{'fraction_non_alphanumeric': 0.03398894551719187, 'fraction_numerical': 0.00660397674251669, 'mean_word_length': 4.665514436762912, 'pattern_counts': {'":': 1, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Network bypasses sustain complexity

Real-world networks are neither regular nor random, a fact elegantly explained by mechanisms such as the Watts-Strogatz or the Barabasi-Albert models. Both mechanisms naturally create shortcuts and hubs, which enhance network's navigability. They also tend to be overused during geodesic navigational processes, making the networks fragile against jamming. Why, then, networks with complex topologies are ubiquitous? Here we show that these models entropically generate network bypasses: alternative routes to shortest paths which are topologically longer but easier to navigate. We develop a mathematical theory that elucidates the emergence and consolidation of network bypasses and measures their navigability gain. We apply our theory to a wide range of real-world networks and find that they sustain complexity by different amounts of network bypasses. At the top of this complexity ranking we found the human brain, what points out the importance of these results to understand the plasticity of complex systems.


Introduction
The advent of Network Science [1,2] was strongly marked by the urgent need to decipher simple mechanisms underlying the formation and growth of natural and artificial networks. These mechanisms should be able to explain the large-scale structural patterns of various complex systems that systematically deviate from those observed from stylized models such as random networks or lattices in a variety of ways. Coining these "complex networks" was not a purely tivity [7] and function robustness against random failures [8], thereby in principle explaining the ubiquity of these mechanisms and the resulting macroscopic patterns. Observe however that the SW mechanism reduces mean path length simply by creating shortcuts, making enhanced connectivity overly dependent -and thus, fragile-on them. Likewise, BA-like networks are robust against random node removal but extremely fragile against targeted attacks of hubs. Now, optimal (geodesic) navigation of SW and BA networks tends on average to "overuse" both shortcuts or hubs, making these the first candidates to experience jamming, a property that induces a failure cascade which can severely harm the macroscopic network's function.
Solving this apparent dilemma between the enhanced communication capacity of real networks and their consequent tendency to congestion requires a finer analysis of the basic mechanisms outlined above.
Our contention in this work is that a network is capable to mitigate the impact of the undesired jamming effects induced by SW and BA-like mechanisms by evolutionarily favouring the consolidation of network bypasses: alternative options to mere geodesic navigation that, if needed, can be used to avoid saturated links and nodes, therefore allowing the overall connectivity to be maintained during congestion and the network to be robust against failure of shortcuts and hubs.
Incidentally, note that the architecture of anatomical networks provide intuitive examples of the relevant role played by network bypasses. A classical example is the case of the so-called "collateral circulation": a system of specialized endogenous bypass vessels present in most tissues providing protection against ischemic injury caused by ischemic stroke, coronary atherosclerosis, peripheral artery disease and other conditions and diseases [9]. Another important example emerges in brain networks: nowadays enough observational evidence supports that they are SW in the Watts-Strogatz sense [10] and possess hubs which create skewness of their degree distributions [11]. At the same time, recent experiments show that the consumption of glucose grows nonlinearly with the degree of the regions (nodes) in the brain [12], i.e. propagating signals using hubs as part of the navigation path might be energetically inefficient. Subsequent studies have therefore started to explore how information can propagate across brain networks by avoiding hubs and using routes alternative to the shortest paths [13,15].
In what follows we start from first principles and develop a theory to define and detect the emergence of network bypasses in both synthetic and real-world networks and quantify their associated gain and impact in terms of network navigability. We will show that the emergence of these network bypasses is an unavoidable (entropic) byproduct of the SW and BA mechanisms themselves and that these bypasses play an optimal role when networks fall in a specific point of SW regime and an intermediate edge density in the sparse regime for BA-like networks.

Results
To fix the intuition, let us begin by illustrating two situations in simple graphs that highlight the importance of bypasses in the operation of a network that harbors transportation and propagation of signals and information. To this aim, we initally consider a particle hopping between the nodes of a network created with the Watts and Strogatz model [3], and we focus on the propagation of the particle between nodes i and j (see Panel (a) in Fig. 1). Starting with rewiring probability p = 0 we have a circulant graph G, and the path P 1 = {i, j − 1, j} of length 2 (highlighted in blue) is the only shortest path connecting i and j. Mimicking the action of SW mechanism kicking in, let us now pick the edge e = (i, i + 1) of G and randomly rewire it, creating G , such that one of the endpoints of e is now j − 1. As a consequence, vertices i and i + 1 drop their degree by one whereas vertex j − 1 increases its degree accordingly. This situation creates a small degree heterogeneity in the graph G which did not exist in the circulant graph G: node j − 1 now participates in many more shortest paths starting elsewhere and ending at vertex j and thus is likely to be often "jammed" with particles. Accordingly, the length-2 path P 1 , in practice, might not be the "best" route to connect i and j, even if is still the shortest path, Figure 1: (Panel a) Illustration of the effects of the edge rewiring process in the Watts-Strogatz model on the paths connecting two arbitrary vertices of the resulting graph: the shortest path P 1 (blue) can by bypassed by the path P 2 (pink), topologically longer but with a lower energetic cost. (Panel b) Similar phenomenon happens when the node h becomes a hub after a rich-get-richer mechanism. Shortest path P 1 (blue) typically cross the hub but, with a sufficiently large mean degree, other paths such as P 2 (pink) can bypass the shortest path P 1 , allowing alternative routes when hubs reach capacity and become saturated or damaged. (Panel c) Navigational dilemma embedded in a network: the green path is the shortest path, but it turns out that the red 'braquistochronic' path is more advantageous as it avoids congestion and is less resistive (see SI section S5 for an explicit calculation). topologically speaking. For instance, the length-3 path P 2 = {i, i − 1, i + 1, j} (highlighted in pink), while being topologically longer, contains nodes i and i + 1 whose degree is at the same time lower than the average and avoids j − 1, hence can be seen as a potentially more ballistic and less-congested route connecting i and j.
A similar situation is depicted in Panel (b) of Fig.1 where node h becomes a hub via a rich-getricher (i.e. BA-like) mechanism. The shortest path between i and j (highlighted in blue) will again be potentially jammed due to a congested h once the BA mechanism enhances such heterogeneity. Now, if the network supports a sufficiently large 1 mean degree -i.e. if the network allows more edges to be formed than a spanning tree-, then other routes can emerge, bypassing 1 yet sufficiently small so that the network is in the sparse regime the hub (red path).
The two examples illustrated in Panels (a) and (b) Fig. 1 raise the question of whether a particle would "prefer" to travel from i to j via the shortest, albeit potentially jammed, path P 1 or along the slightly longer but less congested alternative path P 2 , see also Panel (c) of the same figure for an illustrative cartoon where two alternative routes (a shortest path, in red, and a topologically longer one, in yellow) are highlighted. Observe that an alternative interpretation of this dilemma is that a random walker choosing P 1 might have a higher chance of "diffusing out" -and thus hardly reaching the destination-than choosing P 2 , which yields a longer but more "ballistic" route. Intuition tell us that there should be a trade-off: sometimes P 1 is to be preferred, sometimes P 2 is a contingently better option. Extending this situation to a networkgrowth mechanism, this suggests that the creation of shortcuts (SW) and hubs (BA) should be sustained by the emergence of some low-degree paths bypassing these, and thus the energetic balance needed to produce these bypasses points to the existence of an 'optimal' rewiring probability p (SW) as well as an optimal mean degree for a given hub abundance (BA). In what follows we introduce a formalism that puts these questions and their general solution in a solid grounding.

The concept of Resistive Paths
Starting from first principles, the possible trajectories that a hopping particle can perform over a network G = (V, E) of |V | = n nodes with adjacency matrix A = {A ij } n i,j=1 (whose simplest (unweighted and undirected) definition for its entries is: and A ij = 0 otherwise) can be enumerated by computing the powers of A. A natural way to penalize longer trajectories connecting the same initial and end nodes is to properly weight them where β is an empirical parameter. This expression is known as the communicability function of a graph [16,17]. While originally being a purely combinatorial expression that encapsulate the contributions of different walks in a graph, G(β) indeed appears as a central matrix when analysing a wide variety of dynamics on graphs: in Hamiltonian dynamics, G(β) corresponds to the thermal Green's function of a network of coupled quantum harmonic oscillators [17], and similarly, the so-called self-communicability term G ii (β) quantifies the probability of finding a network, represented by a tight-binding Hamiltonian, in a state with energy E j = −λ j at inverse temperature β [18]. In the context of epidemic processes, G(β) appears in the solution of a linearised upper bound to the susceptible-infected (see SI S1.1) and susceptible-infectedsusceptible models [19,20]. G(β) has also been recently shown to be the the solution a modification of the Kuramoto model [21]. Nowadays communicability is applied across a range of disciplines, from neuroscience [24,25,26,27,28,29,30] or cancer research [31] to ecology [33] or economics [34], to cite a few.
For the sake of exposition, let us from now on consider the setting where particles are hoping across a network, where G ij is the propagator and thus accounts for how much j attracts particles sent from i. Similarly, G ii accounts for how much a node i retains an item at it, as the item returns to i infinitely often. For a particle initially located at the node i the difference.: accounts for the resistance offered by the network to the displacement of a particle sent by node i to the node j, where the smaller the value of R i→j , the higher the probability that the particle does not get trapped at the origin i and can propagate to node j, i.e., there are more conductive walks between i and j than those returning back to the origin. In order to account for the resistance of the displacements between any pair of nodes we should take into account the two possible directions of their mutual communication (i → j and j → i). To this aim, one can symmetrize (2) to define the communication resistance between nodes i and j as: From the definition of the communicability function, and setting β = 1 without loss of generality, we obtain that the communication resistance follows (see SI sections S1.1-3): where (ψ m ) i is the i-th entry of the eigenvector associated to the m-th eigenvalue (λ m ) of A. We rigorously proved that ξ ij is an Euclidean distance (see SI section S2 por a proof). Conceptually, ξ ij is a measure of the network opposition to the flow between i and j, and indeed shares formal similarities to the effective resistance defined in an electric circuit, as we also prove in SI section S1.2.

Network Geometrization and Resistive Shortest Paths
Since ξ ij is an Euclidean distance and particles motion is confined to the network edges, we can proceed to the geometrization of the network. To this aim we consider each edge e = (i, j) ∈ E as a compact 1-dimensional manifold with boundary ∂e = i ∪ j, such that Observe that Eq.5 can have more than one solution, specially for large networks (see SI S4).
Second, and based on the geometrization induced by the communication resistance above, we also define an effective length L p(s→t) by summing the induced length of each of the links involved in p(s → t): At odds with p(s→t) , which blindly assigns the same length (unity) to every edge of the network, L p(s→t) takes into account the topological neighborhoods of each of the nodes in the path and the associated likelihood that the particle might diffuse out of the path, accordingly. Likewise, it penalises paths for which particles take naturally more time to travel due to the structure of the network in which the path is embedded in. The specific path connecting s and t that minimizes this effective length is denoted the Shortest Resistive Path SRP(s, t), defined as: We are now ready to quantify (i) the emergence of potential bypasses -i.e. the proliferation of non-SP paths between any two nodes-and (ii) decide in a principled way when this path redundancy become relevant to the network function -something that, we advance, will happen when SRPs start to differ from SPs-.

Communicability Entropy
To address the first question we quantify, both microscopically and then at the network level, the degree by which, as disorder increases, new routes between edges become available. To this aim, let us return to the WS and BA models that we have considered before. As we have discussed, both the rewiring process and the BA mechanism create degree heterogeneities that intuitively make some a priori 'inefficient' paths (e.g. long ones) to scale up in a pre-defined efficiency ranking: that would indeed be the case of path P 2 connecting nodes i and j (Fig. 1). Now, in practice both WS and BA mechanisms can have heterogeneous effects on this re-ranking, depending on the particularities of the starting and ending nodes i and j (see SI section S3 for an in-depth microscopic analysis on the effect of these local mechanisms on ξ ij and L p(i→j) ) .
We first start by quantifying how these mechanisms generate a richness of possible trajectories connecting any pair of nodes i and j. The probability that a randomly intercepted trajectory indeed corresponds to one connecting i and j is Then the heterogeneity in the different number of choices for the trajectory of a particle, i.e. the trajectory richness of the network is given by the entropy that we call the communicability entropy. From an information-theoretic perspective, this entropy is a measure of the ignorance we have on who is the sender node and receiver node, when intercepting a message navigating the network. Since 0 ≤ S(q) ≤ ln(n(n − 1)/2), the upper bound only reached when the set of probabilties q are uniform, we define a normalized version S(q) := S(q)/ ln(n(n − 1)/2).
Let us now analyze howŜ(q) behaves in our two reference frameworks. Intuitively,Ŝ(q) will increase in the SW model as p increases, since this process increases trajectory richness. Likewise, in a BA model one can vary the network's mean degree: for very small mean degree the resulting BA network is almost tree-like, with no potential bypasses and thus low trajectory richness, whereas when we allow the mean degree to increase, additional routes are formed thus increasing the trajectory richness, henceŜ(q) should also increase. Figures 2.a-b (red axis) confirm our intuitive arguments. In particular, in Fig. 2.a we observe that entropy grows rather quickly in a WS model for small rewiring probability 0 < p ≤ 0.4, reaching a steady maximum afterwards. The impact of rewiring is notably stronger for small p, a region in where few shortcuts are present and each new one makes a difference, than for large values of p, a region for which the entropy saturates very quickly to (Ŝ(q) 1) and the addition of more shortcuts does not make much of a difference beyond a certain p. Figure 2.b reveals a similar behavior ofŜ(q) for the BA model as the mean degree k increases (within the sparse regime for the BA preferential attachment mechanism to hold, see below), reaching full trajectory richness very quickly after a sudden increase in the region of small k values. In short, rewiring an ordered structure and increasing the link density of an heterogeneous network quickly (nonlinearly) boosts the trajectory richness, and thus the amount of potential bypasses to any specific shortest path connecting any pair of nodes.
We now need to quantify when some of these new routes actually may become consolidated bypasses to shortest paths, like the situation illustrated in Fig. 1, where a particle traveling between two nodes i and j "might prefer" to use P 2 , although being longer (in terms of number of edges to be traversed) than the shortest path P 1 .

Bypass consolidation and associated navigability gain
To evaluate the impact of potential bypasses on the actual navigability, using Eq. (6) we say that for any pair of nodes i and j, the SRP between i and j is a consolidated bypass to the shortest path(s) if the effective length of the SRP is smaller than the effective length of the (potentially many) SPs (i.e. L SRP (i, j) < L SP (i, j) for all SPs connecting i and j). Interestingly, this criterion results to be equivalent to check that SRP (i,j) > SP (i,j) (see SI S4 for details). Once bypass detection is done, we need to quantify its impact. A measure that quantifies the impact of bypasses on the network's navigability is the topological length excess (i,j) which indicates that, for a particle traveling between two arbitrary nodes i and j, choosing the consolidated bypass SRP over the SP, while beneficial according to the (hidden) network geometry, leads to an apparent excess of (i,j) % from the topological distance travelled via the shortest path. It turns out that Eq.10 also quantifies the effective distance per link and the resulting gain of using RSP over SP (see SI S4 for a full derivation of these metrics and their interpreration). To extract a global metric for the whole network, we just average (i,j) over all pairs of nodes to define the network navigability gain: Equipped with these metrics, we can now quantify bypass consolidation and its associated navigability gain on relation to both WS and BA mechanisms. When we apply this formalism to the evolving WS graph we obtain the results illustrated in panel (a) of Fig. 2 (left axis). We observe that the navigability gain factor exhibits a clear non-monotonic shape as a function of the rewiring probability p. In fact, our measure detects a maximum for p ≈ 0.15 at which, on average, traveling through the SRP is much more favorable than doing so through the SP. We call this probability the "good navigational point" (GNP) of the network, p GNP . It is interesting to observe that p GNP is a precise location inside the so-called small-world regime. Incidentally, it also appears to coincide with the saturating point of spectral spacing in SW networks [22,23]. Now, note that the SW mechanism consolidates bypasses out of a regular-to-random transition, so comparatively speaking the values of should be typically higher in more structured networks -e.g. in networks with fat-tailed degree distributions like the BA model-where the presence of hubs makes the existence of bypasses even more necessary. This guess is confirmed in Fig.2.b (right axis), in which reaches roughly values one order of magnitude larger in the BA model than those found in a SW model. In this case we observe again non-monotonic behavior of with k , displaying a maximum close to k ≈ 11, i.e. the BA model also has a good navigational point when mean degree is k GNP ≈ 11, where bypassing shortest paths that include hubs is maximally relevant.
To further analyse the impact of bypasses, we compare the values of obtained in a BA model with a given number of nodes N and mean degree k against those obtained for an Erdos-Renyi (ER) graph with the same N and k , this latter being a model with the same number of edges but with a homogeneous (Poisson) degree distribution and thus virtually lacking any hubs.
Results are shown in Panels (c) and (d) of Fig.2 and certify that, in the sparse regime ( k < 35), is substantially larger in BA than in ER, i.e. the effective distance saving of having and using bypasses is considerably more important in heterogeneous networks, as expected [38].
Finally, one can ask about the theoretical upper bound on . Heuristically, the effect of bypasses would be fully maximized in a situation where we add to a given (connected) network a new node that is linked to every other node. Such new node would be a 'super-hub' that makes the network have shortest paths of length ≤ 2 for all pairs of nodes. In this extreme situation, many of the shortest paths will be systematically bypassed, and the effective distance gain would explode, i.e. would explodes: we call this the umbrella effect [39] (see SI S6 for details). Now, is this just a theoretical scenario? It turns out that this situation can take place in an extreme version of the BA model, where k is large enough (compared to the initial seed) so that new nodes entering systematically connect to a large portion of the network. Evidently, in this case there is no preferential attachment anymore, so in some sense the rationale behind the BA model breaks down in this regime of high k . We call this the dense regime of the BA model, and in the inset of panel (c) of Fig.2 we report such explosion of when the network has N = 250 nodes and k > 35.

Effect on Dynamics
So far we have shown that the network growth and complexification induced by WS and BA mechanisms carries as a byproduct the emergence and consolidation of bypasses, which we argue make complex networks more robust against undesirable jamming effects. It is well known that structure affects dynamics, and in this sense we have made preliminary progress on analysing how these bypasses have further effects on the network's function by considering two classical dynamical processes embedded in a network (synchronization and epidemic spreading) and find that their prototypical dynamical fingerprints are affected by bypass consolidation (see SI section S7 and supplementary figure S3 for details).

Empirical networks
To round off, we now consider a total of 37 empirical networks of different nature, including social (4 collaboration networks of different nature, 3 termite mounds), biological (Human brain, C. Elegans, protein-protein interaction, transcription yeast, 15 food webs) and technological ones (air transportation, Internet, 3 electronic circuits, power grid, 5 software networks), see SI section S8.1 for details. In table 1 we present the estimation of the normalized communicability entropyŜ(q) and the values of the navigability gain for all of them. Interestingly, all of them are entropic enough for potential bypasses to have been formed, and in all of them we observe consolidated bypasses ( > 0). At the top of the ranking, the net gain induced by consolidated bypasses reaches over 50% for the Human brain connectome, followed by many other self-organised networks (collaboration, C. Elegans, etc). At the bottom of the list we find designed networks, such as electronic circuits and the power grid, with a discrete 2.6% reduction in effective distance. This is indicative that the power grid, while having hubs to some degree [4,35], has not evolved according to mechanisms such as SW or BA, is not self-organised and as a consequence does not hold the necessary preemptive structural bypasses to avoid systemic failures, as we have seen during blackouts [36].
Our contention is that network bypasses become more relevant as the network complexifies (e.g. via SW or BA mechanisms), as they allow de-saturate shortest paths and hubs and so make networks more robust. In the case of the SW model, bypasses reach an optimal role for a given rewiring probability p GNP ≈ 0.15. We can thus ask to which extent empirical networks are close to their Good Navigational Point, a sort of bypass optimality that in turn induces networks to be Small-World. To address this question, for a given empirical network G with n nodes and mean degreek, we can make the exercise of finding which equivalent SW-generated network G (p) with the same (n,k) better represents G. Technically, we make use of a spectral dissimilarity measure D(G, G ) = n j=1 (λ j (G) − λ j (G )) 2 , where λ j (G) is the j-th eigenvalue of the adjacency matrix of network G. Accordingly, the closest SW-generated network G (p) to a given empirical network G with n nodes is found at p * = argmin p [D(G, G (p)]. In the last column of table 1 we report the values of p * for all empirical networks. Our analysis concludes that networks can be typically clustered in two types: one (which includes the human brain connectome, the protein-protein interaction network, collaboration networks, Roget network and the US air network) where D(G, G (p)) has a non-monotonic shape with a mini-

Discussion
The journey of network complexification is supported by basic mechanisms including the celebrated SW and BA, among others. As the network evolves accordingly, we have shown that it naturally increases its communicability entropy S(q) and, in so doing, it allows for new navigational routes to be built, entropically providing bypass 'candidates' to the network. Our theory allows to detect when some of these new routes consolidate their bypassing property by subsequently getting to be more favorable than the corresponding shortest paths connecting the same pairs of nodes, and we show that consolidation takes place in both SW and BA models. Inter-estingly, we find that the role of bypasses is maximised in a small parameter region -which we call the network's Good Navigational Point-, located in a point inside the Small-World regime and for a specific mean degree in the BA model. We have shown that many empirical networks considered very complex, including brain networks, indeed have Good Navigational Point properties, while those that are not catalogued as self-organised but have been designed tend to not include bypasses in their design, with unfortunate consequences [36].
Our results provide a theoretical and mechanistic support for the role of bypasses in physiological systems such as the ones discussed in the introduction. In this sense, very recently [37] it has been shown that brain function is robust against damage by re-adapting and re-purposing non-damaged links, something that can be interpreted to the brain's ability to re-compute RSPs and thus re-rank bypasses after network damage.
Finally, note that while bypass emergence is contingent on the growth mechanism and thus appears to be a byproduct of it, bypass consolidation however is the effect which probably makes those growth mechanisms to be sustainable in the first place. Simply put, we argue, bypasses sustain complexity. Contributions EE and LL designed the research and EE performed the computational and analytical calculations. EE and LL discussed results. LL wrote the first draft of the paper. Both authors revised the paper.