You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Quantitative descriptions of network structure in big data can provide fundamental insights into the function of interconnected complex systems. Small-world structure, commonly diagnosed by high local clustering yet short average path length between any two nodes, directly enables information flow in coupled systems, a key function that can differ across conditions or between groups. However, current techniques to quantify small-world structure are dependent on nuisance variables such as density and agnostic to critical variables such as the strengths of connections between nodes, thereby hampering accurate and comparable assessments of small-world structure in different networks. Here, we address both limitations with a novel metric called the Small-World Propensity (SWP). In its binary instantiation, the SWP provides an unbiased assessment of small-world structure in networks of varying densities. We extend this concept to the case of weighted networks by developing (i) a standardized procedure for generating weighted small-world networks, (ii) a weighted extension of the SWP, and (iii) a stringent and generalizable method for mapping real-world data onto the theoretical model. In applying these techniques to real world brain networks, we uncover the surprising fact that the canonical example of a biological small-world network, the *C. elegans* neuronal network, has strikingly low SWP in comparison to other examined brain networks. These metrics, models, and maps form a coherent toolbox for the assessment of architectural properties in real-world networks and their statistical comparison across conditions.'
author:
- Sarah Feldt Muldoon
- 'Eric W. Bridgeford'
- 'Danielle S. Bassett'
bibliography:
- 'wswn\_bib.bib'
title: 'Small-World Propensity in Weighted, Real-World Networks'
---

Introduction
============

In the era of big data sciences, the ability to infer fundamental principles of system function from large sets of interconnected variables is of increasingly pressing importance for structural characterization, functional prediction, and novel network design. Traditionally, network science has been posited as a particularly appealing framework in which to form such inferences based on its simplicity and ubiquitous utility [@Newman2010]. Yet, it is exactly this simplicity – arguably network science’s greatest strength – which often ignores critical system details, causing it to dual as the field’s greatest potential weakness [@Proulx2005]. For example, many network-based tools were originally developed to examine isolated networks constructed from binary links. However, storage facilities around the world now house rich new data offering multiple instances of the same or related networks, with finely measured weights on links between nodes. Progress in using these data to understand and controllably manipulate complex systems requires complementary theoretical advances in the realism of network-based analysis tools.

Particularly relevant tools are those used for characterization: quantitative statistics to describe the organization or structure of networks, from which one can infer properties of their function. A quintessential example is the structure of small-worldness, ubiquitous across many real-world networks [@Costa:2011do], whose local clustering combined with the ability to move quickly through the network has been shown to have important implications for functions from synchronizability to information flow [@Nishikawa:2003wy; @Lu:2004uh; @Roxin:2004th; @Oliveira:2014dx]. Watts and Strogatz formalized the concept of a small-world network by describing a simple theoretical model on binary networks [@Watts:1998vz]. Nodes are placed on a lattice and connected to their nearest neighbors within a radius, $r$, giving the network a high clustering coefficient. Shortcuts are then introduced to the network by randomly rewiring each edge with some probability, $p$. For $p=0$ the network is a lattice, for $p=1$ the network is random, and for a range of intermediate $p$, the network is small-world.

While this model has proven useful for theoretical investigations [@Percha:2005tv; @Netoff:2004ub; @Newman:1999cv; @Newman:1999wa], it does not provide a direct metric to assess small-worldness in observed real-world networks. If one has observations of a network over a range of size scales, one can determine small-worldness by asking whether the characteristic path length scales as the $log(N)$ where $N$ is the number of nodes in the network [@Newman:2000hc]. More commonly, however, one has observations of networks at a single size scale, and therefore needs to map real-world data to the theoretical Watts-Strogatz model. Such a mapping can be achieved by a comparison of the observed clustering coefficient and path length to that of random and/or lattice networks [@Humphries:2008bx; @Humphries:2006jy; @Telesford:2011dv].

Critically, these models – and associated tools and mappings – have lagged behind advances in the data sciences enabling more comprehensive network measurement. Specifically, these statistics are dependent on network density and neglect critical variables such as the strengths of connections between nodes, limiting their ability to diagnose and compare small-world structure in different networks. To address these limitations, we introduce a novel diagnostic called the Small-World Propensity (SWP), which quantifies the extent to which a network displays small-world characteristics while accounting for variation in network density. We next present (i) a standardized procedure for generating weighted small-world networks, (ii) a weighted extension of the SWP, and (iii) a stringent and generalizable method for mapping real-world data onto the theoretical model. We verify our methods using standard benchmark networks and use them to examine small-world properties of real-world brain networks in humans and neuronal networks in worms. Surprisingly, we observe that the neuronal network of *C. elegans*, originally touted as a quintessential example of a biological small-world network, shows especially weak SWP.

Results
=======

Small-World Propensity
----------------------

To quantify the extent to which a network displays small-world structure, we define the Small-World Propensity, $\phi$, to reflect the deviation of a network’s clustering coefficient, $C_{obs}$, and characteristic path length, $L_{obs}$, from both lattice ($C_{latt}$, $L_{latt}$) and random ($C_{rand}$, $L_{rand}$) networks constructed with the same number of nodes and the same degree distribution:

$$\label{eq:SWP}
\phi = 1-\sqrt{\frac{\Delta_{C}^{2}+\Delta_{L}^{2}}{2}},$$

where $$\Delta_{C}=\frac{C_{latt}-C_{obs}}{C_{latt}-C_{rand}}$$ and $$\Delta_{L}=\frac{L_{obs}-L_{rand}}{L_{latt}-L_{rand}}.$$

The ratio $\Delta_{C/L}$ represents the fractional deviation of the metric ($C_{obs}$ or $L_{obs}$) from its respective null model (a lattice or random network).[^1] See arrows in Fig. \[fig1\]A for a schematic of the deviation, and see *Materials and Methods* for mathematical definitions of $C$ and $L$.

In a standard Watts-Strogatz model, we observe that SWP is maximal for network configurations with the greatest small-world characteristics (Fig. \[fig1\]A-B). For small $p$, the SWP is low, driven by a high path length compared to that of a random network (resulting in a large $\Delta_{L}$). However, as $p$ increases, the path length quickly becomes closer to that of a random network, while the network retains a high clustering coefficient, similar to a lattice. Thus, when the SWP is maximal for $p\approx0.02$, we see equal contributions from $\Delta_{C}$ and $\Delta_{L}$. As $p$ increases further, the network becomes increasingly random: the path length remains small, and the low SWP is now driven by the lack of local clustering (high $\Delta_{C}$).

As illustrated by the application to the Watts-Strogatz model, the SWP is best utilized as a comparative metric to describe the extent to which a network displays small-world structure. While a definitive hard threshold on this value is meaningful only in the context of a theoretical model [@Telesford:2011dv], we pragmatically choose a reference value of $\phi_T=0.6$ to distinguish a network with a strong small-world propensity from a network with weak small-world propensity (Fig. \[fig1\]B).[^2]

The SWP has the critical ability to discern differences in small-world structure across network densities (Fig. \[fig1\]C). As density increases, the range of values spanned by the clustering coefficient and path length decreases (Supplementary Fig. 1). The SWP is normalized by these ranges, minimizing the effects of network density on the calculation. The utility of this feature is particularly evident in comparison to the commonly used small-world index, $\sigma$, proposed by Humphries et al. [@Humphries:2008bx; @Humphries:2006jy]. As seen in Fig. \[fig1\]C-D, unlike the small-world index, the SWP retains a large dynamic range even as network density is increased, accurately pinpointing a small-world regime. (See the [*SI*]{} for a discussion of small-worldness in high density networks.)

![**Small-World Propensity in binary networks.** *(A)* Clustering coefficient and path length as a function of the rewiring parameter, $p$, for a standard Watts-Strogatz formulation of a small-world network with $N=1000$ nodes and $r=5$. The dashed horizontal lines mark the baseline value of the clustering coefficient in a similar lattice network (top) and the baseline value for the path length in a similar random network (bottom). *(B)* SWP calculated for the same network as in panel *(A)*. Error bars represent the standard error of the mean calculated over 50 simulations, and the shaded regions represent the range denoted as SW if using a threshold value of $\phi_T=0.6$. *(C)* SWP as a function of network density (increasing $r$ for $N=1000$ nodes). *(D)* Small-world index for the same networks as in panel *(C)*. \[fig1\]](wswn_fig1_20150331)

Generating Weighted Small-World Networks
----------------------------------------

To extend the SWP to weighted networks, it is necessary to define models of weighted lattice and random networks. Here, we take inspiration from the importance of physical distance in spatially embedded networks [@Barthelemy:2011dq], and the observation that in many real physical and biological networks, the strength of an edge is inversely correlated with the physical distance between nodes, i.e., nodes located near one another tend to be linked by stronger edges than nodes located far from one another [@Kaiser2006; @Levy2010; @Bassett2015]. By incorporating physical space, we define a re-wiring mechanism that allows a weighted network to be manipulated from a lattice network to a random network while maintaining the distribution of edge strengths:

1.  [Begin with a network of $N$ nodes that are arranged on a lattice and connected to all neighbors within a radius, $r$.]{}

2.  [Assign edge weights $w_{ij}$ according to the distance between nodes $d_{ij}$: $$w_{ij}=D_{max}-d_{ij}$$ where $D_{max}=max\{d_{ij}\}+\delta$ is the maximum possible distance between two nodes plus a single unit of measurement of the lattice spacing.[^3]$^{,}$[^4]]{}

3.  [Re-wire each edge with probability, $p$, retaining the weight of the edge.]{}

Following network formation, we assess small-world structure as a function of the rewiring probability by computing weighted versions of the clustering coefficient, $C_{w}$, and characteristic path length, $L_{w}$, (see Fig. \[fig2\]B and *Materials and Methods* for definitions). These variables monitor a transition of our weighted networks through the small-world regime in a manner that is highly similar to that observed in the original Watts-Strogatz procedure for binary networks (compare Figs. \[fig1\]A & \[fig2\]B). We quantify this transition by computing a weighted version of the SWP using $C_{latt}=C_{w}(p=0)$ and $L_{rand}=L_{w}(p=1)$. We observe that the weighted SWP behaves similarly to the unweighted SWP (compare Fig. \[fig1\]B & Fig. \[fig2\]C), and maintains a large dynamic range as network density is increased (Fig. \[fig2\]D).

![**Small-World Propensity in weighted networks.** *(A)* Generation of weighted small-world networks. After building a lattice, the edges are weighted by distance such that close edges have a higher strength than distant edges. These edge weights are then retained as links are rewired with a probability, $p$, to create a weighted small-world network. *(B)* Weighted clustering coefficient and weighted path length as a function of the rewiring parameter, $p$, for a weighted formulation of a small-world network with $N=1000$ nodes and $r=5$. *(C)* Weighted SWP calculated for the same network as in panel *(B)*. Error bars represent the standard error of the mean calculated over 50 simulations, and the shaded regions represent the range denoted as small-world if using a threshold value of $\phi_T=0.6$. *(D)* Weighted SWP as a function of network density (increasing $r$ for $N=1000$ nodes). \[fig2\]](wswn_fig2_20150331)

Mapping Real-World Observations to Theory
-----------------------------------------

To compute the SWP in a real-world network, it is necessary to determine the appropriate comparable lattice and random networks. To account for the effects of network density, we construct lattice and random networks that maintain the network density – number of nodes and edge weight distribution – observed in the real-world network. Specifically, we construct a comparable weighted lattice by arranging the observed edge weights such that the edges that correspond to the smallest Euclidean distance between nodes are assigned the highest weights (Supplementary Fig. 2). For example, for a $1$D lattice with $N$ nodes and unit spacing between nodes, we have $N$ edges with a Euclidean distance of $d=1$ between nodes. We therefore rank the observed edge weights by decreasing strength and randomly distribute the connections with the $N$ highest weights among the edges representing $d=1$. We proceed to distribute the next $N-1$ edges of highest weight among the edges of the lattice corresponding to $d=2$, and we continue to proceed in this manner until the total number of edges in the real-world network have been placed in the lattice. To create a comparable random network, the observed edge weights are randomly distributed among the $N$ nodes of the network. These reference networks are then used to calculate the SWP of the observed network.

Synthetic Benchmark Networks
----------------------------

To further assess the validity of the SWP, we examine its performance on weighted networks with well studied structure: fractal hierarchical (FH) and modular small-world (MSW) networks (see *Materials and Methods*). While both types of networks have a modular structure and therefore dense local clustering, the networks differ in terms of their path length: the MSW networks incorporate random shortcuts of moderate connection strength between modules, decreasing the weighted path length, while the FH networks are composed of hierarchically organized modules that are weakly interconnected, increasing the weighted path length (see Fig. \[fig3\]A). It follows that MSW networks have greater small-world structure than FH networks.

In Fig. \[fig3\], we compare the ability of three metrics to detect small-world structure in these networks constructed at various densities: (i) the weighted SWP computed on the true networks, and (ii) the un-weighted SWP and (iii) Humphries’ small-world index computed on binarized versions of the true networks. In the FH networks, we observe that both the weighted and unweighted SWP values are relatively independent of density, while the small-world index is strongly dependent on density. Moreover, the small-world index classifies low and medium density FH networks as having SW properties ($\sigma > 1$), despite the long path lengths characteristic of these networks. The SWP instead reflects our expectations, and remains below the small-world threshold ($\phi_T=0.6$) even in the case of binarized networks. Importantly, when edge weights are retained, the SWP reveals that the FH networks are quite different than their MSW counterparts and do not show a small-world structure.

In the MSW networks, we observe that the SWP values are sensitive to a meaningful change in the network structure as density increases. Specifically, the path length becomes closer to that of a random network, but the addition of shortcuts between modules reduces the clustering coefficient thereby increasing the deviation from the clustering coefficient of a comparable lattice network $\Delta_C$. This change is reflected by decreasing SWP. The small-world index is less sensitive to the deviation in clustering, and classifies all MSW networks as having small-world structure.

![**Benchmark networks.** *(A)* Adjacency matrices depicting network organization for increasingly dense Fractal Hierarchical (FH) and Modular Small-World (MSW) Networks. *(B)* Comparison of the unweighted SWP, weighted SWP, and small-world index when applied to the matrices in panel *(A)*. The dashed line represents the chosen threshold for indicating small-world structure ($\phi_T=0.6$ for SWP or $\sigma_T=1$ for the small-world index). Stars denote networks classified as small-world. \[fig3\]](wswn_fig3_20150505)

Real-World Brain and Neuronal Networks
--------------------------------------

The quintessential example of a small-world network in biology is the neuronal network of *C. elegans* [@Watts:1998vz]. Indeed, brain networks more broadly have been described as small-world for the past decade [@Bassett:2006kc; @Stam:2012kk]. Yet, with advanced imaging and methodological techniques that provide more detailed measurements of these data with varying network densities and fine-scale estimates of edge strength, the assumptions of small-worldness have been brought into question [@Hilgetag2015]. Here we aim to resolve this controversy by applying the SWP to a representative set of structural and functional brain networks from several species whose properties have been well-studied in previous literature, and whose edges are both binary and weighted (Fig. \[fig4\]A). The three weighted networks are (i) a structural network representing the number of white matter tracts connecting 83 brain regions obtained from human diffusion spectrum imaging (DSI) data [@Gu:2014vx], (ii) a functional network given by correlations between the blood oxygen level dependent signal of 638 brain regions measured using resting state functional magnetic resonance imaging (rs-fMRI) [@Crossley:2013kl], and (iii) a structural network of the cat cortex obtained from tract-tracing studies between 52 brain regions [@Scannell:1999tg]. The two binary networks are (i) the neuronal network of *C. elegans* representing synaptic and gap junction connections between 279 neurons [@Varshney:2011ju], and (ii) a structural network derived from tract-tracing studies between 71 regions in the macaque cortex [@Young:1993if].

We observed that all networks displayed relatively large SWP values, indicating that brain and neuronal networks in general do indeed display small-world properties (Fig. \[fig4\]B). Surprisingly, the network with the lowest SWP, sitting just below the $\phi_T=0.6$ threshold, is the neuronal network of *C. elegans*, the quintessential example of a small-world network in biology [@Watts:1998vz]. We explored this surprising result by examining the contributions of the clustering ($\Delta_C$) and the path length ($\Delta_L$) to the SWP value (Fig. \[fig4\]C). In contrast to the other brain networks studied, the neuronal network of *C. elegans* displays drastically different contributions: a remarkably high $\Delta_C$ indicating divergence from a lattice network, and an exceptionally low $\Delta_L$ indicating similarity to a random network. The observed SWP is therefore driven almost entirely by the short path length, indicating that, in reality, this canonical example of a small-world network, does not strongly embody small-world principles.

![**Real-world brain networks.** *(A)* Adjacency matrices for three weighted brain networks (upper) and two binary brain networks (lower). All matrices are symmetric. *(B)* SWP for the matrices shown in panel *(A)*. The dashed line denotes the $\phi_T=0.6$ threshold. *(C)* Breakdown of the SWP into the individual contributions from the clustering coefficient and path length. High values of $\Delta_{C}$ or $\Delta_{L}$ indicate a large deviation from the comparable benchmark value, which results in a reduction of the SWP. \[fig4\]](wswn_fig4_20150331)

Discussion
==========

As petabytes of rich new data quickly file onto growing storage farms, complex interconnected systems are being increasingly queried. Using these data to characterize and manipulate systems requires the development of network-based analysis tools that accurately account for the newly measured features of the data. Here we offer a novel network statistic (SWP) that accurately quantifies small-world structure in networked systems, that is agnostic to nuisance variables such as density and exquisitely sensitive to critical variables such as the strengths of connections between nodes. This enterprise has required the development of (i) a standardized procedure for generating weighted small-world networks, (ii) a weighted extension of the SWP, and (iii) a stringent and generalizable method for mapping real-world data onto the theoretical model. We have illustrated the application and utility of these methods in the context of both benchmark and real-world data. The work represents a single effort in the much larger space of meeting the demands of big data with increasingly sophisticated network-based tools.

Small-World Statistics for Real-World Data
------------------------------------------

The small-world index [@Humphries:2008bx; @Humphries:2006jy] is the most common statistic to quantify small-world structure in binary networks, but produces values greater than one for a large range of network topologies, obscuring accurate interpretations. Other measures monitor binary topologies with greater sensitivity but remain dependent on network density and independent of edge weights [@Telesford:2011dv]. More recent work has used weighted clustering coefficients and path lengths to compare observed networks to random networks [@Li:2007gf], and to measure local small-world structure [@Bolanos:2013jd] and topological dimension. However, the generalizability of these methods has been hampered by the lack of a theoretical model to construct weighted small-world networks and to study the transition in and out of the small-world regime. (Although, see [@Zhang:2010iq] for a method to generate weighted scale-free small-world networks.) Our work directly fills this gap, allowing the creation of weighted small-world networks with varied degrees of SWP that can be used to compare and contrast network behavior and dynamics throughout the small-world regime.

Binary Categories vs. Continuous Narration
------------------------------------------

Recent network-focused efforts in the applied mathematics, physics, computer science, and engineering communities evidence the age-old tension to retain model simplicity while maximizing pragmatic utility. Network statistics are no exception: we often wish to obtain binary categorizations rather than continuous descriptions. For example, one might wish to emphatically state that a network does or does not display community structure, rich-club architecture, or small-worldness. Yet, arguably a more interesting and useful statement might assess gradations of these properties in real-world systems [@Rombach2014; @Bassett2013]. The true power of the SWP lies not in its ability to define small-world structure, but instead in its ability to quantify and compare the *continuous degree* of small-world structure between different networks. Our work therefore complements ongoing efforts to extend traditionally categorical distinctions to continuous measurments in the context of core-periphery structure [@Rombach2014] and community structure [@Palla2005].

Pragmatic Utility in Neuronal and Brain Networks
------------------------------------------------

Network statistics that are independent of density are critical for network comparisons in real-world settings. In the context of brain networks, the need for such statistics is underscored by the growing interest in examining neuronal networks across (i) time in development and normal aging [@Shi:2013eb; @Tsujimoto:2008ha], (ii) health and disease [@Hoffman:1997tm; @vandenHeuvel:2010ka], and (iii) different stages of neurodegeneration [@Lo:2010kz]. In these and similar contexts, it becomes very difficult to determine if observed differences in small-world structure are simply a function of the differences in network density, or if they represent a true form of topological reorganization. By minimizing the effects of network density in the computation of the SWP, we allow for a more direct comparison of the topological network structure across time, and between groups.

Network statistics that are highly sensitive to edge weights provide increased sensitivity to network function. In the context of brain networks, the need for such statistics has only recently been actively appreciated, as a growing body of literature demonstrates that both healthy and diseased brain function is differentially driven by strong *versus* weak connections [@Bassett:2012dh; @Khambhati:2014wv]. Weak connections have traditionally been ignored because of commonly applied thresholding techniques [@Bassett:2012dh], but have recently been identified as potential biomarkers in psychiatric pathologies [@Bassett:2012dh] and as predictors of cognitive function and fluid intelligence [@Schneidman:2006he; @Santarnecchi:2014ju; @Cole:2012cd]. In the future, it will be interesting to build on weighted network diagnostics like the SWP to provide novel quantifications of weak connectivity and its role in cognition.

Surprising Biological Insights
------------------------------

The canonical example of a biological small-world network is the wiring diagram of *C. elegans*. However, the observation of small-world structure in this organism has been built on a simplification of the weighted wiring diagram to a binary graph. Using the weighted SWP, we observe that in fact, this network displays very little small-world propensity, predominantly due to a lack of local clustering. We speculate that several key biological feautres of this network may explain this surprising contradiction to the historical literature. The wiring diagram of *C. elegans* is (i) a micro-scale network, (ii) represents neuron-to-neuron connections throughout the entire body of the organism rather than only the head, and (iii) is drawn from a comparatively symplistic organanism, evolutionarily speaking [@Bassett:2010hf]. These features of scale, physical extent, and evolutionary class may drive toplogical properties away from the small-world architecture to enable a different class of neural functions than those associated with the brains of higher-order animals. Future work must more stringently examine other canonical examples of small-world networks in diverse real-world systems, and thereby build a more accurate assessment of the role of topology in complex system function.

Materials and Methods
=====================

All analysis was done using MATLAB (MathWorks) and code to compute the SWP in real-world networks can be accessed at http://www.seas.upenn.edu/$\sim$dsb/.

Clustering coefficient
----------------------

To calculate the clustering coefficient, as in [@Watts:1998vz], we first calculate the local clustering coefficient for each node, $c_i$ and then define the clustering coefficient, $C$, to be the average of the local coefficients $$C=\frac{1}{N}\sum_i c_i.$$ For binary networks, the local coefficient, $c_i$, of each node, $n_i$, is the fraction of closest neighbors of $n_i$ that are also connected. Multiple extensions of $c_i$ to weighted networks have been proposed:

Onnela et al. [@Onnela:2005de]: $$c_{i,O} = \frac{1}{k_i(k_i-1)}\sum\limits_{j,k}(\hat{w}_{ij}\hat{w}_{jk}\hat{w}_{ik})^{1/3},
\label{c_onnela}$$

Barrat et al. [@Barrat:2004bk]: $$c_{i,B} = \frac{1}{k_i(k_i-1)}\sum\limits_{j,k}\frac{w_{ij}+w_{ik}}{2\langle w_i \rangle}a_{ij}a_{jk}a_{ik},$$

Zhang et al. [@Zhang:2005er]: $$c_{i,Z} = \frac{\sum\limits_{j,k}\hat{w}_{ij}\hat{w}_{jk}\hat{w}_{ik}}{(\sum\limits_k\hat{w}_{ik})^2-\sum\limits_k(\hat{w}_{ik}^2)}.$$ In these equations, $w_{ij}$ is the strength of a connection between nodes $i$ and $j$, $\hat{w}_{ij} = w_{ij}/max(w)$, $k_i$ is the number of edges connected to $n_i$, and $a_{ij}=1$ if a connection exists between nodes $i$ and $j$ ($a_{ij}=0$ otherwise). For all results reported in the main manuscript, we used the definition given by Onnela et al. (Eqn. \[c\_onnela\]), but similar results were obtained when using other definitions (see Supplementary Fig. 3).

Path length
-----------

The characteristic path length for a network is given by $$L=\frac{1}{N(N-1)}\sum_{i\ne j}d_{ij},$$ where for a binary network, $d_{ij}$ is the shortest path between nodes $i$ and $j$. For weighted networks, as in [@Newman:2001kc], we define the distance between two nodes to be $d_{ij}=1/w_{ij}$.

Synthetic benchmark networks
----------------------------

The synthetic benchmark networks were created using the Brain Connectivity Toolbox [@Rubinov:2010jd] with modifications to make networks symmetric and with weighting schemes as defined in [@Lohse:2014ws]. All synthetic networks have $N=1024$ nodes.

Fractal hierarchical networks were created from the makefractalCIJ function provided in the toolbox, with $mx\_lv=10$, $E=2$, and $sz\_cl =5, 6,$ or $7$ for low, medium, and high density networks, respectively. This results in networks with 6, 5, or 4 hierarchical levels with a base module size of $n=32, 64$ or $128$. At the lowest level of the hierarchy, modules are fully connected, and connections are placed within each hierarchical level with a probability $p= 2^{-l}$, where $l$ is the hierarchical level. The weight of each connection $w_{ij}  = p_{ij}$, such that the weight of a connection is equivalent to the probability that a connection exists. Because this method creates a directed network, the matrices were symmetrized by selecting the upper triangle of the resultant matrix and using these connections to create an undirected, symmetric network. This procedure resulted in final networks with weighted (binary) densities of $4.5 (10.8) \%$, $9.1 (18.7) \%$, and $17.9 (31.3) \%$ for low, medium, and high density FH networks.

Modular small-world networks were created from the makeevenCIJ function provided in the toolbox, with $N=1024$, $K=65000, 100000,$ or $150000$, and $sz\_cl=6$ for low, medium, and high density networks, respectively. This creates networks with 16 fully connected modules of size $n=64$ and $E=K-64512$ randomly distributed edges between modules. Connection strengths within modules were set to $w_{ij}=1$ while strengths of inter-module edges were set to $w_{ij}=0.5$. As with the FH networks, this method creates a directed network, so the matrices were symmetrized by selecting the upper triangle of the resultant matrix and using these connections to create an undirected, symmetric network. This procedure resulted in final networks with weighted (binary) densities of $6.1 (6.2) \%$, $7.9 (9.5) \%$, and $10.3 (14.3) \%$ for low, medium, and high density MSW networks.

This work was supported by the Alfred P. Sloan Foundation, the John D. and Catherine T. MacArthur Foundation, the National Science Foundation through BCS-1441502 from the ENG, CISE, and SBE directorates, and the Army Research Laboratory through contract no. W911NF-10-2-0022 from the U.S. Army Research Office. The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies.

[^1]: Because it is possible for real-world networks to display path lengths or clustering coefficients that exceed that of a lattice or random network, we bound these values between $0$ and $1$. Thus, if $\Delta_{C/L} > 1$ we set $\Delta_{C/L}=1$ and if $\Delta_{C/L} < 0$ we set $\Delta_{C/L}=0$, which guarantees that $\phi$ is bounded in the range $[0\: 1]$.

[^2]: It is also important to stress that although we have suggested $\phi_T=0.6$ as a potential useful threshold above which a network falls in the range typically identified as SW when applying a Watts-Strogatz characterization, more or less stringent definitions of SW structure could be chosen.

[^3]: For a lattice reflecting real data, $\delta$ would equal the precision of the measurement.

[^4]: The inclusion of $\delta$ ensures that no edges will be assigned a weight of $0$.