Eur. Phys. J. B (2012) 85: 118 DOI: 10.1140/epjb/e2012-30074-6
THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Evolving networks with bimodal degree distribution Abhijeet R. Sonawanea , A. Bhattacharyay, M.S. Santhanam, and G. Ambika Indian Institute of Science Education and Research, 411021 Pune, India Received 31 October 2011 / Received in final form 30 January 2012 c EDP Sciences, Societ` Published online 11 April 2012 – a Italiana di Fisica, Springer-Verlag 2012 Abstract. Networks with bimodal degree distribution are most robust to targeted and random attacks. We present a model for constructing a network with bimodal degree distribution. The procedure adopted is to add nodes to the network with a probability p and delete the links between nodes with probability (1 − p). We introduce an additional constraint in the process through an immunity score, which controls the dynamics of the growth process based on the feedback value of the last few time steps. This results in bimodal nature for the degree distribution. We study the standard quantities which characterize the networks, like average path length and clustering coefficient in the context of our growth process and show that the resultant network is in the small world family. It is interesting to note that bimodality in degree distribution is an emergent phenomenon.
1 Introduction Evolution of networks with complex topologies has been a subject of intense research in the last decade [1]. The focus has mainly been on networks found in natural systems like ecological webs, physical systems like power transmission grids, internet and myriad biological systems, for instance, functional brain networks, metabolic networks and many others [2,3]. The need to optimize natural networks, for a wide variety of cost functions, as well as the quest for creating efficient technical networks are the major motivations behind the intense search for various organizational principles to evolve networks. An important criterion to classify the networks based on its structural property or topology is the degree distribution P (k), i.e, number of nodes that have k links. For instance, Erd¨ osR˝enyi random networks have Poissonian degree distribution, Barab´ asi-Albert scale-free networks which are part of small world family of networks [4], shows a power-law behavior with non-universal degree exponent. Nonlocal rewiring of few links in regular networks leads to WattsStrogatz and Newmann-Watts small world networks [5–7]. Most of the real life complex networks have a heavy-tailed degree distribution and due to this the networks differ significantly from the random graph models. To obtain such heavy tailed distribution, which is typically a power law, numerous strategies are proposed of which preferential attachment [8] is a celebrated method based on the principle of “rich gets richer”. There have been other mechanisms like ‘power of choice’ [9], optimization [10], etc, leading to very interesting behaviors as shown by different degreedistributions. a
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There is inherent limitation on number of hubs in scale free networks to become comparable to the total number of nodes as the network grows to large sizes [11]. This makes them more vulnerable to targeted attacks, whereas these scale free networks are more robust to random attacks because probability of picking hubs as target [12] is very small. In this setting, networks which have bimodal degree distribution are found to be robust to both type of node/edge removal strategies, targeted or random [13–17]. If k¯ represents the mean degree of bimodal network, then all the nodes of this network fall under two modes. For each mode, we denote the local mean degree by k¯loc . Then, one mode involve the nodes whose local mean degree k¯loc > k¯ are called the ‘super-peer’ nodes while the second mode has nodes with k¯loc < k¯ called ‘peer’ nodes. These modes provide the bimodal network with enhanced stability under random or targeted attacks. Further, as the number of modes in the degree distribution approach from two to infinity, the network topology changes from bimodal network to a scale-free network [15]. In this sense, a bimodal network can be thought of as a special case of scale-free topology. In this paper, we propose a realistic growth model for bimodal degree distribution networks based on stochastic process. There are existing algorithms for generating simple random graphs with specific degree sequence d1 , d2 , d3 , . . ., where di is the integer specifying the degree of node i. There are two approaches which differ in abstraction to generate networks with desired degree sequence. One may choose a graph of specific degree sequence randomly with uniform probability from a set of graphs with that degree sequence [18]. The other approach is based on generating functions for the probability distributions of node degrees. Here, one may choose degree
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sequence approximately close to the desired probability distribution and average over all graphs with that particular sequence. The configuration model [19–23] creates nodes and assigns number of connections called ‘stubs’, drawn from a distribution. Using this degree sequence, stubs from a node may be linked to ‘stubs’ from other nodes chosen randomly without forming self links and multiple links. If the graph passes the graphicality criterion, given by Erd¨ os-Gallai theorem [24], it is accepted and the one which fails is rejected. Using these models it is indeed possible to generate networks which have bimodal degree distributions but we propose a simple model of network growth where bimodality is an emergent phenomenon and the network asymptotically attains bimodal degree distribution through a mechanism of separation of timescales based on a immunity score. We also provide a set of tunable parameters which are essential from the perspective of design considerations of efficient networks. We note that, bimodal degree distribution networks occur in many real-life cases. For instance, it occurs in simulated gel network (SGN) in which as temperature increases, the network topology changes from unimodal to bimodal [25]. In the context of neuronal networks, the switching of the network from one attractor or pattern of activity to another requires stability because such attractors correspond to functional states and the switching is used to model cognitive functions. This behavior, called ‘Stable-yet-Switchable’, is found to be most stable in bimodal networks [26,27]. With a proper definition of hubs, bimodality is observed in protein interaction networks [28]. This indicates that the bimodal networks could be identified in many natural systems either by trying out different definitions of links or applying certain criterion to identify the core and the periphery nodes and nature of links between them. Similarly snow flake type networks, which are bimodal networks with high degree node clusters and low degree node clusters, is identified as the optimal network for smart grids for which the synchronization cost is also minimum [29]. In spite of such a wide occurrence of bimodal networks, it is not yet studied in detail and, in particular, there is no known algorithm to stochastically evolve such a network. In next section, we propose our model followed by the numerical results. We then provide design parameters and characterize the networks obtained by us using average path lengths and clustering coefficients. We show that bimodal degree networks thus obtained are part of the small world family. In the concluding section, we make some observation which may be important for understanding bimodality in distribution as an emergent property.
2 The model Now we present the network evolution algorithm that leads to a degree distribution which is bimodal asymptotically. The algorithm proceeds through two broad steps. We start by describing a scheme by which depending on the outcome of the coin toss, links are either made or cut.
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By repeating this process we obtain a network with a unimodal degree distribution. In Section 2.1 we show that by introducing additional constraints in the form of immunity score, the network ultimately shows bimodal degree distribution. For generating a network with N nodes, we start with an initial seed network of m(t0 ) nodes which forms the initial connected component into which subsequent nodes from the set of N −m(t0 ) unconnected nodes are added. A node with at least one link to any other node is called an ‘active’ node as opposed to an ‘inactive’ node which has no links. For each active node i, we choose a random number r(i) ∈ [0, 1] from a uniform distribution. Each active node chooses a node randomly from the network and makes a connection with it with probability p or loses one of its existing links with probability (1 − p). In simple terms, a coin is tossed for each active node i, the output being mapped to a binary outcome as (r(i) ≤ p) → +1, where it gains a link and (r(i) > p) → −1, where it loses a link. We take initial seed network of active nodes m(t0 ) of size 10. At each time step t, the nodes which have at least one link are considered ‘active’ i.e. these nodes participate in the evolution of the network, i.e. in coin-toss. This results in a uni-modal network of size N generated by the following steps: 1. consider a ‘coin toss’ at all the ‘active’ nodes of the set m(t); 2. at each node i ∈ m(t), perform the operation of linking or delinking as follows, (a) each node i makes new link to a node η ∈ (N − 1) nodes (no self linking is allowed) with probability r(i) ≤ p. This might set an ‘inactive’ node η as ‘active’ for next time step, had η not been connected at least by a single link previously but receives the new connection now; (b) with probability r(i) > p, the node i breaks one of its existing connection with other nodes with equal probability which might lead to inactivation of a node η ∈ m(t) which had a single connection with the ith node and no other connections elsewhere; 3. check the degree of all nodes after all coin-tosses, set those nodes as ‘inactive’ with no link to any other node. These ‘inactive’ nodes can again be activated if they gain link from an active node; 4. repeat step 1 to 3. As the network is not directed, when a link between ith and jth node is cut, it leads to decrease in number of connections for both the nodes. Also multiple links between any pair of node are treated as one link. By this scheme, the adjacency matrix A with entries aij evolves as, ⎧ ⎨ aiη + 1 if rt (i) ≤ p, η is new node aij = aij if rt (i) ≤ p, j is existing neighbour of i ⎩ a − 1 if r (i) > p, j is existing neighbour of i. ij t (1) One obtains a random network with unimodal distribution using the above process. For probability p = 0.5, it leads to saturation of the degree distribution except for some fluctuations. If all the nodes are active, there would
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defined as, ⎧ φ (t) + 1 if ⎪ ⎪ i ⎪ ⎨ ri (t), . . . , ri (t − (α − 1)) ≤ p if φi (t + 1) = φi (t) − 1 ⎪ ⎪ r (t), . . . , r (t − (β − 1)) > p. i i ⎪ ⎩
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be equal number of link addition and deletion from the network with N nodes. To ensure the network to grow from the small seed, there should be multiple links for few nodes in the initial network. For p < 0.5, since on average there are more links removed than added, the network will never grow beyond the seed level. When p > 0.5, the network can grow indefinitely with a slowing down of the growth rate with the network size. The properties of network thus constructed for unbiased case p = 0.5 is shown in Figure 1. The average degree k¯ shows a steady increase followed by a saturation value k¯sat . The degree distribution is unimodal and we find that √the value of k¯sat for different network size N scales as N.
2.1 Bimodality in degree distribution In order to obtain a heavy tailed distribution, one needs to have few hubs in the network. One of the possible ways to have redundancy in hubs is to have bimodal distribution of degree of nodes. This bimodality in the degree distribution may be obtained if we device a mechanism which separates one set of nodes with mean degree k¯loc = k1 > k¯ ¯ This may be attained if and others with k¯loc = k2 < k. we manage to increase the time for which a node remains ‘active’ for few nodes over the others. This will enable them to participate in network evolution for longer duration and in turn gain more links which would increase their degree. Thus, these persistent nodes may form a class of their own and emerge as a new mode in the distribution resulting into bimodality. This differentiation cannot be done arbitrarily, hence we introduce an index called as an immunity score φi (t) for ith node, which represents the fitness of the node to remain active up to time t. It is
This definition simply implies that the immunity score φi (t) for each node is incremented by unity when the outcome of α consecutive coin tosses is: +1, +1, . . . , +1α for node i. On the contrary, the immunity score is decremented by one if the outcome of β consecutive coin tosses at i is sequence of −1, −1, . . . , −1β . The parameters α and β represent the relative timescales on which we need to measure performance of each node. The performance here means that a node has made new connections or has lost existing ones. The bias p in the coin toss process provides us with a handle to either increase the number of good performer nodes or bad performer nodes in the network. The value of φi (t) indicates the level of immunity, a node i possesses, i.e., if immunity φi (τ ) = 0, the evolution of the node is frozen for rest of the time t > τ and the node is ‘dead’ in the sense that no coin is tossed for it any more. We check at each time step for nodes whose immunity score is zero. For instance, if the immunity score is φi (t) = f > 0, the node will participate in network evolution for at least f times. Here, consistent ‘good (bad) performance’ is measured in terms of α (β) time-steps of +1(−1) coin toss outcomes. The algorithm has the machinery of rewards in the form of immunity score increment and punishment in the form of decrement in the immunity score. A node with consistent ‘good’ performance gets an opportunity to remain active for longer time and gain more links and this introduces segregation in degree distribution. These punitive responses or positive reinforcements or in more familiar term of ‘carrot and sticks’ is more or less sides of the same coin. A threat to refuse a reward is functionally identical to imposing punishment [30]. We tend to make the distinction more apparent by treating the punishment not just as an absence of reward but by further deteriorating chance of survival of under-performers with decrement in immunity score. The nodes with no immunity are weeded out at each time step.
2.2 Results In Figure 2, the degree distribution for various values of bias p is shown. The network is of size N = 500 and the parameters used are α = 4 and β = 3. The initial immunity score given to all the nodes uniformly is zero. For p = 0.55, the evolved network is not fully connected but there are few nodes which have high degree. Upon increasing value of p to p = 0.6, another mode emerges showing some amount of separation between degree averages. The height of the second mode increases steadily and at point p = 0.655, the height of both the modes become almost equal. This value of p is denoted as pc . Further, the height
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Fig. 3. (Color online) Snapshot of network topologies (a)–(f) for values of the bias p = 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 respectively for N = 50 for α = 4 and β = 3.
of mode with lower average decreases while that of the one with higher average increases continuously with increasing bias p. At higher values of bias p, the network becomes uni-modal at a higher average degree due to the obvious reason that all the nodes are active for a long enough time to get connected to almost all others. Eventually the network becomes all-to-all connected network at p = 1. We notice that the bimodal nature of degree distribution has a mild dependence on α and β but depends substantially on the bias p. As Figure 2 shows, the bimodal degree distribution emerges at an optimal bias of p = 0.655 than at p = 0 or 1. In Figure 3, the nodes are shown as red(black) and the links are shown as grey. The network is visualized, using NWB software1, as a radial graph. The graph shows the network as a tree structure rooted at some representative node at the center. Each node lies on the ring corre1
Network Workbench Tool, Indiana University, Northeastern University and University of Michigan, 2006.
sponding to the shortest network distance from the central node. Immediate neighbors lie on the second ring and so on. In Figure 3a, for p = 0.5 the network is seen to be fragmented and there are few nodes emerging as hubs as seen in long tail in Figure 2. At p = 0.6, the network becomes fully connected. The bimodal distribution emerges as the network starts to separate between nodes around two values of average degree of network as can be seen in Figure 3b. The network is still hierarchical and the nodes have multiple layers of neighborhood. For values of p between p = 0.7 and p = 0.9, the network has become a bimodal with variations in relative height of the modes of distribution. The topological features can be seen in Figures 3c–3e. It is seen that the network gradually separates into a core-periphery network consisting of peers and superpeers. The network has evolved such that the peers and superpeers are connected among themselves and are also inter-connected. The emergence of such bimodality in the degree distribution is due to the additional constraint introduced through the immunity score. In Figure 3f the network finally becomes a globally connected all-to-all network. Formulation of analytical model for the growth process described above is difficult due the fact that the number of active nodes in the network is a dynamical variable so that the number of master equations required will itself change with time. If one decides to look for mean field dynamics of total number of active nodes, those responsible for making new connections and those loosing connections, etc will involve complicated nonlinearities in ordinary differential equation. It would also require that average degree of network is itself a dynamical variable which would couple with node number. Introduction of timescales in form of parameters of α, β will be an arduous task given the analytical complexity of the numerical simple model. However we find that numerical investigations performed here bring out the salient features and mechanisms involved reasonably well.
3 Design parameters We note that there are several parameters of the evolved network which are important from a design perspective of efficient networks. These include locations of mean of the two different modes k1 and k2 in the bimodal distribution and the difference between them. These parameters can be modulated as per design requirements by varying model parameters like pc , α, and β. Although the bimodal nature can be considered with existence of modes of any height, pc denotes a particular case of identical heights in the distribution. This is clear from Figure 4 where the values of these parameters are shown in the α − β plane. In Figure 4a, p = pc is plotted for various values of α−β. It is seen that for lower values of β (immunity decrement more probable) and higher values of α (immunity increment less probable), we need to increase the bias p more in order to allow the network to grow and also to allow some nodes get a higher value of immunity score which is required for distribution to become bimodal. For reverse case of high
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Fig. 5. (Color online) (a) Average path length is shown for different values of bias p for N = 1000 for α = 4 and β = 3. (b) Average clustering coefficient is shown for different values of p for N = 1000 for α = 4 and β = 3.
Fig. 4. (Color online) Phase diagram in α−β parameter space showing different variables on z-axis in color for network size 500. The colorbar in each figure shows representative color for region between two values. (a) pc is the of bias p at which the degree distribution is bimodal with nearly equal modal heights. (b) The value of degree k1 around which the first mode of the distribution is situated and (c) the value of degree k2 around which the first mode of the distribution is situated. (d) The difference between two modal means in the distribution k2 −k1 is plotted for various values of α − β.
β and low α, the bimodality is achieved for even a smaller values of pc > 0.5. Figures 4b, 4c show the average values of degree around which first mode k1 and second mode k2 of the distribution is located respectively and Figure 4d shows the distance between them i.e. (k2 −k1 ) for different combinations of α and β. We have also performed simulation for large network sizes of 10 000 nodes. We still see that the bimodality is obtained for larger networks sizes.
paths between pairs of nodes of a network is calculated and the shortest path lengths are calculated via breadthfirst search method. The average clustering coefficient (cliquishness) for the whole network as given by Watts and Strogatz [5] is the average of the local clustering coefficients of all the vertices N: N 1 C¯ = Ci . (3) N i=1 Figure 5b shows that the average clustering coefficient increases with the bias p as it is clear that the number of links are growing with p. The distribution of clustering coefficient C is also seen to be bimodal (not shown here). We use NWB software to calculate the above quantities1 . As is clear from Figure 5, the shorter average path length and high clustering coefficient clearly shows that the algorithm generates a bimodal network which belongs to the family of small world networks with additional feature of stability to random or targeted attacks. The high number of links in the network shows that there is large redundancy in the connections. The only price with such a construction of bimodal degree distribution network is the cost of network in terms of large number of connections. This can be a problem where there is constraint on number of edges between the nodes. But in social networks and biological networks where redundancy is common, our algorithm may serve as a one of the possible templates for the evolution of bimodal networks.
4 Properties of the evolved network In this section, we discuss the characteristic properties of the evolved network such as the average clustering coefficient and average path length. The average path length L is the average distance between any pair of nodes, averaged over all pair of nodes, and it determines the effective size of the network. In Figure 5a, the average path length is plotted for a network of maximal size N = 1000 as a function of bias p. It is seen that the ‘effective’ size of network decreases with increasing p and corresponds to the network going from being disconnected to all-to-all connected. Between these two extremes, various flavors of bimodal degree distribution networks are obtained. In the intermediate region, the average path length is relatively small and is a decreasing function of p. For calculating average path length, the average length of the shortest
5 Discussion Starting from a small seed, we have been able to stochastically evolve a uni-modal random network of desired size with saturated degree distributions. However, our main purpose in the present context is to identify the minimal conditions that render bi-modality to this stochastically evolving distribution and have some control over the process. For this, we introduce a scheme of rewarding the consistency of performance, somewhat in the line of “rich gets richer” principle used for generating scale free networks which very efficiently leads to a bimodal one. Since evolution is a process that happens over time, consistency of performance over time can be a natural choice for rewards and the opposite being true for inconsistent performers.
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With this principle of biasing, an otherwise completely stochastic process of evolution comes out to be an efficient generator of widely separated classes (modes) and also allows us to have controls to almost continuously move between networks from single to double mode distributions. The principle of “consistent lives longer”, where nodes that get connected with others more frequently live longer than the others can serve as an important point in understanding evolution of many bimodal networks. Not only the bimodality with equal mode heights, at various levels of gross performance (value of p) of the evolving nodes, we see in Figure 3 that interesting clustering occurs in networks with weak bimodality as well. Also it should be noted that the bimodality with equal modal heights occurs at a optimal level of performance and not just for extreme values of p. The evolutionary scenario proposed here, captures the three broad classes of random networks shown in Figure 2, namely unimodal sparsely connected, bimodal and unimodal densely connected as the ones which have evolved for three different characteristic time scales (relatively small, intermediate and large) before saturating. Thus, within our network evolution scheme, a wider class of random networks are captured on the same footing. There are various naturally occurring bimodal networks as has been mentioned above (in the context of biology and elsewhere) where connections are based on functional attachments or other static properties. In understanding the evolution of such networks to their present topology, the principle of “consistent lives longer” – which is being shown here to be at the origin of the emergence of bi-modality – can prove to be an important clue. The authors acknowledge DST, New Delhi, India for support through project SR/S2/HEP-08/2009.
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