A continuous spectrum of routing strategies combining local and global information

We model messages or signals transferred from a source brain region to a target brain region as random walkers traversing a brain network, where network nodes and edges represent small cortical parcels that are connected by bundles of axons. We consider the dynamics of such random walkers (signals/messages) on the network, where walkers must reach an a priori specified target node t. Formally, let X be a random variable indicating the current node of the walker, Y the random variable indicating the node to which the walker will move in the next time step, and T the random variable indicating the target node where the walk will terminate (we assume that all nodes can be reached from all nodes in finite time). For all t, we denote the transition probabilities at X = i as:

= Pr(Y = j|X = i,T = t) where ∑_j = 1, and = 0 when there is no connection between nodes i and j. Finally, the walk ends when i = t, in which case = 1 for j = t and 0 for all other j. Formally, the network dynamics for each separate target t form a Markov chain with state t as an absorbing state (see Methods). The set of transition probabilities for all t express the routing strategy that governs the dynamics of walkers (signals) navigating the network.

We specify transition probabilities at every node using a family of dynamical processes that combine local and global information about the network’s topology. To this end, we define the dynamics of the system by tuning a global information bias using the following stochastic model: (1) where is a normalization factor. Transition probabilities are governed by two sources of information:

• a local source of information d_ij denoting the length of the edge connecting i and j (d_ij ≠∞ if and only if a connection between i and j exists).
• a global source of information d_{ij +} g_jt, denoting the minimum distance from node i to target t though node j. This is the sum of the distance between node i and neighbor node j (d_ij), plus the distance from node j to the target node t through the shortest path (g_jt—note that this term has no dependence on i).

The parameter λ controls the extent to which transition probabilities are shaped by global information. Most importantly, λ gradually changes the dynamics on the network from an unbiased random walk towards a shortest-path routing strategy (see Fig 1):

• When λ = 0, a walker’s motion is driven only by local information. Transition probabilities are simply given by and do not depend on the target node (nonetheless, the walk still terminates when it eventually reaches the target node t). In the case of brain networks, where edge-weights express connection strengths or node proximities in the interval (0,1) (this can always be enforced through a unique linear normalization function [32,33]; See Methods), we apply the proximity-to-distance function d_ij = −log(w_ij) and map all edge-weights onto edge-distances. The resulting dynamics , where s_i = ∑_j w_ij is the strength of node i, defines an unbiased random walk on the network where walkers favor transitions through edges with shorter connection distances (i.e. closer proximities). We refer to the unbiased random walk as the reference navigation strategy, P^ref, as it represents a null model of navigation that would naturally take place on the network if no bias is introduced.
• When λ → ∞ global information governs the model and transition probabilities converge to P^t_ij = 1 if the edge {i, j} lies on the (unique) shortest path between i and t (degenerate shortest paths, i.e. more than one shortest path from i to t, are less common in weighted networks, compared to unweighted networks, but see Methods for the case where degenerative shortest paths exist) and = 0 otherwise. Hence, statistics computed on such walks will correspond to a “shortest-path” routing strategy—in particular average walk length will be equal to shortest path length.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. A spectrum of routing strategies.

The parameter λ controls the extent to which routing strategies (transition probabilities) are reshaped by global information. Toy networks in the top row illustrate how transition probabilities, represented by orange arrows, are reshaped as the parameter λ increases. At each node, the orange arrows are proportional to the probability of a walker moving to a neighboring node via that edge. Blue arrows on the toy networks in the bottom row illustrate a possible walk followed by a random walker (signal) going from node 1 to node t, while operating according to the routing strategy represented by the orange arrows. When λ = 0, transition probabilities at each node are proportional to the strength of its connections. Random walkers operating under this routing strategy (the reference navigation strategy, P^ref) diffuse through the network until they eventually arrive at the target node. When λ → ∞, transition probabilities at each node route walkers through the shortest path to the target node; a walker starting at node 1 will follow the shortest path to node t, as illustrated by the blue arrows. In the middle of the spectrum, walker’s dynamics are influenced by global information but still driven partially by local topological properties. Notice that only transition probabilities vary with λ while the underlying network structure remains invariant.

https://doi.org/10.1371/journal.pcbi.1006833.g001

It is worth noting that the model acts on the routing strategies by changing the transition probabilities at each node, but we assume that the topology and weight structure of the network remain unchanged (see Fig 1). This procedure was formally introduced by Lambiotte et al [27], who showed that a biased random walk on a network A can be interpreted as an unbiased random walk on an appropriately defined flow graph A’, where the weights of the connections of A’ dictate the patterns of flow of a diffusion process at equilibrium.

The cost of reshaping the system’s dynamics

We are interested in characterizing the communication cost of the dynamics generated by our model as we gradually increase λ, therefore increasing the global information bias on the dynamics. Here, we focus on two aspects of the cost associated with a communication process. First, we consider a transmission cost, which is the cost associated with messages being transmitted from one node to another. Second, we consider an informational cost, which is the cost associated with using global information to reshape the system’s dynamics and thus route messages efficiently.

We consider a walker navigating the network and acting according to the routing strategies P_λ(Y| X,T). Let = ∑_j P_λ(Y = j| X = i,T = t) d_ij be the immediate transmission cost at node i for a walker going to node t with routing strategy P_λ(Y| X = i,T = t). The immediate transmission cost quantifies the cost associated with X = i partaking in the communication process by relaying the message to one of its neighbors, and in this setting it is equivalent to the expected distance that a walker at node i has to travel to move to a neighbor of i. Let n^t_λ(i,k) be the mean number of times node k is visited by a walker starting at a source node X₀ = i and acting according to a routing strategy P_λ(Y| X = i,T = t). We define the transmission cost of a walk starting at source node X₀ = i and terminating at the target node t as the sum of the immediate transmission costs accumulated at each visited node along a walk, that is = ∑_k n^t_λ(i,k) . Thus, a walk’s transmission cost is equivalent to the mean walk length between nodes i and t, under the routing strategy defined by λ. Noting that the transmission cost is not a symmetric measure, (i.e. may not be the same as , except for when λ → ∞), we can define the average transmission cost of a node acting as a source as , and the average transmission cost of a node acting as a target as . These measures quantify the source and target closeness centrality of each node under a routing strategy: quantifies the average walk length from a node i to any other target node in the network, whereas quantifies the average walk length from any source node to the target node t.

To quantify the informational cost associated with routing messages to a target node t under the routing strategy P_λ(Y| X = i,T = t), we define = KL(P_λ(Y| X = i,T = t)||P^ref(Y| X = i,T = t)) as the informational cost at node X = i, measuring the Kullback-Leibler divergence between the routing strategy P_λ(Y|i,t) and the null model or reference routing strategy P^ref (Y|i,t). The Kullback-Leibler divergence measures the additional bits of information required to manipulate the outgoing transition probabilities at a given node, and adopt a routing strategy that deviates from the reference routing strategy, P^ref (Y|i,t). Hence,the informational cost quantifies the effect of the global information bias by measuring the extent to which the biased dynamics P_λ(Y|i,t) deviate from P^ref (Y|i,t)at node X = i [34].

Then, the informational cost of routing a message from a starting at node X = i to a target node t is the weighted average informational cost across all nodes in the network, weighted by the frequency with which each node is visited along the walk: = . Finally, we can define the average informational cost of a node acting as a source as , and the average informational cost of a node acting as a target as .

It is important to note that, in order to model the system’s dynamics and construct a matrix of transition probabilities P_λ(Y| X,T), the shortest path-length between all node pairs (i.e. the global information term d_{ij +} g_jt) is required, for any λ > 0, as an input parameter to the model. Once P_λ(Y| X,T) are defined by the model, the knowledge about the global topology becomes “fuzzy” or probabilistic andthe dynamics become autonomous; the walker trajectories on the network will evolve according to the dynamics dictated by P_λ(Y| X,T). There is no need for the neural elements to store the global information in a table, as any information about the global topology is “implicitly encoded” as a bias on the random walk, and the degree of such bias is precisely what we quantify with the informational cost measure.

In the following sections we will study the communication costs of routing strategies generated by our stochastic model applied to the structural brain connectivity matrices of two cohorts of healthy subjects. In the main text, we focus on 173 unrelated subjects from the Human Connectome Project (HCP) dataset [35,36]. The Supplementary S1–S5 Figs show results from the replication dataset (LAU), composed of 40 healthy subjects (see Methods). We first analyze cost measures at the global, nodal, and pairwise level, measured and averaged across all subjects (within each cohort). Lastly, we examine individual subject differences with respect to the proposed cost measures. We emphasize that our aim is not to identify a routing strategy for brain communication, but instead, to expose a spectrum of communication dynamics that unifies the classical shortest-paths routing vs. diffusion dichotomy [12,37].

Brain networks are more efficient within an intermediate region of the communication spectrum

By construction, the transmission and informational cost have a competing relationship (or trade-off) such that as we increase λ in the stochastic model, the mean walk lengths () of messages acting according to P_λ become shorter while the bias effect due to global information () increases. This trade-off is shown in Fig 2A where averages of and across all {i,t} pairs are plotted as a function of λ. It can be seen that , measuring the average walk length, approaches a shortest path-length regime at around λ = 1 (ln(λ) = 0 in Fig 2), suggesting that in this regime messages can be efficiently routed at a low informational cost.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. A spectrum of communication processes.

(a) Averages of (red) and (blue) across all node pairs, as a function of λ. Solid red and blue lines correspond to the median across all subjects, whereas the shaded red and blue regions denote the 95^th percentile. (b) Averages of (red) and (blue) across all node pairs. These curves are computed by normalizing and with respect to the same cost measures computed on ensembles of 500 randomized networks (per subject). Shaded red and blue areas indicate sections of the curves and that are smaller than 1, respectively, indicating the regions in the spectrum where the communication cost of empirical networks (i.e. networks that are derived from empirical data) is smaller than the cost computed on the randomized ensembles. The dashed vertical lines are placed at the minimum and maximum of (λ₂ and λ₃, respectively), and at two points near the extremes of the spectrum (λ₁ and λ₄). (c) pairwise values of for all node pairs. (d) pairwise values of for all node pairs. In all panels, λ₁ = e^-4.49, λ₂ = e^-1.64, λ₃ = e^0.37 and λ₄ = e^1.79.

https://doi.org/10.1371/journal.pcbi.1006833.g002

Next, we consider an ensemble of random networks and compare average transmission and informational costs incurred in empirical brain networks and in randomized ensembles of networks. All randomized networks preserve node degree, node strength (evaluated with respect to the proximity edge-weights), and the network’s weight distribution (see Methods). We generate routing strategies P_λ for all randomized networks and normalize the cost measures and of each subject’s empirical brain network with respect to the average cost measures computed across the corresponding randomized counterparts. Fig 2B shows normalized cost measures (red line) and (blue line) as a function of λ. In accordance with prior work (37–39), we find that average walk lengths are shorter for random networks (i.e. > 1) at the extremes of the spectrum, representing the unbiased random walk (P^ref) and shortest path regimes. Interestingly, our analysis reveals an interval of λ values (shaded region in Fig 2B) for which empirical networks exhibit shorter walk-lengths compared to the randomized counterparts (i.e. < 1). Moreover, the informational cost behaves similarly, although the regions < 1 and < 1 barely overlap. Overall, these results show that the randomized counterparts of empirical brain networks are more efficient only at the extremes of the communication spectrum.

Fig 2C and 2D show pairwise and (median across subjects) computed for routing strategies generated with λ₁ = e^-4.49, λ₂ = e^-1.64, λ₃ = e^0.37 and λ₄ = e^1.79 (see dashed vertical lines in Fig 2A and 2B). These values of λ correspond to two points located near the extremes of the communication spectrum, and two points located at the minimum and maximum of the curve , where the empirical networks are most and least efficient compared to their randomized counterparts. As evidenced by the column-like patterns in the matrices corresponding to λ₁ and λ₂, the dynamics of messages navigating the network are strongly determined by the local connectivity of the target node when the global information bias is small. As the bias increases and routing strategies depart from the reference strategy P^ref, the dynamics of messages are less dependent on the target node only. Finally, as walk-lengths converge towards shortest-path, the transmission cost becomes symmetric, i.e., = .

Source vs. target communication cost

We now analyze cost measures at the nodal level. Fig 3A and 3B show scatter plots of the average source and target transmission costs ( and , respectively), and the average source and target informational costs ( and , respectively) associated to all nodes (median across all subjects) for the same values of λ highlighted in Fig 2. Nodes are colored according to their membership in functional intrinsic connectivity networks (ICNs; see Methods), highlighting a tendency of some ICNs to contain an overabundance of costly sources and/or targets, while other ICNs’ cost varies as a function of λ. Interestingly, nodes belonging to the unimodal networks, namely the visual (VIS, colored red) and somatomotor (SM, colored green) networks, exhibit less variability across cost measures. Nodes belonging to the somatomotor network tend to exhibit a high and low for λ < e^0.37, while they also exhibit a consistent low ; nodes belonging to the visual network exhibit high and for λ>e^-4.49, while and vary as a function of λ. We also note that the dorsal attention regions (DA, colored purple) consistently exhibit low and for λ>e^-4.49.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Nodal average transmission costs for four increasingly biased routing strategies.

(a) Scatter plots show the transmission cost associated to each node when it acts as source () and target () during communication processes taking place under routing strategies generated with the values λ₁, λ₂, λ₃ and λ₄. (b) Scatter plots show the informational cost associated to each node when it acts as source () and target () during communication processes taking place under routing strategies generated with the values λ₁, λ₂, λ₃ and λ₄. Markers in the scatter plots in (a) and (b), representing each node, are colored according to the node’s membership in the 7 intrinsic connectivity networks (ICN) defined by Yeo et al. (2011) [71]: Visual (VIS), Somatomotor (SM), Dorsal Attention (DA), Ventral Attention (VA), Limbic (LIM), Frontal Parietal (FP), and Default Mode Network (DMN). The size of the markers is proportional to node’s strength. (c) Correlations between node strength and (red), (orange), (green) and (blue) as a function of λ. Solid lines show median correlation across all subjects, shaded areas surrounding the lines show 95^th percentile. Shaded colored areas between the vertical dashed lines indicate regions where the correlations were not significant (p > 0.001). (d) Correlation between and (red), and and (blue), as a function of λ. Solid lines show medians across all subjects and shaded areas surrounding solid lines show the 95^th percentile. Shaded areas between the vertical dashed lines indicate areas where correlation values were not significant (p > 0.001). In all panels, λ₁ = e^-4.49, λ₂ = e^-1.64, λ₃ = e^0.37 and λ₄ = e^1.79.

https://doi.org/10.1371/journal.pcbi.1006833.g003

In order to assess to what extent high or low nodal costs are driven by the network’s overall topology, as opposed to nodal degree or strength distribution, we standardize nodal costs with respect to the corresponding nodal cost distributions measured on the randomized network ensembles. Significantly high or low standardized nodal cost measures are indicative of global connectivity patterns that are encountered only in empirical brain networks. Supplementary S6 and S7 Figs show thresholded z-scores (α = 0.01) for all nodal cost measures as a function of lambda. As expected, near the extremes of the spectrum (λ = 0 and λ > 1), most nodes exhibit significantly higher costs, compared to the randomized networks, however, significantly low cost regions are found in the middle of the spectrum. Prominent low regions include the right and left hemisphere frontal, precentral, paracentral and postcentral regions; low regions include the right and left posterior cingulate, the supramarginal gyrus, the superior parietal cortex, the precuneus, and the inferior parietal cortex. Prominent low regions are mainly located in the frontal cortex (frontal pole, medial orbital frontal and rostral middle frontal regions), right and left superior parietal regions, the right and left precuneus, and the left cuneus. Interestingly, no significantly low regions were identified, suggesting that randomized topology favors all regions as routing targets, but not as routing sources.

Our analyses also reveal a varying relationship (as a function of λ) between the nodal cost measures and node strength (see Fig 3C). At the extremes of the spectrum, transmission cost is strongly driven by node strength. When λ = 0, the correlation between node strength and and is r = 0.55 and r = -0.61, respectively (p < 0.001), indicating that high strength nodes (hubs) are costly sources but low cost targets with respect to transmission cost. In other words, when the global information bias is low (or zero), messages can be routed at a low transmission cost from any brain region to a hub; conversely, routing a message from a hub to any brain region incurs a high transmission cost. At the other end of the spectrum (i.e. for large values of λ), hub nodes are low cost sources and targets with respect to transmission cost (r = -0.53, p<0.001; note that the orange and red lines in Fig 3C converge). However, in the middle of the spectrum, the average correlation between node strength and is close to zero, whereas the correlation between node strength and remains significant (r ≈ -0.5, p<0.001) throughout the entire spectrum. This varying relationship between node strength and cost measures as a function of λ highlights a distinction between the dynamical measures proposed here, and static centrality measures that rely only on the network’s structure. Static measures such as betweenness centrality, page rank [38] and communicability [39], are strongly driven by nodal degree or strength (See Supplementary S1 Text showing a comparison between and a set of static centrality measures), but are blind to the patterns of flow imposed by the network structure and the dynamics of the system.

The relationship between source and target costs also varies as a function of λ (see Fig 3D). For low values of λ, both and , and and are negatively correlated. In other words, nodes that are costly sources are efficient targets, and nodes that are costly targets are efficient sources. However, the correlations undergo a sign flip as λ increases and and , and and become positively correlated. Note that the correlation between and converges to 1 as these two measures are identical at the shortest-path extreme (the symmetry between and at the shortest path extreme will hold for any undirected network).

A node’s propensity to be a costly transmission/informational source or target is projected onto the cortical surface in Fig 4, where we show the difference between a node’s source and target costs for λ₁ = e^-4.49, λ₂ = e^-1.64, λ₃ = e^0.37 and λ₄ = e^1.79 (same values highlighted in Fig 2 and Fig 3). Cortical regions that are costly sources (compared to the cost of being a target) are colored red whereas regions that are costly targets (compared to the cost of being a source) are colored blue. This analysis reveals that dorsal portions of the precentral and postcentral gyri are increasingly costlier sources in terms of transmission cost, whereas frontal regions of the temporal lobes and inferior frontal areas are increasingly costlier targets, as λ increases. In terms of informational cost, we see the opposite relationship in the same anatomical regions, but the informational cost differences decrease as λ increases.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. A brain region’s propensity to be a costly source or target.

Cortical surfaces show the difference between a node’s source and target transmission costs. (a) for routing strategies generated with the values λ₁, λ₂, λ₃ and λ₄. (b) for routing strategies generated with the values λ₁, λ₂, λ₃ and λ₄. Red colored areas on the cortical surfaces correspond to nodes whose source transmission/informational cost is higher than their target transmission/informational cost. Blue colored areas correspond to nodes whose target transmission/informational cost is higher than their source transmission/informational cost. In all panels, λ₁ = e^-4.49, λ₂ = e^-1.64, λ₃ = e^0.37 and λ₄ = e^1.79.

https://doi.org/10.1371/journal.pcbi.1006833.g004

Routing strategies for privileged nodes

In this section we will explore a different scenario where, in the interest of economizing on informational cost, we allow only a subset of privileged nodes to be affected by the global information bias. We consider increasingly larger size sets of r privileged nodes that are able to reshape their routing strategies according to the influence of global information. Privileged nodes are selected according to different node centrality rankings. Given a centrality-based ranking of nodes, we generate routing strategies for the r-highest ranked (privileged) nodes according to the stochastic model, where λ is an attribute that only applies to the set of privileged nodes; all non- privileged nodes’ routing strategies remain unbiased and are equal to P^ref(X). The left and middle panel of Fig 5 show network average values of and (median across all subjects) as a function of λ for varying fractions of privileged nodes that are selected according to various centrality-based rankings. The black dotted lines show and , respectively, for the case in which all nodes’ routing strategies are biased.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Routing strategies for privileged nodes.

Network average values of (left panel) and (middle panel) as a function of λ (node medians across all subjects) for 22, 55, 110, and 165 privileged nodes (corresponding to 10%, 25%, 50% and 75% of the network’s nodes) that are selected according to betweenness centrality ranking (yellow line), strength ranking (purple line), shortest-path-based closeness centrality (green line), and random-walk-based closeness centrality (blue line). For comparison purposes, we also show cost measures for randomly sampled nodes (red line represents average across 500 samples). The dotted lines show and , respectively, for the case in which all nodes’ routing strategies are biased (i.e. 100% privileged nodes). Right panel shows node stretch distributions for the different sets of privileged nodes and centrality rankings. Black markers indicate the median of the distributions.

https://doi.org/10.1371/journal.pcbi.1006833.g005

This approach reveals three interesting properties about the routing capacity of the brain. First, the composition of the set of privileged nodes matters, as evidenced by the differences in and that are obtained as the set size and composition is varied. Second, for a fixed number of privileged nodes, the more the system economizes on informational cost, the more it expends on transmission cost. For example, routing strategies where we select privileged nodes according to betweenness centrality ranking yield smaller and larger throughout the entire spectrum, compared to other centrality-based privileged node selections. Conversely, routing strategies where we select privileged nodes according to a random walk centrality ranking are the most costly in terms of , but least costly in terms of . Third, a small number of strategically selected privileged nodes can achieve a that approximates the achieved when all nodes are subject to the global information bias. To show this, we compute the stretch of a walk [25] defined as the absolute difference between optimally shortest path lengths obtained when all nodes’ dynamics are biased by global information, and shortest walk length obtained when only privileged node’s dynamics are biased by global information. Node stretch distributions (medians across all subjects) are shown in the right-side panel of Fig 5. We note that when the top 25% betweenness centrality nodes are selected as privileged nodes, the average stretch is only 4.2, in contrast to a stretch of 12.7 obtained when the top 25% random walk closeness centrality nodes are selected. Overall, these results indicate that efficient routing patterns can emerge even when less than half of the nodes are capable of routing information.

A communication cost trade-off within subjects

Our approach allows us to study the variability of communication cost measures across subjects. We first examine whether subjects who exhibit higher values of at λ = 0 (that is, longer walk lengths for the unbiased random walk) will maintain a high throughout the entire spectrum. Fig 6A shows correlations between all subject’s across all values of λ. These correlations show that subjects who exhibit higher values of at λ < e^-3.1 are also subjects with the highest at λ >1, but the relationship is inverted in the middle of the spectrum.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Communication cost trade-off within subjects.

(a) Correlations between all subject’s across all values of λ. Positive correlations are colored in red, negative correlations are colored in blue. (b) Eight subject’s and curves after normalization with respect to the and , respectively. Notice how some subject’s curves decay faster than others, and how some subject’s curves grow faster than others. (c) Scatter plot of the computed areas under the normalized and curves, sowing a trade-off between the decay of and the growth of (the correlation between and is r = -0.74, p < 0.001).

https://doi.org/10.1371/journal.pcbi.1006833.g006

Finally, we investigate if there are differences in how individual subject’s brain networks take advantage of the global information bias. We address this question by measuring the area under each subject’s curve and curve. Moreover, since we are interested in capturing the rate of decay and growth of subject’s and curves, we first normalize each subject’s curve with respect to at λ = 0 (that is, the average length of unbiased random walks), and we normalize each subject’s curve with respect to at λ = e^3.2 (that is, the max value of ). The normalized and curves of 8 subjects are shown in Fig 6B, illustrating curves that decay/grow faster with λ, which we can capture by measuring the area under the curve. Fig 6C shows a scatter plot of the areas under the normalized and curves of all subjects, exhibiting a strong negative correlation between the normalized areas under and (r = -0.74, p<0.001). This strong relationship indicates that there is a trade-off between a brain network’s ability to take advantage of global information to route messages in a fast manner, and the amount of informational cost required to achieve optimally fast routing. How this trade-off is negotiated varies across individual subjects.

Ethics statement

Informed written consent in accordance with the Institutional guidelines (protocol approved by the Ethics Committee of Clinical Research of the Faculty of Biology and Medicine, University of Lausanne, Switzerland) was obtained for all subjects included in the LAU dataset.

The HCP imaging data in this study are from the data sample labeled 100 Unrelated Subjects in ConnectomeDB (https://db.humanconnectome.org), the database managed by the Washington University-University of Minnesota (WU-Minn) consortium of the Human Connectome Project (HCP; http://www.humanconnectome.org). Participants were recruited by the WU-Minn HCP consortium and provided written informed consent prior to experiments [35]. All experimental procedures were approved by the Institutional Review Board (IRB) at Washington University (IRB number 201204036; “Mapping the Human Connectome: Structure, Function, and Heritability”) and no further IRB approval is required for our data analysis.

Data sets

Defining topological distances for weighted human structural connectivity networks

The edge weights of human brain structural connectivity networks are normally defined in terms of proximity measures such as the number of streamlines or fiber densities. These proximity edge-weights are often interpreted as a measure of information flow or traffic capacity that can travel through a connection (a notion that is analogous to the concept of bandwidth in telecommunication networks). Hence, the proximity between two brain regions is determined by the sequence of edges that maximize the traffic or flow capacity. In order to define topological distances on human brain structural connectivity networks, a proximity-to-distance mapping must be applied over the set of edge-weights, such that large edge-weights (large edge-proximities) are mapped onto small edge-distances, and small edge-weights are mapped onto large edge-distances. The proximity-to-distance mapping can be defined in various ways. Following previous work [6,47], in this study we use the mapping d_ij = log(1/w_ij), where w_ij are edge-proximities (i.e. fiber densities) and d_ij are the resulting edge-distances. This mapping has been shown to be less biased towards using only a small set of strong connections for shortest paths [6], and moreover, it yields edge-distances with a log-normal distribution, which is consistent with evidence showing log-normal distributions of synaptic strengths between cortical cells [65] and cortico-cortical projections [66]. Finally, in order to implement this mapping, we first normalize all edge-weights, to ensure that w_ij are bounded in the interval [0,1]. As shown previously [32], there is a unique linear function that can normalize any weighted graph onto the unit interval without affecting network properties: (2)

Here we use ϵ = MIN(w_ij), in order to obtain normalized edge-weights in the interval (0,1) which allows us to apply the proximity-to-distance map d_ij = .

Computation of

Let M = {S, P_λ } be a Markov chain composed by a set of N states S = {1,2, …, N} that correspond element by element to the set of nodes of a graph G with N nodes and E edges; P_λ is the matrix of transition probabilities characterizing the probability of transitioning from one state to another. Then, P_λ(i,j) ≠ 0 if and only if an edge exists between nodes i and j in graph G.

Let X be a random variable indicating the current state of the chain, or equivalently, the current node where the walker is located; Y is the random variable indicating the node to which the walker will move in the next time step, and T is the random variable indicating the target node where the walk will terminate (we assume that M is an irreducible chain). For a given value of λ, and an specified target T = t, let P_λ be the NxN matrix of transition probabilities where elements of P_λ are defined as (3) where is a normalization factor, d_ij is the distance from i to j and g_jt is the geodesic distance from j to the target node t.

We make M an absorbing chain and t an absorbing state by setting all transition probabilities P_λ(Y = j|X = t,T = t) = 0 for j ≠ t and P_λ(Y = j|X = t,T = t) = 1 for j = t, and define Q^t_λ as the (N-1)x(N-1) matrix of transition probabilities from non-absorbing to non-absorbing states. Then, n^t_λ = (I- Q^t_λ)^-1 is the fundamental matrix for the absorbing chain [67], and the elements n^t_λ(i,j) denote the amount of time that the chain spends in the j-th non-absorbing state when the chain is initialized in the i-th non-absorbing state. In other words, if we take P_λ to represent the transition probabilities for a (biased) random walker on graph G, and going from a source node i to a target node t, then n^t_λ(i,j) represents the number of times that the random walker starting at node i visits node j before it reaches node t.

Transition probabilities for degenerate paths

Let π₁ and π₂ be any two paths going from node i to node t through edges {i,j}, and {i,k}, respectively. The ratio between the transition probabilities P^t_ij and P^t_ik is: (4)

Assume that the length of π₁ and π₂ is equal, so d_ij+g_jt = d_ik+g_kt. Then we can write: (5)

Now, let S indicate the set of edges leaving from node i along which there is a shortest path from node i to node t. Since all edges in S lie on shortest paths, for any pair of edges {i,j},{i,k}∈S, it must be that d_ij+g_jt = d_ik+g_kt. Then, when λ→∞, we can write (6)

If the network is unweighted, then all d_ij = const. In that case, all edges in S will have a uniform transition probability from node i.

Note that in the λ→∞ case, only transitions along shortest paths will be allowed. This means that the random walk path lengths will be equal to shortest path lengths.

Randomized networks

For each subject, we created a population of 500 randomized brain networks, with preserved degree and strength sequence, and preserved weight distribution, following the procedure described in [68], which is a modified version of the randomizations proposed in [69,70]. Specifically, the empirical networks were first binarized and then randomized by swapping pairs of connections as proposed by Maslov and Sneppen in [71], thus preserving the binary degree of each node. In order to approximate the strength sequence of the empirical structural connectivity matrices, we shuffle the empirical weights and randomly assign them to the edges of the randomized network. Then, we used a simulated annealing algorithm that minimizes the cost function C = ∑_i|s_i—r_i|, where s_i is the strength of node i in the empirical network and r_i is the strength in the randomized network. The cost function is minimized by randomly permuting weight assignments across edges and probabilistically accepting the permutations that reduced the energy as the temperature parameter of the algorithm is decreased. The annealing schedule consisted of 123 iterations and a starting temperature of t₀ = 100, which was scaled by 0.125 after each iteration. The result of this procedure was an average final energy of C = 0.2797±0.04, which indicates that the average strength discrepancy per node was between 0.0011–0.0014.

Intrinsic connectivity networks

We mapped the Desikan Killiany anatomical parcels used to construct individual subject structural connectivity networks, onto the seven intrinsic connectivity networks (ICN) defined by Yeo et al. (2011) [72]. This parcellation was derived by using a clustering algorithm to partition the cerebral cortex of 1000 healthy subjects into networks of functionally coupled regions. The clustering procedure resulted in the definition of seven clusters comprising systems previously described in the literature including the visual (VIS) and somatomotor (SM) regions, dorsal (DA) and ventral (VA) attention networks, frontoparietal control (FP), limbic (LIM) and default mode network (DMN). The mapping between the Desikan-Killiany anatomical parcels and the seven ICNs from the ICN parcellation was obtained by extracting the vertices of the brain surface corresponding to each anatomical region in the Desikan-Killiany atlas, and then evaluating the mode of the vertices’ assignment in the ICN parcellation.