Generating Biological Networks: A Software Tool

Transcription

1 Generating Biological Networks: A Software Tool Brian Mulhall July 27, 2012 Abstract The study of biological networks and the representation of those networks as graphs has created an opportunity to better fill the needs of researchers who may possess limited experience creating software but nonetheless require software created for a specific task. A problem we have identified as being worthwhile to address is in the need to standardize the set of graphs used by researchers in wide range of network science applications. Researchers trying to produce a set of consistent and comparable results require a software tool that can generate a graph tailored to their specific needs in both an efficient and accurate manner. This consistency can be achieved by creating a software that tool that allows the end user a degree of control over the networks used in their particular application. The approach taken to solve this problem is to sift through numerous graph generating algorithms and try to find or modify an existing algorithm to best suit our stated purpose. The Erdos-Renyi, Watts-Strogatz, Barabasi- Albert, Klemm-Euguilez, Newman and a number of algorithms based on the configuration model were all considered, but only three of these were chosen to implement and then conduct trials on. The efficacy of this approach was tested by running trials on all three algorithms and comparing the anticipated results to the actual results that were observed using the available network analysis software. The algorithm that initially appears to be the most promising is the Degree algorithm. The results that have been obtained to this point, while preliminary in nature give a good deal of insight into which of the three algorithms chosen are most suitable to the ambitious goal of controlling all four stated properties simultaneously. 1

2 1 Introduction This section will briefly define the properties of biological networks that have been targeted as those that should be reproducible in a random graph created by this software tool. The three algorithms that have been implemented will also be described in detail in the following section. This will serve as a means of orientating the reader to the methods and goals of this project. 1.1 Biological Networks Biological processes are best captured when they are represented in form of complex networks such as protein-protein inter-action and metabolic pathway networks. The study of biological networks, their modeling, analysis, and visualization are important tasks in life science today.[1] An understanding of these networks is essential in order to further the related work in fields ranging form social sciences to electrical engineering. The focus of this paper will be limited to biological networks and making sense of much of the data that is now being generated by the study of such networks. The increasing importance of biological networks is also evidenced by the expansion in the number of analytical tools and the quantifiable metrics used in network science, this fact coupled with the rise in publications and researchers working in the field adds credence to suggestion this area of research has a rising level of importance. This topic remains a rich area of research due to the fact that most biological networks are still far from being complete and it is difficult to fully understand the complexity of the relationships and the peculiarities of the data at the present time.[1] The generation of biological networks that are flexible enough to be used in a wide range of applications in the field of network science is the purpose of the research that we have done up to this point. We have concluded that in order to be useful these generated networks need to exhibit a set of properties that have been deemed critical to being truly representative of a real world network. One such property is the small world effect, which is the name given to the finding that the average distance between vertices in a network is small usually scaling logarithmically with the total number of vertices. Another common property is the power law degree distributions that many networks possess. This type of distribution is found in many real world networks and has been empirically found to have an exponent between 2 and 3. A power law degree distribution can be spotted with a linear best fit 2

3 curve when the degree of each node is placed on a log-log plot.the degree of a vertex in a network is the number of other vertices to which it is connected, and one finds that there are typically many vertices in a network with low degree and a small number with high degree, the precise distribution often following a power-law (heavy tailed).[3] A third property that many networks have in common is clustering, or network transitivity, which is the property that two vertices that are both neighbours of the same third vertex have a heightened probability of also being neighbours of one another. The clustering coefficient is one on a fully connected graph and has typical values in the range of 0.1 to 0.5 in many real-world networks.[3] Many real networks are observed to be fundamentally modular, meaning that the network can be seamlessly partitioned into a collection of modules.[4] The community structure of a any given network is also of importance to biological research. Many networks have been observed to possess subsets of vertices within which connections are dense, but between which connections are less dense.[3] The scale-free property existing simultaneously with the networks potential modularity creates a challenge for any random graph generation technique.in order to account for the coexistence of modularity, local clustering and scale-free topology in real systems, we have to assume that clusters combine in an iterative manner. It is possible that the communities themselves also join together to form larger and larger communities still, and that those larger communities are themselves joined together, and so on in a hierarchical structure. Being able to identify these communities could help us to understand these networks more effectively and therefore it is necessary to incorporate this property into any software tool which has our stated purpose.[3] 1.2 Klemm-Eguilez Model The Klemm-Eguilez graph generating model combines the three properties that are associated with being truly representative of real world networks.[3] This makes this model appealing for the purpose of generating networks that can be tuned by the user to exhibit the desired properties to varying degrees. The algorithm itself incorporates elements of preferential attachment which increases the likelihood of there being some nodes which have a higher degree thus leading to what is referred to as a scale free network (a network with a power law degree distribution). The mean geodesic distance is kept low along with a relatively high clustering coefficient. These properties can be theoretically tuned by altering the input parameter m, which can be seen 3

4 in the pseudo code listed below. The details of how the algorithm functions and what the algorithm offers us as we try to create a software tool that satisfies our stated purpose will be given below. The algorithm employs a list of active nodes in order to ensure a preferential attachment model. This allows for fine tuning of this approach so that the small-world and scale-free topology of the Klemm-Egulez network is maintained. It begins with the creation of a fully connected network of a size that is specified as an input parameter. The remaining nodes in the network are introduced sequentially along with edges that total in number equal to that initial network size. This allows that average degree of nodes to be controlled also by the user. This active node list is biased toward containing nodes with higher degrees. A given parameter mu, is the probability with which new edges are connected to non-active nodes. When new nodes are added to the network, each new edge is connected from the new node to either a node in the list of active nodes or with probability mu, to a randomly selected deactive node. The new node is added to the list of active nodes, and one node is then randomly chosen, with probability equal to the input parameter pd, for removal from the active node list. This choice is biased ability pd is biased toward nodes with a lower degree, so that the nodes with ability pd the highest degree are less likely to be chosen for removal. The combination of preferential attachment with the random selection of nodes outside the active nodes for new edges and the final step of edge switching which creates short cuts between any given node and a randomly selected node outside of its neighbourhood allows for a power law degree distribution as well as the small world property.[3] pd = the probability a node will be deactivated kj = the degree of node j //First create the fully connected initial network, and add its nodes to active nodes For: all nodes i = 1m_ Add: i to Active_Nodes For: all nodes j = i+1m If: Undirected Create a reciprocal edge between node i and node j Else: Create a directed edge from node i to node j 4

5 Create a directed edge from node j to node i EndFor EndFor //Second, iteratively, connect remaining nodes to active nodes or a random node with probability m, then remove one active node For: all nodes i = m + 1N For: all nodes j in Active_Nodes Set: Chance = a randomly chosen continuous variable between 0 and 1 If: m > Chance or the set of Deactivated_Nodes = 0 If: Undirected Create a reciprocal edge between i and node j Else: Create a directed edge from node i to node j Create a directed edge from node j to node i Else: Set: Connected = false While: Connected == false Set: node j = randomly chosen from the set of all Deactivated_Nodes Set: Chance = a uniform random number between 0 Set: E = sum(degrees of all Deactivated_Nodes) / E > Chance If: kj If: Undirected Create a reciprocal edge between i and node j Else: Create a directed edge from node i to node j Create a directed edge from node j to node i Set: Connected = true EndWhile 5

6 //Replace an active node with node i. Active nodes with lower degrees are more likely to be replaced. Set: node i as an Active_Node While: node j is not chosen Set: j = uniformly randomly chosen node from Active_Nodes kj ) kj ) / sum(1 / kj = (1 / kj Set: pd Set: Chance = a uniform random number between 0 and 1 > Chance If: pd Set: node j as chosen Remove: node j from Active_Nodes EndWhile EndFor //Third, if directed, use d to set undirected equivalency For: all nodes i = 1N For: all nodes j = the set of all nodes adjacent to node i Set: Chance = a uniform random variable between 0 and 1 If: d > Chance Set: node h = uniformly randomly chosen from the set of all nodes, excluding i and nodes adjacent to i Disconnect: directed edge from node i to node j Connect: directed edge from node i to node h EndFor EndFor Mathematical verification The undirected network of Klemm and Egulez (2002b) 1.3 Newman Algorithm The Newman algorithm generates a graph with tunable clustering coefficient and given degree sequence. This algorithm is a modification of the configuration which wont be discussed in this paper but is the foundation of two of the three algorithms used by this software tool. In this algorithm, for 6

7 each node v, the number of triangles it is part of and the number of single edges incident upon it that do not participate in any triangle are given as input parameters.the sum of all triangles in the graph in its entirety is a multiple of 3 and the sum of all single edges in the graph is a multiple of 2. As the algorithm is based on the configuration model, each node v can be seen as consisting of a given number of stubs and corners. Then a triangle is created by selecting three corners and the three corresponding nodes at random and joining them. This process is repeated until each corner is part of a unique triangle. An edge is created by selecting two stubs and two corresponding nodes at random and creating an edge between them. This process is repeated until each stub is part of a unique edge.[6] As we choose unconnected stubs or corners to form edges or triangles, we keep track of the number of unconnected stubs and corners for each node selected node. The number of unconnected stubs at a particular node is the difference between the initially given single edges and the current single edges at that node. It is also known as the residual single edge degree. Similarly, the number of unconnected corners at a node is the difference between the initially given triangle degree and the current triangle degree at that node. It is known as the residual triangle degree. Conf-1(n, s, t) 1 input: number of nodes n, single edge and triangle degree sequence arrays s and t 2 output: graph (V,E), where V is a set of nodes and E a is set of edges 3 V {1, 2,... n} 4 E 5 rtt 6 rss 7 T total number of trianlges to create 8 M total number of single edges to create 9 while T > 0 do Choose three distinct nodes u, v, w 11 if rt[u] > 0, rt[v] > 0 and rt[w] > 0 12 then E E {(u, v), (v, w), (u, w)} 13 rt[u]rt[u]1 14 rt[v]rt[v] 1 7

8 15 rt[w]rt[w] 1 16 T T 1 17 while M > 0 18 do Choose two distinct nodes u, v from 1 to n 19 if rs[u] > 0 and rs[v] > 0 20 then E E {(u, v)} 21 rs[u]rs[u]1 22 rs[v]rs[v] 1 23 M M 1 24 return (V,E) Figure 7.1: Algorithm Conf Degree Algorithm The Degree algorithm generates a random graph with a degree sequence and clustering coefficient given as input parameters. It takes the number of nodes, the degree sequence and the clustering coefficient as input and generates a simple random graph with these properties. The next step in the algorithm, which is outlined below in pseudo code, is to estimate the number of triangles needed to achieve the given clustering coefficient and then create those triangles. It is done by selecting three stubs at random and joining them to create a triangle much like the Newman algorithm which was described in the previous section. Once the estimated number of triangles are created, a pair of stubs are selected independently at random and joined to create additional edges. We also avoid any self-loops by discarding any iteration that does not produce distinct nodes for creating edges and triangles.[6] For creating a triangle, this algorithm chooses three distinct nodes with a probability proportional to their residual degree. If these three nodes have residual degrees greater than zero than a triangle is created and there residual degrees are updated appropriately as well a the remaining number of triangles and single edges left to create, otherwise the node selection is ignored. This process is repeated until either there no longer remain any triangles or single edges to create or the counter that is keeping track of the total number of iterations is exceeded and the algorithm is terminated short of the number of triangles and edges needed to create. 8

9 Deg(n,d,C) 1 input: number of nodes n, degree sequence array d and target clustering coefficient C 2 output: graph (V,E), where V is set of nodes and E is set of edges 3 V {1, 2,..., n} 4 E 5 M the numner of single ednges to create 6 rd d 7 T the number of trianlges to create 8 while T > 0 9 do Choose three distinct nodes with probability proportional to their residual degree 13 Check rd[u], rd[v] and rd[w]are greater than zero 14 Add edges needed to create a triangle between u,v and w 15 update rd[u], rd[v], rd[w], T and M appropriately. 16 T T 1 17 while M > 0 do Choose two distinct nodes with probability proportional to their residual degree then E E {(u, v)} 20 rd[u]rd[u]1 21 rd[v]rd[v] 1 22 M M 1 2 Methodology The experimental results were gathered by testing the software tool s ability to generate graphs with the required properties and how close those property values were to the ones that had been specified by the user or by using a 9

10 set of analytically determined values if there is not direct analogue between the input parameters and the property values. These trials were done in order to gain some insight into the tools efficacy and efficiency. The tests included timed trials during the construction of networks of various sizes. The networks were all generated with an expectation of a particular clustering coefficient as well as a particular degree distribution and the analytical results generated by Cytoscape (a software tool used for network analysis) were used to cross reference the empirical results with the anticipated results. The results of these tests would be used to off insight into which of the implemented algorithms is most appropriate to solve the stated problem of creating a set of standardized biological networks. 3 Experimental Results Degree Algorithm Table 1: Clustering Coefficient Initial Triangle Degree Anticipated Clustering Coefficient Newman Algorithm Table 2: Newman Algorithm Clustering Coefficient, Numbers of Nodes = 1000 and the initial single edge degree is for all nodes = 0 4 Discussion The results that have been obtained to this point while preliminary can offer some insight into which of the three algorithms that were chosen for implementation are best suited for creating networks that have tunable global properties. The Degree algorithm is an efficient algorithm being capable of creating graphs of large size in relatively short amount of time. This coupled with its performance when comparing the anticipated clustering coefficient to what was observed empirically as well being able to accept a degree sequence as an input parameter which guarantees a power law degree distribution if that is so desired by the user. 10

11 The Newman algorithm was also able to generate results efficiently and accurately when compared to the set of analytically determined clustering coefficient. The input triangle degree sequences were chosen carefully in order to make it possible to compare the resulting clustering coefficients to those calculated by hand. This does not lend itself to the practical use of this software but a more appropriate test was not devised at this time. The Klemm-Eugilez algorithm has a much more convoluted set of parameters that control the output. This makes it difficult to control any of the global properties. This algorithm seemed at first to have promise as it appeared able to offer control over the global properties of the network but at this point it now appears to have limited use in the formation of a viable solution to creating networks. Their are portions of code form the Klemm-Euguilez algorithm that may be used in a hybrid algorithm that could better preserve the small world properties of the network. The edge swapping portion of code can be appended to the existing Degree algorithm and the Newman algorithm and some short cuts can be made between nodes that otherwise have no edges between them. The use of an active and deactive list in the algorithm is an interesting approach to incorporating preferential attachment, to facilitate the creation of a nodes with a substantially higher degree. 5 Conclusion The algorithms that have been implemented up to this point in time have been optimized to run as efficiently as possible and a set of preliminary results were gathered for the production of this manuscript. The point that this research is at right now is the beginning stages of what is needed to be done in order to achieve a software tool that will be able to be used by researchers across the field of network science. The goal of creating a tool that can allow the user to tune both the clustering coefficient and the degree distribution, while maintaining a small world network with a hierarchical structure has not been achieved and is requires significant work to realize. The work that has been done up until now is a preliminary step towards that stated goal. 6 Future Work The continuation of this work would include the creation of an effective means of translating the set of properties that are interesting an understood 11

12 by a potential end user and the input parameters that are accepted by the algorithms that have been implemented to date. The following stages after that challenge has been overcome is the analysis of the generated graphs to verify that those graphs are indeed being created to reflect those set of inputs. Acknowledgements Brian Mulhall conducted this research as part of the BIOGRID initiatives in the Department of Computer Science and Engineering at the University of Connecticut. It is supported by the National Science Foundation REU grant OCI We are grateful to our mentor Saad Quadar for all his patience, guidance, and support throughout this project and to Professor Chun-Hsi Huang for giving us the opportunity to participate in this project. References [1] C. Bachmaier, Biological Networks [2] C. Herrera and P. J. Zufiria, Generating Scale-free Networks with Adjustable Clustering Coefficient Via Random Walks, pp [3] Girvan2002 M. Girvan and M. E. J. Newman, Community structure in social and biological networks., Proceedings of the National Academy of Sciences of the United States of America, vol. 99, no. 12, pp , Jun [4] S. Wuchty, The Architecture of Biological Networks, pp [5] B. J. Prettejohn, M. J. Berryman, and M. D. McDonnell, Methods for generating complex networks with selected structural properties for simulations: a review and tutorial for neuroscientists., Frontiers in computational neuroscience, vol. 5, no. March, p. 11, Jan [6] G. DAngelo and S. Ferretti, Simulation of scale-free networks, Proceedings of the Second International ICST Conference on Simulation Tools and Techniques, [7] N. Parikh, Generating Random Graphs with Tunable Clustering Coefficient Generating Random Graphs with Tunable Clustering Coefficient,