Routes Obey Hierarchy in Complex Networks
Attila Csoma, Attila K?rösi, Gábor Rétvári, Zalán Heszberger, József Bíró, Mariann Slíz, Andrea Avena-Koenigsberger, Alessandra Griffa, Patric Hagmann, András Gulyás
RRoutes Obey Hierarchy in Complex Networks
Attila Csoma , Attila K ˝or ¨osi , G ´abor R ´etv ´ari , Zal ´an Heszberger , J ´ozsef B´ır ´o , MariannSl´ız , Andrea Avena-Koenigsberger , Alessandra Griffa , Patric Hagmann , andAndr ´as Guly ´as Budapest University of Technology and Economics, MTA-BME Information Systems Research Group, H-1117Budapest, Magyar tud ´osok krt. 2, Hungary E ¨otv ¨os Lor ´and University, Institute of Hungarian Linguistics and Finno-Ugric Studies, H-1088 Budapest, M ´uzeumkrt. 4/A, Hungary Indiana University, Psychological and Brain Sciences, Bloomington IN, 47405, USA Department of Radiology, Centre Hospitalier Universitaire Vaudois (CHUV) and University of Lausanne (UNIL),Lausanne, Switzerland Signal Processing Laboratory 5 (LTS5), ´Ecole Polytechnique F ´ed ´erale de Lausanne (EPFL), Lausanne,Switzerland A. Guly ´as was supported by the J ´anos Bolyai Fellowship of the Hungarian Academy of Sciences * Corresponding author: Andr ´as Guly ´as ([email protected])
ABSTRACT
Various hypotheses exist about the paths used for communication between the nodes of complex networks. Most studiessimply suppose that communication goes via shortest paths, while others have more explicit assumptions about how routing(alternatively navigation or search) works or should work in real networks [1–7]. However, these assumptions are rarelychecked against real data [8, 9]. Here we directly analyze the structure of operational paths using real measurements. Forthis purpose we use existing and newly created datasets having both the topology of the network and a sufficient numberof empirically-determined paths over it. Such datasets are processed for air transportation networks, the human brain, theInternet and the fit-fat-cat word ladder game. Our results suggest that from the great number of possible paths, nature seemsto pick according to some simple rules, which we will refer to as routing policies . First we confirm, that the preference ofshort paths is an inevitable policy element, however the observed stretch of the paths suggests that there are other policies atwork simultaneously. We identify two additional policies common in our networks: the “conform hierarchy”, meaning that thepaths should obey the structural hierarchy of the network, and the “prefer downstream” policy which promotes avoiding thenetwork core if possible. Building upon these simple policies, we propose a synthetic routing policy which can recover thebasic statistical properties of the operational paths in networks. Our results can be helpful in estimating the reaction of complexsystems for stress coming from the outside more accurately than the shortest path assumption permits.
The implicit “shortest path” assumption, meaning that the used communication path in a network is the one with the shortestlength, seems to dominate the network science community and most of the fundamental network metrics (diameter, averagepath length, centrality metrics, [10] etc.) are computed using this assumption. There are other works supposing various models[1], network metrics (e.g. degree, centrality, congestion, homophily [2–4]) and hidden structures (e.g. hidden hierarchies andmetric spaces [5–7]) guiding path selection. The contribution of these works is considerable in the modeling and understandingof simple routing strategies that can recover near shortest paths without requiring global knowledge of the topology. However, alack of confirmation with empirical data leaves an important question open: What kind of paths are actually chosen by naturein real world networks? Here we approach the question of path selection in networks from this lacking empirical angle. Usingexisting and newly created datasets of the traffic flow on real world networks (see Methods), we compare the topology ofthe networks to the structure of empirically-determined paths extracted from these datasets. From this comparison we infercommon characteristic rules of path selection in different networks, which we call routing policies . Our study here presentsthe analysis of empirically-determined paths in air transportation networks, the Internet, the fit-fat-cat word morph game, andempirically-inferred paths in the human brain (see Fig. 1 a for an illustration). For the remainder of this text we will refer toempirically-determined and inferred paths as empirical paths. The main topological features of our networks and the statisticsof the empirical paths are shown in Fig. 2a.Our first finding is that traffic in networks does not necessarily follow shortest paths. Fig. 2b presents the stretch of thepaths which is computed as the length of the empirical path minus the length of the corresponding shortest path having the samesource and destination pair. The figure shows a significant resemblance in the distribution of path stretch across our networks.While around 60-80% of the empirical paths exhibit zero stretch, the remaining paths show path stretch which can exceed up to1 a r X i v : . [ phy s i c s . s o c - ph ] F e b CoreEdge C l o s e n e s s h i e r a r c h y Non-CH pathUpstream CH pathDownstream CH path b Figure 1.
Visualization of the empirical paths in the human brain (panel a ). The illustration of paths conforming to thepolicies identified by our measurements is shown in panel b . A path is hierarchically conform (CH) if does not contain alarge-small-large pattern forming a “ valley” anywhere in its closeness centrality sequence (green and yellow paths in panel b ).An upstream path contains at least one step towards the core of the network (yellow paths), while in downstream paths thecloseness centrality monotonically decreases (green paths). The underlying faded network in panel b is only for illustrationpurposes.4-5 hops. From this result two things follow. First, the plot confirms the shortest path assumption of previous studies in thesense that most of the empirical paths are shortest indeed. In this respect nature’s routing policy definitely “prefers short paths”.However, the non-negligible portion (20-40%) of inflated paths suggests that there may be other policies at use simultaneously.Network Airport Intern. Brain fit-fat-cat (a) Stretch P e r c en t age o f e m p . pa t h s - - - - l l l l ll AirportInternetBrainfit−fat−cat (b)
Figure 2.
Properties of networks and stretch in empirical paths. Panel ( a ) shows the basic structural properties of ournetworks and paths we have analyzed. The stretch of the empirical paths with respect to their shortest counterparts is shown onpanel ( b ) . While most of the empirical paths exhibit zero stretch (confirming the shortest path assumption), a large fraction(20-40%) of the paths is “inflated” even up to 4-5 hops. The plot indicates a stunning resemblance in the distribution of pathstretch in our networks.One such policy our measurements support is the “conform hierarchy” (CH) policy, meaning that the used paths follow thetopological hierarchy of the network. For showing this we have computed the closeness centrality of the nodes comprisingthe empirical paths indicating which (inner or outer) parts of the network the information flows through. We found that mostof the empirical paths do not contain a large-small-large pattern forming a “ valley” anywhere in their closeness centralitysequence. This informally means that higher level nodes do not prefer the exchange of information through their subordinateseven if there are short paths through them. On a CH path the closeness centrality increases monotonically at first up to a point(upstream), then starts to decrease (downstream) until it reaches the destination, or it is just go upstream or downstream allthe way. Fig. 1 b illustrates this graphically. One could argue that maybe short paths on real networks have this property as adefault, but Fig. 3 a - d verify that this is not the case. For comparison we picked random paths between the source-destinationpairs of our empirical paths with the same stretch distribution and plotted the results for that case too. One can see that, whilethe path length distribution is the same for the two datasets, a much larger fraction of stretch-equivalent random paths violatethe CH policy. tretch % o f h i e r . c on f o r m pa t h s ● ● ● ● ●● ● ● ● ●● ● ● ● ●●●● Emp. pathsRandomDS random
Stretch R e l . d i ff a Airport topology policy Stretch % o f h i e r . c on f o r m pa t h s Emp. pathsRandomDS random
Stretch R e l . d i ff b Internet { topology policy Stretch % o f h i e r . c on f o r m pa t h s Emp. pathsRandomDS random
Stretch R e l . d i ff c Brain
Stretch % o f h i e r . c on f o r m pa t h s Emp. pathsRandomDS random
Stretch R e l . d i ff d fit-fat-cat T r a c e CD F ● ● ● ● ● ●● ● ● ● ● ●●● Emp. pathsRandom e Airport T r a c e CD F ● ● ● ● ● ●● ● ● ● ● ●●● Emp. pathsRandom f Internet T r a c e CD F ● ● ● ● ● ●● ● ● ● ● ●●● Emp. pathsRandom g Brain T r a c e CD F ● ● ● ● ● ●● ● ● ● ● ●●● Emp. pathsRandom h fit-fat-cat Figure 3.
Identified routing policies confirmed by our measurement data. Panels a - d show the hierarchical conformity of theempirically-determined paths against stretch. The inset of the plots shows the relative difference between the number of CHpaths in the empirical and the random paths. In the case of small networks there are 15-85% more CH paths in the empiricaltraces but in the case of the large AS level Internet this goes up to 100-500%. The cyan colored data in the plots show thenumber of CH paths in a randomized version of our networks generated with the degree sequence (DS) algorithm whichproduces exactly the same degrees for the nodes but the edges are completely randomized. The plots confirm that thetopological peculiarities of real networks increase the number of CH paths between endpoints with respect to the DS networks(see the explanation brackets between the cyan and magenta colored dots of panel a and b ). However, we argue that the effectof the CH policy is at least that important or even more fundamental (e.g. in case of the Internet). Panels e - h show thecumulative distribution of upstream steps in the traces of our datasets. The empirical paths tend to avoid the core, which isreflected by the much lower number of upstream steps (in comparison with the randomly selected CH paths of the same length)before entering the downstream phase. Figure 3.
Identified routing policies confirmed by our measurement data. Panels a - d show the hierarchical conformity of theempirically-determined paths against stretch. The inset of the plots shows the relative difference between the number of CHpaths in the empirical and the random paths. In the case of small networks there are 15-85% more CH paths in the empiricaltraces but in the case of the large AS level Internet this goes up to 100-500%. The cyan colored data in the plots show thenumber of CH paths in a randomized version of our networks generated with the degree sequence (DS) algorithm whichproduces exactly the same degrees for the nodes but the edges are completely randomized. The plots confirm that thetopological peculiarities of real networks increase the number of CH paths between endpoints with respect to the DS networks(see the explanation brackets between the cyan and magenta colored dots of panel a and b ). However, we argue that the effectof the CH policy is at least that important or even more fundamental (e.g. in case of the Internet). Panels e - h show thecumulative distribution of upstream steps in the traces of our datasets. The empirical paths tend to avoid the core, which isreflected by the much lower number of upstream steps (in comparison with the randomly selected CH paths of the same length)before entering the downstream phase. here can be subtle differences between CH paths of similar length. For example, a path can contain upstream thendownstream steps or downstream steps only. Recall that an upstream step goes towards the core, while a downstream stepgoes towards the periphery of the network. Is there a preference among these? For answering this we plotted the CumulativeDistribution Function (CDF) of CH paths with respect to the number of upstream steps preceding the downstream phase. Forcomparison we have also plotted the results of a random policy which picks randomly from the possible CH paths of the givenlength. The plots confirm that the empirical paths contain much less upstream steps, which means that these paths try to avoidthe core. This finding adds “prefer downstream” as a third identifiable policy component. See Fig. 1 b for an illustration.Our datasets hint at the operation of the “prefer short paths”, “conform hierarchy” and “prefer downstream” policies, withno clear relative priorities among them. In what follows we set up a synthetic toy routing policy which uses these componentsin a prioritized fashion and, we show that using these simple ingredients we can approximate the empirical paths much betterthan simple shortest paths do. According to Fig. 2b the prefer shortest path policy can only have lower priority than the“conform hierarchy” and the “prefer downstream” otherwise we would not experience stretch at all. Since “prefer downstream”implies the conform hierarchy policy, the only reasonable choice is to: prefer hierarchically conform paths at first, then preferdownstream if there is a downstream path and from the remaining paths prefer the short paths. More specifically we define oursynthetic routing policy to ( i ) use CH paths only, ( ii ) use downstream if applicable and from the path remaining after ( i ) and ( ii ) use the shortest one (if there are still multiple choices break ties randomly). We note that such routing policy is not unfamiliar inthe literature [11] [12]. Fig. 4 a shows that this simple routing policy immediately gives very realistic path inflation, close to thestretch computed for the real paths. What this simple algorithm cannot reproduce is that the empirical paths sometimes violatethe “conform hierarchy” and the “prefer downstream” policies. Fig. 4 b shows the CH distribution of the paths generated withour synthetic routing policy over the same network, when the closeness values are slightly randomized. This randomizationcan be interpreted as simulating the case in which nodes do not have full global information about the network and therefore,the precise closeness values of the nodes are not known. Instead the nodes can only have an approximate picture about thecloseness hierarchy. One can see that this modification recovers both the stretch and CH distributions exhibited by the empiricalpaths (see Fig. 2 a - b ).An immediate application of these findings is the more accurate estimation of the network’s response to outer stress (e.g.sudden load increase). Fig 4 c - f show the load of the nodes when carrying the traffic of the empirical paths in comparison withshortest paths and our synthetic toy policy. One can see that real traffic has larger volume (due to the stretch) and it is evenmore concentrated in the core of the network. Our synthetic routing policy recovers this behavior, thus, compared to a simpleshortest path policy, the synthetic policy can more accurately assess the load footprint of real paths on the network. Discussion
Despite the simplicity and intuitive nature of the identified policies, readers may have the feeling that there should be a moresimple explanation out there. For example one could imagine a weighting of the edges over which shortest path computationwill give exactly the same paths that we find in our data. We had two reasons for going this way. First, our data hints for policiesthat are not unfamiliar in the literature. For example, the authors of [11] use a very similar routing algorithm for modeling theflow of information in organizational networks, after an in-depth analysis of such networks. Secondly, we have run experimentsto find an appropriate weighting using edge betweenness (see Appendix A for a brief listing of the results). These experimentssuggest that finding reasonable weights that are able to generate paths matching all of our statistics is far from trivial.
Methods
Collecting or inferring paths in networks is a non-trivial problem. Here we list our methods for every specific networks analyzedin our paper.
Internet AS topology and real AS paths –
The Internet protocols permit the tracing of packets. We have downloadedan AS level Internet topology and full AS level packet traces from CAIDA (Center for Applied Internet Data Analysis, ) recorded on 09/29/2015. The topology contains 52194 nodes and 117251 connections. For this topologywe had around 2 . Air transportation network and flight travels –
The world’s flight map is available from OpenFlights ( ), from which the topology of the air transportation network can be reconstructed. For a realistic estimation of the flightsused by customers, we used the Rome2Rio ( ) trip planner and generated routes between 27444 randomlychosen pairs of airports. From the offered paths we have chosen the cheapest one in the analysis in the paper. However, we notethat picking according to other parameters (lowest number of transfers, lowest travel time) did not qualitatively change ourresults. To achieve a more realistic topology we used airport connections extracted from traceroutes to increase the accuracy ofthe OpenFlight topology. The reconstructed map contained 3433 airports and 20347 flights connecting them. tretch P e r c en t age o f e m p . pa t h s - - l l l l ll AirportInternetBrainfit−fat−cat a Stretch % o f h i e r . c on f o r m pa t h s l l l l ll AirportInternetBrainfit−fat−cat b . . . . . Closeness C u m u l a t i v e l oad llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lll llllllllllllllll lll llll l l Shortest pathEmp. pathsSynthetic c Airport . . . . . Closeness C u m u l a t i v e l oad llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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lllll ll l l l l l l Shortest pathEmp. pathsSynthetic d Internet . . . . Closeness C u m u l a t i v e l oad lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllll ll l l l l lll l Shortest pathEmp. pathsSynthetic e Brain . . . . . Closeness C u m u l a t i v e l oad llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll l Shortest pathEmp. pathsSynthetic f fit-fat-cat Figure 4.
Results of the synthetic routing policy. Our toy policy exhibit very realistic stretch (panel a ) and CH distribution(panel b ) (see Fig. 2 a - b for comparison). Panels c - f present the cumulative load experienced on the nodes as the function ofcloseness, for our four datasets. The blue squares and the red circles show the load footprint of the shortest path policy and theempirical paths respectively. Due to the stretch of the empirical paths, the empirical plots give larger load that is moreconcentrated on the core. Our synthetic algorithm (green triangles) approximates this behaviour better than pure shortest paths. fit-fat-cat word ladder game app and word chains – For collecting paths from word networks we have implemented a wordladder game named “fit-fat-cat” for smartphones. The goal of the game is to transform a source word into a target word throughmeaningful intermediate words by changing only one letter at a time. The word chain fit-fat-cat is a good solution of a gamewith source word fit and target word cat. These word chains, collected anonymously from our users, can be considered asthe footprints of human navigation over the word-maze of the English language. For the reconstruction of the word graphwe have downloaded the official three-letter English Scrabble words from WordFind ( ) and created an edge between all the words differing only in one letter. The collected three-letterword chains were considered as our traces. For capturing only the “working” paths we have filtered out the first 20 games (thewarming up phase) and the games taking more than 30 seconds (when the players are not just using a known path but discoveran unknown one) of every player. After all, we have a dataset of more than 2500 paths from 100+ players.
Human brain and estimated paths –
Getting realistic paths from inside the human brain is extremely hard, if not impossible.As a consequence, almost all studies in the literature concerning path-related analysis assume shortest path signaling paths.Taking into account the extreme non-triviality of path estimation in the brain we ask here if we can use empirical anatomicaland functional data to infer feasible communication traces. Our dataset comprises 40 healthy human subjects who underwentan MRI session where Diffusion Spectrum Imaging (DSI) and resting-state functional MRI data were acquired for eachsubject. DSI data was processed following the procedures described in [13–15], resulting in 40 weighted, undirected structuralconnectivity maps ( GS ) comprising 1015 nodes, where each node represents a parcel of cortical or subcortical gray matter, nd connections represent white matter streamlines connecting a pair of brain regions. Connection weights determine theaverage density of white matter streamlines and here only consider connections with density above 0.0001, resulting in GS with an average of 12596 . ( i ) Identify source-destination pairs with high statistically-dependent brain activity. We searched for pairs of nodes such that thePearson correlation of the BOLD signal time series - without global regression - was above 90%. These nodes were used as thesource-destination pairs of our paths. ( ii ) Determine which nodes are active at every time-step. We say that a node is “active”at a given time-step if the BOLD signal is > γ and “inactive” otherwise. We construct activity vectors for each time-stepindicating which nodes were active. Here we use γ = γ . ( iii ) Constructsubgraphs of active nodes. We constructed a subgraph GS i of GS for each time-step by considering only the nodes that areactive at a given time-step i . ( iv ) Define paths between source-destination node pairs. For all of our source-destination pairs(generated in step ( i ) ), we considered the shortest path in the GS i graphs, if the path existed. If there was multiple shortestpaths between a source-destination pair we choose one randomly. Our source-destination traces include the paths found acrossall GS i subgraphs. It is worth noting that this method assumes that information can only traverse active nodes. Furthermore,we are considering here a model for large spatial and temporal scale communication in brain networks that is not necessarilyapplicable to neural networks at smaller scales. While we cannot validate with empirical data whether these paths are actuallyused for the flow of neural signals, from a path inflation perspective we can consider these paths as a lower bound on the lengthof the real signaling pathways. References Noh, J. D. & Rieger, H. Random walks on complex networks.
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Neuroimage , 2142–2154 (2012). Author contributions statement
A.Cs., A.K., G.R., Z.H., J.B. and A.G. contributed to all of the experiments and the setup of the synthetic routing. M.S.contributed to the fit-fat-cat experiment and A.A-K., A.Gri. P.H. contributed to the measurements on the human brain. Allauthors contributed to and reviewed the manuscript.
Additional information
Competing financial interests
The authors declare no competing financial interests.
A Results with betweenness based egde weights
The researchers’ desire towards simplicity would suggest that there should be a simple weighting of the edges over whichshortest path computation can recover all the observed statistics of real paths. Although we cannot disprove that such weightingexists, we list here some of our experiments to illustrate that obtaining the correct weights is far from trivial. In our first attemptwe set the weights to w i = i , where EB stands for egde betweenness. This setting reflects the intuition that edges withhigh betweenness are “cheaper” or “less congested” or “have more capacity” and using them is better than using hop basedshortest paths (where w i = b ) of Figure 5 shows that computing the shortest paths over this weighting results in theappearance of stretch indeed. However the stretch in this case is way higher than we experience in real networks. In this setting50 −
60% of the paths are inflated in contrary with our real data which exhibit stretch only for ≈
30% of the real paths. Eventhe decay of the stretch distribution is clearly different than the real data replicated in panel ( a ) of Figure 5. We have also triedthe setting w i = √ EB i which produces a very similar plot.We could recover very realistic stretch distribution by applying the weighting: w i = log ( max ( EB ) EB i ) (see panel ( c ) in Figure 5).This setting is based on the observation that the availability of edges with high betweenness is larger i.e. edges in the core aremore reliable. Computing shortest paths over these weights gives a path that is available with the highest probability. Althoughthe stretch distribution is very promising in this case panel ( d ) points that such weighting cannot be the reason for stretch inreal networks. The plot shows ∑ i ∈ SHP w i − ∑ i ∈ RP w i for all source target pairs of our real traces, where SHP stands for the hopbased shortest path using w i = tretch P e r c en t age o f t r a c e s - - - - l l l l ll AirportInternetBrainfit−fat−cat ( a ) Stretch P e r c en t age o f t r a c e s - - - - l l l l l ll AirportInternetBrainfit−fat−cat ( b ) Stretch P e r c en t age o f t r a c e s - - - - l l l l ll AirportInternetBrainfit−fat−cat ( c ) llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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− − − Trace index C o s t ( S H P ) − C o s t ( R P ) ( d ) Figure 5.
Results with betweenness based policies. Panel ( a ) replicates the stretch data from real networks. Panel ( b ) showsthe stretch results of an experiment where all link weights are set to i and shortest paths are generated using these weights.Stretch distribution by using log ( max ( EB ) EB i ) as link weights is presented in panel ( c ). This plot matches better with the real dataon panel ( a ). However, panel ( d ) indicates that this setting cannot be the cause of stretch in real networks, since in thisweighting the sum of weights on the real paths is larger then of the simple hop-based shortest paths where all weights are set toone.) indicates that this setting cannot be the cause of stretch in real networks, since in thisweighting the sum of weights on the real paths is larger then of the simple hop-based shortest paths where all weights are set toone.