) by non elected (NE) or by elected (E) candidates respectively.

<p>Over-expressed (OE) and under-expressed (UE) endorsement of a specific answer (second column) given to the related question (first column) by non elected (NE) or by elected (E) candidates respectively.</p