Complex networks and public funding: the case of the 20072013 Italian program
 Stefano Nicotri^{1}Email author,
 Eufemia Tinelli^{2, 3},
 Nicola Amoroso^{1, 2},
 Elena Garuccio^{4} and
 Roberto Bellotti^{1, 2}
DOI: 10.1140/epjds/s136880150047z
© Nicotri et al. 2015
Received: 27 October 2014
Accepted: 23 June 2015
Published: 9 July 2015
Abstract
In this paper we apply techniques of complex network analysis to data sources representing public funding programs and discuss the importance of the considered indicators for program evaluation. Starting from the Open Data repository of the 20072013 Italian Program Programma Operativo Nazionale ‘Ricerca e Competitività’ (PON R&C), we build a set of data models and perform network analysis over them. We discuss the obtained experimental results outlining interesting new perspectives that emerge from the application of the proposed methods to the socioeconomical evaluation of funded programs.
Keywords
public funding open data complex networks program evaluation1 Introduction
Since the last years of the past century, the importance of basing policies on evidence, data, and analysis has quickly spread all over the world. The EvidenceBased Policy movement [1–5] has grown enormously, and mainly all public administrations are now focused on maximising utility and show a pragmatic problemsolving approach to socioeconomical issues [6]. In this respect, the evaluation of public funding programs is a field of great interest for policymakers and economists. Politicians and technicians need to estimate the impact that funding has on life and society, in order to address future programs and to modify their decisions. Many standard and advanced statistical methods are commonly used for this purpose, such as linear/nonlinear regressions, Bayesian inference, machine learning, data mining, and so on. In this paper we suggest new indicators, coming from network analysis, that can help underlining in a quantitative way important effects that are not usually considered, being them outside the domain of investigation of standard statistical tools. This does certainly not mean that program evaluation cannot be performed without including network analysis, but that valuable insight about public funding programs could hopefully be inferred from such techniques, in order to help increasing objectivity of the extracted results. Recently, a growing interest towards complex network analysis applied to evaluation can be seen both in literature [7–12] and institutional reports [13]. The indicators we suggest can be used by experts in program evaluation for their analyses, giving them the opportunity of considering and quantitatively measuring important features of the funding programs, such as relations between the actors involved in them. Social network analysis is a particularly suitable tool to extract information about relations among the different components of a system. Investigating the relations between the actors participating to a program could be of interest, since can e.g. show structural contradictions in the organisation of the different levels involved [10]. Considering the set of projects, research institutions and enterprises that participate to a funding program as a complex dynamical system, it is possible to identify underlying network structures simply defining the edges according to some relations among the components that are of interest for the evaluator. Once the network is constructed, global and local properties can be evaluated and discussed.
From a data collection perspective, the proposed analysis can profit from current emerging technologies and precise guidelines of European governmental institutions to support initiatives such as Smart Cities & Communities ^{1} and their corelated action goals (Urban and Citizen App, eGovernment, eDemocracy and so on). All this initiatives have produced a large number of freely available datasets containing information, collected by national governments, which third parties are encouraged to use for their scope, analyse and republish as they wish, without restrictions from any copyright. Recently, Open Government Data (OGD) is emerging as a major movement in knowledge sharing. It promotes transparency and accountability, enables collaboration among stakeholders, encourages novel socioeconomic activities and growing of the socalled network economy. Starting from the idea that without sharing information it is not possible to establish a culture of collaboration and participation among the relevant stakeholders, the Linked Open Data (LOD) [14]. Movement, which provides existing data in a machinereadable format, has gained large importance over the last years. From a such perspective, LOD facilitates innovation and knowledge creation from interlinked data, but it also introduces a level of complexity for information management and integration. Considering a good tradeoff between data expressiveness and computational cost for data analysis, we have selected only Open Data repository without linked data and RDF^{2} triples. Despite the main aim of such movement of reaching the largest possible portion of users, our investigation has outlined that such datasets are usually of heterogeneous quality and size, and that their analysis requires efforts in a preprocessing phase composed of typical ETL (ExtractTransformLoad) [15] and data cleaning procedures. It is worth mentioning that problems are commonly encountered while using network analysis for evaluation, like the concern about anonymity of nonaggregated data (and eventual anonymisation), or the fact that making results public usually interferes with the structure of the network itself [9]. These kind of problems are mitigated by using Open Data, since they are public ‘by construction’.
The paper is organised as follows: in Section 2 we introduce the steps composing the schema of the overall analysis process. In Section 3 we describe the structure of the open data repository of the 20072013 Italian Program Programma Operativo Nazionale ‘Ricerca e Competitività’ (PON R&C), in order to keep the paper selfcontained, and introduce the data model for network analysis; in Section 4 we present features and properties of the analysed network. In order to better discuss the experimental results, we distinguish among local properties, global properties and community structure. Section 5 close the paper.
2 Methodology

processing of data sources (grey blocks),

finding analysis models and metrics (blue blocks).
3 PON R&C: from datasets to data models

Projects  10,104 tuples with 52 attributes describing project information about program references, activities, textual description of project scope and objectives, details about partners and so on;

Locations  11,390 tuples with 8 attributes describing details about geographical localisation of project partners;

Budgets  5,670 tuples with 13 attributes describing details about amount and state of project funding
Sketch of the structure of original files from PON R&C
Project  

UPC  title  smart_cities  social_innovation  …  healthcare  FC  name 
PON04a2_A  PRISMA  1  0  …  1  84001850589  INFN 
PON04a2_A  PRISMA  1  0  …  1  84001850589  INFN 
PON04a2_A  PRISMA  1  0  …  1  80002170720  UNIBA 
…  …  …  …  …  …  …  … 
Location  

UPC  FC  name  kind  region  … 
PON04a2_A  84001850589  I.N.F.N.  PRI  Apulia  … 
PON04a2_A  80002170720  UNIBA  University  Apulia  … 
…  …  …  …  …  … 
Budget  

UPC  FC  name  total_cost  total_funded  … 
PON04a2_A  84001850589  INFN  Apulia  2231915.7  1785532.57  … 
PON04a2_A  80002170720  University of Bari  2052539  1642031.2  … 
…  …  …  …  …  … 

PartnertoPartner  the distinct couples of partners involved in the same project;

ProjecttoProject  the distinct couples of projects having at least one partner in common, together with the calculated number of shared distinct partners;

ProjecttoPartner  the set of distinct involved partners for each project;

PartnertoFunding  the funding, for each beneficiary, calculated considering all the PON R&C projects it is involved in;

BeneficiarytoBeneficiary  the distinct couples of beneficiaries having at least one project in common, together with the calculated number of shared distinct projects.
4 Network analysis
4.1 Local properties
Properties of vertices evaluated in the present analysis include degree centrality, betweenness centrality, closeness centrality, eccentricity, eigenvector centrality, radiality centrality and PageRank centrality, based on the Google PageRank algorithm [23].
The highest values of all centralities is found in correspondence to public research institutions, like universities and specific research centres. In particular, the Italian National Research Centre ( CNR ) shows the best values for all the indicators. It is worth saying that it is a peculiar vertex of the network, being composed of 104 institutes spread over geographically distributed sites (in all the biggest cities in Italy), and covering a large spectrum of activities in many fields, from pure research, to applied disciplines. Probably, it would be better to split such vertex and consider each site, or department, separately, but the dataset does not contain such details. On the contrary, dividing the CNR in many entities would in a certain sense spoil its central nature in the Italian panorama. Resolving this controversy is interesting, but is over the purposes of the present paper, and is left for a future work, when more detailed Open Data will be available. Apart from cases like the one described, the central role of public research institution for the network structure is clear from all the centralities.
Degree centralities are discussed in detail in the next Section 4.2, since global properties of the network can be inferred from the distribution of such quantities, despite being them local in nature.
Betweenness [24] measures the importance of a node for traffic of information across the network. Large betweenness centrality of a vertex indicates that many shortest paths between couples of other vertices pass through that node. The relevance of this quantity for program evaluation stands in the possibility of assessing the role of institutions/enterprises for the eventual aggregation of ‘far’ nodes. For example, a policymaker interested in promoting a program aimed at aggregating and consolidating the productive system of a region should pay attention not to spoil the edge betweenness of the network of relations between the actors involved in the program.
Closeness centrality indicates whether a node is at a short average distance from every other reachable vertex, with higher closeness meaning shorter distance. A variant is radiality centrality, which gives higher weight to the neighbourhood of the node. From the social/economical point of view these quantities give indication about how easily an institution/enterprise can connect to all the other members of the network (and, so, of the productive system). For example, an enterprise with high closeness centrality could be the right promoter for initiatives like the creation of technological districts, associations or lobbies. Exploiting the information contained in this quantity, a policymaker could more easily head the productive system in the desired direction with focused regulatory interventions.
Eccentricity is the maximum value of the distances between a node and any other node in the network. It gives an idea of how central a vertex is within the network, with smallest values corresponding to more central nodes.
4.2 Global properties
The first property analysed here is the degree distribution of the vertices, i.e. the frequencies of the degree centralities described in the previous Section 4.1. The importance of such distribution stands in the possibility of inferring from it information about the topology of the graph, and in particular to understand if the network is scalefree [28]. The property of being scalefree is shared by many real networks, showing power lawshaped degree distributions \(P(k)=A k^{\gamma}\), with exponents usually varying in the range \(2<\gamma<3\), which have the same form at all scales.
This is of particular interest since power laws are commonly associated with secondorder phase transitions in dynamical systems. Phase transitions in complex networks represent an interesting research field [29, 30], but the graph considered here is static, so no considerations can be made in this respect. Anyway, this is an interesting perspective for a future work, in which dynamics can be taken into account.
Scalefree networks have an inhomogeneous degree distribution, with many nodes having more connections than the average (hubs). The hubs follow a hierarchy, in which large ones are connected to smaller ones, which are themselves connected to even smaller ones, and so on. This feature makes the network robust against casual failures, since the removal of a random vertex would not systematically affect the main hubs, and connectedness would not be spoiled. Hence, scalefree graphs are a desirable result for policymakers interested in generating a solid network of relationships between productive actors on the territory. Apart from being a strong point for networks, hubs also represent a weakness, since their systematic removal would quickly destroy the network. The property of being scalefree is an important point to be taken into account for an evaluator, as we will show below, in order to monitor and evaluate the results of funding programs. Moreover, it suggests to decision makers that effort should be put in promoting funding program which hubs can profit from.
Moments of the degree distribution of the PON R&C network
Moment  Value 

Expected value  〈k〉 = 12.661 
Mode  M = 6 
Median  6<m<7 
Standard deviation  σ = 23.781 
Skewness  s = 8.876 
Kurtosis  κ = 113.882 
Once assessed such powerlaw nature, it is interesting to identify the main hubs. In the PON R&C network considered here, hubs are public research centres, and this represents a strong point for the relationships of the involved productive system. In fact, it is natural, in the lifecycle of a productive system, that some enterprises rise while others fall, resulting, in the language of networks, in the random removal of vertices described above. Anyway, as previously stated, the random removal of vertices from a scalefree network does not spoil connectivity, which happens with the systematical removal of the main hubs instead. In this case, it is unlikely that one of the main hubs, identified here with large public research centres, could disappear, since this would mean e.g. the closure of a large public university, a quite rare event. This picture was partly expected, since in many cases it was mandatory to involve public research institutions in the TSAs. Nevertheless it still represents a strong indication for a decision maker, suggesting that it is ‘safer’ including public research in a future program, since it is the easiest way to keep a solid relationship network within the productive system.
Another way of assessing if a network is scalefree consists in evaluating the distribution of local clustering coefficients, i.e. the number of edges connecting the neighbours of each vertex v, divided by the number of edges of a complete graph of the same cardinality of the neighbourhood of v [33]. The PON R&C network represents a special case, in which local clustering coefficients are less important, the majority of them being close to 1 by construction. In fact, since the graph is a union of complete graphs, it is likely that the neighbourhood of a vertex is fully connected, implying the closeness to one of the local clustering coefficient. The global clustering coefficient \(\mathcal{C}\), i.e. the fraction of paths of length two that are closed (over all paths of length two), is much more significant instead, and it takes a small value \(\mathcal{C}=0.215\) for the giant component, meaning that the network is not strongly clustered. From the political and sociological point of view, this is an interesting point, since the network is made by ‘scattered’ relationships, despite being composed of ‘closed’ TSAs.
Other important features that can guide the policymaker in evaluating the effects of the program or planning future ones are vertex connectivity \(V_{c}\) and edge connectivity \(E_{c}\), i.e. the smallest number of vertices or edges to be removed in order to disconnect the graph, respectively. For the case under examination such quantities take value \(V_{c}=1\) and \(E_{c}=1\), meaning that the removal of a single node or edge can be catastrophic for network connectivity. Identifying and monitoring such nodes/edges can be very important in case of low values of such parameters, in order to keep the network of relations tightly connected.
Another important property of scalefree networks is that they are small world networks [34]. This means that relatively short paths exist between any two nodes (with respect to the large size of the graph), with an average shortest path length^{6} \(L\sim \mathcal{ O}(\log N)\), N being the total number of vertices. This is due to the existence of links between vertices belonging to farther parts of the graph, having the role of connecting them and reducing distances to few hops. Usually, in scalefree networks such vertices are the hubs and the smallworld property is enhanced when \(2<\gamma<3\) where \(L\sim\mathcal { O}(\log\log N)\) (while it is \(L\sim\mathcal{ O}(\log N)\) when \(\gamma >3\)) [35]. For the PON R&C network, \(\gamma=1.998\pm0.040\), \(L=2.532\), \(\log N=6.645\) and \(\log\log N=1.889\), so the small world property is enhanced, as expected when the vertex distribution follows a powerlaw with \(2<\gamma<3\). Again, this cannot be considered a smoking gun proving that the network is scalefree, but just another indication in addition to the ones mentioned above.
Other global features of the network are the radius \(\mathcal{ R}\) and diameter \(\mathcal{ D}\) of the graph, defined as the minimum and the maximum eccentricity of all vertices, respectively, the eccentricity being the longest shortest path from a source node to every other vertex in the graph. For the PON R&C network \(\mathcal{ D}=5\) and \(\mathcal{ R}=3\), meaning that no vertex is more than 5 hops far from any other node, and that the farthest destination is never closer than 3 hops from any source. From the point of view of program evaluation, this means that PON R&C has been successful in creating (or intersecting) a network of close relationships between the funded actors. Being interested in promoting such a relationship network while defining the program, these could be good ex post indicators of the goodness of the obtained results.
The centre of the graph^{7} is shown in Figure 4. It includes public research centres like CNR (which is also the main hub) and ENEA, all the major Universities involved in the program (Bari, Calabria, Catanzaro, Foggia, Naples, Palermo, Salento, Salerno), private research centres like CETMA, and also some large private enterprises like Avio S.p.A., Engeneering S.p.A., IBM, SELEX S.p.A., and EXEURA S.r.l. This is a strong indication that the network of funded projects gravitates around large poles involving research centres (public and private), which turn out to have a key role in aggregating entities. This can also be an explanation for the scalefree property of the graph, since preferential attachment is known to be a generating mechanism for this kind of networks [36–38], in which nodes prefer to link to vertices with high degree. It is reasonable to imagine that many small actors prefer forming TSAs including large research organisations, which are usually able to get more funds, rather than form TSAs between themselves. From the point of view of social networks and relationships, it is particularly interesting to study such feature sidebyside with assortativity [39], which indicates whether nodes of the graph tend to connect with their connectivity peers (vertices with similar degree) or not. In the first case the network is said to be assortative, while in the second case it is antiassortative. This feature is quantitatively measured through the assortativity coefficient r, whose range is \(1\leqslant r\leqslant1\), \(r=1\) (−1) meaning a perfectly (anti)assortative graph and \(r=0\) indicating no particular preference for the majority of the nodes. In the present network, \(r=0.173\), meaning that the graph is slightly antiassortative. This means that the productive system funded by such program has a little tendency not to form lobbies among important actors, but to associate strongly connected hubs to smaller and less connected enterprises/institutions. From the socioeconomical point of view, it seems reasonable that small enterprises turn to larger ones or to big research centres to benefit from sharing and collaborations. This is an interesting result, since most social networks show assortative mixing by node degree [40], and it also has some implications on the topology of the network. First, antiassortative networks are more susceptible to the removal of highdegree nodes (here represented by universities and research centres), which is an indication for the policymaker of the importance that public research has in the productive system, and of the possible disruptive effect of its underestimation. Second, in antiassortative networks epidemics span to larger portions of the nodes than in similar assortative ones. This means that being antiassortative is preferable for the spreading of knowledge and knowhow in the productive system, making it more efficient. It is worth noting that in a recent work [12] a social network similar to the one studied here, concerning the funding of FP7 (Seventh Framework Programme) European research projects, has been found to be antiassortative as well, and conclusions close to the ones put foward here are drawn. This could be an indication of some structural feature shared by graphs constructed starting from public funding programs, and we plan to further investigate this point in a future work. Lastly, link efficiency is a measure of traffic capacity within the network, representing how efficiently information can be transmitted along the graph. This parameter takes the very high value \(\xi=0.999\) in the PON R&C network, which is a strong indicator of robustness for the relations between vertices, especially in a graph with small density \(\rho=0.017\) as the one under examination.
Global properties of the PON R&C network
Property  Value 

Radius  \(\mathcal{ R}=3\) 
Diameter  \(\mathcal{ D}=5\) 
Density  ρ = 0.017 
Global clustering coefficient  \(\mathcal{ C}=0.215\) 
Vertex connectivity  \(V_{c}=1\) 
Edge connectivity  \(E_{c}=1\) 
Average shortest path length  L = 2.532 (\(\mathcal{ O}(\log\log N)\)) 
Link efficiency  ξ = 0.999 
Assortativity coefficient  r = −0.173 
4.3 Community structure
5 Conclusions and perspectives
In this paper we have used techniques borrowed from complex network analysis to evaluate the effects of a public funding program on the relations between the funded ‘actors’. The PON R&C program involves a large number of actors and is extended over a period of seven years (20072013). The dataset is completely made of Open Data, and we have shown a way of concretely using information made available by Governments, in the spirit promoted by current global guidelines. We have described the full process of knowledge management, from data acquisition, to cleaning, model building and querying. The whole chain is data oriented and is focused in retaining every piece of available information, in order for the output of the analysis to show the highest possible accuracy.
The processed PON R&C data have been used for complex network analysis, and the resulting network has 769 vertices and 4,868 edges. We have evaluated the most important centralities for each node, plus some relevant global properties of the graph. The outcome of our analysis shows a dominant role of public (and, but less importantly, also private) research institutions within the Italian productive panorama, at least for the part portrayed by the program under examination. Universities and research centres play the role of the ‘glue’ for this particular program, i.e. they are responsible of the connectedness of the network, and a failure involving some of them would be disruptive for the whole productive system. This picture was partly expected, due to the way the program has been realised, since in many cases it was mandatory to involve public research institutions in the projects. Nevertheless it can be useful to use such result as an expost indicator. Moreover, we have found that the PON R&C network is antiassortative, an unusual feature of social networks, shared only with other cases involving FP7 public program of European research funding, preferable than the most common assortative mixing for the spreading of knowledge and knowhow in the productive system and for its efficiency.
We have shown that social network analysis can produce useful results for program evaluators, since it allows to consider, in a quantitative fashion, very important aspects that are usually ignored, due to common difficulties in quantifying them. A mathematical description of the structure of relations generated by a national funding program is the example shown in this work. Indicators such as vertex and edge centralities have been used to generate a ranking between the main actors involved in the program, as shown in Figures 5 and 6. We hope that the procedure and the results described in the present paper can help opening interesting new perspectives from new indicators for decision and policy makers and program evaluators, providing them with an useful tool.
Many possibilities are left open by the present work. First of all, as mentioned in Section 3, around ∼78% of the total budget is concentrated into ∼10% of the funded projects. This suggests that introducing information about the financial aspect into the network analysis could be interesting and meaningful for the evaluator. This could be done in many different ways, from simple visualisation techniques in sociograms, like relating the size of the nodes to the total funding received by the actor, to more refined analysis, like defining weighted networks with weights related to budgets. We plan to investigate these directions in future works. Other planned future activities include the introduction of dynamical networks, involving the study of temporal series, refining of network analysis techniques, e.g. by introducing different kinds of weighted networks and related features, and generalising the analysis extending it to different levels [7, 12]. Moreover, the expected improvement in quality of Open Data (for example increasing the level of detail within public research institution, e.g. discriminating among single departments rather than universities) could lead to many interesting improvements of the present analysis.
The path from Open Data to Linked Open Data was firstly introduced by Sir Tim BernersLee in the 5 Stars Model at the Gov 2.0 Expo in Washington DC in 2010, where costs and benefits for both publishers and consumers of LOD are explained.
Representing the probability that a traveler randomly navigating the network continues doing it at a given point.
Declarations
Acknowledgements
We thank N Coniglio for valuable suggestions and discussions. SN thanks R Anglani and V Mariani for discussions. SN, and NA acknowledge funding by the Italian MIUR grant PON PRISMA Cod. PON04a2_A. ET acknowledges partial funding by the Italian MIUR grant PON PRISMA Cod. PON04a2_A.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Smith AFM (1996) Mad cows and ecstasy: chance and choice in an evidencebased society. J R Stat Soc, Ser A, Stat Soc 159(3):367383. doi:10.2307/2983324 View ArticleMATHGoogle Scholar
 Nutley SM, Davies HTO, Smith PC (2000) What works?: evidencebased policy and practice in public services. Policy Press Google Scholar
 Solesbury W, Policy EUCfEB, Practice (2001) Evidence based policy: whence it came and where it’s going. ESRC UK Centre for Evidence Based Policy and Practice Google Scholar
 Young K, Ashby D, Boaz A, Grayson L (2002) Social science and the evidencebased policy movement. Soc Policy Soc 1:215224. doi:10.1017/S1474746402003068 View ArticleGoogle Scholar
 Kay A (2011) Evidencebased policymaking: the elusive search for rational public administration. Aust J Public Adm 70(3):236245. doi:10.1111/j.14678500.2011.00728.x View ArticleMathSciNetGoogle Scholar
 Legrand T (2012) Overseas and over here: policy transfer and evidencebased policymaking. Policy Stud J 33(4):329348. doi:10.1080/01442872.2012.695945 View ArticleMathSciNetGoogle Scholar
 Provan KG, Milward HB (2001) Do networks really work? A framework for evaluating publicsector organizational networks. Public Adm Rev 61(4):414423. doi:10.1111/00333352.00045 View ArticleGoogle Scholar
 Ferlie E, Lynn LE, Pollitt C, Klijn EH (2007) Networks and interorganizational management: challenging, steering, evaluation, and the role of public actors in public management. Oxford University Press, London http://www.oxfordhandbooks.com/10.1093/oxfordhb/9780199226443.001.0001/oxfordhb9780199226443e12 Google Scholar
 Penuel WR, Sussex W, Korbak C, Hoadley C (2006) Investigating the potential of using social network analysis in educational evaluation. Amer J Eval 27(4):437451. doi:10.1177/1098214006294307 View ArticleGoogle Scholar
 Horelli L (2009) Network evaluation from the everyday life perspective: a tool for capacitybuilding and voice. Evaluation 15(2):205223. doi:10.1177/1356389008101969 View ArticleGoogle Scholar
 Ploszaj A (2011) Networks in evaluation. In: Olejniczak K, Kozak M, Bienias S (eds) Evaluating the effects of regional interventions. A look beyond current structural funds practice. MRR, Warszawa Google Scholar
 Tsouchnika M, Argyrakis P (2014) Network of participants in European research: accepted versus rejected proposals. Eur Phys J B 87(12):292. doi:10.1140/epjb/e2014504504 View ArticleGoogle Scholar
 Regione Puglia, Area Politiche per il Lavoro Sviluppo e Innovazione Servizio ricerca Industriale e Innovazione, InnovaPuglia S.p.A. (2014) Agenda Digitale Puglia 2020. Bollettino Ufficiale della Regione Puglia (BURP) 128:3342333502
 Bizer C, Heath T, BernersLee T (2009) Linked datathe story so far. Int J Semantic Web Inf Syst 5(3):122 View ArticleGoogle Scholar
 Lenzerini M, Vassiliou Y, Vassiliadis P, Jarke M (2003) Fundamentals of data warehouses. Springer, Berlin Google Scholar
 Abiteboul S, Hull R, Vianu V (1995) Foundations of databases, vol 8. AddisonWesley, Reading MATHGoogle Scholar
 Janowicz K, Hitzler P, Adams B, Kolas D, Vardeman C (2014) Five stars of linked data vocabulary use. Semant Web 5(3):173176 Google Scholar
 Newman MEJ (2001) Scientific collaboration networks. I. Network construction and fundamental results. Phys Rev E 64:016131. doi:10.1103/PhysRevE.64.016131 View ArticleGoogle Scholar
 Newman MEJ (2001) Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys Rev E 64:016132. doi:10.1103/PhysRevE.64.016132 View ArticleGoogle Scholar
 Olejniczak K, Bienias S, Kozak M (2012) Evaluating the effects of regional interventions. A look beyond current structural funds practice. Ministry of Regional Development, Republic of Poland Google Scholar
 Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and dynamics. Phys Rep 424(45):175308. doi:10.1016/j.physrep.2005.10.009 View ArticleMathSciNetGoogle Scholar
 Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:4797. doi:10.1103/RevModPhys.74.47 View ArticleMATHGoogle Scholar
 Brin S, Page L (1998) The anatomy of a largescale hypertextual Web search engine. Comput Netw ISDN Syst 30(17):107117. Proceedings of the Seventh International World Wide Web Conference. doi:10.1016/S01697552(98)00110X View ArticleGoogle Scholar
 Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 40(1):3541 View ArticleGoogle Scholar
 Page L (2001) Method for node ranking in a linked database. Google patents. US Patent 6,285,999. http://www.google.com/patents/US6285999
 Page L (2011) Annotating links in a document based on the ranks of documents pointed to by the links. Google patents. US Patent 7,908,277. http://www.google.com/patents/US7908277
 Page L (2014) Scoring documents in a linked database. Google patents. US Patent 8,725,726. http://www.google.com/patents/US8725726
 Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509512. doi:10.1126/science.286.5439.509 View ArticleMathSciNetGoogle Scholar
 Albert R, Barabási AL (2000) Topology of evolving networks: local events and universality. Phys Rev Lett 85:52345237. doi:10.1103/PhysRevLett.85.5234 View ArticleGoogle Scholar
 Iglói F, Turban L (2002) First and secondorder phase transitions in scalefree networks. Phys Rev E 66:036140. doi:10.1103/PhysRevE.66.036140 View ArticleGoogle Scholar
 Avriel M (2003) Nonlinear programming: analysis and methods. Dover books on computer science series. Dover Publications, New York Google Scholar
 Eadie WT, Drijard D, James FE, Roos M, Sadoulet B (1971) Statistical methods in experimental physics. NorthHolland, Amsterdam MATHGoogle Scholar
 Watts DJ, Strogatz SH (1998) Collective dynamics of ‘smallworld’ networks. Nature 393(6684):440442 View ArticleGoogle Scholar
 Milgram S (1967) The small world problem. Psychol Today 2(1):6067 MathSciNetGoogle Scholar
 Chung F, Lu L (2002) The average distances in random graphs with given expected degrees. Proc Natl Acad Sci USA 99(25):1587915882. doi:10.1073/pnas.252631999 View ArticleMATHMathSciNetGoogle Scholar
 Barabási AL, Albert R, Jeong H (1999) Meanfield theory for scalefree random networks. Physica A 272(12):173187. doi:10.1016/S03784371(99)002915 View ArticleGoogle Scholar
 Dorogovtsev SN, Mendes JFF (2002) Evolution of networks. Adv Phys 51(4):10791187. doi:10.1080/00018730110112519 View ArticleGoogle Scholar
 Krapivsky PL, Redner S, Leyvraz F (2000) Connectivity of growing random networks. Phys Rev Lett 85:46294632. doi:10.1103/PhysRevLett.85.4629 View ArticleGoogle Scholar
 Newman MEJ (2002) Assortative mixing in networks. Phys Rev Lett 89:208701. doi:10.1103/PhysRevLett.89.208701 View ArticleGoogle Scholar
 Newman MEJ (2003) Mixing patterns in networks. Phys Rev E 67:026126. doi:10.1103/PhysRevE.67.026126 View ArticleMathSciNetGoogle Scholar
 Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69:026113. doi:10.1103/PhysRevE.69.026113 View ArticleGoogle Scholar
 Newman MEJ (2006) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74:036104. doi:10.1103/PhysRevE.74.036104 View ArticleMathSciNetGoogle Scholar
 Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exp 2008(10):P10008 View ArticleGoogle Scholar
 Reichardt J, Bornholdt S (2006) Statistical mechanics of community detection. Phys Rev E 74:016110. doi:10.1103/PhysRevE.74.016110 View ArticleMathSciNetGoogle Scholar
 Lancichinetti A, Fortunato S (2012) Consensus clustering in complex networks. Sci Rep 2. doi:10.1038/srep00336