# Complexity and Vulnerability Analysis of the C. Elegans Gap Junction Connectome

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Investigation of Small, Random Networks

#### 2.2. Application to C. Elegans Data

#### 2.3. Complex Structure and the Rich Club

#### 2.4. Complex Structure beyond Degree

#### 2.5. Robustness to Specific Elimination Order

#### 2.6. Robustness to Initial Edge Choice

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Network Complexity $\Psi (G)$

#### 4.2. $\Delta \Psi (G)$ and Iterative Edge Removal

#### 4.3. Illustrative Example: 12 Nodes of Four Degrees

#### 4.4. The C. Elegans Connectome

#### 4.5. Comparison of C. elegans to Random Connectivity

#### 4.6. C. elegans Rich Club

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) To illustrate our method, we generated 10,000 random Erdős-Rényi networks, all with 12 nodes of four degrees. The distribution of our complexity measure $\Psi (G)$ is shown, along with the networks having the lowest and highest $\Psi (G)$ values; (

**b**) $\Delta \Psi $ of an edge is the amount by which $\Psi (G)$ is reduced if that edge is removed. We progressively delete the edges with the highest $\Delta \Psi $ at each step. In this example, most of the complexity in the graph is contained within the upper pentagram-shaped connection structure; the elimination ordering reveals these “complexity-causing” structures.

**Figure 2.**The C. elegans gap junction connectome has a complexity score of $\Psi ({G}_{gap})=0.00143$. For comparison, we generated 10,000 random networks with the same degree distribution and calculated $\Psi (G)$ for each. The distribution of $\Psi ({G}_{rand})$ is approximately normal, with an average score of $\Psi ({G}_{rand})=0.001173\pm 0.000015$. Thus, the actual C. elegans gap junction network has a complexity 16.5 standard deviations above the mean value for its degree distribution.

**Figure 3.**Distribution of $\Delta \Psi $ values for the intact C. elegans gap junction connectome. The link between the command interneuron pair AVAL/AVAR is a clear outlier, causing by far the largest drop in network complexity. It is notable that every edge below the red line at $\Delta \Psi =-1.083$ involves at least one interneuron. Another notable feature is that some deletions will actually increase the complexity: deleting the edge between the motor neurons VB03/VA07 causes $\Psi (G)$ to increase slightly.

**Figure 4.**(

**a**) We iteratively delete connections with the highest $\Delta \Psi $, causing the complexity to decay as shown by the blue curve. At each point, we calculate $\Psi ({G}_{rand})$ for 256 random graphs with the same degree distribution. The red line shows the mean $\Psi ({G}_{rand})$, with the red band showing the range $\pm 2\sigma $; (

**b**) the same data converted to z-score (i.e., the number of standard deviations by which the actual network differs from random). The C. elegans gap junction network is initially much more complex than randomly expected, but as we successively delete edges, it reveals an underlying network that is much less complex than random.

**Figure 5.**(

**a**) As in [24], the C. elegans synaptic Connectome can be understood to have a “rich club” structure. Edges are classified as “Club” (if between two rich nodes), “Feeder” (if between a rich and poor node), or “Local” (if between two poor nodes); (

**b**) the same curve as Figure 4b, with each deleted edge labeled by class. In the region where the graph is much more complex than average, the procedure disproportionately targets Club and Feeder edges; (

**c**) the fraction of each class which has been deleted at each iteration.

**Figure 6.**(

**a**) the initial degree distribution of the graph. Each node has one of 16 unique degree values, which we label by their relative rank from highest to lowest. The first implicated edge $({i}_{0},{j}_{0})$ connects nodes with degree ranks 1 and 2, such that its “Maximum Degree Rank” is 1; (

**b**) the maximum degree rank of the subsequently targeted edges, plotted in blue. The green line indicates the number of unique degrees, which changes as the degree distribution is altered. The procedure does tend to trim edges to relatively highly connected nodes, but this relationship is not the driving criterion, and it does not simply choose edges based upon connectivity level alone.

**Figure 7.**In 50 trials, we deleted all club edges in a random order, then all feeder edges, and then all local edges. The red/green/blue bands show the average resulting complexity curve within one standard deviation. This was repeated for 50 trials in which we instead deleted local edges, then feeder edges, and then club edges. The resulting distribution of complexity curves is indicated by the blue/green/red bands. The edge deletion order prescribed by our algorithm (i.e., the blue curve in Figure 4a) is shown by the black dotted line. A random Club/Feeder/Local deletion order results in a significantly slower complexity decay than our algorithm prescribes, but leads to a much larger decrease in complexity than the Local/Feeder/Club ordering. Thus, the results implicating the rich club are robust to the specific edge deletion order.

**Figure 8.**The greedy edge-elimination procedure was repeated for all possible choices of initial edge deletions (i.e., the edge chosen for deletion at the first step). Each row corresponds to a different choice of initial edge, with each column showing the class of the edges subsequently deleted by the iterative procedure. Club and Feeder edges are targeted disproportionately regardless of the initial edge choice.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Kunert-Graf, J.M.; Sakhanenko, N.A.; Galas, D.J. Complexity and Vulnerability Analysis of the *C. Elegans* Gap Junction Connectome. *Entropy* **2017**, *19*, 104.
https://doi.org/10.3390/e19030104

**AMA Style**

Kunert-Graf JM, Sakhanenko NA, Galas DJ. Complexity and Vulnerability Analysis of the *C. Elegans* Gap Junction Connectome. *Entropy*. 2017; 19(3):104.
https://doi.org/10.3390/e19030104

**Chicago/Turabian Style**

Kunert-Graf, James M., Nikita A. Sakhanenko, and David J. Galas. 2017. "Complexity and Vulnerability Analysis of the *C. Elegans* Gap Junction Connectome" *Entropy* 19, no. 3: 104.
https://doi.org/10.3390/e19030104