I Introduction
In studying complex systems via the interconnection of their elements, network science has emerged in the last two decades as an insightful approach for understanding collective behavior in brains, societies, and physical infrastructures. Common network science analysis techniques draw on dynamical systems theory [1, 2] and many universal properties of disparate networks have been found [3]. Other prevalent analysis techniques are based on network flow [4].
We take the flow perspective and introduce a novel notion of network flow that arises in many biological, social, and technological networks, yet has not previously been studied. Consider a network where a commodity to be transmitted from a source node to a destination node can be split into pieces in time and sent in a pipelined fashion over (possibly) several hops using as many time slots as needed. The commodity, however, must go over a single route rather than being split over several routes to be recombined by the destination [5, 6]. This is in contrast to the maximum capacity problem [7, 8]
where flow between two nodes may use as many different routes as needed. This “circuitswitched” model with a single route is prevalent in systems without the ability to split and recombine, e.g. signal flow in simple neuronal networks, message flow in social networks, and the flow of train cars in railroad networks with few engines.
We will see that the optimal paths for the pipelined network flow problem must not only be short in terms of number of hops but also wide in terms of the bottleneck edge in the route. That is, to maximize flow requires finding the single best route between the two nodes: the route that minimizes the weight of the maximumweight edge in the route and yet is short in path length. Finding allpairs shortest paths in weighted networks can be accomplished in polynomial time using Floyd’s dynamic programming algorithm [9]. This is an optimization problem in a metric space. Finding widest paths in weighted networks can be accomplished by taking paths in a maximum spanning tree [6, 5], also in polynomial time. This is an optimization problem in an ultrametric (nonArchimedean) space [10]. As far as we can tell, our problem of finding paths that minimize the widthlength product between two nodes has remained unstudied in the literature. We develop efficient algorithms for finding short and wide paths between two given nodes, as well as for all pairs. As part of the development, we prove correctness and also characterize the computational complexity as polynomial time. Depthfirst search strategies that enumerate all simple paths between two nodes [11], would check a factorial number of paths in the worst case.
Efficient algorithms enable us to characterize the allpairs distribution of short and wide paths for many complex networks. Note that traditional notions of network diameter and average path length are studied extensively in network science [2], but the allpairs geodesic distance distribution of unweighted graphs of fairly arbitrary topology is also starting to be of interest [12, 13], essentially building on results for ErdösRényi random graphs [14, 15]. As far as we know, this distance distribution for weighted graphs remains unstudied, as does the distribution of our shortandwide path lengths. In studying the shortandwide path length distribution, we do not see universality across networks.
To demonstrate the detailed structurefunction insights that pipelined flow gives, we consider neuronal networks. Indeed with advances in experimental connectomics producing wiring diagrams of many neuronal networks, there is growing interest in informational systems theories to provide insight [16, 17, 18]. For concreteness, we focus on hermaphrodite nematode Caenorhabditis elegans, which is a standard model organism in biology [19, 20, 21] and has exactly neurons [22]. We consider three scientific questions in asking whether information transmission through the nervous system is a bottleneck that limits behavior. (Neural efficiency hypotheses of intelligence also argue information flows better in the nervous systems of bright individuals [23, 24].)
Question I.1.
Do neuronal circuits allow behaviors to happen as quickly as possible under information flow limitations imposed by synaptic noise properties and neuronal connectivity patterns?
Question I.2.
Are information flow properties of neuronal networks significantly different from random graphs drawn from ensembles that match other network functionals? That is, are networks nonrandom [25] in allowing information flow that is faster or slower than other networks?
Question I.3.
Does the synaptic microarchitecture of functional subcircuits optimize information flow under constraint on number of synapses?
Since the exact computations performed by the nervous system are unclear, we use general informationtheoretic methods to lower bound optimal computational performance of a given neural circuit in terms of its physical noise and connectivity structure. This approach is inspired by informationtheoretic limits in distributed computing and control [26, 27]. If the performance of a neural circuit is close to the lower bound, then it is operating close to optimally.
We specifically consider gap junctions in C. elegans
, where neurons are directly electrically connected to each other through pores in their membranes. There can be more than one gap junction connecting two given neurons. We, for the first time, model and compute the Shannon channel capacity of gap junctions. Channel capacity used together with the network topology of the system in the shortandwide path computation, and with an estimate of the informational requirements to perform computations, yields a bound on the minimum time to perform biologically plausible computations. Remarkably, when this lower bound is applied to
C. elegans, the result is rather close to behaviorallyobserved timescales. This suggests nematodes may be operating close to the behavioral limits imposed by physical properties of their nervous system.In asking whether the network is nonrandom [25] in allowing behavior that is faster or slower than other networks, we surprisingly find that the complete C. elegans connectome has greater distances and is therefore slower in supporting global information flow than random networks. Contrarily, we prove that the hubandspoke architecture of C. elegans functional subcircuits [28, 29] optimizes computation speed under constraint on number of synapses. As such, global information flow may not be a relevant criterion for neurobiology, at least for a small invertebrate system like C. elegans. Rather, functional subcircuits may be a primary organizational principle.
Ii Pipelining Model of Information Flow
Consider a network where a commodity is to be sent from a source to a destination in a manner that can be split into pieces in time and sent in a pipelined fashion over (possibly) several hops using as many time slots as needed. The commodity must go over a single route rather than being split over several routes to be recombined by the destination. As an example, in C. elegans, each neuron is identified by name and is different from any other neuron [22]; computational specialization may arise from neuronal specialization, which in turn may require specific paths for specific information.
In this model, maximizing flow requires finding the single best route between the two nodes: the route that minimizes the weight of the maximumweight edge in the route and yet is short in path length. In the context of network behavior, note that since we adopt the shortandwide path view of information flow rather than the maximum capacity view [7, 8], bounds on computation speed will be governed by an appropriate notion of graph diameter rather than by notions of graph conductance [27]. Since diameter provides weaker bounds than graph conductance, this is without loss of generality. Our notion of graph diameter is defined in the next section.
Iia Distance and Effective Diameter
Consider the following standard definitions of graph distance for undirected, weighted graphs.
Definition II.1.
Let be a weighted graph. Then the geodesic distance between nodes is denoted and is the number of edges connecting and in the path with the smallest number of hops between them. If there is no path connecting the two nodes, then .
Definition II.2.
Let be a weighted graph. Then the weighted distance between nodes is denoted and is the total weight of edges connecting and , in the path with the smallest total weight between them. If there is no path connecting the two nodes, then .
Another notion of distance arises from the pipelining model of flow. We want a path between two nodes that has a small number of hops but is also such that the weight of the maximumweight edge is small; we measure path length weighted by this bottleneck weight.
Definition II.3.
Let be a weighted graph. Then the bottleneck distance between nodes is denoted and is the number of edges connecting and , scaled by the weight of the maximumweight edge, in the path with the smallest total scaled weight between them. If there is no path connecting the two nodes, then .
Proposition II.1.
If weights of all actual edges are or less, geodesic distance upper bounds the bottleneck distance:
(1) 
Proof.
Consider the path between and that governs . Since the weights of all actual edges are or less, the maximumweight edge weight is or less. Hence the bottleneck weight of this path must be less than or equal to the number of edges connecting and in that path. Since by definition minimizes the bottleneck weight among paths connecting and , . ∎
Proposition II.2.
Weighted distance lower bounds the bottleneck distance:
(2) 
Proof.
Consider the path between and that governs . Let be the maximumweight edge weight and the number of edges in it. Since other weights in the path are only less than , the total weight of this path is less than , but by definition is less than or equal to this quantity. Hence . ∎
It is convenient for the sequel to write these distances as constrained optimization problems. We first define a set of constraints on the decision variables that indicate how different edges are used in optimal paths, and indicates neighborhood.
(3)  
(4)  
(5)  
(6) 
These constraints simply enforce the movement of one unit of flow from node to node and maintain flow balance at all other nodes. Integrality constraints ensure that exactly one path is chosen. These constraints are the same as for standard shortest path problems [4] and therefore the constraint matrix is totally unimodular. The distance expressions use the notation for the edge weight between nodes and .
(7) 
(8) 
(9) 
Notice the objective functions are nonlinear; yet we develop efficient algorithms in Section III.
Any of these distance functions can be used to define the allpairs distance distribution of a network, which is just the empirical distribution of distances among all pairs of vertices, for a graph of size
. These distance distributions have various moments and order statistics, such as the average path length and the diameter.
Definition II.4.
The graph diameter is
(10) 
We also define a notion of effective diameter where node pairs that are outliers in the allpairs distance distribution do not enter into the calculation. Recall that the quantile function corresponding to the cumulative distribution function (cdf)
isfor a probability value
.Definition II.5.
For a network of size , let be the empirical cdf of the distances of all distinct node pairs. Then the effective diameter is:
(11) 
This definition is more stringent than others in the literature [30]. Of course, . Moving forward, we use effective diameter rather than diameter since it characterizes when most of a commodity would have reached its destination. Thresholds other than can be easily defined.
IiB Unique Property of Bottleneck Paths
In this subsection, we discuss a property of shortandwide paths that is distinct from geodesic and weighted paths and that has algorithmic importance.
A key attribute of shortest paths exploited by many shortest path algorithms is the optimal substructure property, that all subpaths of shortest paths are shortest paths [4]. This follows from metric structure, but for the ultrametric space induced by shortandwide paths, the property does not hold. We prove this through a counterexample.
Consider the network in Figure 1, with the network weights shown on the edges. For source node and destination node , the geodesic, weighted, and shortandwide paths are, respectively: (3 units), (1.367 units), and (or , 2 units). Instead, if we compute the same paths with as the source and as the destination, path is the geodesic path (6 units) is the weighted path (3.367 units) and path is the shortandwide path. Notice if we compute the shortandwide path from to , this does not guarantee that its subpath up to node is optimal for source and destination .
This implies shortandwide paths do not form trees, unlike geodesic or weighted paths. Hence, classic treebased shortest path algorithms [4] must be modified significantly for this setting, as we now show.
Iii Efficient Algorithms for Computing Short and Wide Paths
In this section, we present algorithms for computing shortandwide paths.
Iiia OnetoAll Bottleneck Distance Algorithm
We present Algorithm 1 to compute the bottleneck distance. The algorithm maintains label sets at each node , denoted by , where the label sets include: (i) , the index of the label set, (ii) , the value of label in label set at node , = 1, 2, 3; where: (a) tracks the number of edges traversed from until the current node, (b) tracks the maximum width along a path from the origin until the current node, (c) tracks the product of the maximum width and the number of edges traversed from until the current node ; (iii) : predecessor node for label set at node , (iv) : index of predecessor’s label set for label at node . We also let denote the number of nondominated label sets for node .
Definition III.1.
A label set is strictly dominated by label set at node , if and (and consequently, ). Label set is dominated by label set at node , if either: (a) and or (b) and (and again consequently, ). A label set that is not strictly dominated or dominated, is nondominated.
Remark III.1.
Observe that each node can have at most nondominated labels, where is the maximum number of discrete values of weights , over all edges in the network. This is a consequence of the fact and are the only two quantities being tracked (and is the product of and ); and the possibilities for nondominated labels are if and (or vice versa) in labels and of node . Also note that all nondominated labels must be maintained since the optimal substructure property does not hold, and each such label must be propagated to the downstream nodes because it may dominate after propagation. Since is always an integer, the number of discrete weights upper bounds the possible number of nondominated labels that can exist at each node.
Lemma III.1.
Along a given path, the value of the labels on the path monotonically (but not necessarily strictly monotonically) increases.
Proof.
Note that the values in labels and are always nonnegative because the labels measure the number of edges and the max width thus far, respectively. The UpdateLabels operation defined below can only increase label values, and thus, on a given path (including if is the same as ), the label values only increase. ∎
Lemma III.2.
The path corresponding to bottleneck distance will not contain any cycles.
Proof.
Proof is by contradiction. Suppose the path with the bottleneck distance contains a cycle, that is the path is , with cycle . However, by Lemma III.1, we know that as we travel along , the value of only increases. Therefore the path that excludes the cycle will have a smaller value of , contradicting that is the shortest path. ∎
Theorem III.1.
When Algorithm 1 terminates, all nodes have the bottleneck distance from to as labels.
Proof.
The correctness of the algorithm follows because the label sets keep track of , and for each node and each label corresponding to that node . At each iteration, each nondominated label is explored, in increasing order of . That is, each label set from each edge out of the selected node is propagated downstream during the UpdateLabels step. Due to the ConsolidateLabels step at each node in each iteration, a maximum of labels can be present at each node (Remark III.1). All nondominated labels are explored before termination, and due to Lemma III.1 and III.2, all possible paths are explored, resulting in the shortandwide path. ∎
Having established correctness, we also analyze the computational complexity, in terms of the number of nodes , number of edges , and the number of possible discrete edge weights .
Lemma III.3.
The running time of Algorithm 1 is , which is pseudopolynomial.
Proof.
The algorithm is structured like Dijkstra’s algorithm which has complexity [4], but during each iteration, since the UpdateLabels step propagates from each label out of each edge and at each node, a consolidation step must be performed. With a discrete number of weights over all edges in the network, the possible labels at each node is upper bounded by . Hence, the order of the algorithm is . ∎
IiiB AllPairs Bottleneck Distance Algorithm
Now we consider the computation for allpairs, rather than separately computing for each sourcedestination pair. This is Algorithm 2.
Theorem III.2.
When Algorithm 2 terminates, all edges have bottleneck distance from to as labels.
Proof.
Note that the shortandwide path between any nodes and will not contain any cycles, and therefore have at most edges. In the th iteration, we consider adding node to each edge along the path connecting and , from each label set at to each label set at . This algorithm is equivalent to enumerating all possible combinations of the label sets and . If the shortandwide path between and contains node , then the condition will be violated in iteration , and is added to one of the edges on the path. If there is an improving or nondominated label that can be added to a node, it is added even if is not on the bottleneck path between and . If the bottleneck path does not contain node , then the condition will not be violated and the current best path and distance are retained, leading to correctness. ∎
After establishing correctness, we characterize computational complexity in terms of the number of nodes , number of edges , and the number of discrete edge weights .
Theorem III.3.
The worstcase runtime for Algorithm 2 is pseudopolynomial .
Proof.
Note that the structure of the algorithm is similar to the FloydWarshall algorithm for shortest paths, which is [4]. Rather than one comparison inside of this, the bottleneck distance algorithm has two subalgorithms, Algorithm 3 and Algorithm 4, to compare all of the label sets. Each subalgorithm at worst iterates through two label sets in parallel with size if there are levels of weights based on Remark III.1 (or maximum size ). Therefore their runtimes are (or if all edges have different weights). Combining this with the whole algorithm gives the total runtime as (or worstcase of ). ∎
Iv AllPairs Bottleneck Distance Distribution in Complex Networks
With efficient algorithms in place, we can compute the allpairs bottleneck distance distribution for several undirected, weighted, realworld networks drawn from the Index of Complex Networks [31] database, which is publicly available. We choose networks that span a variety of systems, including transportation networks, biological networks, and social networks that may naturally support pipelined flow. Table I details the size, type, and sources of each network.
Name  Nodes  Edges  Type  Source 

US airports  1574  28236  Transportation  US airport networks (2010) [31] 
Mumbai bus routes  2266  3042  Transportation  India bus routes (2016) [31] 
Chennai bus routes  1009  1610  Transportation  India bus routes (2016) [31] 
Author collaborations  475  625  Social  Social Networks authors (2008) [31] 
Freeranging dogs  108  1296  Social  WilsonAggarwal dogs [31] 
Game of Thrones  107  353  Social  Game of Thrones coappearances [31] 
Resting state fMRI network  638  18625  Biological  Human brain functional coactivations [31] 
Human brain coactivation  638  18625  Biological  Human brain functional coactivations [31] 
In Figure 2 we plot the survival functions for the alltoall geodesic, weighted, and bottleneck distances , , and . For each network, we find that the inequalities hold, as required by Propositions II.1 and II.2. If all edge weights were , all distance metrics are equivalent, i.e., . For example, note that in the Authors’ collaboration network (Figure 1(d)), the values of and very nearly coincide because the maximum weight along nearly every path is . However, because a significant number of weights are far from , the values of and do not coincide. Additionally, we observe that the geodesic, bottleneck, and weighted distances diverge as the weights are distributed away from .
Previous investigations of the geodesic distance distribution of random graphs had suggested good fits by Weibull, gamma, lognormal, and generalized threeparameter gamma distributions
[12] as well as basic generative models to explain these distributions. We consider the same parametric families to understand if shortandwide path length distributions are welldescribed by such parametric forms. For each network in Table I, we fit the survival functions from Figure 2 and find goodnessoffit for several different parametric families using the KolmogorovSmirnov test. No single parametric family was best for all networks, but the gamma distribution provided reasonable fits for several networks, as shown in Table II which provides the fitted distribution parameters, along with the and values from the test. As far as we can tell, there is no universality in the bottleneck distance distribution across networks.Name  Shape  Location  Scale  value  

US airports  0.27  0  0.64  5.80  0.63 
Chennai bus routes  0.54  0  0.22  1.03  0.06 
Mumbai bus routes  0.41  0  0.23  21.25  0.095 
Author collaborations  0.83  0  0.32  8.19  0.22 
Freeranging dogs  0.67  0  0.40  3.56  0.61 
Game of Thrones  0.55  0  0.18  4.71  0.008 
Resting state fMRI network  0.61  0  0.19  4.91  0.23 
Human brain coactivation  368.63  7.47  0.02  8.756  0.85 
V C. elegans Neuronal Network: Global and Local Flow
We turn attention specifically to the C. elegans gap junction network, to investigate how information flow limits behavior. A bound on the Shannon capacity [bits/sec] of a single gap junction is developed in the Appendix. Although there is no reason to believe capacityachieving codes are used in neural signaling, Shannon capacity provides bounds on the information rate for any signaling scheme.
The topology of the gap junction network has been characterized in some detail in our prior work [22]. The somatic network consists of a giant component comprising neurons, two small connected components, and several isolated neurons. Within the giant component, the average geodesic distance between two neurons is . Since this characteristic path length is similar to that of a random graph and since the clustering coefficient is large with respect to a random graph, the network is said to be a smallworld network. Moreover, the C. elegans network overall has good expander properties [32, App. C].
Here we compute the effective diameter of the giant component of the gap junction network, with respect to the bottleneck distance. We also find upper and lower bounds. Figure 3 shows the survival function of the empirical allpairs geodesic, weighted, and bottleneck distances. As can be observed, the effective diameter for bottleneck distance is between and . The upper and lower bounds are close to one another since, as shown in Figure 4, the minimax width of paths in the C. elegans network when ignoring path length is almost always one inverse gap junction rather than smaller. Taking path lengths into account only increases the bottleneck width.
Va Limits in Computation Speed
Having characterized the channel capacity of links and the topology of neuronal connectivity, we now develop an informationtheoretic model of computation, which in turn yields a limit on computation speed derived from information flow bounds.
Consider the chemosensing problem faced by an organism like C. elegans. It has different types of chemoreceptors [33] and must take behavioral actions based on the chemical properties of its environment [34, 28]. Suppose that differentiating between chemicals requires bits of information, which we call the message volume . To perform an action, the neurons must reach consensus among themselves based on the sensory neuron signals [28]. This may, in principle, need transport of information between all parts of the neuronal network. We make the following natural assumption: the consensus time is bounded by the amount of time it takes to transport bits of information across the network (maximum over any two pair of neurons).
The intuition behind this assumption is the following, which follows from the sparsity of sensory response in C. elegans [35]. Suppose one part of the neuronal network has “strong” sensing information about an event in the environment, e.g. worm near a chemical, but the other part has little or no such sensing information. Then effectively all information is communicated from one part of the network to the other. Accounting for such instances naturally justifies the above assumption. Note that the actual computational procedures used by C. elegans may require several sweeps of signals through the organism, but for bounding purposes, we assume that one sweep is enough to spread the requisite information.
In effect, the time for transportation of information across the network and hence to reach consensus is bounded below by
This bound uses the geodesic effective diameter and assumes bandwidth Hz or equivalently a refractory period of ms (see Appendix). There is evidence suggesting that the C. elegans refractory period is likely to be near ms instead. In that case, the above bound of ms would become ms. The bound of ms or ms applies to the whole giant component.
There may be smaller subcircuits within the neuronal network responsible for specific functional reactions, within the giant component. If consensus is to be reached only in those functional subcircuits, we should utilize their diameter in place of effective diameter . As explained in Section VB, they may have extremal diameters of . Then the information propagation time would be bounded by ms (or ms under ms refractory period).
Using circuittheoretic techniques, the predicted timescale of operation of functional circuits was between ms and ms [22]. This is clearly an excellent match to the range predicted by our bounds: ms to ms (or ms to ms under the ms refractory period). This is rather surprising given that our technique is only attempting to derive fundamental lower bounds using anatomical information.
Now we compare our lowerbound predictions to experimentally observed behavioral times. This is a true test of our methods in bounding the propagation time for decisionmaking information. The behavioral switch times in response to chemical gradients as fast as ms have been observed in the literature [36, Supp. Fig. 1]. Since the time required for motor action like turning around must also be taken account—the worm can straighten itself in viscous fluids within to ms [37, Fig. 4B]—the lowerbound results are in agreement.
Collectively, these agreements with the model calculation bounds suggest that information propagation is likely to be a primary bottleneck in the behavioral decision making of C. elegans.
VB HubandSpoke Architecture
As mentioned earlier, there are smaller subcircuits of the neuronal network that are responsible for certain functional reactions. For such subcircuits, the hubandspoke architecture has optimality properties. The basic premise is that the diameter of such a subcircuit should be small for computational speed; a hubandspoke network structure provides the smallest possible diameter of with the constraint on the number of edges as well as connectivity requirement. Formally, we state the following easy fact.
Proposition V.1.
Given a connected graph of nodes and edges, the smallest possible diameter is and is achieved by the hubandspoke structure.
Proof.
Clearly for a connected network with nodes and edges, it is a tree (no cycles). Further if there are nodes and edges, there must be a pair of nodes not connected to each other through an edge. Since the graph is connected, they must be at least hops apart. That is, the diameter of such a graph must be at least . A hubandspoke network, by construction has diameter . ∎
Indeed, as shown in Figure 5, certain known functional subcircuits in the C. elegans neuronal network do indeed follow the hubandspoke architecture (or nearly so) [28, 29]. Note that other arguments also suggest the benefits to neuronal networks of small diameter [38], but not the optimality of hubandspoke architectures.
VC Comparison to Random Networks
Now we wish to study whether the bottleneck diameter of the C. elegans gap junction network is more than, less than, or similar to the bottleneck diameter of random graphs that have certain other network functionals fixed.
To evaluate the nonrandomness of the bottleneck diameter of the C. elegans network giant component, we compare it with the same quantity expected in random networks. We start with a weighted version of the ErdösRényi random network ensemble because it is a basic ensemble. Constructing the topology requires a single parameter, the probability of a connection between two neurons. There are gap junction connections over somatic neurons in C. elegans, and so we choose the probability of connection as . After fixing the topology, we choose the multiplicity of the connections by sampling randomly according to the C. elegans multiplicity distribution [22, Fig. 3(B)], which is wellmodeled as a powerlaw with parameter . Note that in general the giant component for such a construction will be much larger than that of C. elegans.
Figure 6 shows the survival function of the empirical allpairs geodesic, weighted, and bottleneck distances of one hundred random networks. A random example is highlighted. As can be observed, the effective diameter bottleneck distance is roughly , significantly less than that for the C. elegans network.
Now we consider a degreematched weighted ensemble of random networks. In such a random network, the degree distribution matches the degree distribution of the gap junction network; the degree of a neuron is the number of neurons with which it makes a gap junction. Such a random ensemble is created using a numerical rewiring procedure to generate samples [39, 40]. Upon fixing the topology, the multiplicity of connections is sampled as for the ErdösRényi ensemble. Note that in general the giant component for such a construction will be much larger than that of C. elegans.
Figure 7 shows the survival function of the empirical allpairs geodesic, weighted, and bottleneck distances of one hundred random networks. A random example is highlighted. As can be observed, the effective diameter for bottleneck distance is just below , quite significantly less than that for the C. elegans network. Comparing Figures 6 and 7 to Figure 3, note that the results hold for many defining thresholds for effective diameter, not just .
These results reveal a key nonrandom feature in synaptic connectivity of the C. elegans gap junction network, but perhaps contrary to expectation. The network has a nonrandomly worse bottleneck diameter compared to basic random graph ensembles. It enables globally slower behavioral speed than similar random networks. In contrast, Section VB had found that at the microlevel of small functional subcircuits, the C. elegans gap junction network has several hubandspoke structures [28, 29], which are actually optimal from an information flow perspective. Thus, these results lend greater nuance to efficient flow hypotheses in neuroscience.
Vi Conclusion
We have modeled pipelined network flow, as arises in several biological, social, and technological networks and developed algorithms to efficiently find shortandwide paths that optimize this novel network flow. This new model of flow also specifically provides a new approach to understand the limits of information propagation and behavioral speed supported by neuronal networks. Within the field of network information theory itself, it is of interest to study the novel notion of bottleneck distance in detail from a theoretical perspective. Since we did not universal scaling laws across several realworld networks, it is also of interest to analytically characterize its distribution in random ensembles such as WattsStrogatz small worlds, BarabásiAlbert scalefree networks, Kronecker random graphs, or random geometric graphs.
Beyond our general study, we also specifically considered circuit neuroscience and connectomics, where the overarching goal is try to understand how an animal’s behavior arises from the operations of its nervous system. The nervous system must transport information from one part to another, whether engaged in communication, computation, control, or maintenance. This paper proposes a way to characterize information flow through the nervous system from detailed properties of anatomical connectivity data and to use this characterization to make lower bound statements on the behavioral timescales of animals.
The efficacy of these techniques was demonstrated by explaining the communication bottlenecks in the gap junction network of the nematode C. elegans. Remarkably, the timescale lower bounds are predictive of behavioral timescales.
In considering the possibility of changing the network topology itself, we discovered that the network has much worse bottleneck distance than similar random graphs (whether ErdösRényi or degreematched). The network does not seem to be optimized for global information flow. On the other hand, we noted the prominence of hubandspoke functional subcircuits in the C. elegans gap junction network and proved their optimality for information flow under number of gap junctions constraints. In terms of neural organization, this suggests that smaller subcircuits within the larger neuronal network are responsible for specific functions, and these should have fast information flow (to quickly achieve the computational objective of that circuit, such as chemotaxis). Behavioral speed of the global network may not be biologically relevant.
As more and more connectomes are uncovered and more details of the biophysical properties of synapses are determined, these information flow techniques may provide a general methodology to understand how physical constraints lead to informational and thereby behavioral limits in nervous systems.
Authors’ Contributions
LM developed and implemented efficient algorithms for computing bottleneck distance, proved their correctness and characterized their complexity, participated in the design of the study, and carried out analysis; LRV conceived of the study, designed the study, coordinated the study, proved certain results, participated in data analysis, and drafted the manuscript; DS conceived of the pipelining model of network flow and proved certain theoretical results; NAP and MEG implemented algorithms and participated in data analysis. All authors gave final approval for publication and agree to be held accountable for the work performed therein.
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Appendix: Gap Junction Links
Although nearly all informationtheoretic investigation of synaptic transmission has focused on electrochemical synapses [41, 42], purely electrical gap junctions are also rather ubiquitous, not only in C. elegans but also in the mammalian brain [43]. Among the somatic neurons in the C. elegans connectome, there are gap junctions [22]. As such, it is important to provide a mathematical model of signal flow through gap junctions, along with the noise that perturbs signals.
Via Thermal Noise
A gap junction is a hollow protein that allows electrical current to flow between cells, and thereby allows signaling in both directions. In previous modeling efforts, it has been determined that gap junctions can essentially just be modeled as resistors with thermal noise that is additive white Gaussian (AWGN) [44]. Noise due to stochastic chemical effects or due to random background synaptic activity [45] need not be considered.
The electrical conductance of a C. elegans gap junction is pS [22]; a resistor with conductance pS corresponds to a resistance of M. In order to compute the root mean square (RMS) thermal noise voltage , we use the JohnsonNyquist formula [46, 47]:
(12) 
where is Boltzmann’s constant J/K, is the absolute temperature in K, is the resistance in , and is the bandwidth in Hz over which the noise is measured. We assume room temperature ( K) and for reasons that will become evident in the sequel, we take the bandwidth to be Hz. Then
(13)  
We consider the thermal noise RMS voltage of a C. elegans gap junction to be this mV value.
ViB Plateau Potentials
Although one might hope that the signaling scheme that can be used over a gap junction is unrestricted, there are biophysical constraints that impose limits. We describe these signaling limitations for C. elegans.
Signaling arises from regenerative events in neurons. Electrophysiologists have described four main types of regenerative events: action potentials, graded potentials, intrinsic oscillations, and plateau potentials. Plateau potentials are prolonged allornone depolarizations that can be triggered and terminated by brief positive and negativecurrent pulses, respectively [48]. Plateau potentials are the biological equivalent of Schmitt triggers [48].
It is thought that plateau potentials may be used by many neurons in C. elegans, and that they may arise through synaptic interaction [48]. RMD neurons in C. elegans have two stable resting potentials, one near mV and one near mV [48], and these values are thought to hold across the nervous system. These two levels can be thought of as the two possible input levels to a gap junction channel:
(14)  
(15) 
as part of a random telegraph signal.
Switching between the two levels cannot happen arbitrarily quickly, as there is an absolute refractory period between regenerative events due to biochemical constraints. Unfortunately, the absolute refractory period for C. elegans is not known due to the difficulty in performing the requisite electrophysiology experiments [email communication with S. R. Lockery (14 Dec. 2010) and later confirmation].
The fastest impulse potentials observed in neurons in common reference are, perhaps, in the Renshaw cells of the mammalian spinal motor system, and have been reported as high as per second [49, p. 47]. Moving forward, we use this as a bound, however the typical value for the absolute refractory period across the animal kingdom is ms; C. elegans may be even slower. This is where the noise bandwidth value of Hz also comes from.
ViC Capacity
Having determined the noise distribution and the signaling constraints, we aim to find the capacity of this continuoustime, intertransitiontimerestricted, binaryinput AWGN channel. This capacity computation problem has been studied by Chayat and Shamai [50], assuming no timing jitter. Rather than using those precise results, we take a simplified approach through the discretization of time. We assume slotted pulse amplitude modulation, which is robust to any timing jitter that may be present in C. elegans signal propagation.
In particular, we consider a discretetime channel with 1700 channel usages per second, two binary input levels of and
, and AWGN noise with standard deviation
. As can be noted, the signaltonoise ratio is rather high,
, and so the capacity will be approximately one bit per channel usage.Just to be sure, we compute this more precisely. The capacity is
(16) 
where is the differential entropy of the distribution:
(17) 
with , see e.g. [51]. Performing the calculation demonstrates the rate loss below bit per channel usage due to noise is negligible. Thus we assume that the capacity of a C. elegans gap junction is bits per second, or equivalently seconds per bit.
Synaptic connection between two neurons may contain more than one gap junction.^{1}^{1}1The mean number of gap junctions between two connected neurons is ; see [22, Fig. 3(b)] for the distribution, which is wellmodeled by a power law with exponent . Although it is difficult to maintain electrical separation between individual gap junctions, for the purposes of this paper we assume that each gap junction can act independently. Hence the channel capacity of parallel gap junction links is simply taken to be the number of gap junctions between the two neurons multiplied by the capacity of an individual gap junction.
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