transportation infrastructures are of crucial importance to the development of a country  and are important indicators of its economic growth .they form the backbone of tourism industry , support movement of goods and people across the country , thereby driving the national economy .roadways , railways , and airways are the major means of transport in india , although contribution of airways is small compared to that of the other two .the civil aviation sector in india has been developing steadily and is expected to grow at an accelerated pace with the shift in the policy of the government , and the addition of several low - cost private air service providers  .these private players offer competitive pricing , provide more region - oriented service , and thereby help to increase the air traffic .understanding of these transportation systems is important for reasons of policy , administration and efficiency .  during the past few years, complex network analysis has been used to study transportation systems ( railways  , airlines  ) , which are man - made infrastructures , from different aspects , apart from many other systems of diverse origins  . the world - wide airport network ( wan ) has been studied from topological as well as traffic dynamics perspective .wan has been studied  for its degree distribution and a model with constraints ( such as cost of adding links to the nodes or the limited capacity of nodes ) was proposed to account for the truncation for high - degrees in its scale - free cumulative degree distribution .it has been observed  that in wan , most connected nodes are not necessarily the most `` central '' nodes through which most of the shortest paths go .a model , incorporating the geo - political constraints , has been proposed  to explain this apparently surprising result . beyond the topological features , wanwas studied  in more detail by considering the traffic strength of interactions between nodes dynamics on it .a model was proposed  for the evolution of weighted evolving networks in an effort to understand the statistical properties of real - world systems .airport network of china ( anc )  , a network much smaller than wan , was also analyzed for its topology and traffic dynamics .its topology was found to be having small - world network features and a two - regime power - law degree distribution .we investigate india s domestic airport network which comprises air services of all major civil air service providers .we study the network for its topological features and for its traffic dynamics , by considering the intensity of interactions .first , we study the network as an `` unweighted network '' to investigate its topological properties .this network is formed by considering whether or not a pair of airports are connected by an airline . as in many other complex networks , the details of flow of informationis a crucial factor for airport networks .hence we study `` weighted network '' , in which we consider the number of flights plying between any two airports per week .`` unweighted network '' represents the architecture ( topology ) of airports connectivity , whereas `` weighted network '' represents traffic dynamics in the network .our analysis shows that while ani is similar to the wan in some aspects , it has differences in some features as reflected in its network parameters .we find that ani has small - world  network features and has a scale - free  degree distribution .the traffic in ani is found to be accumulated on interconnected groups of airports and is concentrated on trunk routes between large airports .it is found to be having a disassortative topology , in contrast to wan .the airport network of india ( ani ) comprises domestic airports of india and airlines connecting them .the air services provided in ani include all major domestic air service providers in india , including some international airlines which ply domestic routes .the data was obtained from a timetable ( excel s timetable for air services within india ) of neighboring countries in the close vicinity of india . ] , as on 12th january 2004 ) of air services , which includes domestic services provided by following air - travel service providers : indian airlines , alliance air , jet airways , air sahara , air deccan , jagson , druk air , air india , bangladesh biman , royal nepal and srilankan airlines .the `` unweighted ani '' is a directed network with ( = ) 79 nodes ( airports ) and 442 directed links ( flights going from one airport to another ) .it is represented by a binary adjacency matrix , ( ) , whose elements take value if there is a flight by any service provider from airport to airport on any day of the week and otherwise .the asymmetric adjacency matrix ( ) was used to find properties ( in - degrees , out - degrees , and shortest paths ) which are sensitive to the direction .certain properties ( degrees , degree - correlations , clustering coefficients ) were calculated after symmetrizing the adjacency matrix , which is justified , as out of total flight routes are bi - directional .we needed to add only 14 fictitious flights so as to symmetrize .the total number of edges in the symmetrized adjacency matrix is ,    to include the information about the amount of traffic flowing on the network , the `` weighted ani '' is defined by considering the strengths of the links in terms of number of flights per week .it is represented by a weight matrix , , where each element stands for the the total number of flights per week from airport to airport .since majority of the nodes are symmetric in ani , in terms of traffic flowing into and out of it ( ) , we symmetrize the weight matrix which is used for all the weighted analyses .degree of a node is the number of nodes it is directly connected to .degree of a node is defined as , in a directed network , in - degree ( out - degree ) of a node is the number of in - coming ( out - going ) links . in ani , in - degree ( ) and out - degree ( ) of an airport stand for the number of flights terminating - into and number of flights originating - from that airport , respectively .we observe that for a very large number of nodes in ani the average degree of symmetrized ani was found to be the average shortest path length ( ) for a directed network with nodes is defined as , where shortest path length from node to . clustering coefficient ( ) of a node is defined as the ratio of number of links shared by its neighboring nodes to the maximum number of possible links among them . in other words, is the probability that two nodes are linked to each other given that they are both connected to .the average clustering coefficient is defined as , where clustering coefficient of node .small - world  networks are characterized by a very small average shortest path length ( ) and a high average clustering coefficient ( ) .shortest path length from node to , , is the number of flights needed to be taken to go from to by the shortest route .we found the average shortest path length of ani to be ( = ) , which is of the order of that of a random network ( ) of same size and average degree .the average clustering coefficient of ani was found to be ( = ) , which is an order of magnitude higher than that of the comparable random network ( ) .these two properties indicate that ani is a small - world network .wan  and anc  have also been found to be small - world networks . in particular, anc , which is of comparable size of that of ani , has average shortest path length and clustering coefficient , and , respectively .shortest path distribution in ani ]   ani ( a directed connected network with nodes ) has distinct paths node - to - node flight routes .shortest path analysis is intended to give us an idea about the ease of travel in the network .[ fig : shortest_path_dist ] shows the distribution of the shortest paths in ani .table [ table : shortest_paths ] summarizes the results of shortest paths analysis .apart from other statistics , we also show number of flights needed to be changed , which is an indicator of the convenience of travel in the network ; the lesser the better . around % of pathsare reachable by changing a maximum of flights . around % of pathsare reachable by changing maximum flight .whereas about % paths are connected by direct flights .we also deduce that the diameter ( defined as the longest of all shortest paths ) of ani is , which means that one needs to change at most flights to reach from any airport to any other airport in ani .  from the passengers point of view , best is as few change of flights as possible , to reach from any airport to any other airport in the network .though this is an ideal situation for the passengers , it is not economically viable for the air service providers .we infer that the airport network has evolved , with the efforts from the service providers , to cater for the convenience of passengers , thus acquiring the small - world topology ..[table : shortest_paths]break - up of the number of flight routes having a certain shortest path , their percentage and corresponding number of flights needed to be changed by the shortest route . [ cols="^,>,>,^ " , ]   degree , as defined in eq .[ eq : degree ] , is one of the measures of centrality of a node in the network .degree symbolizes the importance of a node in the network the larger the degree , the more important it is .the distribution of degrees in a network is an important feature which reflects the topology of the network .it may shed light on the process by which the network has come into existence .the networks in which the links between two nodes are assigned randomly have a poisson degree distribution  with most of the nodes having a typical degree .a large number of networks ranging from internet to protein - protein interaction network in yeast , have been found  to be having scale - free degree distributions .the scale - free distribution , characterized by a power law , with a scaling exponent  can be explained with the help of a growing network model with preferential attachment of the nodes which are being added to the network  .since ani is a small network , we analyze the cumulative degree distribution , , whose scaling exponent is related to that of by .we find that the cumulative degree distribution of ani follows a power law as seen in fig .[ fig : cum_degree_dist ] for a wide range of , although it deviates from it for large degrees .the deviation for large degrees can be attributed to the cost of adding links to nodes or to the limited capacity of nodes . with the help of numerical simulations, it was shown  that the cost of adding links leads to a cutoff of the power - law regime for large in the cumulative degree distribution , as is the case with ani .we find the scaling exponent of ani to be ,   cumulative degree distribution in ani .it can be approximated by the power - law fit , analyzed the weighted ani , by considering the flow of information ( traffic ) on the topology of the network .the statistical analysis of weights between pairs of nodes indicates the presence of right - skewed distribution as shown in fig .[ fig : cum_weights_dist ] .this shows a high level of heterogeneity in ani , as also found in the case of airport network of china  and wan  .cumulative weights distribution in ani . ]it has been observed that the individual weights do not provide a general picture of the network s complexity  .we use a different measure of importance of a node which considers the flow of information in a network , following barrat et al .the weighted counterpart of degree , _ strength _ , is defined as , strength of an airport represents total traffic handled by it per week . fig .[ fig : sk ] shows the correlation between degree ( ) and the average strength of vertices with degree , .we find that increases with as , if the strength and degree of a node were uncorrelated , then , where , which will yield this is an uninteresting case , as in this situation weights do not provide any better information than degrees . for ani , we find that the exponent to be this implies that the strengths of nodes are strongly correlated to their degrees in ani .larger is an airport , the more traffic it handles , in contrast to what would be expected if the topology and the traffic were uncorrelated .this feature of ani is similar to that of wan , where it was found that properties of network are decided , among many other things , by the way the network has evolved , the inherent interdependencies of the nodes , and the architectural constraints .network parameters are defined so as to be able to capture such features of networks .we study few such parameters related to the local cohesiveness and connection tendency of nodes , with and without consideration of weights to gain better insight of traffic dynamics on the architecture of ani .average strength as a function of degree ( k ) of nodes. ]   clustering coefficient as defined in section [ sec : uw - ani ] , captures the local cohesiveness of a node .average clustering coefficient ( ) measures the global density of interconnected nodes in the network .additional information at the intermediate level can be obtained by defining , average clustering coefficient of nodes with degree .these topological parameters overlook the flow of information on the network and hence may not present correct information about the network dynamics .the clustering coefficient could be redefined  , to incorporate the weights of the edges , as , this parameter measures local cohesiveness , by taking into account the interaction intensity found on the local triplets .the unweighted and weighted clustering coefficients could be compared ( the normalization ensures that ) to assess the tendency of accumulation of traffic on interconnected triplets . weighted clustering coefficients averaged over all nodes ( ) and over all nodes with degree ( )are defined analogous to their topological counterparts .it has been shown  that hierarchical networks are expected to have a non - trivial , power - law decay of as a function of , which means low degree nodes belong to interconnected communities .many real networks have been found  to be having such nontrivial decay .we find that in ani , shows a power - law decay with this points toward an inherent hierarchy in the architecture of ani .this is an interesting result as it has been found  that some other infrastructure networks ( internet at the router level and power grid network of western united states ) which are constrained by `` geographical organization '' do not show such hierarchy .this could be because , as opposed to the above - mentioned networks , where the cost of having a link grows in proportion to the distance between two nodes , the geographical constraint is not so strong in the case of airport networks .it is relatively easier to add an air connection to the existing network , than to add a data cable or an electrical cable for comparable geographical distances .average unweighted ( ) and weighted ( ) clustering coefficients of nodes with degree . ]  as seen in fig .[ fig : cc ] , after remaining almost constant for , falls rapidly thereafter .this indicates that large airports provide air connectivity to far - off airports , which themselves do not tend to be connected . in ani ,regional and national hubs provide air connectivity to airports in their domains , with the latter providing it to a much larger number , which reduces the of these hubs .we find that , which means that the traffic is accumulated on interconnected groups of nodes which form high traffic corridors , also known as _ trunk routes_. we find that is restricted in its range across and is consistently higher than the corresponding the higher the , more pronounced is the difference between and .this implies that with increasing degree , airports have progressive tendency to form interconnected groups with high - traffic links .this _ rich - club phenomena _  , in which high degree nodes tend to form cliques with nodes with equal or higher degree , is also observed in wan  .another parameter which is used to investigate the networks architecture is the average degree of nearest neighbors , , for nodes of degree  .it is convenient to calculate , which is related to the correlations between the degrees of connected vertices as where is the conditional probability that a given vertex is connected to a vertex of degree for a topology with no correlations among nodes connectivity , the conditional probability as well as are independent of on the contrary , explicit dependence on is a signature of non - trivial correlations among the nodes connectivity and the possible presence of a hierarchical structure in the network topology . in the presence of correlations ,a network can be classified as one having `` assortative mixing '' or `` disassortative mixing '' , based on whether is an increasing or decreasing function of , respectively . in the former class , high degree nodeshave tendency to be connected to nodes with large degrees , whereas in the latter class high degree nodes have a majority of low degree nodes as their neighbors . for a weighted network the _ weighted average nearest neighbors degree _  could be defined as , which measures the effective _ affinity _ to connect with high- or low - degree neighbors according to the magnitude of the actual interactions .the trend of indicates the weighted assortative or disassortative nature of the network considering the strength of interactions among the nodes .average unweighted ( ) and weighted ( ) degree of nearest neighbors of nodes with degree . ]we find ( fig .[ fig : degree_corr ] ) that ani has uncorrelated structure for small degrees ( ) with no definite assortative or disassortative mixing . but for larger degrees ( ) ani shows a clear disassortative mixing .this property of ani is drastically different from that of wan which was found to be having assortative mixing for small degrees ( ) beyond which it approaches a constant value .the consistent disassortativity in ani for higher degrees could be attributed to the political compulsions on regional and national hubs to provide connectivity to a large number of low - degree destinations .a hub in wan is connected to many other hubs ( in other countries ) , thus compensating for the large number of connections with low - degree airports .since the number of hubs is limited in a national airport network , such as ani , it could lead to disassortative topology .this feature of ani is consistent with the nature of airport network of china  which is reported to be disassortative .this could well be a feature of national infrastructure networks which have similar geo - political compulsions .the normalized correlation function ( )  ( ) is another global quantitative measure of degree correlations in a network , which is zero for no correlations among nodes connectivity and positive or negative for assortative or disassortative mixing , respectively . for ani we find , indicating disassortative mixing which is consistent with the observation in fig .[ fig : degree_corr ] .many technological networks and biological networks are found to be disassortative , while social networks are found to be assortative  .indian railway network  has also been found to be disassortative .we find that weighted degree correlation , , is restricted in its range across and is consistently higher than its unweighted counterpart , , indicating toward bias in the traffic in ani. the larger the degree of a node , the more pronounced is the difference .this implies that despite disassortative topology , the traffic is concentrated between high degree nodes .this is understandable as most airline service providers concentrate on certain profitable , so called , _ trunk routes _ , thereby creating high traffic corridors  .we find that ani , despite being small in size , has complex dynamics similar to those of bigger air transportation networks .it represents an evolving transport infrastructure of a developing nation .it presents a case of a national airport network which has a different mechanism of formation than the global network .ani , whose topology has a signature of hierarchy , has small - world network features and is characterized by a truncated scale - free degree distribution .analysis of weighted ani reveals clearer picture of the network dynamics . the traffic in aniis found to be accumulated on interconnected groups of airports .it has disassortative mixing in contrast to wan , with hubs having large number of low - degree neighbors .the topological disassortativity is offset by the traffic dynamics as the traffic is concentrated between high - degree nodes .ani is expected to grow at a rapid speed with addition of airports , several low - cost air services  , and importantly by increase in strength and complexity of interactions .it will be interesting to study the evolution of this air transportation network , as it offers an excellent example to understand and model the topology , traffic dynamics and geo - political constraints which shape a national infrastructure of economic importance .the author is grateful to somdatta sinha , marc barthlemy , and s s manna for valuable discussions , and madhavi for help in obtaining data .the author is thankful to deepak dhar , r e amritkar , v suresh , and p k mohanty for their comments on the manuscript .author thanks council of scientific and industrial research ( csir ) , govt . of india , for the senior research fellowship .

Abstract: transportation infrastructure of a country is one of the most important indicators of its economic growth . here we study the airport network of india ( ani ) , which represents india s domestic civil aviation infrastructure , as a complex network . we find that ani , a network of domestic airports connected by air links , is a small - world network characterized by a truncated power - law degree distribution , and has a signature of hierarchy . we investigate ani as a weighted network to explore its various properties and compare them with their topological counterparts . the traffic in ani , as in the world - wide airport network ( wan ) , is found to be accumulated on interconnected groups of airports and is concentrated between large airports . in contrast to wan , ani is found to be having disassortative mixing which is offset by the traffic dynamics . the analysis indicates toward possible mechanism of formation of a national transportation network , which is different from that on a global scale .


often one would like to create beam - matched maps of iras images in order to study the properties of the emitting grains and the incident radiation field .for example , the ratio between the iras 60 and 100 m bands provides a measure of the equilibrium dust temperature and is often used as a proxy for the intensity of the radiation field .the 12/100  m ratio and 12/25  m ratio can be used to trace abundance variations in different grain populations ( e.g. , boulanger et al .1990 ; ryter , puget , & prault 1987 ) . in order to use any such ratio image for quantitative purposesit is important that the beam shape of the two images be well matched .the most commonly used iras image data product for such studies is the iras sky survey atlas ( issa ) . with the issathe construction of ratio maps does not pose a significant problem since all four iras bands have been brought to a common resolution of .drawbacks to using these images include the presence of striping and the relatively low resolution .hires processing in size ) through the infrared processing and analysis center ( ipac ) ] ( aumann , fowler , & melnyk 1990 ) can yield much higher spatial resolution and , with the addition of new destriping and zodiacal subtraction algorithms , superior image quality .however , there are a number of characteristics of the resulting images that make the creation of an accurate ratio map non - trivial : the achieved resolution is different in each band for a given number of iterations , the beam shape varies between bands primarily because of the difference in the detector sizes , and within each band the beam shape also varies as a function of position because of the varying iras scan pattern on the sky .these image characteristics mean that a simple ratioing of the hires images is not satisfactory in the majority of cases .various techniques have been developed to create accurate high resolution ratio maps with hires data . as will be discussed, most existing techniques for creating hires ratio maps tend to be time consuming , and in some cases also require access to the raw iras data . the creation of a new fast technique is primarily motivated by the existence of two very large hires data sets that cover parts of the galactic plane .the iras galaxy atlas ( iga ; cao et al.1997 ) consists of 1 and 20 iteration far - infrared ( 60 and 100  m ) images and ancillary files covering the entire galactic plane between .7 .the mid - infrared galaxy atlas ( miga ; kerton & martin 2000 ) is the equivalent mid - infrared data set ( 12 and 25  m ) covering  ; .both of these data sets have been incorporated as large mosaics into the canadian galactic plane survey ( cgps ; taylor 1999 ) an international project that is conducting a multiwavelength ( 21-cm hi line , 1420 and 408 mhz continuum , co ( j ) , and infrared ) survey of the galactic plane at a common spatial resolution of .the existence of this large uniform data set means that high - resolution studies of dust in the ism can now be easily done over very large spatial scales .  in section[ sec : previous ] we review the various techniques that have been used to create ratio maps of hires images . a new technique developed specifically for the fast creation of large - scale hires ratio mapsis presented in section [ sec : algorithm ] and the resulting ratio maps are examined in section [ sec : compare ] .section [ sec : apply ] presents some sample applications of the new technique , and conclusions are given in section [ sec : conclude ] .the similarity in the size and layout of the 12 and 25 m detectors in the focal plane array of iras ( iras explanatory supplement 1988 ) means that the resulting beam shapes are very similar and two - dimensional gaussian fits to the beams will often agree to within the fitting errors .beam maps showing the hires beam shape at various locations in the image and tables of two - dimensional gaussian fits to the beam shape are available as part of the hires service at ipac and are also included in the iga and miga releases .this similarity in beam shape means that , with care , high resolution ratio maps can be created by simply dividing the 12 and 25 m images .care is required because , although the beam shapes are very close , the actual beam shapes are not two - dimensional gaussians and can vary in detail because of differences in the ratio between the point source flux and the background level during hires processing between the two images ( see section 4.1 of kerton & martin 2000 for details ) . as an example ,figure [ fig : s151 ] shows two ratio maps of the region around the hii region sh 2 - 151 , chosen because of the unusual irregular cross - shaped beam pattern caused by the significant difference in the scan angle between the two iras coverages of the region ( this effect is most noticeable at high ecliptic latitudes ) .the top image was constructed by simply dividing a 12 m image from the miga by the corresponding 25 m miga image , while the bottom image was constructed using the cross - band simulation technique ( see section [ sec : crossband ] for details ) to bring the two images to a common beam shape . in generalthe structure shown in each image is the same , but note that the point sources are more clearly defined in the bottom image because of the matching beam shapes .for example , the feature associated with sh 2 - 151 is more clearly shown to be a point source in the beam - matched image than in the simple ratio image .the lower image also has a less mottled background due to the matched beams and the slightly lower resolution .the beneficial effect of bringing the two images to the same beam shape is seen in the way that the cross - shaped beam pattern of the point sources seen in figure [ fig : s151 ] is undistorted in the cross - band simulator ratio image while in the simple ratio map the point sources show some non - physical structure .most of the point sources ( stars ) in the ratio image saturate and appear as black structures since they are brighter at 12 m than at 25 m .the two noticeable exceptions are the point source associated with sh 2 - 151 at , and the point source at , ( the planetary nebula pk 107.1 ) .for some purposes a simple 12/25 m ratio map , such as that shown in figure [ fig : s151 ] , may suffice .the user needs to closely inspect the associated beam maps to determine if the beam shapes are close enough for their purposes .if so , the similar beam shape and resolution of the mid - infrared bands provides a quick and easy way to obtain high resolution mid - infrared ratio maps .note that any simple ratio map involving the 60 and 100 m bands will not be satisfactory because of the large difference in beam resolution and shape , and some form of beam matching will be desirable .this is the focus of the rest of this paper .one simple technique that is often used is to smooth the hires images to a common lower resolution .for example , , in their study of m31 , convolve their hires images to a common circular beam ( their lowest original hires resolution ) in order to construct ratio images .this technique is fast and does not require access to the original iras data ( just the hires beam tables ) , but suffers from the disadvantage that one is losing much of the resolution gained by the hires processing .this can especially be a concern for the mid - infrared bands where one can achieve very high resolution ( the original resolution of the 12 and 25 m images from was ) .over very large areas , where the beam properties are expected to change significantly , one also has to take care to use the correct convolving kernel to smooth the beams correctly .if the smoothing is very large , this is not a major concern , but again one loses some of the advantage of using hires in the first place .this technique takes advantage of the fact that the resolution achieved by hires processing improves as the number of iterations increases . by stopping the hires processing at different iterations for data at different wavelengths one can obtain approximately matching resolution in different bands ( e.g. ,see the wang 1994 study of the supernova remnant ic 443 ). the exact beam shape can still be slightly different so the final images are often convolved to a circular beam to reduce this effect .since this technique requires some degree of trial and error to match the beam resolution in the different bands it is really only suitable for mapping small areas and for users that have easy access to both the raw iras archive and the hires processing software ( yoric ) .the most accurate technique for matching the various hires beams is called cross - band simulation .this technique makes use of the hires iras simulator that is part of the yoric software package . using this technique ,if one wanted to create a ratio map of a 12 and 60 m image , the simulator would be used to scan the 12 m hires image with the 60 m detector pattern to create simulated data for the 12 m sky .these `` observations '' would then be hires processed as if 60 m data to create a somewhat lower resolution version of the 12  m image .the same process is followed for the 60 m data .the final result is two images at the same resolution ( in this example , each slightly poorer than the original 60 m hires resolution ) .the major advantage of this technique is that because the resulting beams are very well matched at every location in the image , spurious results are not introduced into the ratio maps .this technique is very time consuming since the hires algorithm is essentially run twice for each field that needs to be processed .simulator images can be requested from ipac for single fields so a user in principle does not need direct access to the yoric software or the iras data archive for single field images .if one is interested in creating large scale images ( such as the large mosaics that can be created with the iga and miga data ) then access to the yoric software and the iras data archive is essential since large scale mosaics can not be created by simply patching together smaller images obtained from ipac because of the local destriping algorithms that are used .both the miga and iga are on a common grid and were created using a special pre - processing technique that allows the easy creation of high - quality large - scale mosaics ( e.g. , see cao et al .1997 or kerton & martin 2000 ) .since the typical user of the miga , iga , and/or cgps data will not have access to either yoric or the raw iras data it is important to have a robust technique that users can apply to the data that they do have in order to construct ratio maps .the criteria used in the development of this new technique are that it should make use of only those data available to the typical user , it should be fast , and it should result in images that are close to what the cross - band simulator technique would produce .  as mentioned above , the resolution and position angle of the hires beam varies as a function of position over an image . as part of the standard miga and iga releasestwo - dimensional gaussian fits were made to the beam shape and recorded in accompanying files .this information is also available for individual hires processing requests .the basic assumption underlying the new ratio map technique is that the hires beam is well described by the two - dimensional gaussian at the various points in the images .clearly this assumption works best in those regions where the beam shape is `` well - behaved '' . in the worst case , in regions of high ecliptic latitude , the beam shape can even have x - shaped wings due to very large variations in the scan angle of the iras satellite . at these locations , while the assumption is clearly dubious , the resulting ratio maps turn out to still be useful .this is because the central part of the x - shaped beam is still well described by the gaussian approximation .we describe the operation of the algorithm on a single standard hires field that one would obtain from either the iga or the miga . in order to build up a large scale image, one would repeat the technique for a number of fields then mosaic the images together . in a standard field ,beam information is reported in a array of points spread evenly over the image .this sampling interval does greatly undersample the beam variation across the image , but finer sampling was not possible because of the way the beam maps are constructed .the algorithm assumes that the beam information describes the beam shape over the region surrounding each beam .the validity of this assumption depends upon how little the scan pattern changes over the .users can determine this by either looking at the beam maps or the detector track maps that are included in the standard miga and iga releases and in a typical hires request .the algorithm runs as follows .to start , one has two images , image 1 and image 2 , at iras band 1 and band 2 respectively .the major and minor fwhm and orientation for the beams in image 2 is read from a file the user has prepared from the hires beam tables . for each beama two - dimensional gaussian kernel is created and convolved with a region centered on that beam that has been clipped out of image 1 .the size of the area clipped out was chosen to provide ample room to avoid edge effects when doing the convolution .the center of the convolved region is clipped out and placed in the final convolved image .the procedure is repeated until all 25 beams in the 1 center of the image have been processed and the convolved image has been built up .beams centered 0.1 from the outer edge of the image are ignored since they correspond to a 0.2 border that should be clipped off the final images before mosaicing , or making ratio maps ( this buffer zone was included in the hires processing to avoid problems with artificially low detector coverage ) .the entire procedure is then repeated using image 2 and the beam data for image 1 .essentially the new algorithm is a combination of the smoothing technique discussed in section [ sec : smooth ] and the cross - band simulator technique described in section [ sec : crossband ] .however , instead of scanning the image with the iras beam shape and detector scan pattern it uses a two - dimensional gaussian convolution to match the beam shapes based upon the information provided in the beam maps and tables . also , instead of smoothing to a markedly lower resolution it convolves the image only to the degree necessary to match the two images . the algorithm was implemented using idl because of its matrix manipulation capability , but could easily be modified to any computer language .  while this algorithm was developed for and initially applied to iras hires data there is nothing about it that restricts its use solely to infrared data .the algorithm could be easily applied to , for example , an infrared and a radio data set provided sufficient information about the beam shapes was available ( see section [ sec : apply ] ) .large - scale radio spectral index maps can also be created using the algorithm .a specialized version of the algorithm ( called mosconv , for mosaic convolve ) has been developed independently by l. higgs at drao to create large - scale ( ) spectral index maps using the 1420 mhz and 408 mhz data and ancillary files contained in the cgps .the algorithm was tested on two regions with markedly different hires beam shapes . in each casethree sets of images and beam maps were created : regular hires , cross - band simulated hires , and hires processed with the new algorithm .the images allowed us to compare the relative quality of the resulting ratio maps while the beam maps allowed us to compare in more detail how the beam shape changes in each case . for the beam comparisons we focused on the 12 and 100  m hires beams since they have the greatest difference in resolution and beam shape .first we looked at a field ( centered at , near ic 443 , see figure [ fig : ic443 ] ) where the hires beam is `` regular '' , i.e. it is very close to a two - dimensional gaussian .figure [ fig : regcomp ] shows one - dimensional cuts through the minor axis ( at the same position angle ) of the 12 and 100  m beams located at , along with the absolute value of the difference between the two beams .the beam profiles have been normalized to the same central intensity so that differences in the beam shapes can be highlighted .notice the large difference in the beam shapes in the raw images ( top profile in figure [ fig : regcomp ] ) .negative rings are evident in the 100  m beam ; this is a well known artifact of hires processing .the middle profiles show the beams that are created using the algorithm presented in this paper .the match between the beams is greatly improved from the original images .finally the lower cuts show the beams that are formed in the cross - band simulation process .there is a significant improvement over the regular hires beams and even the ringing around the center of the beam is well matched in this case .  to illustrate the quality of the match in two dimensions , the upper row of figure [ fig : allsur ] shows surface plots of the absolute difference between the 12 and 100  m beams for the regular hires , convolution , and cross - band simulation processes for this region .again there is a substantial , comparable improvement in the match between the beams evident in the convolution and cross - band simulation plots .the differences between the two beams for the cross - band simulation and convolution beams are of the same ( small ) magnitude .the same tests were repeated in another field ( centered at , , see figure [ fig : xbeam ] ) where the hires beam has x - shaped wings ( especially evident at 12  m ) .the one - dimensional cuts through the various beams are shown in figure [ fig : xcomp ] .again we see a significant comparable improvement in the agreement between the beam shapes both in the cross - band simulation and our new technique .the bottom row of figure [ fig : allsur ] shows the surface plots of the difference in the beam shape .these plots show that the convolution and cross - band simulation technique will provide similar degrees of improvement to the quality of the beam match .figure [ fig : allsad ] shows the sum of the absolute difference between the 12  m and 100  m beams for all of the beams in the two areas we tested .the beam that has been discussed in detail is indicated on the figure , and clearly is a representative result . in all cases we see that the convolution algorithm yields a substantial improvement in the match between the two beams that is comparable to the results obtained using the cross - band simulation technique .  the resulting ratio maps created using our convolution technique and the cross - band simulation technique are shown in the bottom two frames of figures [ fig : ic443 ] and [ fig : xbeam ] .the only major difference between the two results is the detailed structure close to the point sources .the convolution map is much faster to create than the cross - band simulation map and results in a final ratio map that is comparable in achieved resolution and overall image quality .  in order to compare the various ratio maps the rms fractional difference between the cross - band simulated andthe unprocessed hires or convolution ratio image was calculated .this was also carried out for four subregions corresponding to various features in each image .results from the comparison are presented in tables [ tbl : xb1 ] and [ tbl : xb2 ] for various iras band combinations . as would be expectedthe convolved ratio map is a much better match to the simulated ratio map as indicated by the lowering of the rms fractional difference for all of the various iras band combinations .this quantitative assessment was combined with a visual inspection of the convolved ratio images and the simulated ratio images to make sure they were comparable overall .in figure [ fig : w5 ] we show an example of how this technique can be applied to the creation of large - scale infrared ratio maps .the upper image is the 12/100  m ratio map of the w5 hii region created using issa images .the poor resolution and image quality ( striping ) referred to in section [ sec : intro ] are apparent .the lower image is the same image constructed using hires data from the iga and miga processed using our algorithm to match the 12 and 100 m beam shape at all locations in the image .clearly this image is more useful for any scientific investigation .the region with a low 12/100 m ratio that dominates the center of the map corresponds to the ionized gas in w5 .the drop in this ratio relates to the destruction of pah dust grains in the ionized regions which in turn causes a reduction in the 12  m emission .the various dark points visible in this map are all hii regions as confirmed by inspection of the 1420 mhz continuum images in the cgps database .maps like this one can be easily compared with large scale data products at other wavelengths in order to investigate the properties of interstellar dust in different environments .note that whenever anything unusual is found in a hires color map , no matter how the map was created , a detailed examination of the separate - band images is called for ( e.g. , aperture photometry , checking the noise and coverage maps , etc . ) .although we have focused on the construction of infrared ratio maps in this paper , another application of the new algorithm is the comparison of multiwavelength data sets using difference images or multi - colour imaging . in these applicationsthe two datasets are first scaled and then either subtracted or displayed as different colours .this allows the relative distribution of the emission to be easily studied .figure [ fig : mw1 ] shows two of the typical mosaics from the cgps database in the infrared ( 12  m hires ; left panel ) and in 1420 mhz continuum ( right panel ) .the object that dominates the center of the mosaic is lbn .comparison with co and hi images of this region in the cgps data base show that this region is an edge - on view of an hii - hi - molecular gas interface ( a photodissociation region or pdr ) .we were interested in investigating the relative position of the 12  m emission and the 1420 mhz continuum emission at this interface .the beam shape in the 1420 mhz mosaic is also variable across the mosaic , so we created beam - matched images using the beam information provided in the cgps database and the algorithm presented in this paper .lower and upper data limits ( peak 95% of image histogram ) were selected that resulted in similar visual contrast in each image ( as shown in figure [ fig : mw1 ] ) .each image was then rescaled from 0 to 256 and the 12  m image was subtracted from the 1420 mhz continuum image .the center panel of figure [ fig : mw1 ] shows the resulting difference map .dark areas are regions where the ir emission dominates , light areas are where 1420 mhz continuum emission dominates .instead of subtracting the images one could also construct a two - colour image with the infrared data in the red channel and radio in the blue channel ( for example ) . in this casewe see that the intense 12  m emission is coming mostly from a region just to the east of the intense continuum emission with a very slight overlap area .the pdr nature of the object is best seen by examining a cut through the region from east to west .figure [ fig : mw1_cut ] shows a cut ( background subtracted and normalized ) from , to , .the 12  m emission peaks just inside of the 1420 mhz emission and then falls off in intensity as one moves farther into the neutral / molecular material to the east .this rapid increase in infrared emission at the ionized - neutral interface followed by a slow decline in emission as one moves into the neutral / molecular material is what one expects to see in a cross - section of a pdr .beam - matched maps of hires iras data products have been shown to be useful both in studies of dust properties and in multiwavelength analyses of the dust and gas phases of the interstellar medium .  for images in any of the iras bands , simple ratio maps are seldom sufficient for any application beyond superficial inspection . for quantitative studies involving hires datasome degree of extra processing is required .  to create high - resolution infrared beam - matched maps a simple , fast ,convolution technique that makes full use of the beam shape information reported in the miga and iga releases and also available as part of the standard hires request output has been developed .the algorithm provides a good match of the hires beams combined with a minimal loss of the resolution gained from using the hires processing technique .we expect that this technique will be quite useful for the creation of very large scale ( over many square degrees ) infrared ratio maps for studies of the properties of dust in the ism .the technique is also useful for the construction of multiwavelength difference or multi - colour images , and large - scale radio spectral index maps .c.r.k . would like to thank the ontario graduate scholarship program for financial support during part of this study .the research of p.g.m . is supported through grants from the natural sciences and engineering research council of canada ( nserc ) .lccccc b1/b2 & 0.075 & 0.078 & 0.096 & 0.065 & 0.080 + b1/b3 & 0.084 & 0.139 & 0.162 & 0.094 & 0.100 + b1/b4 & 0.088 & 0.232 & 0.247 & 0.110 & 0.130 + b2/b3 & 0.061 & 0.074 & 0.128 & 0.105 & 0.085 + b2/b4 & 0.058 & 0.077 & 0.112 & 0.077 & 0.090 + b3/b4 & 0.042 & 0.047 & 0.150 & 0.090 & 0.072 + b1/b2 & 0.041 & 0.050 & 0.064 & 0.040 & 0.049 + b1/b3 & 0.033 & 0.067 & 0.078 & 0.073 & 0.079 + b1/b4 & 0.013 & 0.056 & 0.061 & 0.071 & 0.098 + b2/b3 & 0.032 & 0.049 & 0.094 & 0.084 & 0.055 + b2/b4 & 0.013 & 0.030 & 0.051 & 0.057 & 0.044 + b3/b4 & 0.021 & 0.023 & 0.074 & 0.057 & 0.038 +  lccccc b1/b2 & 0.030 & 0.109 & 0.191 & 0.030 & 0.060 + b1/b3 & 0.038 & 0.161 & 0.280 & 0.043 & 0.087 + b1/b4 & 0.046 & 0.305 & 0.431 & 0.058 & 0.131 + b2/b3 & 0.020 & 0.093 & 0.247 & 0.030 & 0.054 + b2/b4 & 0.022 & 0.165 & 0.396 & 0.034 & 0.085 + b3/b4 & 0.017 & 0.025 & 0.138 & 0.065 & 0.032 + b1/b2 & 0.016 & 0.089 & 0.139 & 0.015 & 0.041 + b1/b3 & 0.012 & 0.085 & 0.272 & 0.021 & 0.055 + b1/b4 & 0.010 & 0.074 & 0.241 & 0.016 & 0.049 + b2/b3 & 0.010 & 0.035 & 0.192 & 0.022 & 0.037 + b2/b4 & 0.007 & 0.032 & 0.160 & 0.015 & 0.031 + b3/b4 & 0.007 & 0.008 & 0.044 & 0.017 & 0.011 +

Abstract: hires processing provides a significant improvement in both resolution and image quality over previous iras image products , but the characteristics of the hires beam make accurate comparisons between the various iras bandpasses and between hires data and data at other wavelengths non - trivial . we present a new , fast technique for the construction of hires beam - matched images that is especially well suited for the creation of large - scale ( several square degrees ) ratio maps . other techniques for the construction of ratio maps are discussed and compared with the new algorithm . examples of the large - scale ratio maps that can be constructed using this new technique are presented . the algorithm s application to the construction of multiwavelength difference images and multi - colour images is also demonstrated .


recent years have seen the growth of complex networks theory , a research area concerned with the general theory of systems of interacting elements  .its relevance has been illustrated in a number of problems , such as infectious disease propagation  , the strength of social ties  , data routing in technological networks  , and motifs in biological networks  .an underlying driver for the growth of this field has been the increased availability of digitized information , which can be efficiently analyzed to uncover relations between system elements .a simplifying assumption that is made in networks theory is to characterize interactions as being exclusively pairwise ( each interaction represented by a link between two nodes ) , often with an associated interaction intensity or weight , generating so - called weighted networks ( also known as weighted graphs in mathematics ) .the reason for this approach is that usually the information available for real systems is relatively limited . despite these limitations , weighted networkshave proven very useful , as a number of measurable network quantities have shown their relevance in application .examples of these quantities are the distribution of node degree  ( number of links connecting to a node ) , optimal path lengths between network nodes  , and node clustering  ( a measure of loops of length three ) .other properties that depend on specific groups of links ( e.g. , network communities ) have also proven quite useful  .there are situations , however , where it is known that interactions extend to groups larger than two ( multiway interactions ) , and one can use such information to create more accurate models , avoiding the possibility of oversimplified or misleading results . examples of these situations are , for instance , networks of affiliations  , where nodes representing individuals connect to each other by virtue of their membership to a group such as their family or workplace colleagues ; another example are folksonomies  , systems that encode information of triplets of the following three ingredients : objects , descriptors of the objects , and the individuals making the descriptions .characterizing these examples by avoiding the pairwise simplification should lead to more informative and reliable results .  through various independent approaches ,researchers focusing on problems of multiway interactions have proposed mechanisms by which pairwise network weights are generated as a consequence of these interactions ( see , e.g.  and ) .for instance , in affiliation networks , when two nodes belong simultaneously to multiple groups , a feature called co - membership , it is assumed that their relationship intensity is equal to the number of groups they both belong to .perhaps the most appealing feature of these ideas is that they provide a unifying principle to the structure of some interacting systems : the presence of a group generates links , and being part of multiple groups generates weights .surprisingly , these unifying ideas have received limited attention , perhaps because some of the mathematical models that are required are less straightforward than typical networks .here , i focus on a systematic approach grounded in statistical mechanics to relate multiway interactions to weighted networks .to model multiway interactions , it is appropriate to use hypergraphs , which are generalizations of networks  .they are composed of a set of nodes and a set of hyperedges .each hyperedge is a group of interconnected nodes ( a clique ) , and the hypergraph is the collection of all the hyperedges and isolated nodes ; networks are the specialization of hypergraphs in which all hyperedges are cliques each with only two nodes , i.e. , links .the size of a hyperedge is called rank . in a statistical mechanics formulation ( random )hypergraphs are called homogeneous when all hyperedges are equally likely to be present , or heterogeneous when each hyperedge has its own ( possibly unique ) probability to appear . for the examples mentioned above : in a folksonomy , for instance , hyperedges are all of rank three , whereas in affiliation networks , in principle , hyperedges can have different ranks ; both examples are likely to be heterogeneous hypergraphs .the notion of hypergraphs generating weights is equivalent to constructing networks that represent a projection of a hypergraph . in other words , starting from a hypergraph , one can create an associated set of links that form a _ weighted projected network _, where each link weight is given by the structure of the hypergraph and a projection rule .this construction suggests some intriguing possibilities : some data that is typically studied as a network may in fact emerge from underlying hypergraphs .if that is the case , it should be possible in principle to construct hypergraph models and accompanying projections that can fit observed data and narrow down its origins .  in this article, i study homogeneous and heterogeneous entropy maximizing hypergraph ensembles of arbitrary uniform rank and define general projections of hypergraphs that lead to ensembles of weighted projected networks .some specific projection examples that have been used in the literature are explored  , the properties of their respective projected networks calculated , and their interpretations briefly discussed .the percolation threshold and size of the largest connected component of hypergraphs of arbitrary uniform rank are also derived by use of the mapping between the potts model and percolation theory  , and the results are then translated into the percolation properties of the projected networks .these results show that the transition is of second order .i find that , as a function of size , the link weights on weighted projected networks can display a signature of the presence of hidden multiway relations : when faced with a weighted network , this signature could provide indications that there is an associated hypergraph hidden in the data .  the article is structured in the following way : sec .[ max - ent ] focuses on the general definitions of projections of hypergraphs onto networks , and on models of entropy maximizing ensembles of hypergraphs . with these results , in sec .[ applications ] i study in greater detail the statistical properties of general projected networks , as well as some concrete examples .these results suggest how to explore network data for possible signatures of multiway relations .completing the results , sec .[ percolation ] focuses on the percolation properties of hypergraphs and their projected networks , and explores the general notion of sparsity .i finalize the article in sec .[ conclusions ] with some discussion and conclusions .consider a hypergraph , represented by , consisting of a set of nodes , and for each possible hyperedge of nodes , an indicator equal to 1 if the hyperedge is present and 0 if it is absent ; all subindices take non - repeated values from the set . in general , a hypergraph does not require to be the same for all hyperedges .however , for the sake of simplicity , i focus on single rank ( all hyperedges have the same ) undirected hypergraphs , with the indicator symmetric under permutations of ( if one is interested in studying combinations of rank , one merely requires the introduction of the proper parameters for this , but the qualitative nature of the problem is the same as that studied here ) .unweighted undirected networks correspond to .the general hypergraph projection onto a network is defined as a function applied over hyperedges of that produces the adjacency matrix for the weighted projected network .network is formed by the same node set as , and its adjacency matrix is .if a node does not belong to any hyperedge , it is isolated in both and . for given , one can define the subset of its hyperedges that include simultaneously nodes and .the kinds of projections studied here are of the type where is the size ( cardinality ) of .thus , the weight of link in only depends on the number of hyperedges that contain and , an intuitive choice , although certainly not the only possible model ( in the literature , all examples i have found are limited to this kind of projection  ) .  on a concrete empirical case, the projection should reflect the understanding of the relation between and .here , i present results for some reasonable sample choices of , namely ( additive projection ) and ( nominal projection ) , where is the heaviside step function ( if the argument is 0 or less , and 1 otherwise ) .in addition , i show some features satisfied by the projected networks generated by a large class of projections with the general form of eq .( [ wij - p ] ) . to perform calculations , note that the additive projection can be written in terms of as whereas the nominal projection is represented by an illustration of for the case of shown in fig .[ wij - project - illustration ] .  in the literature , both hypergraphs and projections have been used to study interaction data qualitatively embedded in complex networks theory , but without a sense of unification .for instance , the choice is implicit in work such as  ; there , if is interpreted as a specific motif ( structural pattern ) , the model generates unweighted networks guaranteed to posses those motifs . in another approach , found in refs . , each hypergraph ( containing and 3 only ) treats each rank separately in that the interactions of nodes by way of pairs is counted independently to the triplet interactions , with no notion of projection onto a simple network . do consider projections in some form , but are limited by rank of hypergraph and by the nature of the projection .projection is in fact common  , and it is often used as a way to characterize the one - mode networks that emerge from bipartite graphs  ( recent work also uses the notion of projection in the context of time evolving networks  ) .( [ wij - p ] ) offers a unified way to relate networks and hypergraphs , which can be applied to the models cited above to develop additional understanding of the problems .  to build unbiased statistical models , i adapt to hypergraphs the canonical ensemble approach developed in ref .the set of all possible hypergraphs is given by ( the ensemble ) , or in other words , is the union of all possible unique hypergraphs . to analytically formulate the ensemble problem ,consider the entropy , defined as where represents the probability of a given configuration within the hypergraph ensemble , and the sum over configurations is equivalent to summing over all hyperedge combinations , or .the canonical ensemble approach finds the distribution that maximizes while satisfying conditions that define the ensemble of interest .such conditions , say , with an enumeration index , are taken to be of the form finally , since are probabilities , one must guarantee normalization , which translates into the solution to this problem ( satisfying the conditions above ) is obtained via lagrange multipliers .each condition is related to a multiplier , and one solves the equations =0 , \label{lagrange - problem}\ ] ] for , with the lagrange multipliers .the solution to the problem can be expressed as the partition function , and ( _ defined _ as the hamiltonian ) , are respectively given by and   among the simplest non - trivial problems one can address is that of the fully random hypergraph with equal probability for any hyperedge to exist .the constraint associated with this example is the requirement that there is a given average number of hyperedges , , over the hypergraph ensemble . since for a given configuration is given by , the set of constraints reduces to two lagrange multipliers , one for the normalization , and another parameter , labelled , for . introducing this in eq .( [ lagrange - problem ] ) generates the hamiltonian and the partition function where is the set of all possible hyperedges , i.e. , the complete hypergraph of single rank and size . the last equality can also be obtained from the symmetry of the hamiltonian over exchange of indices among .the result expresses that there are possible hyperedges among the nodes . using this resultone can show that the constraint is satisfied for the proper choice of , as seen from averaging in the ensemble and is the probability for a hyperedge to be present , which is evident from writing ; also corresponds to the expectation value of any hyperedge , i.e. , the probability for any hyperedge to exist .the fact that all hyperedges are equally likely suggests referring to this case as the _ homogeneous hypergraph ensemble_. the probability of a specific hypergraph configuration to be observed is given by eq .( [ lagrange - solution ] ) , which in this case yields where the relations and have been used .the application of the and to the homogeneous ensemble is tackled below in a more general ensemble .the solution to the simple homogeneous problem above , helps to identify some basic features of the canonical approach , including quantities such as the probability of a hyperedge , and of a specific hypergraph .building on this , one can construct the more general _ heterogeneous _ case , where each hyperedge has its own expectation .thus , the hamiltonian of eq .( [ hamiltonian ] ) becomes in analogy with the homogeneous case , one defines .the partition function becomes where represents the hyperedge expectations .the probability of a hypergraph configuration is then which is the joint probability that hyperedges with are present , and those with are absent . if for all , , one recovers the homogeneous case .the heterogeneous ensemble possesses the most degrees of freedom among non - interacting undirected hypergraph models .if more specific constraints are imposed such as , for instance , conditions on the average number of hyperedges visiting a node , they would translate into additional constraints on the values of the set ( see e.g. discussion at end of sec .[ applications ] ) .since only projections of the form are considered here , the statistical properties of the projected networks depend on the statistical properties of .it is most useful to focus on the distribution of in the heterogeneous ensemble , , and determine how this translates into the homogeneous case ( table [ table - phi_oij ] summarizes the notation used to compute ) .let us define as the set of all hyperedges on the complete hypergraph that visit and simultaneously .i also define , the complement of with respect to .in addition , for each configuration , is the set of hyperedges visiting and which may or may not have cardinality ( if it does , it is represented as before with ) .the complement of with respect to is , and thus . taking this into account, can be calculated through the expression where is the kronecker delta which can have two or more arguments , and is equal to 1 if all the arguments are equal , and 0 otherwise . in the sumabove , only those configurations for which delta is 1 contribute to , and this occurs only when there are exactly hyperedges in that include .  to perform the calculation , note the independence of each component of in eq .( [ psigma_hetero ] ) .this allows factoring the sum over configurations in eq .( [ phi_oij_def ] ) into a product of i ) the configurations of hyperedges , which can not affect the delta , and ii ) the configurations of hyperedges , which can .the hyperedges each contribute a factor .therefore , the remaining factors of eq .( [ phi_oij_def ] ) lead to where and are the unions of all possible sets and , respectively , and is the complement of with respect to .equation ( [ phi_oij_final ] ) has been expressed in a way that makes it straightforward to explain and convert into an algorithm for calculation , as i attempt to explain now .the expression can be described in the following terms : i ) separate the hyperedges from into two groups , one that can influence over all possible configurations , namely , and another that can not ( ) , ii ) identify out of the hyperedges of visiting and , , iii ) only when , contributes to , and iv ) since there are numerous ways to choose from , one requires the set , which contains all those choices , i.e. , is the ensemble of allowed configurations .consequently , the last line of eq .( [ phi_oij_final ] ) can be read as the sum of probabilities over all possible configurations of hyperedge sets , where each hyperedge belongs to .note that there are hyperedges to choose from and each ( configuration ) picks of them , and therefore .it is worth mentioning that is used in to emphasize its origin as a particular hypergraph configuration , but that it becomes redundant when the meaning of is fixed as the collection of configurations ( the ensemble ) contributing to ; at this point , each specifies a unique configuration and no further reference to is necessary .  in fact , dropping from offers a combinatorial picture for the last line of eq .( [ phi_oij_final ] ) and other distributions in this section .since each is a set of hyperedges connecting non - repeated nodes in cliques of rank , one can think of each hyperedge as an -tuple of non - repeated indices taken from , and a configuration as a collection of non - repeated -tuples .therefore , is the collection of all possible -tuples , the subset of containing all -tuples that have indices and simultaneously , each a sample without replacement of -tuples taken from , and the collection of all possible samplings .this way to think about eq .( [ phi_oij_final ] ) transfers the emphasis from a graph theoretic problem to a purely combinatorial one .the cardinalities of all the sets calculated before follow naturally .  the average of can be determined from eq .( [ phi_oij_final ] ) , through , or by calculating .the result is which fits intuition , stating that the expectation of the number of hyperedges visiting the pair , is the sum of expectations of each hyperedge that can visit to be present over the ensemble .  in the homogeneous case , where for all , becomes a binomial distribution with ( fig .[ distributions](a ) ) . both this average , and more generally eq . ( [ avg_oij_final ] ) , have an interesting interpretation explained below regarding signatures of multiway interactions in observational studies . another noteworthy fact exhibited by eq .( [ phi_oij_homo_final ] ) , even in this very simple case of homogeneous , is that does not have a fixed value but instead follows a probability distribution consequence of the projection process .some general features of can now be described if one conditions it to be a monotonic smooth projection , satisfying the inverse function theorem .this condition offers a way to formally write the distribution of from the distribution of because there is a one - to - one relation between the two quantities .furthermore , one can make use of the monotonicity in both the discrete and continuous variable cases . for the discrete case , defining the distribution of weights , it is simple to see that and the set of possible is obtained by applying eq .( [ wij - p ] ) to the domain of . in the continuous case ,introducing the densities and , the change of variables theorem for probability distributions implies where is the derivative of .the additive projection satisfies monotonicity in a trivial way because it is just the identity function .however , a large class of functions also satisfy monotonicity , including all power law and logarithmic growth functions . the nominal projection , on the other hand ,does not satisfy the condition because any value of leads to the same weight , and thus the inverse of is not uniquely defined . regarding the influence of ,if this distribution is sufficiently narrow in comparison to the shape of , asymptotic estimates of and it properties ( e.g. , moments ) can be straightforwardly obtained .  as i now describe, carries a signature of multiway interactions ( ) that can emerge in a large class of weighted projected networks ; if such signature is observed in independently collected weighted network data for which there is no direct observation of multiway interactions , it would be reasonable to suspect the presence of these interactions hidden underneath the weighted network . to explain this , consider eq .( [ avg_oij_final ] ) for the ensemble of hypergraphs ( although the following results are also valid for a single typical large enough hypergraph ) and label the set of probabilities of the hyperedges that visit , with the average and the variance over the elements . for simplicity ,let us assume that increases to , and each new node connects to via new hyperedges .thus , e.g. for nodes , and there are new hyperedges which can potentially occur in a hypergraph , and similarly for all other new nodes .let us also assume that the total set of hyperedges after the addition of has the same average as , i.e. , and variance ; this is satisfied if the in our model are all drawn from the same distribution .both and are finite because . under these conditions , the central limit theorem  applies to the elements of and .hence , asymptotically in .in fact , if every ( large enough ) subset of is such that and are constant , .the set of conditions stipulated above are naturally attained in the heterogeneous hypergraph model , where the elements of can be looked at as independent random variables .the homogeneous case , with all , satisfies this average exactly even away from the large limit as seen from the average of eq .( [ phi_oij_homo_final ] ) .the relevance of this result lies in the fact that there is a qualitative change in the behavior of for and . in the former which does not scale with , butin the later it scales as  ; i.e. , makes monotonically increasing with . to appreciate the importance of this , consider a weighted network where one observes that link weights and size evolve together with positive correlation ( say , both and the set of grow together ) .two possible qualitative pictures come to mind to explain this behavior : i ) the underlying system is in fact composed of a number of multiway interactions that superimpose to produce a weighted network , roughly as described in this article , or ii ) the system is pairwise , but the addition of nodes not only introduces edges between each new node and an old existing node , but also the existing nodes tend to strengthen the connections between each other by some additional mechanism . the later model ,although possible , is much less natural , and represents a less economical explanation of the correlation between and link weights .therefore , being conservative in our interpretation , if is positively correlated with some or all of the , this should be taken as a first indication that multiway interactions can be present .note that the absence of correlations between and does not exclude the presence of multiway interactions , as specific cases may have additional effects not covered here that may obscure the signatures , e.g. , local multiway relations , interactions between hyperedges , etc .certainly , one could argue that for there to be a weighted network with the behavior described here , it is also necessary to impose conditions on the type of projection that applies .this is indeed true , but the conditions necessary to obtain correlation between and are relatively modest and well justified in numerous circumstances .for instance , here i focus on a monotonically increasing , which proves sufficient .furthermore , it is even acceptable to have a that decreases with , just as long as the decrease is slower than .the level of detail known about and goes hand in hand with the detail that can be learned about the multiway interactions .for instance , if ( homogeneous ) and , then vs. would yield the value of .if , on the other hand , all one knows is that is monotonically increasing , the vs. would offer an estimate of the combined effects of and .overall , the present discussion suggest that _ in the case of evolving networks with correlations between and the set of , it is reasonable to suspect multiway interactions active in the background , and further exploration for evidence of such interactions is well justified . _these results are general , and do not need to be specialized into a particular example .however , if one is interested in using the results of this article as a method to attempt to determine quantitative details of the multiway interactions that may be present , additional work is needed to extend the results to more detailed and perhaps slightly different situations .the two projections and can now be explained further . for ,the properties of are those of , and thus already calculated . the other property to describeis the so - called strength of node , equal to .it is intuitively helpful to calculate the distribution of strengths by making use of the relation between and , the number of hyperedges visiting .these two quantities relate via , and one can determine the distribution of and from it compute .note that while is a property of the graph , is a property of the hypergraph . once again, the independence of the components of simplifies the sum over configurations ( notation in table [ table - zeta_elli ] ) .the hyperedges that could affect belong to , the collection of all hyperedges visiting in , and is the set of hyperedges from in configuration ( when one writes it as ) . from the definition , one can quickly conclude that where is the ensemble of configurations , and .then where takes values . once again, an equivalence between hyperedge sets and combinatorics can be drawn : is the union of all -tuples drawn from with one element always , and thus there are -tuples in total . each is a distinct choice of of these -tuples ; clearly .the sum is a sum over all choices of -tuples from .the averages of these quantities are given by and for the homogeneous case , , with average .therefore , and ( see fig .[ distributions](b ) ) .the nominal interaction needs a different treatment . note that under this projection , can be either 0 or 1 . to determine the probability for , , one merely needs to determine the probabilities that is either 0 or , that is or .therefore , in the homogeneous case , both eqs .( [ pi_wij_final ] ) and ( [ pi_wij_homo_final ] ) are closely related to the average number of connections for each node of a projected network , as explained next .  for any weighted projected network generated from a satisfying and , such as and ,the number of connections visiting node are characterized by , the distribution of ( an expanded and pedagogical exposition of the calculation and its consequences can be found in  ) .the degree can be either 0 or take any value from . to determine ( notation in table [ table - psi_ki ] ), one can proceed in a similar way as before : in configuration , the set of hyperedges visiting and producing degree is .this means that hyperedges in visit exactly nodes and node .it is interesting to note that another configuration , associated with , with a different set and/or number of hyperedges can lead to the same , because these hyperedges still visit the same number of nodes ( see fig .[ ki - illustration ] for an illustration ) . with this definition, one can write where is the union of all possible sets , and the complement set satisfies . since the number of hyperedges is not fixed across members of , one can further organize the by their numbers of hyperedges .the bounds of are dictated by the following : for degree , a minimum of hyperedges is required ( represents the ceiling function ) , and there can be no more than hyperedges . using this organization , and introducing the notation and to represent , respectively ,the sets involving exactly hyperedges and their unions , one can write the sets are only subsets of in which the hyperedges involve exactly and other nodes .finally , it is possible to exploit one more symmetry that facilitates an algorithmic understanding of : the sets that make up involve several possible distinct node sets .however , one can further segregate these sets by the specific nodes in them .hence , if one takes a set , , of specific nodes and , there are several configurations in which their associated contain hyperedges visiting only those nodes .thus , a configuration with specific nodes connected to , using hyperedges is labelled , and the union of configurations is labelled .the union of all sets ( which are non - intersecting ) is equal to .this leads to the final expression where is the complement of with respect to , and is the union of all possible , each one a distinct -tuple taken from the set with one choice always being .the sizes of sets are : , and ; the later is the result of a combinatorial problem that can be defined in terms of general graph theory .specifically , corresponds to the number of distinct hypergraphs that can be constructed with nodes and hyperedges of rank , and each node belongs to at least one of the hyperedges  .in fact , each can be mapped to each one of these hypergraphs . to determine , it is convenient to use the relation by first summing over a single , one notices that only hyperedges in must be considered in detail .given that one arrives at .\label{avg_ki_final}\ ] ] when compared with eq .( [ pi_wij_final ] ) , it becomes evident that each link contributes to independently .  in the homogeneous case , making use of the combinatorial results presented , one obtains ( fig .[ distributions](c ) ) without diving into too much detail , can be calculated via the inclusion - exclusion principle of combinatorics  , which produces among the identities satisfied by  , one finds that , which is used to show normalization of .another identity ,  ] ( fig .[ fig - dilute](c ) ) . from these results ,the meaning of emerges as the parameter that measures the average node strength of the projected network .finally , the degree distribution can be calculated if one keeps in mind that in the sparse limit , the probability that hyperedges overlap is minimal , and therefore , one expects that only the minimum number of hyperedges contribute to the distribution .there are subtleties present in explicitly calculating and when is not a multiple of because hyperedges are forced to overlap in this case , and thus to avoid further details , i only write the unevaluated result ( fig . [ fig - dilute](d ) ) .however , the calculations are not prohibitive , and are derived in detail in  .the sparse regime close to percolation is not the only possible sparse regime . to be concrete ,note that for close to , the average node strength is constant , but the average overlap on projected links scales as , so the larger the network , the less interaction present along the links .however , one can consider a regime in which is constant , and in this regime node strength increases with .both of these regimes are `` sparse '' in the sense that vanishes asymptotically , but each regime has specific properties .generally , these sparse regimes can be defined based on any sensible property , and lead to interesting behavior .finally , for the dense regime ( constant ) , the interesting effect of growth of vs. emerges , which is a unique feature of this model , and the signature that multiway interactions are potentially present .in conclusion , in this article i present a model of hypergraphs and associated weighted projected networks that offers a concise and intuitive picture of hypergraphs , networks , and weights . by using statistical mechanics concepts , together with combinatorial tools ,i have been able to determine some basic features of homogeneous and heterogeneous projected networks that offer concrete tests to determine whether a network that has been empirically measured may bear the signature of multiway ( group ) interactions .the general idea of using the projection of a hypergraph onto a network , which has not been studied systematically to the author s knowledge until this article , deserves a close look to determine further properties that can help give a better understanding of the genuine limits and virtues of pairwise simplifications in network research .the author thanks l. roberts , a. gerig , f. reed - tsochas , and o. riordan , for helpful discussions , and tsb / epsrc grant saturn ( ts / h001832/1 ) , ict ecollective eu project ( 238597 ) , and the jamesmartin 21st century foundation reference no : lc1213 - 006 for financial support .99 r. albert and a .-barabsi , rev .* 74 * , 47 ( 2002 ) ; r. pastor - satorras and a. vespignani , _ structure and evolution of the internet : a statistical physics approach _ ( cambridge university press , cambridge , 2004 ) ; s. n. dorogovtsev and j. f. f. mendes , _ evolution of networks : from biological nets to the internet and www _ ( oxford university press , oxford , 2003 ) .v. colizza , a. barrat , m. barthlemy , and a. vespignani , proc .usa * 103 * , 2015 ( 2006 ) .onnela , j. saramki , j. hyvnen , g. szabo , d. lazer , k. kaski , j. kertsz , and a .-barabsi , proc .usa * 104 * , 7332 ( 2007 ) .s. sreenivasan , r. cohen , e. lpez , z. toroczkai , and h. e. stanley , phys .e * 75 * , 036105 ( 2007 ) .s. s. shen - orr , r. milo , s. mangan , and u. alon , nature gen .* 31 * , 64 ( 2002 ) .r. albert , h. jeong , and a. l. barabsi , nature * 406 * , 6794 ( 2000 ) ; * 406 * , 378 ( 2000 ) .r. cohen , k. erez , d. ben - avraham , and s. havlin , phys .lett . * 85 * , 4626 ( 2000 ) .y. chen , e. lpez , s. havlin , and h. e. stanley , phys .96 * , 068702 ( 2006 ) .d. j. watts and s. h. strogatz , nature * 393 * , 440 ( 1998 ) . s. fortunato , phys* 486 * , 75 ( 2010 ) .m. a. porter , j. p. onnela , p. j. mucha , notices of the ams * 56 * , 1082 ( 2009 ) . ,s. wasserman and k. faust ( cambridge university press , cambridge , 2005 ) .s. p. borgatti and d. s. halgin in _ the sage handbook of social network analysis _ carrington , p. and scott , j. 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Abstract: many natural , technological , and social systems incorporate multiway interactions , yet are characterized and measured on the basis of weighted pairwise interactions . in this article , i propose a family of models in which pairwise interactions originate from multiway interactions , by starting from ensembles of hypergraphs and applying projections that generate ensembles of _ weighted projected networks_. i calculate analytically the statistical properties of weighted projected networks , and suggest ways these could be used beyond theoretical studies . weighted projected networks typically exhibit weight disorder along links even for very simple generating hypergraph ensembles . also , as the size of a hypergraph changes , a signature of multiway interaction emerges on the link weights of weighted projected networks that distinguishes them from fundamentally weighted pairwise networks . this signature could be used to search for hidden multiway interactions in weighted network data . i find the percolation threshold and size of the largest component for hypergraphs of arbitrary uniform rank , translate the results into projected networks , and show that the transition is second order . this general approach to network formation has the potential to shed new light on our understanding of weighted networks .