Routes Obey Hierarchy in Complex Networks

TL;DR

This paper analyzes real-world paths in complex networks, revealing that routing obeys simple hierarchical and downstream policies beyond just shortest paths, which improves understanding of network behavior under stress.

Contribution

It identifies and validates simple routing policies like hierarchy and downstream preference in real networks, and proposes a synthetic policy to replicate observed path properties.

Findings

Paths tend to follow hierarchy and avoid network core.

Shortest path preference is combined with other policies.

Synthetic routing policy reproduces real path statistics.

Abstract

Various hypotheses exist about the paths used for communication between the nodes of complex networks. Most studies simply 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. However, these assumptions are rarely checked against real data. Here we directly analyze the structure of operational paths using real measurements. For this purpose we use existing and newly created datasets having both the topology of the network and a sufficient number of empirically-determined paths over it. Such datasets are processed for air transportation networks, the human brain, the Internet and the fit-fat-cat word ladder game. Our results suggest that from the great number of possible paths, nature seems to pick according to some simple rules, which we will refer to as routing policies. First we confirm, that the preference of short paths is an inevitable policy element, however the observed stretch of the paths suggests that there are other policies at work simultaneously. We identify two additional policies common in our networks: the "conform hierarchy", meaning that the paths should obey the structural hierarchy of the network, and the "prefer downstream" policy which promotes avoiding the network core if possible. Building upon these simple policies, we propose a synthetic routing policy which can recover the basic statistical properties of the operational paths in networks. Our results can be helpful in estimating the reaction of complex systems for stress coming from the outside more accurately than the shortest path assumption permits.

Full text

The implicit “shortest path” assumption, meaning that the used communication path in a network is the one with the shortest length, seems to dominate the network science community and most of the fundamental network metrics (diameter, average path 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, 3, 4]) and hidden structures (e.g. hidden hierarchies and metric spaces [5, 6, 7]) guiding path selection. The contribution of these works is considerable in the modeling and understanding of simple routing strategies that can recover near shortest paths without requiring global knowledge of the topology. However, a lack of confirmation with empirical data leaves an important question open: What kind of paths are actually chosen by nature in real world networks? Here we approach the question of path selection in networks from this lacking empirical angle. Using existing and newly created datasets of the traffic flow on real world networks (see Methods), we compare the topology of the networks to the structure of empirically-determined paths extracted from these datasets. From this comparison we infer common characteristic rules of path selection in different networks, which we call routing policies. Our study here presents the analysis of empirically-determined paths in air transportation networks, the Internet, the fit-fat-cat word morph game, and empirically-inferred paths in the human brain (see Fig. 1a for an illustration). For the remainder of this text we will refer to empirically-determined and inferred paths as empirical paths. The main topological features of our networks and the statistics of the empirical paths are shown in Fig. 2(a).

Our first finding is that traffic in networks does not necessarily follow shortest paths. Fig. 2(b) presents the stretch of the paths which is computed as the length of the empirical path minus the length of the corresponding shortest path having the same source 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 to 4-5 hops. From this result two things follow. First, the plot confirms the shortest path assumption of previous studies in the sense 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.

One such policy our measurements support is the “conform hierarchy” (CH) policy, meaning that the used paths follow the topological hierarchy of the network. For showing this we have computed the closeness centrality of the nodes comprising the empirical paths indicating which (inner or outer) parts of the network the information flows through. We found that most of the empirical paths do not contain a large-small-large pattern forming a “ valley” anywhere in their closeness centrality sequence. This informally means that higher level nodes do not prefer the exchange of information through their subordinates even 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 all the way. Fig. 1b illustrates this graphically. One could argue that maybe short paths on real networks have this property as a default, but Fig. 3a-d verify that this is not the case. For comparison we picked random paths between the source-destination pairs of our empirical paths with the same stretch distribution and plotted the results for that case too. One can see that, while the path length distribution is the same for the two datasets, a much larger fraction of stretch-equivalent random paths violate the CH policy.

There can be subtle differences between CH paths of similar length. For example, a path can contain upstream then downstream steps or downstream steps only. Recall that an upstream step goes towards the core, while a downstream step goes towards the periphery of the network. Is there a preference among these? For answering this we plotted the Cumulative Distribution Function (CDF) of CH paths with respect to the number of upstream steps preceding the downstream phase. For comparison we have also plotted the results of a random policy which picks randomly from the possible CH paths of the given length. The plots confirm that the empirical paths contain much less upstream steps, which means that these paths try to avoid the core. This finding adds “prefer downstream” as a third identifiable policy component. See Fig. 1b for an illustration.

Our datasets hint at the operation of the “prefer short paths”, “conform hierarchy” and “prefer downstream” policies, with no clear relative priorities among them. In what follows we set up a synthetic toy routing policy which uses these components in a prioritized fashion and, we show that using these simple ingredients we can approximate the empirical paths much better than simple shortest paths do. According to Fig. 2(b) 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 prefer downstream if there is a downstream path and from the remaining paths prefer the short paths. More specifically we define our synthetic 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 in the literature [11] [12]. Fig. 4a shows that this simple routing policy immediately gives very realistic path inflation, close to the stretch computed for the real paths. What this simple algorithm cannot reproduce is that the empirical paths sometimes violate the “conform hierarchy” and the “prefer downstream” policies. Fig. 4b shows the CH distribution of the paths generated with our synthetic routing policy over the same network, when the closeness values are slightly randomized. This randomization can 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 the closeness hierarchy. One can see that this modification recovers both the stretch and CH distributions exhibited by the empirical paths (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 4c-f show the load of the nodes when carrying the traffic of the empirical paths in comparison with shortest paths and our synthetic toy policy. One can see that real traffic has larger volume (due to the stretch) and it is even more concentrated in the core of the network. Our synthetic routing policy recovers this behavior, thus, compared to a simple shortest 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 more simple explanation out there. For example one could imagine a weighting of the edges over which shortest path computation will give exactly the same paths that we find in our data. We had two reasons for going this way. First, our data hints for policies that are not unfamiliar in the literature. For example, the authors of [11] use a very similar routing algorithm for modeling the flow of information in organizational networks, after an in-depth analysis of such networks. Secondly, we have run experiments to find an appropriate weighting using edge betweenness (see Appendix A for a brief listing of the results). These experiments suggest 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 analyzed in our paper.

**Internet AS topology and real AS paths – ** The Internet protocols permit the tracing of packets. We have downloaded an AS level Internet topology and full AS level packet traces from CAIDA (Center for Applied Internet Data Analysis, www.caida.org) recorded on 09/29/2015. The topology contains $52194$ nodes and $117251$ connections. For this topology we had around $2.5$ million traces.

Air transportation network and flight travels – The world’s flight map is available from OpenFlights (www.openflights.org), from which the topology of the air transportation network can be reconstructed. For a realistic estimation of the flights used by customers, we used the Rome2Rio (www.rome2rio.com) trip planner and generated routes between $27444$ randomly chosen pairs of airports. From the offered paths we have chosen the cheapest one in the analysis in the paper. However, we note that picking according to other parameters (lowest number of transfers, lowest travel time) did not qualitatively change our results. To achieve a more realistic topology we used airport connections extracted from traceroutes to increase the accuracy of the OpenFlight topology. The reconstructed map contained $3433$ airports and $20347$ flights connecting them.

fit-fat-cat word ladder game app and word chains – For collecting paths from word networks we have implemented a word ladder game named “fit-fat-cat” for smartphones. The goal of the game is to transform a source word into a target word through meaningful intermediate words by changing only one letter at a time. The word chain fit-fat-cat is a good solution of a game with source word fit and target word cat. These word chains, collected anonymously from our users, can be considered as the footprints of human navigation over the word-maze of the English language. For the reconstruction of the word graph we have downloaded the official three-letter English Scrabble words from WordFind (http://www.wordfind.com/scrabble-word-list/) and created an edge between all the words differing only in one letter. The collected three-letter word chains were considered as our traces. For capturing only the “working” paths we have filtered out the first $20$ games (the warming up phase) and the games taking more than 30 seconds (when the players are not just using a known path but discover an 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 anatomical and functional data to infer feasible communication traces. Our dataset comprises 40 healthy human subjects who underwent an MRI session where Diffusion Spectrum Imaging (DSI) and resting-state functional MRI data were acquired for each subject. DSI data was processed following the procedures described in [13, 14, 15], resulting in 40 weighted, undirected structural connectivity maps ($GS$) comprising 1015 nodes, where each node represents a parcel of cortical or subcortical gray matter, and connections represent white matter streamlines connecting a pair of brain regions. Connection weights determine the average density of white matter streamlines and here only consider connections with density above 0.0001, resulting in $GS$ with an average of $12596.2$ connections per subject. Functional MRI data was processed following state of the art pipelines described in [16, 17], yielding a BOLD signal time-series per node, each with 276 points that were sampled every 1920 ms. The magnitude of the BOLD signal is an indicator of the degree of neural activity at a node. Combining structural and functional data, we infer feasible structural pathways through which neural signals might propagate using the following process. $(i)$ Identify source-destination pairs with high statistically-dependent brain activity. We searched for pairs of nodes such that the Pearson correlation of the BOLD signal time series - without global regression

• was above 90%. These nodes were used as the source-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 $>\gamma$ and “inactive” otherwise. We construct activity vectors for each time-step indicating which nodes were active. Here we use $\gamma=0$ but we get qualitatively similar results for near zero $\gamma$. $(iii)$ Construct subgraphs of active nodes. We constructed a subgraph $GS_{i}$ of $GS$ for each time-step by considering only the nodes that are active 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 shortest paths between a source-destination pair we choose one randomly. Our source-destination traces include the paths found across all $GS_{i}$ 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 necessarily applicable to neural networks at smaller scales. While we cannot validate with empirical data whether these paths are actually used for the flow of neural signals, from a path inflation perspective we can consider these paths as a lower bound on the length of the real signaling pathways.

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. All authors contributed to and reviewed the manuscript.

Additional information

Competing financial interests

The authors declare no competing financial interests.

Appendix 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 which shortest path computation can recover all the observed statistics of real paths. Although we cannot disprove that such weighting exists, we list here some of our experiments to illustrate that obtaining the correct weights is far from trivial. In our first attempt we set the weights to $w_{i}=\frac{1}{\text{EB}_{i}}$, where EB stands for egde betweenness. This setting reflects the intuition that edges with high betweenness are “cheaper” or “less congested” or “have more capacity” and using them is better than using hop based shortest paths (where $w_{i}=1$). Panel (b) of Figure 5 shows that computing the shortest paths over this weighting results in the appearance of stretch indeed. However the stretch in this case is way higher than we experience in real networks. In this setting $50-60$% of the paths are inflated in contrary with our real data which exhibit stretch only for $\approx 30$% of the real paths. Even the decay of the stretch distribution is clearly different than the real data replicated in panel (a) of Figure 5. We have also tried the setting $w_{i}=\frac{1}{\sqrt{\text{EB}_{i}}}$ which produces a very similar plot.

We could recover very realistic stretch distribution by applying the weighting: $w_{i}=\log(\frac{\max(\text{EB})}{\text{EB}_{i}})$ (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 are more reliable. Computing shortest paths over these weights gives a path that is available with the highest probability. Although the stretch distribution is very promising in this case panel (d) points that such weighting cannot be the reason for stretch in real networks. The plot shows $\sum_{i\in SHP}w_{i}-\sum_{i\in RP}w_{i}$ for all source target pairs of our real traces, where SHP stands for the hop based shortest path using $w_{i}=1$ and RP stands for the real paths. In $80$% of the cases the real path is longer that the hop based shortest path (where the plot is negative). This means that the hop based shortest path is actually better in this weighting that the real paths.